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‘¢ 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?to'x = cotr (¢, x) + a tres (¢, x) + a? tres (x, ¢), 
ev +x = cotr (p, x) + a®tres(¢, x) + a tres(x,¢), (2) 
ex =cotr(¢,x) + tres(¢,x) + tres(x, 9), 
Adding these equations, and dividing by 3, we get 
cotr (9; x) =i + {e ap +ax, + ewortax aE ex), 
In like manner we should find 
tres (¢, x) = } (aresPtx + aewwiax 4 eh*x}, = (8) 
and tres (x, ¢) = 4 {aere* re + azer? tax te e*x} ‘ 
Multiplymg together the first and second of equations (2), 
we get 
e~?-X = cotr?(, x) +tres*(p, x) + tres*(x, p)—tres (¢, x) tres (x5 $) 
— tres (x, ¢) cotr (¢, x) - cotr (¢, x) tres(¢, x); 
and multiplying this again by the third, we find 
__ cotr? (g, x) + tres? ($s x) + tres® (x; ) 
, — 3cotr (¢, x) tres (¢, x) tres (x, ¢) = L. 
This equation, which holds good whatever be the values of 
the variables ¢ and y, corresponds in this calculus to the well- 
known relation between the sine and cosine. 
<< In trigonometry we have 
cos (— 8) = cos 8, and sin (- 9) = — sin 8. 
The corresponding formule in this calculus are the following : 
cotr (ad, a’y) = cotr(¢s x), 
tres (ag, a?x) =atres(¢, x), (4) 
tres (a’x, ap) = a*tres(y, >). 
*¢ The relations between the cotresine or tresine of two va- 
riables, and the same functions of these variables with their 
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