Mr. Herschel on circulating functions , &c 
159 
ar+'H.ct ™- 1 + . . . . m U = o 
Q m + 1 H . /3 m “ 1 + m H = o, &c. 
The above mentioned equations are therefore equivalent 
only to ( n — 1) . m distinct ones, which as we have already 
seen, is the number of relations which ought to subsist 
between them. Hence then, all the constants except those 
which remain arbitrary may be eliminated from the expres- 
sion of u x , and the result will be the complete integral 
required. 
( 10). Having thus determined the value of u x , when the 
last term of the proposed equation, or m + is zero, if we 
would then extend the integration to cases where it has any 
given form, the usual theory of linear equations will afford 
the requisite formulae, and their application will be attended 
with no other embarrassment than what may arise from the 
integration of explicit functions of the forms 
SS ,./(*) andsP ,./(*) 
Into this part of the subject, however, we need not enter 
at any length, because it may be avoided by pursuing the 
process originally laid down in (9) without neglecting the 
last term. We shall confine ourselves to the remark, that 
both the above expressions are reducible to the form 
a .S + b .S 4- 
X X • X X -I “ 
x— n - f 1 
and to the developement of one interesting case, viz. that in 
which/(.r) = 1, or the function to be integrated is simply S^,. 
Let us then denote 2 S x by « , and it is remarkable that in 
the deterniination of u x all the ordinary methods are unavail- 
ing. It is true that since is of the form 
