104 Sir G. B. Airy, On a special algebraic function and [Oct. 31, 
The following communications were made: 
(1) Ona special algebraic function and its application to the 
solution of some equations. By Sir G. B. Arry, K.C.B., M.A., LL.D. 
In studying the elegant resolution, by Professor Adams, of the 
expression 2” +—,—2cosna into Factors, it has occurred to me 
x 
that an equivalent solution may be given, by a somewhat different 
process, of which the first step has not been usually recognized, 
but which may possibly be useful in other cases. 1 propose to 
apply it, in the first instance, to the equation a*—-1=0. A new 
function, with a new symbol, will be employed. 
We shall use the symbol 
aw (6) for the expression, cos 6 + /—1.sin@; 
and therefore «fp (=) 252.02. 5.20. - siseee et cos 0 — /—1. sin 0. 
Now the function w possesses these properties :-— 
(1) FO@x~(-O=1 
(2) hp) xv@ =v (pt 9) 
(3) (r(p) =p (np). 
(These properties are analogous to those which apply to the 
function e®.) 
We shall now consider the application of these equations to the 
solution of the equation w"—1=0, or «*=1, where n is an odd 
prime number: (the consideration of the prime 2 will quickly be 
found to be unnecessary). 
The equation (3) above will evidently give 
z=cosp+,/—1.sinp 
for the solution required, provided that we make 
vr (np) = 1, 
or cos (np) + / Salisin (np) = 1. 
This implies two conditions. First, np must be some angle 
which produces sin (np) =0, that is, np=0, or =7, or = 27, or 
= 37, &e. Secondly, we must preserve the condition cos (wp) = + 1. 
This excludes the values a, 37, 57, &c.; and therefore we must 
u 
retain only np=0, or =2m, or =47, &. to (Qn —2)7; alter 
