427 



Multiplying the two latter equations together, we have 



8"C#*«+*6t*» = cotr ((j>, x ) cotr (h,k) + tres (<j>, x ) tres (k, h) 



+ tres (x, <p) tres (A, A) 



+ i (tres (x> 0) tres (k, h) + cotr (<p, x) tres (A, A) 

 + tres (0, x) cotr (h, k) ) 



+ t* {tres (<p, x) tres (A, k) + tres(x> <p) cotr (h, k) 

 + cotr (^, x) tres (h, h) } . 



Now the left-hand member in this equation being also equal to 



cotr(0 + /«, x + ^) + «tres(0 + A, x + ^) + t2 tres(x + *j # + ^)j 



we may compare the similar parts of the two expressions, and 

 thus get at once the three formulas of which we were in search, 

 viz. : 



cotr (^ + h, x + h) = cotr (<j>, x) cotr (h, k) 



+ tres (0, x) tres (&, h) + tres (x, 0) tres (A, k), 



tres (<p + h, x + A) = tres (x? 0) tres (A, A) 



+ cotr (<p, x) tres (A, k) + tres (0, x) cotr (A, k), 



tres (x + h, $ + A) = tres (0, x) tres (A, A) 



+ tres (x ? 0) cotr (A, k) + cotr (0, x) tres (A, A). 



From these equations, combined with (4), we obtain 



cotr (0 + A, \ + k) + cotr (0 + ah, x + a 2 ^) 



+ cotr (<j> + a-h, x + ah) = 3 cotr (f, x) cotr (A, A), 



which is obviously analogous to the formula 



cos (0 + h) + cos (0 - h) = 2 cos cos h ; 



and we might obtain similar formulas for the tresines. 



" In many investigations great convenience arises from the 

 peculiar way in which the functions of our new calculus are 

 affected by differentiation. 



" From formulae (1) or (3) we obtain the following : 



