26 
MR. W. SPOTTISWOODE OX AX EXTEXDED FORM OF 
Let, then, . . Up^ be arbitrary homogeneous functions of x, y, of the degrees 
respectively; then, since 
/(VK=/(i>K=0, because /(^) = 0, 
0 
we may make 
and consequently 
_1^ c, 1C, o_ 1^ C, 
V,/i~ 
fi 
fi 
or replacing C, by its value. 
U= — 17-2 
fi 
■«n,. 
_l2 
_kA 
^ (/-/i){/-/2) Vi 
fiiPih 
This expression, however, leads only to illusory results, for ffpi) = 0 ; and consequently 
on developing y ^pj^p y even in descending powers of the first m terms vanish, 
and all the terms after the (?7i+l)th become infinite; the (??iT-l)th alone being finite. 
The following method is, however, free from this difficulty. Select any second suffix ; ; 
then 
_ N n 
f 
fr' 
_L 
fr^ 
1 
f 
fr^ 
:f-fj ^ 
^m — l 
1 
<3 
o 
1 
1 
<1 
14 -v.)- 
And if c4;y, /S^y, . . be the roots of 
/;-/=« 
when solved with respect to V, we shall have 
and 
- Ivl 
?/_ — k 2 
.(y;-K-)-/:K-)) .. - Vi) .. (/,K-) -/,„(%•)) 

[fMij) -fMj)) • • (//%■) -Vy) .. -/j%)) 
+ similar terms in B,^-, /3,y; ...|. 
tj j 
