MR. A. R. FORSYTH OH A CLASS OF FUNCTIONAL INVARIANTS. 
81 
X and y nearly equal to Y. Hence, we take to be nnity, and ySg and yg small, say, 
— e and — 6 respectively ; and a and /3 are to be considered small, quantities of the 
first order being retained. Thus, we have 
J = 
0 , 0 , 1 
1, — e 
(1 _ eX - dY)-3, 
a, 1, — 6 
= 1 + 3eX + 3dY, 
so far as quantities of the first order. Also 
= — - A vgY = X+aY + eX-+SXY =<k{X,Y), 
U = ^ f ,x -\y = 'SX 4- Y +£XY+SY^ = ^(X, Y), 
to the same order ; and therefore 
(X + p, Y + a-) - (/> (X, Y) 
= p occr + e -f" 2Xp) + 9 (per + pY + crX), 
^ = rP{X + p, Y+cT)-rA(X, Y) 
= (T + ySp + e {per + pY + crX) + 6 (cr^ + 2Ycr). 
Hence, to the first order inclusive, we have 
_ pin'll' _j_ pm' 1 (j-"' + e {p^ + ‘YXp) + 9 (per -j- pY + erX)] 
+ n'p”''(r"'~^ {/3p -j- e (per + pY + crX) + 9 {cr^ + 2Ycr)], 
and therefore 
a, = I + m' (2eX + 9Y) + n' (eX + 29Y), 
>i' + l = (a + ^X), Om' + \, )1'-\ = R (^ + eY), 
+ n> = (w^' + n') e, n'+\ = (w + R) 
All other coefficients are negligible, being of a higher order of small quantities or 
zero (non-occurring); and these give all the combinations of values of m and n for 
« (^“'qr"'). Therefore, for all values of m and n, we have 
3X'« 0Y“ 0r-™ "bif 
[l+m (2eX + 9Y) + n (eX -f 29Y)} 
+ 
+ 
0wi + n ^ 
„_i R (a + 9X) + 
z 
dx"‘ ^ 02 /"+ 1 
0m + 
0*™ 02/'^ 
71 
0^m + l 0^; 
0;rt + « — 1 ^ 0r?^ + « — 1 
l/i (m + W — i ) € + - 
^ ' nr'P^ 
(/3+eY) 
71 {711 n — \) 9. 
MDCCCLXXXIX.-A. 
M 
