Mr. Herschel on circulating functions , &c. 145 
produced from one another by the following regular law, 
u Q — u 0 , u l == au 0 ,u 2 = bu l ,u 3 = au 2 ,u^ — bu 3 , &c. 
It is evident then that, were the coefficients a, b, equal, the 
series would be a recurring one of the simplest kind, viz. a 
geometric progression, and might be represented by a single 
• equation of differences of the first order 
u z-{- 1 === au z 
If this, however, be not the case, the series will consist of two 
distinct geometric progressions, the terms of which alternate 
with one another, thus 
u = u Q u 2 =2 abu Q u 4 = a*b*u 0 &c, 
u t = au Q u 3 =a*hu Q a 3 b 2 u 0 , See. 
It would seem then that no single equation of differences of 
the first order could comprehend all the terms of this series, 
so as to pass uninterruptedly from one to the other; and 
were this really the case, the method which has hitherto been 
always followed, of actually resolving it into the distinct 
series of which it consists, and instituting a separate process 
for the odd and the even values of x, so as to get the two 
equations of differences 
^ 2x 4- i ^ ^ 2x j a nd u 2x 2 ~~ b . u 2x -j- 1 
would be the only course we could pursue. That this, how- 
ever, is not the case, at least in the instance before us, is 
evident, if we consider that both these equations are included 
in the equation of the second order, 
u x + 2 — ab.u x 
the first integral of which will be an equation of the first 
order comprehending the whole extent of the series, provided 
the constant be properly determined by the equation 
MDCCCXVIII. U 
