146 A NEW METHOD OF DETERMINING 
For the angle og — og. 
W 
D1 0.0637n ee 8.3825n 8.4462 .-. + 2029794 
+104.75727 
4104.78521 
For the angles represented by (7g — g’), there may be cases when there are sensi- 
ble terms arising from g + H’, g + 2H’, etc.; if so, we use the column for h’ = — 1, 
and apply the proper numbers of this column to the coefficients of the angles named. 
Likewise in the case of (¢g + g’), there may be terms arising from the product of the 
numbers in the column h’ = 1 and the coefficients of the angles g + ZH’, ete. This 
will be made clear by an inspection of the two expressions 
©) a) 
(@g—g))= dv SG — B) — 2S, sn. (1g — 2H") ete. 
(2 (3) 
—— Jy % (ig + E’) — 2S, & (tg — 2H’) — ete, 
sin 
2) 
2 > C OS y i) 8) COSIN (Ey U 
(ig + J) = —dy & (ty — H') — 2Sy S&S (tg — 2H’) — ete. 
(0) ) 
+ Jy % (ig + Bl) — Wy % (ig + 2B’) + ete.; 
where ((ag — g’)), ((¢g + g’)) represent not the angles but their coefficients. 
In retaining the form (¢g + 7H’) instead of the form (— 7g — 7H’) we can per- 
form the operations indicated without any change of signin case of the sine terms. 
Making the transformations as indicated above, we obtain the following expres- 
a\3 
sions for the functions ue) and ua?(“) § 
