130 
[740 
740. 
ON CERTAIN ALGEBRAICAL IDENTITIES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xvi. (1879), 
pp. 281, 282.] 
If P 0 , P u P, are points on a circle, say the circle a? + y* = 1, then it is possible 
to find functions of (P 0 , P x ) and of (P 1; P 2 ) respectively, which are really independent 
of P 1} and consequently functions of only P 0 and P 2 : the expression “function of 
a point or points ” being here used to mean algebraical function of the coordinates of 
the point or points. Thus the functions of (P 0 , P x ) and of (P 1} P 2 ) being x Q x x +y 0 y x , 
x 0 y 1 — x x y 0 > and x x x 2 + yiy-z, x x y<, — x.,y Y , we have 
{X&* + Ms) (Mi + Ml) + (Ms - Ml) (Ml - Mo) = M2 + Ms. 
and another like equation. This depends obviously on the circumstance that the 
coordinates of a point of the circle are expressible by means of the functions sin, 
cos, x = cos u, y = sin u ; and the identity written down is obtained by expressing the 
cosine of Un — m 0 , = (m 2 — u-i) + (u x — u 0 ), in terms of the cosines and sines of u^ — u x 
and u x — u 0 . 
Evidently the like property holds good for a curve, such that the coordinates of 
any point of it can be expressed by means of “ additive ” functions of a parameter 
u; where, by an additive function f{u), is meant a function such that f(u + v) is 
an algebraical function of /(u), f(v); the sine and cosine are each of them an additive 
function, because 
sin (u + v) = sin n V(1 — sin 2 v) + sin -y \/(l — sin 2 u), 
and, similarly, for the cosine. But it is convenient to consider pairs or groups f{u), 
<£(«),..., where f (u + v), cf)(u + v),... are each of them an algebraical (rational) function 
of f(u), cf> (u),, f (v), the sine and cosine are such a group, and so also are 
the elliptic functions sn, cn, dn; but the H and ©, or say the ^--functions generally, 
are not additive.