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William Froude

William Froude (1810–1879) was an English engineer, hydrodynamicist and naval architect. He was the first to formulate reliable laws for the resistance that water offers to ships (such as the hull speed equation) and for predicting their stability. He was the brother of James Anthony Froude, a historian, and Hurrell Froude, writer.

1810 November 28th. Born at Dartington, Devon, England, the son of Robert Froude, Archdeacon of Totnes. Educated at Westminster School.

1832 Graduation. Attended Oriel College, Oxford, graduating with a first in mathematics.

His first employment was as a surveyor on the South Eastern Railway which, in 1837, led to Brunel giving him responsibility for the construction of a section of the Bristol and Exeter Railway. It was here that he developed his empirical method of setting out track transition curves and introduced an alternative design to the helicoidal skew arch bridge at Rewe and Cowley Bridge Junction, near Exeter.

Married Catherine Henrietta Elizabeth Holdsworth daughter of Dartmouth Governor, mercantile magnate and member of Parliament Arthur Howe Holdsworth.

c1843 Birth of son R. H. Froude.

1851 Living at The Parsonage, Dartington: Robert H. Froude (age 80 born Aveton Gifford, Devon), Rector of Dartington. With his son William Froude (age 40 born Dartington) and his wife Catherine H. Froude (age 41 born Dartington) and their children (his grand-children) Eliza M. Froude (age 11 born Brixham); Richard H. Froude (age 8 born Dartington); Arthur H. Froude (age 6 born Dartington); Robert E. Froude (age 4 born Dartington); and Mary C. Froude (age 2 born Dartington). Three servants.^[1]

At Brunel's invitation, Froude turned his attention to the stability of ships in a seaway and his 1861 paper to the Institution of Naval Architects became influential in ship design. This led to a commission to identify the most efficient hull shape, which he was able to fulfill by reference to scale models: he established a formula (now known as the Froude number) by which the results of small-scale tests could be used to predict the behaviour of full-sized hulls. He built a sequence of 3, 6 and 12 foot scale models and used them in towing trials to establish resistance and scaling laws; Raven's sharp prow followed the "waveline" theory of John Scott Russell, but Swan's blunter profile proved to offer lower resistance. Froude's experiments were vindicated in full-scale trials conducted by the Admiralty and as a result the first ship test tank was built, at public expense, at his home in Torquay (Admiralty Experiment Works). Here he was able to combine mathematical expertise with practical experimentation to such good effect that his methods are still followed today.

1877 Froude was commissioned by the Admiralty to produce a machine capable of absorbing and measuring the power of large naval engines. He invented and built the world's first water brake dynamometer, sometimes known as the hydraulic dynamometer, which was later exploited by his son's business, Heenan and Froude, in Birmingham and Worcester.

1879 May 4th. He died while on holiday (as an official guest of the Royal Navy) in Simonstown, South Africa, and was buried there with full naval honours.

1879 May 30th. Obituary.^[2]

1879 June. A series of articles were published about the nature and results of his scientific labours.^[3]

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1880^[4]

WILLIAM FROUDE, LL.D., F.R.S., was born in 1810, at Dartington Parsonage, Devonshire, the house of his father the Ven. R. H. Fronde, Archdeacon of Totnes.

He was educated at Westminster School and at Oriel College, Oxford, where he took a first-class in mathematical honours in 1832.

He then became a pupil of Mr. Henry Robinson Palmer, Civil Engineer.

In 1837 he became an assistant of Mr. Brunel, and was engaged on the works of the Bristol and Exeter Railway until the completion of the line in 1844, being, during the latter portion of the time, Resident Engineer of the line on the Devonshire side of the summit tunnel.

For family reasons he shortly afterwards retired from the active pursuit of his profession ; but on occasion he assisted his friend Mr. Brunel in engineering matters, of which perhaps the most important was the investigation concerning friction, which he made in reference to the launch of the Great Eastern steamship. By trial of a specimen of the iron sliding surfaces, and by automatic records of the movements of the ship herself, Mr. Froude proved that the friction of the sliding surfaces was not independent of the velocity as commonly supposed, but became much less as the velocity increased.

It was also in connection with the Great Eastern that he undertook the enquiry into the causes of the Rolling of Ships, which he continued during the subsequent twenty years. The mechanical possibility of the trochoidal theory of ocean waves, the effect of the cumulative action of more or less synchronous waves upon a ship, and the mortifying effects of the resistance which a ship offers to rolling, were worked out by him. He also devised apparatus for determining the characteristic qualities of different ships, by recording their behaviour when set rolling in still water, or when rolling in actual waves at sea.

During the same time he was actively engaged in enquiry into another subject most important to naval architecture, namely the resistance of ships, or in other words the force necessary to propel them at various speeds. In the course of this enquiry he determined the law which governs the relation between the resistance of a ship and that of her model: he explained the application of the doctrine of stream lines to the resistance of ships: he put into quantitative form the several elements of use and waste in the employment of steam power for the propulsion of ships: he measured the action of the screw propeller in its effect on the stream lines around the stern of a ship: finally he investigated the circumstances under which a screw propeller operates, and the loss of power duo to its friction, when rotating in the water. It was in connection with this subject of the Resistance of Ships that be undertook for the Admiralty, in 1870, the construction and management of an experimental establishment at Torquay, for the trial of models of ships. Here he instituted a series of exhaustive experiments on the forms of ships, and also determined the resistance of models of most of the ships recently built for the Royal Navy.

Mr. Froude was a member of the Committee on Designs of Ships of War in 1870, and of the Committee on the Inflexible in 1877. He was an able and exact workman, and his mechanical and inventive powers are exhibited by the designs of his experimental apparatus, especially of the rolling-recording instrument, of the machinery for making models (see Proceedings, 1873, p. 202), and of the several governors and dynamometers used in various experiments; also in a marked degree by the conception of the principle of the marine-engine dynamometer, which he explained at the Bristol Meeting of the Institution, 1877 (see Proceedings, p. 237).

In addition to the great services he rendered gratuitously to the country, mention should be made of the important aid he gave to his neighbours, the townsmen of Torquay, in demonstrating the cause of the defective delivery in the water supply of the town, and in carrying to a successful issue Mr. Appold's ingenious suggestion for scraping the pipe (see Proceedings, 1873, p. 210). He also assisted in the trials of implements of the West of England Agricultural Society, and devised a simple dynamometer for recording the power delivered to machinery (see Proceedings, 1858, p. 92).

Mr. Froude became a Member of the Institution in 1852. In 1876 he received the gold medal of the Royal Society for "his researches, both theoretical and experimental, on the Behaviour of Ships, their oscillations, their resistance, and their propulsion."

His death, caused by an attack of dysentery, took place on 4th May 1879, at Admiralty House, Simon's Town, Cape of Good Hope, where he had gone on a pleasure trip in H.M.S. Boadicea for the benefit of his health.

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1880^[5]

William Froude, LL.D., F.R.S., the fourth son of the Ven. R. H. Froude, Archdeacon of Totnes, was born at Dartington Parsonage, on the 28th of November, 1810.

He was educated at Westminster School, and went thence to Oriel College, Oxford, being for some time a pupil of his elder brother, R. Hurrell Froude, an advantage to which he often referred. He took a first class in Mathematical Honours in 1832.

In the beginning of the year 1833, he became a pupil of Henry Robinson Palmer, V.P. Inst. C.E., then Resident Engineer of the London Docks. Mr. Froude was afterwards employed under Mr. Palmer on some of the early surveys of the South Eastern Railway and on other undertakings.

In 1837, Mr. Froude joined the engineering staff of Mr. Brunel, V.P. Inst. C.E., upon the Bristol and Exeter railway, where he had charge of the construction of the line between the Whitehall Tunnel and Exeter, and remained until it was opened in May, 1844.

Here was evinced that painstaking attention to detail which ensured the truth of his later work. As one of many instances it may be mentioned that, in two elliptical brickwork skew-bridges, he introduced taper bricks so arranged as to make correct spiral courses. It was while employed on this line that Mr. Froude propounded 'the curve of adjustment,'many years afterwards described by him in a Paper read and discussed before the 'Institution of Engineers in Scotland,' on the 28th of November, 1860, and on the 23rd of January, 1861.

In the autumn of 1844, he was engaged on the Parliamentary surveys of the Wilts, Somerset, and Weymouth railway, but shortly afterwards he gave up the active pursuit of the profession in order to live at Dartington with his father, who was then in failing health.

As one of Mr. Brunel’s most intimate friends, Mr. Froude had still many opportunities of being associated with important engineering work. The progress of railways in Devonshire, and the trial and failure of the Atmospheric System, were subjects which Mr. Froude studied carefully. A noteworthy instance in which Mr. Froude gave valuable help to Mr. Brunel was at the launch of the 'Great Eastern' steamship, in 1857. The friction of the iron sliding surfaces had been tested by means of an experimental cradle, loaded so as to represent a portion of the actual cradle. To this experimental cradle Mr. Froude fitted a simple self-recording apparatus. He made a powerful pendulum, of very short period of vibration, by hanging a piece of double-headed rail length-ways on centres. A paint brush attached to this pendulum marked every quarter-second of time on a tape attached to the moving cradle, thus furnishing a record of motion, from which the exact amount of the retarding force of friction could be determined for each moment of the motion. Mr. Froude’s experiments showed conclusively, that the friction was not, as stated in test-books, independent of the velocity, but that it became much less as the velocity increased.

The same recording apparatus was applied to the movements of the ship while being launched, and similar phenomena were observed. During the discussion of this question at the launch of the 'Great Eastern,' Mr. Froude pointed out how the fact of friction varying with the speed of the surfaces explained the analogous circumstance, that as soon as the action of a railway brake-block reduced the speed of the wheel below that of the speed of the train, skidding ensued, and then there was no alternative but to ease the brake till the wheel turned freely, and to re-apply it judiciously.

During the period immediately following his retirement from professional work, Mr. Froude was by no means idle. As Justice of the Peace, and as Trustee for several Turnpike Trusts, he had a good deal of general occupation. In these and similar positions his engineering and scientific knowledge naturally threw upon him the burden of advising his colleagues on technical matters. He also for many years took an active part in the Bath and West of England Agricultural Society, as one of the judges of the machinery exhibited.

He constructed a dynamometer for testing the engines and machines, and, with the view of rendering its employment acceptable to competitors, published in the Transactions of the Society an explanation of the principles of the instrument and of the machines whose behaviour it was intended to record. He also contributed valuable reports to the Society on the machines exhibited and on the principles involved in their action.

In connection with the waterworks of Torquay, Mr. Froude rendered important service. Within a short time after the works were completed, the delivery of water through the supply main, 15 miles long, proved to be only about one-half of what it should have been. All kinds of theories and explanations were propounded. Mr. Froude, being called into council, proceeded, as was his habit, to make a scientific examination of the circumstances of the case. By careful measurements of the pressure at various points throughout the main, he found that the defect in the power of the pipe to carry the water was manifested equally throughout the whole length. It thus became evident that the defective delivery was due to the increased friction of the pipe, caused by rust on its interior surface. This fact having been proved, J. G. Appold, Assoc. Inst. C.E., just before his death, suggested that a piston, propelled along the pipe by the pressure in the main, might carry knives to dislodge the rust, which would then be cleared away by the flow of the water. Mr. Froude adopted the suggestion, and constructed a machine, by which the delivery of the main was raised even above the calculated delivery, and the town of Torquay was thus saved for nearly twenty years the expenditure of about £30,000 on a new main.

In his leisure Mr. Froude occupied himself a good deal in mechanical handiwork. His skill as a worker in material was great, and resulted from the educated knowledge of what should be aimed at, rather than from any particular excellence in that kind of aptitude which artisans acquire from practice. Even in ordinary work, he made use of well directed refinements of measurement for saving time.

He was free from superstitious belief in the automatic accuracy of machine tools, and preferred to trust principally to gauges and surface plates, having a maxim that any error which could be detected could always, with proper care, be corrected. At the same time he did not neglect to employ all the advantages that good tools could afford. His lathes were kept in perfect order; there was no slackness, nor, what he seemed still more to dislike, any unnecessary tightness. Nothing was suffered to remain wrong.

It must not, however, be supposed that he was a slave to nicety of work and of fit. He knew well, where and when this was important; and he was never content with a suggested cause for the defective working of any machinery, until, by putting the matter into quantities, he had satisfied himself that the cause was not only right in kind, but was also sufficient to produce the observed effect. He would often caution others against the temptation, as he expressed it, 'to over-estimate tendencies.' His experimental apparatus always exhibited excellently finished work where finished work was necessary, and sufficiently though less finished work in other parts.

Among the more remarkable scientific inquiries undertaken by Mr. Froude, on subjects other than those relating to naval architecture, was that 'On the Law which governs the discharge of Elastic Fluids under pressure,' as to which he presented a Paper to the Institution in 1847. In that communication he pointed out certain misconceptions in the received theory as to the flow of gases, and suggested important improvements in the treatment of the question.

Another subject of general scientific interest which he investigated theoretically, and which it was always his hope still further to work out experimentally, was the resistance experienced by a plane moving obliquely through a fluid, especially as regards the practical exemplification of the theory in the flight of birds.

Before describing his later work it should be mentioned that on the death of his father, in 1859, Mr. Froude left Dartington and went to live at Paignton in Torbay. He afterwards built a house, known as Chelston Cross, on the hill immediately above the Torquay railway station, and upon the design and construction of this house, which he first occupied in the year 1867, much care was bestowed.

About the same date came his opportunity of employing his talents to a pursuit of inestimable value to the nation, one for which he was eminently fitted, and to which thenceforward his whole time and powers were devoted. From his earliest days at school at Westminster, and again while at Oxford, Mr. Froude had been well known for his. management of boats; and later on as a skilful yachtsman he paid frequent attention to many of the problems, such as the resistance of the hull, the effect of the wind on sails, the action of the rudder, and the like, which became matters of more careful study in later years. It was partly with an intention of determining the best form for a yacht which he proposed to build, that he commenced a series of experiments on the resistance of models, which were the germ of the Admiralty experiments he afterwards conducted at Torquay.

Having friends in the Royal Navy, he had many opportunities of becoming familiar with the progress of naval architecture, and shortly after the introduction of the screw propeller into the navy, he was struck by the disadvantageous position in which the screw was placed, especially in the converted line-of-battle ships, and he thereupon made some instructive experiments with a model. He urged the consideration of this detrimental action of the screw for twenty years, but with little effect.

In his younger days, when at Bristol, he saw much of the 'Great Western' and 'Great Britain' steamships, and during the construction of the Great Eastern he, at Mr. Brunel’s request, undertook the investigation of the rolling of ships. His researches on this subject at once attracted attention. The behaviour of waves and of a ship among waves had hitherto been looked upon either as an insoluble problem, or as one in which the solution arrived at would have no real counterpart in the actual circumstances of practical experience. Mr. Froude showed how both the motion of the waves and of vessels could be reduced to rule, and could be mathematically, and indeed mechanically, explained. He investigated the matter in its general features, and also in many of the intricacies involved in the behaviour of abnormal forms of ships; and further, by his aptness in experimental inquiry and by his mechanical skill, he was able to devise apparatus which measured quantitatively the behaviour and characteristics of ships rolling in still water and among waves, and gave at the same time an accurate record of the form of the waves in which the vessel was at the time oscillating.

It has been said that experiments on the yacht model forms were the germ of Mr. Froude’s experiments on the resistance of ships. His first step, in connection with this subject, was to enunciate the true principle of the relation of the resistance of a ship to that of her model, namely, that the resistance is in the proportion of the cube of the linear dimension at speeds proportional to the square root of the linear dimension. He demonstrated this mathematically, and by experiments with different-sized models, some of which were nearly 0.5-ton in displacement.

E. J. Reed, M. Inst. C.E., when Chief of the Constructive Staff of the Navy, encouraged Mr. Froude to propose to the Admiralty to conduct a series of experiments on the resistance of models. The offer was accepted in the year 1870, and from that time, except when occupied on other work for the Admiralty, Mr. Froude devoted his energies to the conduct of experiments for the Government on the resistance of ships, and on the cognate subject of their propulsion.

The Admiralty establishment at Torquay, erected by Mr. Froude for carrying out these experiments, contains a covered tank 250 feet long, 33 feet wide, and 10 feet deep. Above this tank there is a suspended railway, on which runs a truck drawn at any given speed with great exactness, and beneath this truck the model is drawn through the water, and its resistance is measured by a self-acting dynamometer on the truck. There are also arrangements for testing the effect of screw propellers behind the models. The machinery for manufacturing models, and the various governors for regulating and recording speed, are evidences of Mr. Froude’s scientific skill.

The establishment has also been used for other inquiries allied to its original purpose; but that purpose has been at the same time steadily pursued, and an exhaustive series of experiments on the forms of ships has been in progress, from which valuable results have been obtained; and for the Royal Navy all the proportions and forms have been subjected to the investigation given by the experimental apparatus at Torquay.

In these inquiries Mr. Froude appreciated and demonstrated the true bearing of the doctrine of stream-lines, and the qualifications that had to be introduced to reconcile the simpler forms of that doctrine with the condition of a ship moving at the surface of the fluid, and he was thus able to establish the true methods of research which he pursued. His general conceptions of the bearing of the stream-line theory on the resistance of ships were described in his address as President of the Mechanical Section of the British Association at Bristol in 1875, an address which was afterwards delivered as a lecture at the Royal Institution on the 12th of May, 1876.

His knowledge of pure mathematics was considerable, and he was also especially skilled in employing graphical methods for the solution of the large class of problem in which algebraic expressions become inconveniently complicated. An instance of this method may be found in his Paper 'On the Graphic Integration of the Equation of a Ship’s Rolling, including the Effect of Resistance,' read at the Institution of Naval Architects in 1875.

Mr. Froude’s researches into the expenditure of power in screw ships, the proportions of screw propellers, and the information to be deduced from the speed-trials of ships, are of immense importance, not only to the Royal Navy, but also to the Mercantile Marine.

In connection with this subject the Admiralty asked him to design a dynamometer capable of determining the power of large marine engines. The very remarkable machine which he devised to meet the requirements of this problem was not tried on a large scale in his lifetime. He finally inspected the completed machine just before he last left England, and its experimental trials have since taken place with great success. Mr. Froude explained the principle of the invention at the meeting of the Institution of Mechanical Engineers at Bristol in 1877, and subsequently referred to it at the Institution of Civil Engineers.

Mr. Froude’s value as an adviser in naval architecture was publicly recognised by his appointment as a member of the 'Committee on Designs,' in 1870, and as a member of the 'Inflexible' Committee in 1877; but it was shown still more by the friendly confidence accorded to him by the Constructive Staff and by the successive heads of the Admiralty. Nothing perhaps demonstrated more strongly the endearing qualities which Mr. Froude possessed than the manner in which he was treated wherever his work for the Admiralty took him. In the dockyards, and on board the many ships of the Navy where he conducted experiments, his work was necessarily an interference with the regular routine, and frequently with the habits of thought of those among whom he came; but so fully did he feel and express his appreciation of this, and such was his unfailing tact and consideration for others, that the desire of all was to assist and welcome him; indeed, it was a pleasure to go an errand for him, so cordial was the reception which the mention of his name ensured. Wherever Mr. Froude went he was beloved, but nowhere perhaps will his memory be more cherished than among the officers of the Admiralty and of the Royal Navy.

Mr. Froude was elected a Fellow of the Royal Society in 1870, and in 1876 he received the honorary degree of LL.D. from the University of Glasgow. In the same year he received the Royal Medal from the Royal Society, on which occasion the President, in his address, said:-

"A Royal Medal has been awarded to Mr. William Froude, F.R.S., for his researches, both theoretical and experimental, on the Behaviour of Ships, their oscillations, their resistance, and their propulsion.

"It is generally admitted that Mr. Froude has done more than anybody else towards the establishment of a reasonable theory of the oscillation of ships in wave-water, as well as for its experimental verification. The very accurate instruments which he has contrived for the measurement of a ship's oscillation at sea have even permitted him to measure (as a differential phenomenon) the mean wave acting upon the ship with a degree of exactness exceeding that with which it has hitherto been possible to ascertain the profile of the surface-wave of the sea.

"He was also the first to establish on thoroughly sound principles the mechanical possibility of that form of motion known as the trochoidal sea-wave, which more nearly than any other appears to represent the shape of smooth ocean-wave, and which now forms the groundwork of all useful theories of the oscillation of ships.

"He has also conducted a series of experiments, extending now over many years, on the Resistance, Propulsion, and Form of Ships, and on the very important and little-understood question of the law connecting the behaviour of ships, in all these respects, with that of models of ships on a much smaller scale. These experiments have been conducted partly for the Government, and with public money; but they have also very largely taxed Mr. Froude’s own private resources, the sums repaid to him by no means representing his whole expenditure on these matters, and including no compensation whatever for his own time or labour.

"The amount of mechanical skill, as well as of theoretical acuteness, which has been exhibited in all this work has placed Mr. Froude in the foremost rank of all investigators on this subject. No one, indeed, has ever done more, either theoretically or practically, for the accurate determination of a ship’s motion, whether in propulsion or in waves, than Mr. Froude. Without undervaluing other modern writers, it is not too much to say that his investigations at present take completely the lead in this very important question-most important to a maritime nation."

Mr. Froude became a Member of the Institution of Civil Engineers in 1846, and in 1877 he was elected a Member of Council. The serious illness and death of his wife almost entirely prevented his attending the meetings of the Council.

In the winter of 1878, on the invitation of Commodore Richards, he went on a cruise to the Cape, in H.M.S. Boadicea, and was about to return to England, refreshed in body and mind, when he was seized with an attack of dysentery, and died at Admiralty House, Simon’s Town, after a short illness, on the 4th of Ma.y, 1879. His body was followed to the grave in the Naval Cemetery by the officers and men of Her Majesty’s ships then in Simon’s Bay, in recognition of the great services he had rendered to the Royal Navy.

The following is a list of the more important Papers, arranged in chronological order, contributed by Mr. Froude to various societies:-

1         "On the law which governs the discharge of elastic fluids under pressure, through short tubes or orifices."- Minutes of Proceedings Inst. C.E., vol. vi.,

2         "Remarks on mechanical power and description of a new dynamometer."- Bath Soc. Journal., vol. v., 1857, pp. 216-237

3         "On a new dynamometer and friction break."- Inst. of Mech. Eng. Proceedings, 1858, pp. 92-110.

4         "On the rolling of ships."- Trans. Inst. Naval Architects, vol. ii., 1861, pp. 180-227; vol. iii., 1862, pp. 45-62.

5         "On isochronism of oscillation in ships."- Trans. Inst. Naval Architects, vol. iv., 1863, pp. 211-215.

6         "Remarks on the differential wave in a stratified fluid.”-Trans. Inst. Naval Architects, vol. iv., 1863, pp. 216-218.

7         "Remarks on Mr. Scott Russell’s Paper on Rolling.- Trans. Inst. Naval 1847, pp. 356-384. Architects, vol. iv., 1863, pp. 232-275.

8         "Remarks on the mechanical principles of the action of propellers."- Trans. Inst. Naval Architects, vol. vi., 1865, pp. 35-39.

9         "On the practical limits of the rolling of a ship in a sea-way."-Trans. Inst. Naval Architects, vol. vi., 1865, pp. 175-184.

10      "Apparent negative slip in screw-propellers."- Trans. Inst. Naval Architects, vol. viii., 1867, pp. 70-81.

11      "On some difficulties in the received view of fluid friction."- Brit. Assoc. Rep., vol. xxxix., 1869 (Sect.), pp. 211-214.

12      "On the action of the screw-propeller."- Minutes of Proceedings Inst. C.E., Vol. xxxii., 1870-71, pp. 232-244.

13      "On the influence of resistance upon the rolling of ships."- "Naval Science," vol. i., 1872, pp. 411-429; vol. iii., 1874, pp. 107-121 and 312-330.

14      "Experiments on the surface-friction experienced by a plane moving through water."- Brit. Assoc. Rep., vol. xlii., 1872, pp. 118-124; "Nature," vol. vi., 1872, p. 387.

15      "Description of an apparatus for automatically recording the rolling of a ship in a sea-way."- Brit. Assoc. Rep., vol. xlii., 1872 (Sect.), pp. 243-245.

16      "Description et usage d’un pendule b tres-longne poiode pour la mesure du roulis absolu." (in English.)-Cherbourg, Mkm. Soc. Sci. Nat., vol. xvii., 1873,

17      "Considerations respecting the effective wave slope in the rolling of ships at sea."-Trans. Inst. Naval Architects, vol. xiv., 1873, pp. 96-108 ; “Naval Science,” vol. ii., 1873, pp. 215-239.

18      "Description of an instrument for automatically recording he rolling of ships." -Trans. Inst. Naval Architects, vol. xiv., 1873, pp. 179-184.

19      "Apparatus for automatically recording the rolling of a ship in a sea-way, and the contemporaneous wave-slopes." Journal of the R. U. S. Inst., vol. xvii. 1853,

20      "On experiments with H.M.S. 'Greyhound.' "- Trans. Inst. Naval Architects,

21      "On Stream lines.”--“ Naval Science, vol. iii., 1874, pp. 504-507.

22      "On the graphic integration of the equation of a ship’s rolling, including the effect of resistance."- Trans. Inst. Naval Architects, vol. xvi., 1875, pp. 57-71.

23      "Address as President of the Section of Mechanical Science of the British Association".-Brit. Assoc. Rep., vol. slv., 1875 (Sect.), pp. 221-239.

24      "The fundamental principles of the resistance of ships."- Royal Institution. Proceedings., vol. viii., 1875-78, pp. 188-213.

25      "Experiments upon the effect produced on the wave-making resistance of ships by length of parallel middle body."- Trans. Inst. Naval Architects, vol. xviii., 1877, pp. 77-97.

26      "On the elementary relation between pitch, slip, and propulsive efficiency."- Trans. Inat. Naval Architects, vol. xix., 1878, pp. 47-57.