1000 
PROFESSOR A. CAYLEY ON THE SINGLE 
whence also 
crdu-\-Tdv= 
dx 
dy_\ 
y/V’ 
7 , 7 J xdx ydy \ 
o>du+pdv--^y X -^ Y J, 
which are the required relations, depending on the square roots of the sextic 
functions X=abcdef, and Y=a / b / c / d / e / f of x and y respectively; but containing’ the 
constants nr, p, cr, r, the values of which are not as yet ascertained. 
146. I commence the integration of these equations on the assumption that the 
values u = 0, v= 0 correspond to indefinitely large values of x and y. We have 
X=z«(l-f+..),Y=/(l-|+.. ). 
where &=a-\-b-\-c-\-d-\-e-\-f; and thence the equations are 
hence integrating 
o-du+Tdv- if (l + f • ■ j-ifh + f • 
ndu+pdv=-i^ (l+^ . . )+if( 1+ 7 • 
a-u+TV— —■1( j —-„) + . . , 
nr 
u+pv- i(^~ 
and thence 
nril-{-pV-\-^((ru-\-Tv)— ~j-f . . , 
1 1 
where the omitted terms depend on —, q &c. 
Hence neglecting these terms 
ait + tv /I 1 . 
mt + pv + jS(cr?6 + tv) \x'y > 
an equation connecting the indefinitely small values of u, v, with the indefinitely 
large values of x, y. 
147. From the equations A = / n w\/ o. B = Ic~oj\/b, taking (u, v) indefinitely small and 
therefore ( x , y) indefinitely large, we deduce 
