Mr. Herschel on circulating functions , &c. 165 
tion. These properties are easily demonstrated as in the 
case of a single independent variable, and by their aid all 
circulating equations of partial differences may be cleared of 
their circulating form. 
(13). Let the proposed equation of partial differences be 
U 4-( I ' o) P ( m > n ). u . (°, i)p (m,n) , 
U x,y r x,y l x—i, y T l x , y ' U x, y—i * ’ * 
(r-— 1 5 5 — 1) p(m, n) p (*»,«) 
x,y ‘ x — r»y — s A x, y 
To integrate it, we assume 
11 — A ( ° ,o) S (w) S (w) 4 - A (l * o) S (m) S lw) 4 - A ( ° J l) S (w) S (n) 4 - 
u x, y 1 x,y * ’ °i-i * x,y ’ ^x i r 
1 A (/n— 1,71— 1) c(m) c (») 
"T* * a? — ■ m + I M + 1 
and supposing the general representation of any one of the 
coefficients as ( r > s )p( m > M ) to be 
x,y 
(r,s) (o,o) q(w) q(«) , (r, 5) (1, o) o(m) o(«)i (>•>*) 0 qC^OqO) _L & c 
£>;y x y 1 • ^—1 ! a?, 7 - x _y — 1 1 
Then, if the above expression be substituted for u in the 
proposed equation, it becomes 
. o(m) 0(71) (4(0. °)4_( , ,o) ( o ,°) A(m— 1,0) , (o,i) (o,o) a (o, n — 1) _i o. 
* l x,y ‘ x, y ' x — i,y ~ x,y ’ x, y — 1 * 
• s£!, 1 A *.f + (I ’ 0) 4:f ■ k°~z+ + &<=. - 
•s«s« I q A ^ ,) + (, ’ o) ^ . A<rr;; ■>+<«• ■><*;>. A<*± t + & c . _ 
c(») I A (W — 1, W — 1) , (i,°) (m—1, n— 1) A (m— 2, 1) , & _ (m- 
m+i °,y— «+i • f 1 a x , y ’^V-i ,y T otu ^ 
Let now each term of this, enclosed in the brackets, be 
separately made to vanish, and we shall obtain a system of 
re-entering equations of partial differences, for the determina- 
tion of the unknown functions A^°’ °\ See. The number of 
x * y 
these equations is my.n, which being also that of the functions 
to be determined, they suffice for the purpose. 
