318 



LIPKA. 



m 



2fQ.?^ +P. ?^) = 0, (r= 1,2, ...,n-l); 



dur du 



= 0, {r,k=l,2, ...,7i-l). 



y. \dVkdUr dUrdUk 



Applying our condition of orthogonality, we get 



dt 

 dX 



= Px + /i^x+...; 



dX 



dt 



^ = P, + tF,+ ....; 



du 



. =|f^_|.,^+ . . 



du, 



du, 



duk duk duk 



r = S ax, 1\ P,-^t^ ax, (Px F, + P, Fx) + . 



X, X, 



C7. = 2 flx, Px ^ + < 2 ax, f Px ^ + Px ^"l + 



X, dUr X, V dZ<r dUrJ 



\ II OUr X, dUrJ 



1 + 2 « 2 r?,P^ 

 + ....; 



f7. = i(2^,^^ + 2ax,Px^') + 



dUk 



= t 



^ ^*' ^ — Z •" ^~ ^— + -^ axM ^ X 7 — r- 



diikdUk duk dUr J X, dukon, 



+ 



6m r 



= t 



^ djax, Px) 5/m' 



+ ....; 



, \ dUrdVk dUr dUkJ X, dUr 



duk 

 ^ a (ax, Px) c)/, 



X, ^Wr 6w/c 



+ 



The vanishing of the coefficient of the first power of t in the condition 

 of orthogonality reduces to 



(91)2 



- X, 



5(ax, Px)5/, ^(ax, Px)5/, 



dUr duk 



dUr dtti 



+ 2 



dQ.dP, dQ.dP, 



= 0, 



d Ur duk dUk du 

 {r,k= 1,2,.... ,71- 1) 



