106 Sir G. B. Airy, On a special algebraic function and [Oct. 31, 
&c., In one series; and 
x (cos > — ./=T.sin 7), 
n n 
== . 4G 
L— (cos — /=1.sin =) ; 
n n 
&c., in the other series. 
Uniting, by multiplication, the homologous terms from the two 
series, after «—1, and referring to equation (1) above, we find for 
the terms required, 
xz—l, 
2 
a* — 2x. cos— +1, 
4, 
a* — 2x. cos— +1, 
&e., 
at — 2.003 "= 7) 5 ithe 
whose continued product will form 2” — 1. 
We will now apply the same principles to Professor Adams’ 
Formula, or (2" + =) — 2 cos n4. 
I premise that 
2 cos na = (cosna +/— 1. sin a) + (cos nz —/— 1. sin na) 
=p (nz) + 4p (—na); 
and the Formula becomes 
a” +o” — abr (na) —  (— 2), 
or x” —W (na) +a” —(— na). 
This expression at once suggests that, to make the Formula = 0, 
x2” must = (na), and a” must = (— nz); (which two relations, 
by virtue of Equation (1), are equivalent). And, if 
a™ = cosm+/—1.sinm+(m), 
the Formula becomes 
(wr (mm) — yp (n2)} + [xe (— m) — be (— na), 
