1180 
MR. R. F. GWTTHER ON THE DIFFERENTIAL 
2 yph^ ^ \]i p {B]\p — (2AX + BY) h — BXi/; — Ah^ — Bhxp] ]lp ! 
+ A j 
+ b{ 
rxY„ 
n\ 
2YY„ 
nl 
+ 
71 — 1! 
Y Y 
J-p -L )i—p 
+ s 
2}\n — p\ 
= 2/„{1-7i(2AX + BY)}/^! 
+ Bi S ypY,,_p+Jp — l\n—p A- 1! 
+ A 
+ B 
XY„ 
71 ! 
2YY 
+ 
Y 
n-\ 
Y„i 
71 — 1\ 
n 
2 ' 
--(p-i)sYAfL-x2 
y^n— 
p+l 
7l\ ' 'p \ 71 — p\ — 1\ 71 — 27 + l.'J ’ 
In the small terms we may substitute Y for y, and thus obtain 
y>i — ^11 ' 1^1 —< n—p+l 
+ A {(2n — 1) XY„ + n {n — 2) Y,„_i} 
-|- B [(?! — 2) YYa + S [^p — l) «CjuY^Y„_^ “b X S n^p-iYn-p+i] • • (5), 
where by S we denote that all possible values of y) are to be taken from 1 upwards. 
5 . Determinatio7i of tt — p and ^ — x. 
TT — p) stands in place of y—y — — x), and, upon transformation, y — y 
nPPOTTl 
^ _ Y - A X) (r; - Y) + X(.7 - Y) + Y(£- X)} 
-B{(77-Y)2+2Y(.y-Y)]. (6), 
f — X becomes 
^-X + B, (^-Y) 
-A{(^-X)^^ + 2X(^-X)} 
-B{(^-X)(^-Y) + X(^-Y) + Y(^-X)} . . . (7), 
and y-^ becomes 
Yi - + A (XY, - Y) + BYi (XY^ - Y).(8), 
therefore tt — p becomes 
^ _ p _ BYi {rr-B)- A {(I _ X) (tt - P) + X (tt - P)} 
~B{(7r-P)^ + Y,(7r-P)(|-X) + (2Y-XY,)(7r-P)] . (9), 
and ^ — X becomes 
^-X + B,{(7r-P) + T(^-X)} 
-A{(^-X)2+ 2X(^-X)} 
-B[(^-X)(7r- P) + Y,(^-X)"^+X(7r-P) + (Y + XY,)(^-X)] . (10). 
