154 Proceedings of Royal Society of Edinburgh. [sess. 
B 3 = — C(r — 1, s)a 2 $'a 1 % — C(r, s)Y.a^Sfa 1 x + C (r, s)Ya 2 a 1 Bx 
4- C(r — 1, s)Y.a 2 a 1 Y^ 
= C(r — 1, s)(Y^Sa 1 a 2 — a^Sc^a?) + C(r, s)(a 2 Sa-|£C — Y ,a 2 afhx) ; 
B 4 = "V a 3 a 4 . . . a 2r 4-4Sa 1 a 2 a 2r _|_g 
= B"(2 r + 2, 2s - 2)Sa 1 a 2 + Ya 3 a 4 . . .a 2r+4 S£a 2r+5 
B^(2r + 2, 2s - 1) = Y.£a 3 a 4 . . .a 2r+3 Sa 2r+4 - ••-•*• + 
Y . a 3 a 4 . . . a 2r+4 S£a 2r _|_ g 
O = Y.f{S"(2r 4 1, 2s - 1) + Z"(2r + 1, 2s - 1)] + 
Y . a 3 a 4 . . . a 2r+4 S£a 2r+5 
B 4 = C (r, s — l)Ya?Sa-|a 2 \ 
Bi + B 2 + B 3 + B 4 = C(r - 1, s)YxSa^a 2 + C (r, s - l)Y^Sa 4 a 2 4 
C(r, s)(Va 1 a 2 x — Y^Sa 4 a 2 ) ; 
B(2r 4 2,2s) = C {r, s) Y H a 2 x • (13) 
So too, B(2r, 2s + 2) = C(r-l, s + ^Ya^x; . . . (14) 
Proceeding in the same way, take 
Y(2r+1, 2 s+1) = Y 1 + Y 2 + Y 3 + Y 4 ; 
Y 4 + Y 2 = Y.a 1 {Z'(2r, 2s + 1) 4 Y'(2r, 2s + 1)} 
= 3C (r— 1, s)a 4 Sa 2 ^ + C(r, s)Y.a,Ya 2 x 
Y 1 = Y.a 1 a 2 {Z"(2r-l, 2s 4 1) 4 Y"(2r - 1, 2s+l} 
= 2C(r - 1, s) Y.a^Yx; 
Y 3 = — 3C(r — 1, s)a 2 $a l x - C (r, s) Y. a 2 Ya^ + 2C (r - 1, s)Y . a 2 afhx * 
Y 4 = Y. a 3 a 4 . • • a 2r+3 Y a 1 a 2 a 2r+4 . ....... 
= Y"(2r+ 1, 2s- l)Sa 4 a 2 + Y.a 3 a 4 .... a 2r+3 Y£a 2r+4 . . .. - 
Yf(2 r 4 1, 2s) = Y. £a 3 a 4 . ..a 2r+2 Ya 2r+3 . Y.a 3 a 4 ... 02 r+3 Y^a 2r+4 . . . 4 . . 
O = Y.£{Z"(2r, 2s) 4 Y"(2r, 2s)}-V.a 3 a 4 ...a 2r+3 Y^ r+4 .... + ... 
Y 4 = 2C(r, s - l)Y^Sa 4 a 2 + C(r, s - 1)Y .£Vx ; 
Y 4 4 Y 2 4 Y 3 + Y 4 = C(r — 1, s)(3a 1 Sa 2 a? — 3a 2 Sa 4 ^ + 2Y a 2 afh x) 4 
+ C(r, s)(Y. a x Ya 2 x - Y. a 2 Ya 4 x) + C(r, s - l)(2YtfSa lC i 2 + Y.gVx) 
= {C (r — 1, s) +C(r, s - l)}(a 1 Sa 2 » - a^So^a? + 2Va?Sa 1 a 2 ) 4 
+ C(r, s)(Y. a x a 2 x — afa 2 x + a 2 Ba 1 X — Y a 2 a 1 x) . 
