68 U. S. COAST AND GEODETIC SURVEY 
a=(n—1) a=(n—1) 
+C, >) cos2aucosapu+sS, >) sin 2aucosa pu 
= a=o0 3 
a=0 
a=(n—1) a=(n—1) 
+C, >5 cosak u cosa put+S; >) snalucosapu 
a=0 a=0 
a=(n—1) m=k a=(n—1) 
—H, >> cosa p u+ Cn >) cosamucosapu 
a=0 In=1 a=0 
m=l1 a=(n—1) ; 
-{- Sn D>) Shamucosapyu (287) 
m=1 a=o 5 
201. Examining the limits of (287), it will be noted by a reference to 
page 63 that k, the maximum value of m for the C terms is - when n 
is even and 1 when n is odd; also, that / has a value of 57! when 
n is even and —— d when n is odd. The limits of p, which is a partic- 
ular value of m, will, of course, be the same as those of m. 
=(n—1) 
By (273) the quantity = cos a p u becomes zero for all the 
3 a=(n—1) 
values of p, and the quantity >} cosa mucosa p u becomes zero 
a=0 
for all values of m and p except when p equals m. By (273), (278) 
a=(n—1) 
and (281) the quantity 24 sin a mu cos a p u becomes zero for all 
values of m and p 
Formula (287) may therefore be reduced to the form 
a=(n—1) a=(n—1) 
>> hs cosa pu=C,. >3 cos* a pu (288) 
a=o0 a=0 
For any value of p less than 5 
a=(n—1) 
De costa pu — ne ae) 
a=0o 
but when P=5) this quantity becomes equal to n (280). 
Therefore for all values of p less than a 
2 
2 a=(n—1) 
Q=- h, cosa pu (289) 
NM a=0 
but when 7 is exactly > 
1 a=(n—1) 
Oy >> hacosapu (290) 
a=0 
Since in tidal work 7 is always taken less than ~, we are not especially 
9? 
concerned with the latter formula. 
