128 Mk. stokes, on some CASES OF FLUID INIOTION. 



Fourthly, the discrepancy alluded to may be due to the mutual friction, or imperfect fluidity 

 of the fluid. 



12. Motion about an elliptic cylinder of small eccentricity. 



The value of (p, which has been deduced, (Art. 8), for the motion of the fluid about a circular 

 cylinder, is found on the supposition that for each value of r there exists, or may be supposed 

 to exist, a real and finite value of <p. This will be true, in any case of motion in two dimensions 

 where udx + vdy is an exact differential, /or tliose values of x for which thejiuid is not interrupted, 

 but will be true for values of r for which it is interrupted by solids only when it is possible to 

 replace those solids at any instant by masses of fluid, without affecting the motion of the fluid 

 exterior to them, those masses moving in such a manner that the motion of the whole fluid might 

 have been produced instantaneously by impact. In some cases such a substitution could be made, 

 while in others it probably could not. In any case however we may try whether the expansion 

 given by equation (3) will enable us to get a result, and if it will, we need be in no fear that it 

 is wrong, (Art. 2). The same remarks will apply to the question of the possibility of the ex- 

 pansion of (p in the series of Laplace's coefficients given in equation (10), for values of r for 

 which the fluid is interrupted. They will also apply to such a question as that of finding the per- 

 manent temperature of the earth due to the solar heat, the earth being supposed to be a homogeneous 

 oblate spheroid, and the points of the surface being supposed to be kept up to constant temperatures, 

 given by observation, depending on the latitude. 



In cases of fluid motion such as those mentioned, the motion may be determined by conceiving 

 the whole mass of fluid divided into two or more portions, taking the most general value of (p 

 for each portion, this value being in general expressed in a different manner for the different 

 portions, then limiting the general value of (p for each portion so as to satisfy the conditions 

 with respect to the surfaces of solids belonging to that portion, and lastly introducing the con- 

 dition that the velocity and direction of motion of each pair of contiguous particles in any two 

 of the portions are the same. The question first proposed will afford an example of this method 

 of solution. 



Let an elliptic cylinder be moving with a velocity C, in the direction of the major axis of a 

 section of it made by a plane perpendicular to its axis. The motion being supposed to be in two 

 dimensions, it will be sufficient to consider only this section. Let 



r = c(l + e cos 20) 



be the approximate equation to the ellipse so formed, the centre being the pole, and powers of e 

 above the first being neglected. Let a circle be described about the same centre, and having a radius 

 ■y equal to (l + k)c, k being <t e, and being a small quantity of the order e. Let the portions of 

 fluid within and without the radius y be considered separately, and putting 



r = c + jr, 



let the value of (p corresponding to the former portion be 



P + qz+ Rz-, 



P, Q and R being functions of 0, and the term in z^ being retained, in order to get the value of —i- 



dr 



true to the order e, while the terms in «', &c. are omitted. Substituting this value of (p in 



equation (2), and equating to zero coefficients of different powers of z, we have 



2c 2c- dO'' 



which is the only condition to be satisfied, since the other equations would only determine the co- 

 efficients.of ar^, &c. in terms of the preceding ones. We have then 



I 



