425 



" The exponential values of the sine and cosine are most 

 useful in all parts of analysis. Analogous values may be as- 

 signed to the cotresine or tresine of two variables. Since 



e '* +a2 * = cotr (<p, x) + a tres (0, x ) + a 2 feres (x, <£), 

 e**+*x = cotr (<j>, x ) + aHres (<p, x ) + a tres ( x , <£), (2) 

 e* + * = cotr (<j>, x) + tres (f, x ) + tres ( x , tf>), 



Adding these equations, and dividing by 3, we get 

 cotr ($, x) = i {e a * + ° 2 * + (£?*<** + e* + *}. 



In like manner we shoidd find 



tres (<p, x) = 3 {a 2 e a * + a2 x + ae a "f +a ^ + e* + "), (3) 



and tres (x ? i>) = 3 {ae a * +a2 *' -f a 2 e a2 * +a * + e***} . 



Multiplying together the first and second of equations (2), 

 we get 



e -*-* = cotr 2 (>, x ) +tres 2 (0, x ) + tres 2 ( x , $) -tres (<p, x) tres ( x , tf>) 

 - tres ( x , (j>) cotr (<p, x) - cotr ($, x ) tres(>, x ) ; 



and multiplying this again by the third, Ave find 



cotr 3 (<j>, x ) + tres 3 (<p, x ) + tres 3 ( x , <p) 



~ 3 c °tr (0, x) tres (tf,, x) tres ( x , 0) = 1. 



This equation, which holds good whatever be the values of 

 the variables (j> and x 5 corresponds in this calculus to the well- 

 known relation between the sine and cosine. 

 " In trigonometry we have 



cos (- 6) = cos 9, and sin (- 9) = - sin 9. 



The corresponding formula? in this calculus are the following : 



cotr (a<j>, a 2 x) = cotr (0, x ), 



tres (a<f>, a 2 x) = a tres (tj>, x), (4) 



tres (a 2 x? cup) = a 2 tres (x, #). 



" The relations between the cotresine or tresine of two va- 

 riables, and the same functions of these variables with their 

 2t 2 



