924 
PROFESSOR A. CAYLEY ON THE SINGLE 
V,= -4( BWm'+C^CV), =-\.(Ah(Aht+Dh<Dht), 
at at 
V,= ChiCht — A z uA~u') , = :-(B 2 «BV-D 2 ?/DV), 
bg A ’ bg 
V3= _^(_AWw'+BW«'),= ^(CWk'-D^D^'). 
51. Hence v, Vu Vo, V 3 denoting these functions of x, x or of u, u', we have 
A(u-\-u)A(u u) Vi, 
B(«-H/)B(w— r')=^ Vo, 
C(?f + w')C(w— u') = ~ Vo, 
D(// + h')D(w— ?{') = J3v. 
The square-set u-^u', u indefinitely small: differentialformidce of the second order. 
52. I consider the original form 
C 2 0C(w+ u')C(u — u')=ChiC z u' — D~uD~u', 
(which is of course included in the last-mentioned equations). 
Writing- this in the form 
n „ C (u-i-u 1 )C (u—u') _ n2 , D%DV 
C -n Kj~u 
and taking u' indefinitely small, whence 
C(ii-\-u')-=Cu-\-u'C'u-\-\u' i C"n, CV=C0, 
C(u — u')=Cv — u'Cv. -\-^u' z C"u, D»'= u'D'O, 
C(?/-h^0C(w-^0 = Ch^+w 3 {C?rG^-(C / «) 2 } 5 
the equation becomes 
C20 ( 1+ “1S-(^)1)= C20 +“1 C0C "°-( D '0)^}> 
that is 
C"u_ /CW_ CO) _ /HO\ 2 D fi 
Cu \Cu ) CO \ CO J C he 
