64 U. S. COAST AND GEODETIC SURVEY 
iSeate Son San+ SSSR 2e5 | sin 0 
+[Si +S ei) tSenpy+ Sa aoocse : 
er Se ae (Ol me Gal) aan eee ] sin a u 
sh [So-+ S riz) + S en+2) a= -------- ; 
— @—2) — @n=2) S Gn=2) a = - = -- == ]sin2au 
+[Si+Sesryp+tSenty+ ; 
1D) a (on—l) — Sign— 1) — Ab Lads ae ] snlau (262) 
The first term in the above equals zero. The remaining terms will 
take complete account of the series = S,, sin m a u, if a when 
n is even, or oe when n is odd. 
From the foregoing it is evident that the limit of m will not exceed 5 
192. If we let u and a represent any angles with fixed values, m and p 
any integers with fixed values, and a an integer having successive 
values from 0 to (n—1), it may be shown that 
a=(n—1) , nsnmu . 
>> sin (am tee = sin [3 (n—1) m uta] (263) 
a=0 2 
a=(n—1) inl m ; 
>) cos (am a-+-a)= 2, — cos [4 (n—1) mu+al] (264) 
oe sin 4 m u 
a=(n—1) int ‘y—m , i ae, pea 
>> smavusinam Ge pe ee eee aD i) bs cost Ati ieee 
a—o sin 2 (p—m) WU 
ts sin 3 2 (pm) L COs 3 (n—1) (p+m) u (265) 
sin 5 (p—m) u 
a=(n—l) sin 3 n (p—m) u cos 3(n—1) (p—m) uw 
beats 2 2 
Za cosa pucosamu=%z ann (ee 
1 sin 3.7 (p+m) u cos 3 (n—1) (P+m) u 
3 sin 4 (p+m) u RD) 
a ine ol) ee sin 4 n (p—m) u sin 4 (n—1) (p—m) u 
Balas 2 \ 2 
2a sn apucosamu=%z Ss ee 
u sin 4” (pm) i sin 3°(n—1) (ptm) u (267) 
sin $ (p—m) wu 
193. If we let a=0 and we", or 7 u=27, then formulas (263) to 
(267) may be written as follows: 
a=(n—1) | 
> sin a m U= 
=O Sat 
SIO SS 15 
Nn 
sin m 7m sin (1 1" r) 
———— (268) 
F m 
sin mm cos{ m r—— 
a=(n-1) 5 ( n ) 
>) cos am Y= (269) 
73 on th 
ane sin — + 
n 
