the curved lines indicate the paths of particles. Between any two of these stream-lines, the same particles would be found throughout the motion, and these would form a "stream" of which the stream-lines mark the boundaries. It will be noted that, as the streams approach the bow, they broaden, their speed being checked, and the particles diverted laterally; the amount of this diversion decreases as the athwartship distance of the stream from the keel-line increases, and at some distance athwartship the departure of the stream-lines from parallelism with the keel, even when passing the ship, would be very slight indeed. As the streams move aft from the bow, they become narrowed, having their minimum breadth amidships, where the speed of flow is a maximum. Thence, on to the stern, the streams converge, broaden, lose in speed, and finally at some distance astern resume their initial direction and speed. Since there is no friction, there can be no eddying wake.

So much for a vessel wholly submerged; a ship only partly immersed would be differently situated, because even in a frictionless fluid she would produce surface disturbance. At the bow, where the streams broaden and move more slowly, a wave crest will be formed, of the character shown in Fig. 120; amidships, where the conditions are reversed, some depression below the normal water-line will probably occur; and at the stern, where the conditions resemble those forward, another wave crest will be formed. Between the bow and stern waves a train of waves may also exist, under certain circumstances. The existence of such waves, when actual ships are driven through the water, is a well-known fact; every one readily sees why, at the bow, water should be heaped up, and a wave formed, but the existence of the stern wave is more difficult to understand. As remarked above, there is but one reason for both phenomena. A check to the motion of the particles is accompanied by an increase of pressure; the pressure of the atmosphere above the water is practically constant, and hence the increase of pressure in the

water must produce an elevation above the normal level, that is to say, a wave crest. Conversely, amidships, accelerated motion is accompanied by a diminution of pressure, and there is a fall of the water surface below the still-water level, unless the intermediate train of waves should somewhat modify the conditions of the stream-line motion.

These waves require the expenditure of force for their creation, and, when formed, they may travel away into the surrounding

FIG.120

fluid, new waves in the series being created. In the case, therefore, of a ship moving at the surface of frictionless water, the only resistance to be overcome will be that due to surface disturbance. For the wholly submerged body which creates no waves there will be no resistance, when once the motion has been made uniform; the stream-lines once established in a frictionless fluid will maintain their motion without further expenditure of power. This remarkable result follows directly from a general principle, which is thus stated by Professor Rankine :-" When a stream of

FIG.121

"water has its motion modified in passing a solid body, and re"turns exactly to its original velocity and direction of motion "before ceasing to act on the solid body, it exerts on the whole "no resultant force on the solid body because there is no per"maneut change of its momentum." In every stream surrounding the submerged body in Fig. 119, this has been shown to hold; each stream regains its initial direction and velocity astern of the body. The partially immersed ship in the frictionless water differs from the submerged ship in producing surface disturbance.

Perhaps the general principle will be better understood if we borrow one of Mr. Froude's many simple and beautiful illustrations. Taking a perfectly smooth bent pipe (Fig. 121), he supposes it to be shaped symmetrically, and divides it into four equal and

similar lengths, AB, BC, CD, DE. The ends of the pipe at A and E are in the same straight line; a stream of frictionless fluid flows through it, and has uniform speed throughout. From A to B. may be supposed to correspond to the forward part of the entrance of a ship, where the particles have to be diverted laterally, and react upon the inner surface of the pipe, as indicated by the small arrows f,f,ƒ, the resultant of these normal forces being G. At the other end of the pipe, from D to E may be taken to represent the "run" of a ship, where the stream-lines are converging and tending to resume their original directions; on DE there will be a resultant force J equal to G. Similarly, the resultant forces on the other two parts BC and CD are equal. The final result is that the four forces exactly neutralise one another, and there is no tendency to force the pipe on in the direction of the straight line joining A to E, although at first sight it would appear otherwise. The same thing will be true if, instead of being uniform in section, the pipe is of varying size; and if instead of being symmetrical in form, it is not so: provided only that at the end E the fluid resumes the velocity it had at A and flows out in the same direction. The forces required to produce any intervening changes in velocity and direction must have mutually balanced or neutralised one another, as in the preceding example, before the stream could have returned to its original velocity and direction of motion.

Applying these principles to the stream-lines surrounding a ship, it will be possible to remove one or two difficulties which have given rise to erroneous conceptions. It has been supposed, for example, that a ship in motion had to exert considerable force in order to draw in the water behind her as she advanced. As a matter of fact, however, the after part of a ship has not to exercise "suction" at the expense of an increased resistance, but sustains a considerable forward pressure from the fluid in the streams closing in around the stern. Any cause which prevents this natural motion of the streams, and reduces their forward pressure on the stern-such as the action of a screw-propeller— causes a considerable increase in the resistance, because the backward pressures on the bow are not then so nearly balanced by the forward pressures on the stern. Again, it will be evident that apart from its influence on surface disturbance-the extent of the lateral diversion of the streams, in order that they may pass the midship part of the ship, does not affect the resistance so much as might be supposed; since the work done on the foremost part of the ship in producing these divergences is, so

to speak, given back again on the after part where the streams converge. Very considerable importance attaches, however, to the lengths at the bow and stern over which the retardations of the particles extend; since these lengths exercise considerable influence upon the lengths of the bow and stern waves created by the motion of the ship. And, further, the ratios of these lengths of entrance and run to the extreme breadth of the ship must be important, as well as the curvilinear forms of the bow and stern, since the extent to which the particles are retarded in gliding past the ship must be largely influenced by these features; and, as we have seen, the heights of the waves will depend upon the maximum values of the retardations. In other words, with the same lengths of entrance and run, differences in the "fineness" of form at the bow and stern may cause great differences in the heights of the waves created, as well as in the energy required to create and maintain such waves.

Such are the principal features of the stream-line theory of resistance for frictionless fluids and smooth-bottomed ships. The sketch has been necessarily brief and imperfect but it will serve as an introduction to the more important practical case of the motions of actual ships through water. Between the hypothetical and actual cases there are certain important differences. First, and by far the most important, is the frictional resistance of the particles of water which glide over the bottom; secondly, friction of the particles on one another in association with certain forms, especially at the sterns of ships, may produce considerable eddymaking resistance, although this is not a common case; thirdly, friction may so modify the stream-line motions as to alter the forms of the waves created by the motion of the ship, and somewhat increase the resistance.

First, as to frictional resistance. Its magnitude depends upon the area of the immersed surface of the ship, upon the degree of roughness of that surface, or its "coefficient of friction,” upon the length of the surface, and upon the velocity with which the particles glide over the surface. From what has been said above, it will appear that this velocity of gliding varies at different parts of the bottom of a ship, being slower at the bow and stern than it is amidships. Professor Rankine endeavoured to establish a simple formula for computing the resistances of ships when moving at speeds for which their proportions and figures are well adapted. Under these circumstances he considered that "the whole of the appreciable resistance" would result from the for

mation of frictional eddies: in other words, that the wave-making factor in the resistance might be neglected. It is now known that this assumption was not a true one except for moderate speeds; whereas it was applied by Rankine to considerable speeds. On the other hand, his method of approximating to the frictional resistance and attempt to allow for variations in the velocities of gliding of the particles over the surface may still be studied with advantage. Rankine supposed that the wetted surface of a ship could be fairly compared with the surface of a trochoidal riband having the following properties: (1) the same coefficient of friction as the bottom of the ship; (2) the same length as the ship; (3) a uniform breadth equal to the mean girth of transverse sections of the wetted surface: (4) an inflexional tangent, making an angle with the base of the trochoid, of which the value was to be deduced from a process of averages applied to the squares and fourth powers of the sines of the angles of greatest obliquity of the several water-lines in the fore body. For any trochoidal riband in which the angle made by the inflexional tangent with the base was, Rankine had previously obtained the following expression for the resistance due to frictional eddies.

Resistance Length × Breadth × Coefficient of Friction

=

x (Speed)2 × (1+4 sin2 + sin*).

The last term was styled the "coefficient of augmentation." Hence

Resistance Coefficient of Friction × (Speed)2

66

x" Augmented Surface."

And his supposition was that for ships of good forms a similar expression would hold, within the limits of speed usually attained. For clean-painted iron ships the formula was very simply stated:Resistance Length x Mean Girth of Wetted Surface x Coefficient of Augmentation x (Speed in knots)÷100

=

=

Augmented Surface × (Speed in knots)2
100

This method of estimating the probable resistances of ships has been extensively employed by some shipbuilders, and is undoubtedly of use when the speeds to be attained are comparatively moderate. As the speeds increase, and the wave-making resistance assumes importance, the method necessarily fails; the total resistance then varies with a higher power of speed.

Mr. Froude investigated the frictional resistances of ship-shaped models, and as the result of a series of experiments came to a