424 



be expressed by means of the primitive ones; and a number 

 of useful and elegant formulae may be established, connecting 

 the different trigonometric functions of the same variable, or 

 of variables related to each other in particular ways. 



" In the new calculus we have to consider three primitive 

 functions, each of two variables, <j> and x? to which I have 

 ventured to give the names of cotresine ($, x)> tresine (<f>, x)> 

 and tresine (x> <f) I and I have found that the two functions 

 obtained by dividing the two latter by the first possess pro- 

 perties analogous to those of the trigonometric tangent, and 

 are sufficiently remarkable to entitle them to a particular de- 

 signation. I therefore propose to call them tritangents. 



" In the present paper I mean to give a few of the for- 

 mulae which result from a comparison of the functions already 

 noticed. 



" Employing the exponential development, putting a for 



~~ ^ -, the cube root of + 1, and writing n\ for the 



product 1.2.3 ... w, we find 



e a< l> = A + afi + a?v ', 

 where 



X=1+ 3! + 6! + &C - 



^ = ^ + 4! + 7! + &C - 



<j> 2 d> 5 8 , 

 v= ll + 5! + 8! + &C - 

 Again, 



e o2 * = Ai + a 2 fii + av\, 



where Ai, fiu vi are the same functions of % that A, /x, v are of 

 <(>. Hence e°* +ol!x = cotr (<j>, \) + a tres (<£, x) + a 2 tres (x, <f) if 

 we agree to put as definitions, 



cotr (</>, x) = AAi + /i/ui + wi, 



tres ($, x) = A Vl + juAi + v/ui, (1) 



tres (x? #) = A/xi + fivi + v\i. 



