428 Mr. K. Hargreaves 



o 



on 



§21. The solution of these equations can be shown in a 

 constructive manner with much less labour than the original 

 straightforward process by which it was obtained. 



Denote by A a the determinant of the ellipsoid u a with 

 reference to which a solution is sought, use A... A'... for 

 the minors of A a , and P... P'... for the minors of A p . Then 

 consider the equations 



«(A + P \A a ) + ytC' + ^+^jB^ q'\A a ) =A a . 



a {G' + r'\A a ) + y '{B + q\A a )+/3'(A'+p'\A a )=0 , . (81) 



a (B , + ^XA a ) + 7 , (A / +y\A a )+/3'(C + rXA a )= J 



and two other triads, the first of which has the variables 

 y'fia' on the left, and A a on the right transferred to the 

 second equation ; the other has /3Vry ? and A a appears in the 

 third equation. With A(X) for the determinant of (81), 

 and A(X)... for its minors, the solution is expressed by 



oA(X)=A„A(\), a / A(X)=A a A'(X). . . . (82) 

 Differentiate (81) with regard to \, and write it 

 i(A+l?XA fl ) + 7'(C , + r'\A tt )+4 , (B' + ff / XA ) + A a {px + r'ry ! + q'l3') = 

 a (C' + r / XA cr )+7 / (B + y\AJ +fi(A' +p , \A a )+A a (r' a + qry'+p't3') = 

 a(B' + r/\A tt )+7'(A / +/y\A„)+^ / (C + rXA c ;) +A a {q'a+p / n/' + r/3') = 0.. 

 Now solve these as a system to determine otj'0: for a we get 

 aA(X) + A*[A(X) {pot + r'y + q'ff) + C'(X) ( i-'a + ^/ +/Z/3') 



+ B'(X)(^«+jpV + r/80] = O. 



When A(X),... are cleared by the use of (82) the resulting 

 equation is the first of (80) . In the same way the solution 

 for 7' gives one form of the second of (80), and the solution 

 which belongs to the second triad gives the other form. The 

 triads are consistent but not wholly independent. 



When A(X), A(\), A'(X) are fully evaluated we find 



A(X)=^[l + X^(^ + 2 / ya')H-X 2 S(PA-h2P'A')+X 3 A^A a ], 



A {\)=Aa[a + \(qC + rB-2p'A') + X 2 PAJ, 



A'{\) = A a [a[ + \(gV' + r'B'-p'A~pA') + X 2 P'AJ ; 



and the solution, written at length, is 



J a =a + \(qC + rB-2p'A , )- r \ 2 FA a | 



J*' = a ! +\{q ! C' + r'B'-pA'-pA)+\ 2 F'A n . (83) 



J =l+xS(pa + 2p'a') + X 2 2(PA-f 2P'A') +X 3 A p aJ 



