IN THE MOTIONS OF THE EARTH AND VENUS. 
75 
will occur, the same rule must be followed, namely, that the only useful terms 
in the products are those in which the arguments are added. And whenever 
sin 2 -5- occurs in the coefficient, — 2 6 occurs in the argument; so that there 
will be no possibility of mistake in using the notation described in (9). 
11. On examining the third line in the last expression of (4), it will be 
seen in the same manner that the only part of-sin 4 -+ ^ cos (v' — v) 
— cos (v f + v — 2 0) > to be preserved is — — sin 4 . cos (2 v + 2 v — 4 6). 
The same remarks apply to this term as to the last; and for a similar reason 
the notation of (9) may be used without fear of mistake. 
12. By the use of this notation we may in some instances materially shorten 
our expressions. For instance, we might have the terms 
Fe'e 4 . cos ( rit + 4 n t + s' + 4 s — ns — 4vr) 
-j-Ge'e 2 sin 2 . cos ( n't + 4w£ + s' + 4s — cr — 2 — 2 0) 
■fHe 1 sin 4 . cos (n't + 4 nt i 4s — + — 4.0) 
All this would be expressed without the possibility of mistake by the follow¬ 
ing term, 
d e 4 + G e' e 2 sin 2 + H e' sin 4 . cos (1+4). 
The utility of such abridgments, and the quantity of disgusting labour which 
they spare, can be conceived only by those who have gone through the drudgery 
of performing the actual operation. 
13. It is only necessary to add that when we have, for the coefficient of a 
cosine or sine, a series proceeding by powers of e', e, sin 2 -^, &c. we may always 
neglect all after the lowest power. For instance, the correct expression for 
nt-\-z 
+ ^2 e —e3+ A e 5 — &c.^ sin (n t + s— zs) 
l 2 
V IS 
