642 



SCIENCE. 



[N. S. Vol. VIII. No. 202. 



only, while Vortex Motion will be omitted 

 altogether. 



The application of mathematics to the 

 solution of many natural problems in fluid 

 motion possesses one difBculty which is not 

 ■common to most of the problems which 

 confront phj'sicists. It arises from the fact 

 that we are frequently unable to apply the 

 method of approximation. It usually hap- 

 pens that when a problem arises from some 

 natural phenomenon it is not capable of 

 direct solution. But the mathematician is 

 generally able to consider a simpler problem 

 which more or less closely corresponds to 

 the given conditions. Having solved it, he 

 is able to take into account the conditions 

 of the actual problem, and so to obtain a 

 solution to any degree of accuracy which 

 may be desired. This is frequently not the 

 case with problems in fluid motion. The 

 difierential equations of motion may, per- 

 haps, be written down, but the limitations 

 which have to be imposed before a solution 

 can be discovered are so numerous that the 

 solution, when found, often gives no ap- 

 proximation at all to the real circumstances. 

 To illustrate this, we need only mention 

 the case of a sphere moving through water. 

 If we neglect the friction between the 

 water and the surface of the sphere, the 

 viscosity and compressibility of the water, 

 it is easy to find all the circumstances of 

 the motion. But when the velocity of the 

 sphere is not small, and we take these 

 neglected circumstances into account, the 

 motion, as any one will agree, quite changes 

 its character. Instead of the stream lines 

 — that is, the lines followed by the molecules 

 of the fluid — being regular curves, eddies 

 are formed along the surface of the sphere, 

 and the motion in the rear of the sphere 

 becomes turbulent and seems to defy all at- 

 tempts at calculation. Further, the resist- 

 ance to motion caused by the fluid, instead of 

 being zero, as in the simplified problem, act- 

 ually becomes very large. And here, it is to 



be remembered, we are dealing with a simple 

 case of a class of problems which has a 

 high practical interest — the resistance ex- 

 perienced by a ship moving at a speed 

 which we are accustomed to expect, say, 

 from ten to twenty miles an hour. The 

 engineer now knows fairly well the resist- 

 ance by experiment. But neither he nor 

 the mathematician can calculate the resist- 

 ance when the speed is forty miles an hour, 

 and this speed has already become an ac- 

 complished fact. 



I shall first deal with problems in which 

 the motion is irrotational, that is, where the 

 separate molecules of the fluid are not sup- 

 posed to possess any rotation of their own 

 independently of the rotation of the whole 

 mass. In a fluid which is non-viscous such 

 a molecular rotation can never be set up 

 by any conservative system of forces. If 

 we neglect the viscosity, and the skin fric- 

 tion of the solid with which it is in con- 

 tact, all motions considered will be of the 

 irrotational class. This class can be again 

 divided into two others: first, that known 

 as continuous, in which the pressure never 

 comes out to be negative ; and secondly, 

 that known as dscontinuous, in which we may 

 have a negative pressure or a surface across 

 which there may be a finite change of ve- 

 locity. In the first case the fluid com- 

 pletely occupies the spaces around the solid 

 with which it is in contact. In the second 

 case hollows may be formed and there may 

 be a free surface, or the fluid in motion may 

 be in contact with other fluid at rest. After 

 treating these two classes of problems I 

 shall go on to mention the advances made 

 in various kinds of wave motion, including 

 the tides. Then will follow the motions 

 and forms of masses of fluid rotating about 

 an axis under their own gravitation only. 

 Finally, the influence of viscosity will be 

 considered. 



We shall first consider those problems in 

 which the fluid is incompressible and with- 



