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



Upon equating coefficients of the terms of degree 1 and 3 in x, y, z, we obtain 

 exactly the same equations as were obtained before when the terms in e 3 were 

 omitted. Thus no alteration is produced in the cubic terms until the solution is 

 carried as far as e 3 . 



The equations obtained by comparison of the coefficients of x 3 , y 2 , and z 2 are : 



W A W O 7t- ~T C/ IA/1 VI y I *> /-* . /* 



\cr ov 



and two similar equations. In these the terms in e 2 and the terms independent of e 

 must be equal separately. The latter terms again give equations (70), while the 



terms in e 2 give 



a^ 



(99) 



(100) 

 (101) 



Filially, upon equating terms of the fourth degree, we obtain the six equations : 



(107) 



We have seen that equations (70) to (73) must still be true, so that a, b, c, a, /3, y, K 

 will be the same as before. 



26. At present there are eleven quantities to be determined or eliminated, namely, 

 L, M, N, I, m, n, p, q, r, s, and n". The equations' giving these quantities can be 

 simplified in the following way. 



By an argument already used in 17, it appears that the terms in e 2 on the right 

 hand of equation (96) must be harmonic. Thus we must have 



6c u + c 12 + c, 3 = .......... (108) 



6c 22 + c 23 + c 21 = .......... (109) 



6C33 + <%1 + Cg., = .......... (110) 



d l +d 2 + d 3 = .......... (Ill) 



These are, of course, not new equations to be satisfied ; they reduce to identities 

 when the calculated values are inserted for o n , c 12 , &c. 



