.928 
PROFESSOR A. CAYLEY OR THE SINGLE 
the form clx \ 'a — x.b — x.c — x.d—x obtained in my paper “On the Double £-F unctions 
(‘ Crelle-Borchardt,’ tom. 87 (1879), pp. 74-81), viz. : for the differential equation 
dx t dy dz 
0, 
to obtain the particular integral which for y=d reduces itself to z=x, we must, in the 
formulae of the paper just referred to, interchange a and d : and writing for shortness 
a, b, c, d=a—x, b—x, c—x, d—x, and similarly a /; b /; c /5 d =a — y, b—y, c—y, d—y , 
then when the interchange is made, the formulae become 
I V— 
_y 7 d — b.d — e{y / adb,c / -f \/ a/l, be} 
(be, ad) 
_y d—b.d — c.(x—y ) 
y 7 adb,c, — y/ a d^bc 
_\/ d—b.d—c{ ybdc^H ybAca} 
(d— c)y abaA — (b — a) ^/cdC/d, 
a/ (A- < c) v/ aba,b, + (b-d)^- ede/f } 
(be, ad) 
A/^^{x/bda/;- \/hA ac ) 
A adb / c / — y 7 a Abe 
x /d—b.d—c{\/ cdajb, + y'abc A} 
(d — b) Aaca / c / — (c — a) Abdb / d / ’ 
(d—c)^y aba,b, —(5 —«)\/cdc ( d, 
/ \f C JZI7 l £ ~ ft ) -s/beb^c + (&—c) y 7 aba,b, 
(i d—b)\/ aca / c / — (c—a) v / bdb / d / 
C — 3 
a—z 
V 
V c l _ c _ _ 
, { (d-b) Acac,a, + (c-a) y / bdb J d,} 
V 
(be, ad) 
' d ^ZT a { V 7 cda A - y / abc 7 d / } 
y 7 adb,c ( — y 7 a Abc 
A/ T-cM d ~ a?> xAch/b-O-c) x/ada.d, 
(d — c)^/ abaA - (&—a) y 7 cdc / d / 
r d—c 
A 
<7—« 
(<x&, cd) 
(d—b) s/ aca / c / — (c—a)y/bdb,d/ 
