26 A NEW METHOD OF DETERMINING 
n being the number of divisions, we find 
R (c) 8 GC) ree 
() = A,,.cost(g,—eé,) —A,,.8int(g,—€&’,) (36) 
If now, for the purpose of multiplying the series together, we put 
(c) (c) 
A,, => C,,.cosvg + >. Co (37) 
(s) (c) (s) 
A, =2>S,,.cosvg +> S,,.sinrvg 
we have 
G) = Ps C,, cos vg +> C,, sin vg] cos 7(g—e’)—[= S.. , cos vg+> e ‘sin vg | sint (g—e’) 
(38) 
Performing the operations indicated we get - 
e e (¢) (¢) e e (c) . . 
=> cos (tg —te’).C;,, cosryg = =4C,,cos[(t+vr) g—te’ ]+35 4C,, cos [(t@—v) g—ae'] 
° . (s) . (s) . . . (s) . e . 
=> cos (4g —ze’).C;,, sinvg= =4EC,, sin[(¢+v)g—ce']—s>4 3 sin [(¢—v) g—“#’ | 
(c) 
—22 sin (tg —ite') S;, cosvg =—S> 1g. ‘sin [i i+v) g—ts' |— > 1s, sin [(¢—v) g—ve' ] 
e fe . (s) e (8) . . . e 
— sin (7g—ze’) S,, sinvg= ZS, cos[(¢+v) g—te’]— S518.,¢ cos [(¢—v) g—%’] 
Summing the terms we find 
(‘)'= SE1(0,, + S,.) ) cos | (¢=F v)g g—te |F4SS(C,48,) ) sin | (Fv) g—te' | (39) 
(c) 
From the formula of mechanical quadrature just given, we have C;,o, S;,o. when 
(c) (c) 
vy =0; but we know that they are $. C,,, § S,., as shown by their derivation. 
Thus 
(c) 
Jal = Le. ye Ci cos g + C., .cos 2g + ete. (c) (s) 
ea : = >C,, cos 1g + =C,, sin vg 
+ 6, sin g + é. . sin 2g + ete. 
(s) (c) 
(¢) (ce) 
A,=4 8. + S,, cos g + S,. cos 29g + ete. | 
io) s = SS, » GOS VG + SS. , sin 7 
+ S,, sing + S. sin 2g + ete. j J I: 
Hence where vy = 0, each series is reduced to its first term. 
