i£>4 Mr. Herschel on circulating functions, &c. 
(12). A circulating function with two independent varia- 
bles may have a double period of circulation, and may in 
general be represented thus : 
pO» ») Jo, o) 
x,y u x ,y 
S‘ ml Sf + 4;-; 1 S< 1 \ s'"> + > sf 1 s<l,+ 
_L 1) o(m) e(«) 
— rn -f 1 — n -f 1 
one with three may have a triple one, and its expression 
will be 
x,y. 
X x, y, z ^x ^y ^z 
and so on, and these functions possess general properties 
analogous to those of one variable. To avoid complication 
we will confine ourselves to the case of two variables, a 
similar mode of treatment being applicable to any number. 
First then, any function whatever of x, y, S^, &c. 
and their successive values is reducible to the form n ) 
x, y 
and thus the sums, products, quotients, or powers of two or 
more such functions are reducible to a single one, as in the 
case of one variable. 
sdly. Any symmetrical function whatever of the values 
p(m, n) P (m, n) p(m, n) p (m, n ) 
1 x, y » * x — 1, y » r x,y — 1 * x — nz-f 1, y— «-(-i 
is invariable, provided the coefficients of each term in the 
expression of Pj ™’ y n ^ are so; and it is equal to a function 
similarly composed of those coefficients. 
gdly. Any circulating function of the form may be 
reduced to another of the form P^ ,NJ , M and N being any 
multiples of m and n respectively, and thus any number of 
such functions may be reduced to the same period of circula- 
