414 
SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
which being- multiplied by their respective coefficients, w^+w^_i+&c., we get for Z 
as follows : — 
^ f 1 
Z = + (46.) 
— { 1 + :^2 • w^-2 + . . • • (•? term s) } 
— {Xms-n^+Xms+i-ff^-i-i- (2^ terms)} 
— { X— 2S • n^+Xm-2s + 1 • + (35 terms) } 
— &c. 
The first line of this is a circulating function, linear in x in all cases except when 
^;=l,orm and 5 are prime to each other, in which case it loses its circulating 
character and becomes simply 
oc 
^(%0+Zl+)C2+ (47.) 
As regards the second line, we have 
—p.=Wn-n,-{-n— 1 ... 1 j; 
and since n is a multiple of 5, and therefore of v, the multiplier within the brackets is 
readily reduced to a periodic function having n for its period, such as 
&c., which, multiplied by — jo,, gives 
^|wa.w^+w— 1 .6.w^_i+&c.|, 
except when v—\, in which case this expression reduces itself to 
-(%0 + %l +•• --Xm-l) {^•W.r + W - 1 .W.._l+ .... 1 J . 
(48.) 
(34.) To apply the foregoing formulae to the expression of particular cases as 
2 3 4 1 2 
n(x), n(x), n(a?), &c., we begin with n(.r)=l. Therefore, to find H{x), we have 
(p(x) — l, (p'(x) 8 cc.= 0 , \}/:(s)=s—l, 5=2; 
consequently 
and therefore 
X=|; v('^+0*'!'') = -^; Y=-^.2,_„ 
n(j:) = 2(j^-2,_,). 
(35.) For the case of 5=3, we have 
(49.) 
; '4'i(>'’)=|; "4^2(5)=^ 4 ' 2 {^‘) = ‘‘^ 
2 
