424 
be expressed by means of the primitive ones; and a number 
of useful and elegant formule 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, ¢ and x, to which I have 
ventured to give the names of cotresine (¢, x), tresine (¢, x); 
and tresine (xy, @); 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- 
mule which result from a comparison of the functions already 
noticed. 
«‘ Employing the exponential development, putting a for 
=a?) 
2 
product 1.2.3...2, we find 
e?=A+ au t+a’v; 
, the cube root of + 1, and writing 2! for the 
where 
3 6 
d=14+5 4s &e. 
4 ¢" 
R= Gt tit he 
Ae ae 
veott et ele 
Again, 
evx =X, + a1 + ans 
where Ai, pu; v1 are the same functions of x that A, u, v are of 
o. Hence es? = cotr (p, x) +a tres (p, x) + a” tres (x, ¢) if 
we agree to put as definitions, 
cotr (¢, x)  =AAL + py + Vn, 
tres(¢, x) =A t+ Ar + vo, (1) 
tres(x,¢) =Ami+ prt vA. 
