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 <p and x hi the first of formula (3), 

 we have 



3 cotres (- 6, - v) =-, — —7 r ; -, r -, ^ 



v y ' Ay {cotr(^x) + atres(0 >x ) + a 2 tres( x ,0)) 



1 



cotr (<p, x) + a 2 tres (0, x) + « tres (x, <p) 



1 



+ cotr (0, x ) + tres (0, x ) + tres ( x , <f>) 



and on adding these fractions, we get 



cotr (- 0, - x) = cotr 2 (0, x ) - tres (<j>, x ) tres ( x , <j>). 



By a similar process we should obtain 



tres (- 0, - x) = tres * (x> 4) ~ cotr ifr X) tres (0> x)> 

 tres (- x, - i>) = tres 2 (>, x ) - cotr (tf>, x ) tres ( x , tf>). 



These last expressions are particularly useful in geometrical 

 applications of this theory. 



" The known formulae for the sine and cosine of the sum 

 of two arcs may be most readily derived from the equation 

 ep-i = cos 9 + sin 6 . \f- 1. 



In like manner we may obtain formulas for the tresines and 

 cotresine of <j> + h and x + ^ from the equations (2). Thus, 

 if 1 denote a symbol of distributive operation such that t 3 = 1, 

 whilst 1, t 2 , and 1 are absolutely heterogeneous, we shall have 



ef* = A + ifx + i 2 v, 



e* 2 * = A'+ *>' + iv ; 

 whence 



e«*+«*x = cotr(0, x) + ttres (<£, x) + t 2 tres ( x , <£). 



In like manner, 



e h»ik = cotr ^ ^ + ( tre8 ^ %) 4. t 2 tres ^ /^ 



