9 90 
PROFESSOR A. CAYLEY ON THE SINGLE 
We have II ,.9,,— 9 (r 9- t standing for 
3y{ii+u)3 Q {u—u')—3 0 (ii+ii)3y{u—ii), 
and here 3y(u±u) = 3 4f ±b3 4; , 3 {) (u^z u ') = 3 0 ~izb3 0 ; the function in question is thus 
(^+^)(^ 0 -hd 0 )-(^-hd 4 )(^ 0 +h.%) = 2{3 0 b3,-3,b3 0 }, 
where the arguments are (u, v), and the b denotes u'-rA-v'-'--. 
& \ c i u d v 
d 3 di + d r 9- 5 , that is d 5 (?6d-?d)d 1 (M — id) + -9- 1 (w+w'),9 5 (w — u), becomes simply = 2d 5 d l5 
and similarly 3- igdg + dgdis becomes = 2d 13 d 9 ; and the equation thus is 
— 2a ] yi(d 0 hd 1 —d t bd l) ) + (a 1 b8 1 +y 1 b/3 1 )d- / 9-id-( —+ 
where the proper suffix 1 is restored to the a, b/3, y, and bS. 
139. The equation shows that the differential combination 3 0 b3 4 — 3 4 b3 0 is a linear 
function of 3-3 1 and the coefficients of these products being of course linear 
functions of u' and v'; writing the equation 
l 0 M ( -S ( M t =AV,+B>iA, 
we can if we please determine the coefficients in terms of the constants c', c", c'", c iv , c Y ; 
viz., taking u, v indefinitely small, we have 
d 0 =c 0 , b3 { = u(c"'u+cj y v) + A(c 4 W +cfA), 
d 4 =c 4 , b% = u'(cq'"u + c Q iv v) + v(c 0 iw u + c 0 Y v), 
&l = c l> ^5 = Cs'u + C 5 "v, 
r9-g = Cg, Cy^U-^-C-^ V, 
or substituting, and equating the coefficients of u and v respectively 
c 0 (c 4 "V+c 4 V) — c 4 (c 0 "V+CqV)=A cic/ +Bc 9 c 13 ', 
c 0 (c 4 iv w'+c 4 V) — c 4 (c 0 iy ?d+c 0 V) = Ac 1 c b "+Bc 9 c 13 " 
which equations give the values of A, B. 
140. Disregarding the values of the coefficients, and attending only to the form of 
the equation 
»,M,-J 4 »,=A3A+B3A 
