922 
PROFESSOR A. CAYLEY ON THE SINGLE 
(a 3 - j8 2 ) 2 XX' = a 3 . ChiChi' +/3 3 . D 2 wD V - afi(C 2 uT>' 2 u' + BhtChc), 
,i YY'=fi 2 „ fi-av „ a(3 ,, , 
whence 
(a 2 - / 8 a ) 2 (XX , +YYO = (a 3 +/3 3 )(C%C V-f DhtDht) - 2a/3(C^DV+DfitCV), 
(a 3 - /3 3 ) (XX 7 -YY') = {C 2 uC V - DW), 
(where observe that in taking the difference the right hand side becomes divisible by 
a 3 — /3 2 , and therefore in the final result we have on the left hand side the simple factor 
a 3 — /3 2 instead of (a 3 —/3 3 ) 3 ). 
Similarly 
(a 3 — /3 3 )YX / =a/3(C 3 liC 3 M / d-DhtD'hd) — a?Q~uG 2 u — /3 2 C' 2 uD' 2 u', 
„ XY'=afi „ -F „ -« 3 „ , 
and thence 
(« 2 -/3 3 ) 2 ( YX'+XY') =2« j 8(C 3 ttC¥+D%D%V( a2 +^)(C 3 MD%'+DWtt / ), 
(a 3 —/3 2 ) (—YX'+XY') = D+CV-CW+, 
48. Hence recollecting that 
A 3 0 = a 3 +/3 3 , 
B 3 0 = 2a (3, 
C 3 0 = a 3 —/3 3 , 
the original equations become 
C 4 0.A(M+u , )A(w-tf)=A 3 0(C 3 wCV+D 3 MDV)-B 3 0(C 3 uD 3 n'+D 3 wC 3 M , ) ; 
CfiO.B(« + +)B (w— w') = B 3 0(ChtCV+D 3 itD 3 w / )—A 2 0(C 3 wD% , d-D 2 wC 3 w / ), 
C 3 0.C(w+w')C (u-u')= C^W-DW, 
C 2 0.D(w+m')D(w — +) = D 3 wC 3 w'—C 2 mDV. 
49. It will be observed that the four products A(u-\-u)A(u — u'), &c., are each of 
them expressed in terms of C ~u, D 2 u, Chi', D 3 +. Since each of the squared functions 
A ht, B+, Chi, D 3 w is a linear function of any two of them, and the like as regards 
A hi, BV,CV, ~D 2 u', it is clear that in each equation we can on the right hand side 
introduce any two at pleasure of the squared functions of u, and any two at pleasure 
of the squared functions of u. But all the forms so obtained are of course identical if, 
taking x the same function of u that x is of u, we introduce on the right hand side 
x, x instead of u, u ; and the values of A(u-j~u').A(u — u) are found to be equal to 
multiples of v, Vu Vo, V 3 , where 
