154 A NEW METHOD OF DETERMINING 
In multiplying two trigonometric series together, called by HansEN mechanical 
multiplication, 
let «a, the coefficients of the angles 2a in case of the sine, 
2, those of the angles ux in case of the cosine, 
y, those of the angles vy in case of the sine, 
and 4, those of the angles py in case of the cosine. 
The following cases then occur : 
ad, Sin (Aw + py) + 30,0, sin (Ax— py) 
| 
a, sin A@ . 6, Cos py = 
tol 
6, cos ue. y, sin vy = 4 B,y, sin (ue + vy) — $6,y, sin (ue — vy) 
GB, cos ua .d, cos py = & 3,0, cos (ux + py) + 3,6, Cos (ua — py) 
a Sin Aw. y, Sin vY = — $ ~My, Cos (Aw + vy) + Fay, cos (Ax — vy). 
In every term of the second members the factor } occurs. Hence before multiplying 
we resolve the coefficients of one of the factors into two terms, one of which is 2. 
3 : ozo d2 »6dQ 
Performing the operations indicated, we have the values of aQ, ar As that 
* follow : 
