﻿44 Mr. R. Hargreaves on Ellipsoidal 



and -~^ is a sum of terms of the form f^-fn-f^^ thus a 



condition of solution is that the square bracket must contain 

 the factor [lx-\-my-\-nz), or 



a(lp + mr' + nq') + y' (lr' + mq + np f ) + fi r {lq' ■+ mp' + wr) = /cl, 



v'( „ )+£'( » )W( » )=^,1 : - 



£'( „ ) + «'( „ )+7( » )=™- 



Moreover, the ratios I :m:n must be independent of the 

 particular # underlying a, while /e may be expected to con- 

 tain it. Supposing a to depend on the general parameter X, 

 treat (14) as a system of equations to determine the brackets : 

 the solution gives 



(Ip + mr' + n/)A a = k(IA u + mC a ' + nB a '). 

 But A a /A a = (A+pAA a )/A B [>/. JE. (85)], 



and therefore 



{lp + mr f + nq')& a = K{lA+p\A a + mC' + r'\A a ±nB' + g'A.A.), 

 or 



Z^pA^-*" 1 ) + ro(C' + r'A B A-*- 1 ) + w(B' + g'A a \-*- l ) = ; 



and there are two others. These equations are independent 

 of \, if 



\-k- 1 = o or *=l/(\-0o); • . • (15) 



and the system is then 



/(C' + ^ A a )+m(B^^ A a ) +n(A'+^A)=0, [ - (16) 

 l(B' + q'0 o A a )+m(A'+p'0 Q A a )+n(C + r0 o A a )=O. J 



The determinant here vanishes, and O is therefore a root of 

 A(\)=0 or J(\)=0, and there are three types of linear 

 factor corresponding to the several roots. Referring to (1), 

 and the equation defining the a's, viz. M. (81), we see that 

 the quadric u eo is proportional to 



{ X-0 o ){fi-e o )(y-0 o ), 



and as the quadric is then the square of a linear term, this 

 latter is proportional to 



Thus by Cartesian work we infer the existence of Lame's 

 functions with the radicals \/\ — O , . . . 



