Al. a. Tciiouproff 
205 
Putting ?•] = r.> — r in this, we find (cf. (13) of Chapter III) : 
+ f- 2?- fx.,r + r (2r - 1) fx-.r^. + r (r - 1) /a^+o /t^_2 
K'35). 
+ 4<r" fj,^+, yU,r_l - rt-'l /i^+i fMr-S fJ-i 
+ r(r+ 2) - 2r (r- -I) fir - 3?'' A^i - ''"^ ('' - 1 )y^r-i /^/•-2 /^s 
+ (r - 1 ) (r - 2) Mr_3 + I r"" (r - 1)- fi\_n /a./} + . . . ; 
(3) To find accurately the first term (of order 1/^-) in the development of 
we must start with 
r {r -1) 
2 
Using the relations (30), (31) and (32), we find without difficulty: 
^(v'ri - fJ-n) {v'r., - fir J Wts " f^>J 
+ {riV., (/i^g^o - fi^^ /zo) + /A^3.i (/z-r^+o - M2) 
+ '■•2^:i /*r2-i /^ra-i (/*ri+-2 - /^n /^2)] - '"i^a'":! /^n-i A^ra-i A^rj-i Mi! 
~ H'Ti /^ra+i /^r3-i ~ Mrs /^J-j) 
+ '2 (Mri+i 
+ (/^r3+i A^ri+r2-i + A'-ra+i A''j-i+?-3-i + f^ri f^r^ + ra ~ /^ri-i f^r^ l^r^+i 
+ [1\ /Z^j_i (/tr2+r3-i + 2/i^2+i /^'■a - f^r-i /^rj-i ^^2) 
+ nr-iflv^-i {l^r^+ra-^ H-2 + 2m,i+i /i/j - /^n Mr3-i M2) ; 
+ (^r2f ?-3-i /^2 + 2/^r2 /^r3+i - A^ra-i f^r^ 
+ ''2^':! /"?-3r-i (/^ri+rj-i /i2 + 2/A^^+i /x,^ - //,) 
, + (/X^j+rj-i /A, + 2/1,^^ - ^^j^, ^o) 
+ r^Vs/X^^^j (fJ'ri+T^-i M-i + 2/Ari /ir2+i - /*ri-i /^r2 /"i)] 
+ (rs - 1) /^r3-2 (f^n+r^ M2 + 2/Ari+i /"r2+i " H'Ti P-r^ H-'i) 
+ {l\ - 1) /ir2-l iP-n+ra + 2/i^j4i /X^g+i - /i^, /^r3 /^2) 
+ ('"i - 1) /^ri-i (iWr2+r3 /^2 + 2/i^^+, /i^^+j - fJ,,.., f^r, ^^2)] 
(36). 
