150 Mr. Herschel on circulating functions , &c. 
If x be of the form mn 1 , or if oc — * 1 be a multiple of n, 
we have in like manner 
q>(x) = i,o, 0} 
and so on. If then we take 
*,==/{#» 1, o, o, ... o } 
K = f{x,o,o,o,...i } 
the whole series of values of <p(x) will coincide, term by 
term, to infinity, with the corresponding values of 
P x = S x 4- V s *— i + ••••• V S x— « + 1 
and therefore <p(x) = P x , integer values of x only being 
considered. 
For example, 
a x = V S * + a x * S*— 1 + * a x • V — n + 1 » 
{ <v s ,+ V s x-.+ --VS,_. +I } x{^.s. + i',s,-,+.... *VS M+1 } 
= «, . y. . s, + b, . b' x . s, K- V, ■ s* -n + , ; 
{ a. S.+ ».S ,_,+ ••• k. S. _ „ + , }-= a*. S.+ S_, + ... T. S, _ „ + 
It should be remarked, that any number of the coefficients 
a x , b x , &c. may be zero, without infringing on the truth of 
this or the following propositions. 
(6). Every symmetrical function of S a , S^, S x _ w+i 
is invariable. 
Let / |x,y, z, &c.j denote a symmetrical function of 
any number of elements x, y, z, &c. (in the manner proposed 
