216 Dr. L. Isserlis on the Variation of 



So that (22) becomes, correct to 1/n, 



X-X = £ 



w(l — u 2 ) uX 2 — vw v(l — V 2 ) {v — wu) 

 = 2n~ X(l-u 2 ) 2n~ X(l-u 2 ) 



w(l — w 2 )(w — uv) 

 2nX(l-w 2 ) 



j(l-u>) 2 X\l + 2u 2 )-2uvivX 2 -v 2 w 2 j(l-v 2 ) 2 w* 

 + • n X\l-u 2 ) 2 + nX 3 (l-w 2 ) 



^(1-^ 2 )V (1-m 8 )(1-p») / uvA \ 



+ tiX 3 (1-w 2 ) + n \ W 2(l-w 2 )(l-t? a )/ 



vio(r — ?to) — (i# — uv)X 2 

 X X 3 (l-^) 2 * 



(1-^)(1-^ 2 ) / ivuA \ 



+ n V 2(1-™ 2 )(1- W 2 )7 



vw(iv-uv) — (v — wu)X* 

 * X 3 (l-w 2 ) 2 



(1 — w 2 )(l — v 2 ) / _^ wA \ iw 



~nT ~~\ u 2{l-v 2 ){l-w 2 ))x 3 {l-u*)' 



or 2fX 3 (l-w> 



= X 2 {-u 2 (l--u 2 )X 2 + uvw(l-u 2 )-v(v--wu)(l--v 2 ) 



— w(w — uv){l — w 2 )} 

 -\-X\l-u 2 )(l + 2u 2 )-2uvw(l-u 2 )X 2 -v 2 io 2 (l-u 2 ) 



+ w 2 (l-v 2 y + v 2 {l-w 2 y 

 -i {2w(l — v 2 ) -uv(l — X 2 )}{vw(v— wu) — (w — wv)X 2 } 

 + {2v{l-w 2 ) -wu(l-X 2 )}{viv(iv-uv) — {v — ivu)X 2 } 

 + {2(l-v 2 )(l-io' 1 )u--viu{l-ii 2 )(l-X 2 )}(-vw) 

 = ^X 4 + ^X 2 + ^say. 



Collecting the coefficients of the powers of X, we have first 

 (£■=. 1 — w 4 -f u 2 v 2 -f u 2 w 2 — 2uvw 

 = (l-u 2 )(l + n 2 + u 2 X 2 -2uvw). 



