Mr. Herschel on circulating functions , &c. 163 
coefficients of which, taken in their order, form subordinate 
periods within it, each consisting of the n coefficients of Pj^* 
Thus suppose n = 2, and N — 6 then 
n 2 ) = vsf+^-s'i 
The identity of these two expressions is easily recognised : 
when a? is a multiple of 6 , and therefore of 2, they both 
reduce themselves to a x ; when a?-— 1 is such a multiple, to b x ; 
when x — 2 is a multiple of 6 , x — 2 is also one of 2, and 
therefore x is so ; consequently both functions reduce them- 
selves to a x , and so on for every form of x. 
Thus every coefficient of the proposed equation may be 
reduced to one whose period of circulation is N ; and this 
being done, the equation comes under the general form inte- 
grated in (9). The second series whose law is stated in 
Art. (2) leads to an equation of this kind. In fact the 
equation 
(" s ?’+ b s“,+ * s<£ 2 ) - (« S« + a s2i,) *_ 2 = o 
coincides in succession with the whole series of equations 
there assigned ; and if we write instead of the coefficients of 
this, the following, 
" S (6) + b s£>. + c SW. + a S(f> 3 +bS^ + c S^. 
• sf + /S s<2, + - S?i 2 + /? s<» + « s + 0 sj« 
they are reduced to a common period of circulation, and the 
equation may then be integrated as above. It remains to 
consider equations with more than one independent variable 
whose coefficients are circulating functions. 
+ ", • s£ 2 + K- sj2, + ", • s£> + *>,. sW* 
