40 MR. J. H. JEANS ON THE POTENTIAL OF ELLIPSOIDAL BODIES, AND 



or 



since v must vanish when X 0. 



3 



Remembering that -^ = 0, and that f = eu } + ... , the terms in e~ in (37) now 

 ( X 



gve 



. en., 



giving on integration 



( ,,(^,^) ! . . . (40) 



in which o> is again the most general function possible off,?/, f (enabling us to carry on 

 the distortion to the second order in any way we please), and the lower limit of the 

 integral is taken to be zero simply as a matter of convenience. 



The addition of a perfectly general function would be equivalent to the superposition 

 of a perfectly general distortion (proportional to e 2 ) on to the distortion already under 

 consideration. The real object of the present analysis is to be found in its ultimate 

 application to the problem of the rotating fluid, and to solve this problem, it will be 

 found that need contain no terms of degree higher than 4 in f, tj, f, this being also 

 the degree of the other terms in .,. Hence in what follows it will be supposed that 

 it., contains no terms of degree higher than 4 in f, >], . 



A value of r 2 is obtainable from equation (38), but there are, as has been seen, many 

 possible forms for r,, and the most convenient is, in point of fact, obtained by going 

 back to equation (;J4), which in x, y, z co-ordinates is 



where x '" !i y be any function of .r, y, z, and X which vanishes (to the power of e we are 

 now concerned with) both for X and 0. 



Let two new functions w and w' be introduced, defined by 



rv, /A0\ 



- Vw " ........ (42) 



_iV^ ........ (43) 



