IN THE MOTIONS OF THE EARTH AND VENUS. 
73 
8. Supposing then k to be not less than 8 nor greater than 13, the term 
sin (k'rit'— kd — k s^ must be multiplied by ^(13 — k) {rit + s' — w') 
{k — 8) {nt + s — vr)^ in order to produce a term of the form A cos (13 n't 
_ s n t + B) whose coefficient is of the 5th order. The latter factor must have 
arisen from the product of two such terms as e' 13 “ * . (13 — k) {n't + s' — sr') 
and/” 8 .^ s (* — 8) (nt + s — w). The expansion of such a product will always 
produce two terms, one of which has for argument the sum of the arguments 
of the factors, and the other has the difference of the same arguments. The 
point to which I wish particularly to call the attention of the reader is this : 
The term of the product depending on the sum of the arguments is the only 
one which is useful to us. For instance ; the product of e' 2 . sin 2 {n't + s' — ®r') 
and e 3 . sin 3 {nt-\-z — w) will be — \ e ' 2 e 3 . cos (2w^ + 3 n /-}-2s' + 3s — 2sr'— 3 to-) 
. cos (2 n't — 3 n t + 2 s' — 3 s — 2 m' + 3 ■&) ; the combination of the first 
term with cos {Unit — Wnt 11s'— 11s) will produce a term of the form 
A cos (13 rit — 8 n t + B) whose coefficient is of the 5th order: the second 
term will not produce a term of that form. We might choose terms, as 
e' . sin {nit + s' — ns') and e 6 . sin 6 {n t + s — w) such that the part of the product 
depending on the difference of the arguments, or \ e e 6 . cos {n't — 6nt s'— 6s 
-<*' + 6 w) combining with such a term as cos(14/?V — 14w^ + 14s' — 14s), 
would produce a term of the form required : but its coefficient would not be 
of the 5th order. It is equally necessary to remark that, in multiplying the 
term thus selected by ^ (kn't — knt-\-ki — k s^, we again preserve only that 
part of the product depending on the sum of the arguments. 
9. On the circumstance that, in taking the product of two circular functions, 
we have to retain only the term whose argument is the sum of the arguments, 
depends the principle of our notation. For whenever (in an advanced stage 
of the operations) such a term as ^ ^2 rit + 3 n t + 2 s' + 3 s — 2 w' — 3 oc- 
curs, we shall know that, being formed in accordance with this rule, it must 
MDCCCXXXII. 
L 
