Mr. Herschel on circulating functions , &c 157 
however, to examine the integrable cases above alluded to 
-rather more minutely. 
If we leave out the consideration of the last term m + *p > 
which is allowable, because by the well known theory of 
linear equations, the part of the integral dependent on the 
last term can be supplied afterwards, we have I == o, and 
the rest of the coefficients of the equation (D) constant. 
Denoting them then by *H, 2 H m H, and supposing 
u, jG, <y, See. to be the roots of 
m . 1 xt m — 1 , 2 rj m — z . m rr 
z -f- H.3 -f- H . % +.... H = o 
let us take s to represent the function 
( v: r + cvjy + ( v; y + & c . 
m 
and the integral of the equation 
+ wm + * H * nA x+mn — n + • • • W H . “A, = O 
may be universally expressed as follows : 
"A x = S x { *C, .s t +‘ C, . s x _ n+ *C t .s x _ 2n + - C| . 
+ S »— 1 { ' C 2 ■ s i+ 2C a • s *_« + ■ • 
m C s 
2 ' x—mn-i-n 
+ S x-n+ 1 { ' C « * S x + 2 °n • S x _ n + • m C n . S x _ mn +n ] 
This may be proved without difficulty from the known 
form of the integral of the equation of a recurring series. 
Now since the values of 1 A X , 2 \ , &c. are all expressible by 
means of n A x as above described, it is easily seen that these 
are all in like manner circulating functions of a similar form, 
and since 
= ’A x .S x + 2 A x -S t _,+ 
„ + 
