The parabolic completions thus applied to Mr. Denny's thrust curves are found not only to meet them at a common tangent, but to "osculate" them for a considerable distance, which confirms the belief that the operation is not merely a geometrical one but expresses an action which is dynamically real.

Fig. 3 (Plate IX.) shows the process as completed from the records of the trials of Mr. Denny's ship, the Merkara. In this, as in the other similar curves, the black crosses show the points deduced from the steam pressures recorded in the individual trials. The curves of indicated horse-power and the curves of slip are also given.

A curious confirmation of the soundness of the method adopted, and of the exactness with which it determines the initial friction, is supplied by the diagram, Fig. 4 (Plate IX.), which represents the thrusts of the Taupo and the Hawea, which as sister ships ought to have experienced the same resistances throughout. The curves of indicated horse-power for these ships however exhibit a disagreement, and this of a kind which certainly does not at once suggest its true origin; but the corresponding indicated thrust curves, when completed in the manner described, show at once that the disagreement lies in a difference of indicated thrust which is throughout practically constant, and probably expresses in a constant difference between the initial frictions of the two ships' engines. The slight exception to the constancy of the difference which appears at one point of the comparison is manifestly due to an abnormal feature in the Hawea's curve, and this is such as to suggest that there has been some small error in the determination of the speed at the point which occasions it, for at this point there is a corresponding irregularity in the slip curve, and one and the same correction would obliterate both irregularities.

Moreover, through Mr. Denny I have received from his friend Mr. Inglis, the reports of the trials of two ships built by the latter-the Arbutus and the Pachumba, Figs. 5 and 6 (Plate X.); and these also, when similarly analysed, give a precisely analogous result. On comparing these five curves thus analysed and several others in which the analysis was not quite so simple but was I think equally conclusive, it appears that the constant friction is equivalent to from one-eighth to one-sixth of the gross load on the engine when working at its maximum speed and power. And it is not irrational to accept this relation provisionally as the basis of an empirical formula, since the constant friction depends to a large extent on the diameter and weight of the working parts of the engine, and these must be approximately proportionate to the intended maximum strain, subject of course to some allowance for the variation which exists in the types of engine in use. I must admit that the proportion appears to me to be unexpectedly large, but the process by which it is determined is I think so certain

and definite that I cannot doubt the general soundness of the conclusion deduced by it; and that conclusion seems to me to be one of very high importance and significance-namely, that a screw engine when working at even its most moderate and economical speed, must be understood to be throwing away in the one element of this friction alone, not indeed one-seventh of its maximum power, for the engine may be now working at reduced speed, but a power due to one-seventh of its maximum load. Thus in the case of the Merkara, when the ship is steaming at 5 knots in a smooth sea one-half of her whole expenditure of power is due to this circumstance. The question of the apportionment of this large amount of inevitable friction between the several working parts of the engine, and of the proportionate degree in which it attaches to different types of engine as well as of the extent to which the evil is remediable, are enquiries of great importance, but they are more or less out of my reach, and are at all events beyond my present purpose, which is satisfied by the proof-an irresistible proof as it appears to me-that the evil does exist to about the degree named.

But the discovery of the actual amount of power thus expended has been of great assistance to me in the attempt to account for the fact, of which accumulated proof exists, that the total power employed in a screw-ship's propulsion greatly exceeds that required by the nett resistance; and I venture to call the attention of the Meeting to my investigation of this subject as far as I have yet carried it.

The greatness of the excess has become manifest wherever it has been possible to compare a ship's actual resistance at a given speed with the indicated horse-power required to drive her at that speed; and abundant data for the comparison have been supplied in the first place by the dynamometric trials of H.M.S. Greyhound, and in the next by the experiments on the resistances of ships of various forms which I am carrying out for the Admiralty by careful dynamometric trials of their models; for among the latter it naturally happens that many forms which have had their nett resistance thus determined, have also been tried on the measured mile, and to these must be added Mr. Denny's ship, the Merkara. The result of the comparison shows that as a rule only from 37 to 40 per cent. of the whole power delivered is usefully employed, and it will be seen that, using the constant friction of the engines as an index of the scale of the friction generally, the 60 per cent. or more of loss can be fairly accounted for.

the

In the earlier part of this Paper I enumerated the principal elements constituting gross load of the engines. I repeat the list here. (1) Ship's nett resistance. The power

* In Mr. Denny's ship the usefully employed power is as much as 42 per cent. This however is at the light or trial draught.

due to this I shall designate effective horse-power, or E.H.P. (2) Augment of nett resistance due to negative pressure created about the ship's stern by the action of the screw. (3) Water friction of screw. (4) Constant friction, or friction of engine as without external load. (5) Friction due to external load. (6) Air-pump and feed-pump resistance. speed of ship in feet per min. The six elements are force factors, and when multiplied by 33,000 constitute the ship's horse-power as fundamentally due to her progress, and I shall designate this S.H.P. in making up the account, which will consist of the several elements all ultimately reduced into terms of E.H.P., which is in a sense the origin of them all.✨ The horse-power due to slip has to be added subsequently.

I proceed to quantify each element in detail, making up the account in the first instance for the highest speed as a point of departure.

(1) This is to represent the ship's true resistance whatever it be.

(2) The augmentation of the ship's resistance by the induced negative pressure under the stern consequent on the thrust of the screw is a circumstance on which I have often laid great stress at the Meetings of this Institution. There is abundant proof of its existence and approximately of its magnitude in the records of the trials of the Rattler and Alecto, Niger and Basilisk. Here I will only refer to the more crucial proof of its existence and of its magnitude which has been obtained by our experiments with models. After a model's nett resistance at all speeds has been determined in the usual manner, the resistance is again determined under the conditions imposed by screw thrust. There is brought up behind the model, and quite independent of it, a screw shaft carrying a screw of the required pitch and suitably speeded; the shaft being bracketed out forward from and beneath a horizontal frame which possesses a delicate dynamometrically governed horizontal fore-and-aft mobility, and is suspended below a truck, which at a definite distance astern follows the dynamometric truck, by which the model's resistance is determined. The screw, placed exactly where it would be if it were driving the model, rotates with a speed sufficient to drive it, but without touching it or affecting it except through the hydro-dynamic action which it is sought to measure. The force of rotation employed in driving the screw and the drag or thrust it exerts on the frame which carries it are both automatically recorded, and the speeding is varied until a speed is

* To defer thus the introduction of this form is equivalent to substituting the speed of the ship for the speed of the screw in all the power terms-in the E.H.P. as well as the rest. This arrangement is preferred on the ground that it appears to bring out with special distinctness the circumstance that all the elements enumerated, except slip, are alike virtual additions to the ship's resistance, and would equally exist if there were no slip, while the slip, taken separately and subsequently, represents the additional power expended on all the force elements alike, in consequence of the partial yielding of the point of reaction from which the propulsive force is taken. In virtue of the slip there are so many more revolutions per minute performed by the engine, with its total load-its superfluous load as well as its useful load.

found at which the total thrust equals the model's total resistance, that is to say its nett resistance + the augment in question. These experiments show that with ships of ordinary form the augment is from 40 to 50 per cent. of the ship's nett resistance, and in making up the account I shall rate horse-power due to (2) as = 0·4 E.H.P.

(3) Water Friction of Screw.-The Greyhound experiments showed that the additional resistance caused by the screw when it was allowed to rotate freely as the ship went ahead considerably exceeded 0.1 of the ship's natural resistance. Now the speed with which the screw revolved was less than that due to the speed of the ship, and to have driven it at a higher speed would certainly have required more force, or in other words would have been a greater drag on the ship; it cannot therefore be unfair to rate this as 0·1 of the natural resistance. Horse-power due to (3) is therefore = 0·1 E.H.P.

(4) Constant Friction due to Dead Weight and Tightness of the Moving Parts.-It has already been shown that this is at all speeds equal to about one-seventh of the total load on the engines when working with the maximum intended speed and pressure. And since the account is in the first instance taken as it applies to the highest speed, I take the horse-power due to (4) as = 0·143 total S.H.P.

(5) Friction due to Working Load of Engine.-This at the maximum speed can hardly be taken as being less than the dead-load friction, when it is borne in mind that the forces to which it is due exceed by many times the dead weights of the moving parts, to which principally the dead-load friction is due; I therefore rate it at the same amount, so that S.H.P. due to (5) is = 0·143 total S.H.P.

(6) Air-Pump Resistance.-According to Tredgold the load on the engine due to the air pump is between one-tenth and one-twentieth of the whole load on the engine. I shall set it down as 0.075. Thus the S.H.P. due to (6) is = 0·075 S.H.P.

The horse-power due to the several elements, worked out on the foregoing basis and combined, may be tabulated as follows:

And this conclusion agrees very fairly with what, as I have already pointed out, more general experience has led me to adopt as an average expression of the relation between indicated and effective horse-power-namely, that at high speed the former is about 2.7 times the latter, or the latter 37 per cent. of the former.

To convert the formula from one adapted to high speed only to one adapted to all speeds it is necessary to keep the term involving constant friction separate from the rest, for it represents simply the effect of a constant resistance operating with the existing speed of the engine. In shaping the formula I shall adhere to the co-efficient 2.7, derived from rather broad experience, in preference to the co-efficient 2.582 just now built up on somewhat hypothetical data, assuming however that the constant friction is equal throughout to one-seventh of the maximum load. Of the 2.7 E.H.P. which make up the I.H.P. at the maximum speed V, one-seventh part or 385 is the part due to constant friction, leaving 2·315 as due to the other sources of expenditure of power. And to express the I.H.P. due to constant friction at any other speed v, we must alter the co-efficient in the direct ratio of the speed. So that the term becomes 385 × E.H.P. at designed maximum speed. Thus the formula for I.H.P. at any speed v is as follows:

I.H.P. 2·315 E.H.P. +385 × (E.H.P. due to V);

Or if we finally sever the useful from the collateral expenditure of power, it stands thusv I.H.P. = E.H.P. + 1·315 E.H.P+385 × (E.H.P. due to V).

V

The several elements thus calculated are shown in combination in Fig. 7 (Plate XI.) approximately to fit the case of the Merkara, but the figure mutatis mutandis will represent pretty nearly their relative value in the case of other ships. The results are given not as expressing a complete solution, but as a well-considered step towards it.

*See foot note, p. 173.