Mr. Herschel on circulating functions, &c. 153 
u 2 = au t + u Q , u 3 = bu 2 + k, , w 4 = au 3 + , 
u 5 = -f u 3 > &c. 
The period of the coefficients of the second terms of all 
these equations being a, b, if we take S to represent the sum 
of the <2?th powers of the roots of 2* — 1=0, the circulating 
function # + b S t will express the general term of the 
series a, b, a, b, &c. and the equation 
u x =(aS x +b S_.) + u t _ 2 
will coincide in succession with each of the given ones, by 
giving x every integer value from z to infinity. This then 
is the equation of the series, which it only remains to inte- 
grate. To this end assume 
U x = v x . VTs' x + b s ~ = v x . VT X 
putting P^for -f- b S x _ i . Then, by (7) we have P x _ 2 =P ;r 
and the equation becomes, by substitution, 
v . -t/P =P . VP v v . V P 
X V X x * AT— I* X— I < 2* V X 
that is, 
v ,= v ^,-^ P x- P x~, + V x-z 
but, VP x • P x _ t being a symmetrical function of P x and 
P x _ t is by (7) invariable, and equal to Vab, whence we have 
v = V a b, w r 4 - v 
an ordinary equation with constant coefficients, and conse- 
quently integrable by the usual methods. Similar considera- 
tions will enable us to arrive at the equations of all periodical 
series, and we shall therefore confine our attention to the 
equations alone, in the most extended form of which they 
are susceptible. 
MDCCCXVIII. 
X 
