998 
PROFESSOR A. CAYLEY ON THE SINGLE 
142. The left hand side then is 
= P(Vaa^ x /bb / — x/bb^v/aa^+Q^/iaa /rf ~Abb -Abb,^Aaa }j ; 
the coefficients of P and Q are at once found to be 
= - i (a- b V5\ respectively, 
V ab A a A 
or observing that a—b, =a y — b /} —a — b, the equation becomes 
= _ A( X +/P/) ^ cdefa / b / + ( X +/A> v 7 cAe^ab} ; 
or multiplying by Aaba / b / and writing for shortness abcdef=X, a / b / c / d / e / f / =Y, this 
becomes 
AX] +ab{ -pQ^(X+/ra:) AY} = 0. 
143. There are, it is clear, the like equations 
b 7 c {P+^(X +A/)AX } +bc{Q-j--(X'+AA AY {■ — 0, 
c / a /(P“k^(k +/r y)AX} +ca{Q+^(X' +/x x)\Z\} = 0, 
and it is to be shown that X=X'=X" and /r=//= /T'. For this purpose recurring to 
the forms 
AaahAbb —Abb / 6Aaa / =^-^{(X+p.y)Acdefa / b / + (X + /x.r )\/cdef&h}, 
Abb/)Acc~—A ( cc / bAbb / = ^-^{(X'+P- , y) A adefb / c / + (X' + / u',r)A a / d / e / fbc}, 
Acc,d Aaa,— \/aaTbAcc / = ~{(X"-|-/A/) A bdeft^a, + (X''-f-AA Ab d / e / f ca}, 
multiply the first equation by Acc /5 the second by Aaa v , and the third by Abb}, and 
add : the left hand side vanishes, and therefore the right hand side must also vanish 
identically. 
