30 
Proceedings of Boyal Soeiety of Edinhurgh. 
SESS. 
4. 4(^2 _ + 2 ( 3 a 4 - 1 + 1 m)xhf + i(a^ - 2hy-xhf + aY | 
- _ 4(3^6 _ 8^452 _ 8^254 + iQjje)jcY - 4(3a® - - Sa%^ + I6¥)x^y^ - 4:a^i/ : 
+ 6^4 + 4(3a^ - 4a^&^ - 1 2a%^)xY + QaY 
- — 4a4^y2 
+ ai2 = 0. 
A suspicion attaches to the coefficient of which, judging from 
tlie forms of the other coefficients, we should consider to be incorrect 
in not having a term in a%^. This suspicion is intensified on pro- 
ceeding with the simplification of the equation. In the first line we 
observe that the terms of the expansion of a4(^r^ + ^2)4 occur, and 
following up this observation we transform the equation into | 
u4(^2 + ^2)4 + 16&2(^2 _ ^ 2)^6^2 + 32/, 2(^2 _ ^2)^4^4 + 16/,2(52 _ ^2)^2^6 pf 
- 4a«(a2 + - 2h^)z*y^ + 32b%a* + a?h'^ - 2b*)x‘i/ pf 
+ 6a8(*2 + y^f - 1 &a?b\a* + Za?b‘^)xh/ ^ * 
- 4a40(i|j2 ^ ^2^ 
+ a42=:0. ' 
But the sum of the first terms of all the lines here is evidently ' 
-I consequently we make the further transformation 
a\x^^ + - a^)4 + - aPfXp-ij^iY + vY 
4- 32&2(a2 + W){<P - W)xY\^^ + y^) I 
- 1 ^cdh^icd -f 2>P‘hY^y‘^ = ^ • ' 
Now, were it not for the absence of the term above referred to, a 1 
most important simplification could have been effected. This is j 
best seen by actually supplying such a term, viz., — 4/^4 the | 
bracketed portion of the coefficient of x^y^. We then obtain ij 
- (py — 1 - hfxYf Y + y^y 
-2(a‘^+2b‘^){x‘^ + i/) 
\ + a\cd+Ab^-) 
and since the quadratic expression here in x^ -f- resolves into two fj 
factors {x^ + y^)-d^ and + ^2^ _ ^^2^ 4^2^^ former factor, ' !l 
x^ + y^- a^, becomes a factor of the whole of the left-hand side, and |j 
would with a more complete-looking coefficient of xYj have disap- | 
peared altogether and left for this particular case of the glissette an j' 
equation of the degree. As a matter of fact the equation takes | 
the form i 
j = 6ia%^xY ; 
