1182 
MR. R. F. GWYTHER ON THE DIFFERENTIAL 
Multiplying factor 
A only. 
= 1 + — d — cZ;r) AX — dA — X), 
Vn = Y,, + A {{2n — 1 ) XY„ + n {n — 2 ) Y„_i}, 
^ ^ = I _ X - A - X)2 + 2X (^ - X)}, 
= 77 - P - A - X) (tT - P) + X (tT - P)]. 
Hence 
{(^-X)^+ 2 X(^-X)} 3 ^^^+ {(^-X) (77-P) + X(.-P)}g^^ 
— S {{2n — 1 ) XY„ + n (w — 2 ) Y„_i} 
= {d (I - X) - {2iu -d- cZ,) X}/ 
The only new condition arising from these terms is 
(f-X){(f-X)g^ + (.-P)3^-c?/}=S»(..-2)y._,|_ . (13). 
Lastly, 
B only. 
Multiplying factor 
= 1 + B{(i^; — 2 cZ — 2cZ;y) Y + (w + cZ + XY^ — d [t] — Y)} 
yn = Y,j + B {{n — 2 ) YY,, + S(y> — 1 ) «C^YpY;i_^ + XS„C^_iYpY.„_jB+i} 
^_^ = f_X-B{(^-X)(7r-P) + Yi(^-Xr+(Y + XYi)(|-X) + X(7r-P)] 
^ ^ _ P _ B {(tt - P)2 + Yi(7r - P) (^- X) + (2Y - XYi)(7r - P)} 
therefore 
{(f- X) (tt - P) + Y, (f - X)^ + (Y + XYi) (f - X) + X (vr P)j 
+ {{n - P)= + Y, (^ - P) (f - X) + ( 2 Y - XY,) (^ - P)} 
- 2 {(h - 2 ) YY, + 2(j.> - 1 ) .C^Y,,.^ + X 2 „C,-iY,Y„.„i} ^ 
= {(Z (17 — Y) + { 2 d + 2cZ,: — ?(;) Y — (2t; + cZ + d^) XYJ f. 
We have again repetitions of previous conditions with the sole ne^v condition 
(^_P) |(, _ p)^^ + (f_x)g^-c(/-|= SS(p-l)„C,Y,Y„.,|- (14). 
