156 Proceedings of Royal Society of Edinburgh. [sess. 
Y x + Y 2 = Y. a 1 {Z'(2r f 2s + 2) + Y'(2r, 2s + 2) } 
= C(r , s) y. a^SfatfC j 
y i = y.a 1 a 2 {Z'(2 r-l, 2s+2) + Y'(2r- 1, 2s + 2)} 
= 3C(r- 1, s)ya 1 a 2 SiC; 
y 3 = - C(r, s)Y. a 2 Y a i x + 3C(r - 1, s^a^Sa? ; 
y 4 = Y. a 3 a 4 • • • a 2r+3^ ra l a 2 a 2r+4 • • • — • • • 
= y. "(2 r - 1, 2 s + 2)8^2 + V. a 3 a 4 . . . a 2r+3 Y£a 2H _ 4 ...... 
y^(2r+l, 2s + l) = Y. £a 3 a 4 . . . a 2r+2 y a 2r+3 Y. a 3 a 4 . . . a 2H . 3 Y . . 
y 4 = - 2C (r, s)Y£e + y. £{Z"(2r, 2« + 1) + Y"(2r, 2s + 1)} 
= - 2C(r, s)Y& + 3C(r - 1, s)£$x + C (r, s)Y. £Va? ; 
Yj + y 2 + y 3 + y 4 = c(r, s)( y.^y^x - y.^Ya^ - 2Y^ + y. £Ye) + 
+ 3C(r - 1, s)(Ya 2 a 1 S^ + £Sx) = 
= C(r, s)(Y. a x a 2 x - Y. a 2 a x x - 2Y £x) = 0 ; 
or, Y(2r + 1, 2s+2) = 0 ; (21) 
Again, if Y(2r + 2, 2s -f 1) = Y 4 + Y 2 + V 3 + Y 4 , 
Vi + y 2 -y.ai {Z'(2r+1, 2s + 1) +• V / (2r+ 1, 2s +1)} 
= 2C(r,s)y.a 1 Ya^; 
y i = Y.a 1 a 2 {Z"(2r, 2s + 1) + Y"(2r, 2s +1)} 
= 3C(r - 1, s)Y. a 1 a 2 Sa? + C (r, s)Y. ai a 2 Ye ; 
Y 3 = — 2C(?*, s)Y. a^a Y x + 3C(r - 1, s)Y. a 2 a 1 Sa? + C(r, s)Y. a^Yx ; 
Y 4 = Y. a 3 a 4 . • •a ar HY a i02 a 2r+6 •••-•• 
= Y"(2r + 2, 2s - l)Sa 1 a 2 + Y. a 3 a 4 . . . a 2r+ ^V £a 2r+5 ....... 
Y^(2r + 2, 2s) = Y. £a 3 a 4 . ..a 2r+3 ya 2r+4 . ...... + Y. a 3 a 4 . . .a 2r+4 y£a 2r+5 . . . - 
So that, 
Y 4 = C(r + 1, s - l)y^S ai a 2 + C(r + 1, s - 1)Y& 
- Y. £{Z"(2r + 1, 2s) + Y"(2r + 1, 2s)} 
= C(r + 1, s - l)y ai a^ - 3C(r, s - l)£Sa; ; 
Yi + y 2 + y 3 + y 4 = c(r, s)(2Y. ^Ya^ - 2Y. a 2 y a ^ + y. a^Va?) + 
+ 3C (r - 1, s)Y. a 2 afix - 3C(r, s - 1)£S® + C(r + 1, s - l)Yc 
= C(r, s)(2Ya 1 a 2 x - 2Y a 2 a 4 ^ - 2Y. ife + Y. a 2 a x Yx - 3£Sa;) + 
+ C(r + 1, s - ^Ya^a? 
