420 SIR J. F. W. HERSCHEL ON THE ALGEBRAIC EXPRESSION OF THE 
= 1 0x®4- lOx^— 30^— 90:c.2^| 
+ ^{o.60,+9.60._i+104.60,_,-387.60,_3-576.60,_,+905.60,_5 
— 216. 60^_e— 351 .60^_7— 256.60^_8+9.60^_9 + 360.60^_, 0—31 .60^_„ 
-576.60,_,3+9.60,_, 3+104. 60,_h+225.60,_, 0-576. 60,_, 6+329. 60._„ 
-216.60,_i 3-351.60,_, 9+320. 60,_,o + 9.60,_, 1-216. 60,_32-31.60,_33 
-576.60,_34+585.60 ,_,o+104.60,_,6-351.60,_,7-576.60,_38+329.60,_39 
+360.60,_3o-35I.60,_3i-256.60,_33+9.60,_33-216.60,_34+545.60,_3o 
-576.60,_36+9.60,_37+104.60,_38-351.60,_39+0.60,_4o+329.60,_« 
-216. 60,_42- 35 1 . 60,_43— 256 . 60,_44 + 585 . 60,_4o — 216. 60,_46- 3 1 . 60,_4; 
-576.60,_48+9.60,_49+680.60,_oo-351.60,_oi-576.60,_o,+329.60,_o 3 
-216.60,_o4 + 225.60,_oo-256.60,_o6+9.60._o7-216.60,_o8-31.60,_o9j. . 
(40.) The periodic function 0.60^+ ....&c. maybe somewhat simplified byresolving 
it into the sum of three others, having respectively 10, 20 and 30 for their periods. 
For on inspecting its coefficients, we find that the differences of any two, distant from 
each other by 30, are alternately +360 and —360. Now if we suppose, generally, 
any such function as 
flo.60^+a,.60^_i+&c. 
to be made up of the sum of three others, 
;?o-30^+/?i.30^_i + &c. 
9o-20^+^i.20^_, + &c. 
ro.lO^+ri.lO^_, + &c., 
we shall have, supposing i any number of the series 0, 1, 2, ... 9, 
7^i + ?t + ^’t = ^iJ /^i + 10 + 7i+I0 + ^i — ®i+105 /^i + 20 + 9’i + ^’i— ®( + 20 
J»i + ?;+10 + n = «i+30, Pt+10+9i + ri = ai + 40, Pi + 20+?i+10 + n = «i + 50 
which give the following equations of condition among the coefficients a, 
^30 + i ^i— (®40 + j ^lO + i) ®50 + i ^20 + i 
And if these be satisfied (as in this case they are), we have only further to establish 
the following relations between p, q, r, viz. 
Pi+qi+ri=ai 
Pi+lO Pi+(^i+10 ^! + 3 o) ? Pi + 20 Pi~^ {^i + 20' 
9i+10 — §'i+ (^t + 30 ^i) • 
Among the sixty coefficients therefore which this assumption places at our disposal, 
twenty remain arbitrary, and may be put =0. 
Suppose, for example, 
qi=0, Pi+.2o=0, 
