426 
signs changed, are not quite so simple. They are, however, 
easily established ; and instances frequently arise in which it 
is necessary to avail ourselves of them. 
*‘ Changing the signs of # and y in the first of formule (3), 
we have 
es oe Ee SONS 8: 
{cotr(¢,x) + atres(¢,x) + a’tres(x, $)} 
1 
* cotr ($s x) + a’ tres ($, x) + a tres (Go) 
1 
* cotr (ds x) + tres (¢, x) + tres (x; ) 
and on adding these fractions, we get 
cotr (— ¢, — x) = cotr? (p, x) — tres (p, x) tres (x; ¢)- 
By a similar process we should obtain 
tres (— $, - x) = tres* (x, ¢) — cotr (5 x) tres (4; x), 
tres (— x, — ¢) = tres* (¢, x) — cotr(¢, x) tres (y; ¢)- 
These last expressions are particularly useful in geometrical 
applications of this theory. 
«© The known formule for the sine and cosine of the sum 
of two arcs may be most readily derived from the equation 
e® = cos 0+ sin@. Y-1. 
In like manner we may obtain formule for the tresines and 
cotresine of ¢ + h and x +4 from the equations (2). Thus, 
if . denote a symbol of distributive operation such that “= 1, 
whilst 1, .2, and 1 are absolutely heterogeneous, we shall have 
3 cotres (— $5 — x) = 
em=A+m +2, 
ex=Ni Ow tw’; 
whence 
eorex = cote (g, x) + etres (gp, x) + A tres (x, ¢). 
In like manner, 
eho = eotr (h, kh) + etres (h, hk) + & tres (A, h). 
oe el 
