168 Mr. Herschel on circulating functions , &c. 
Eliminating (c c - % ) from the two first of these, and 
C 1 
( c +7) from the two last we find 
+ £).{p'-(ab-«-e)f + «e} =0 
y«-( c +f- ).{?*— (ab— *— /3) q‘ + */ 3 } = 0 
but the values of p and q are such by hypothesis, that the 
latter factors of these equations vanish separately of them- 
selves : the equations are therefore verified independently of 
any particular values of C °r C +^r, which therefore 
remain arbitrary. C and C then being arbitrarily assumed, 
and c, and c ' determined from the above equations, which 
give 
c {p r + q 1 + pq + a.) + C'. ( pq — aQ 
a-pq>(.p+q) 
_ C' {(^) 4 + “</»* + ?* + pq )} ~ C. (pq? . (Pq-«) 
a.pq.{p+q) 
the second member of the equation (F) will be the complete 
value of u , and it will be noticed that we have 
X* 
pq = V 7 p,p t + Cf = ab— {a + / 3 ) 
(P + q) = V'ip- + + 2 p q 
whence the whole is readily reduced to functions of the pro- 
posed coefficients a , b , a, / 3 . 
JOHN F. W. HERSCHEL. 
Slough, Nov. 1817. 
