428 



d cotr (A, v) . . dcotr(<h,\) 



-^Ui> _tres( x ,0), Jjpii .tres(0, x ), 



II^X) = cotr(*, x ), ±^LX) , tres(x , rt , 



<*!&£ rtas^x); *^M> -co* (*,*). 



It appears, then, that the symbol — operating upon any one 



dtp 



of the functions cotr (0, x ), tres (0, x ), tres ( x , 0), changes 



it into the preceding function in that cycle ; whilst the symbol 



— changes any one of these functions into the succeeding one 

 d(p 



d 2 

 in the same cycle. It follows, therefore, that _ is inope- 



d<pdO 



rative upon each of these three functions. These results are 

 analogous to those which we are familiar with in trigonometry, 

 where we find sines and cosines reproduced by differentiation. 

 " In discussing the properties of the surface whose equa- 

 tion in rectangular co-ordinates is 



x 3 + y 3 + z 3 - 3 xyz = 1, 



as the co-ordinates x, y, z are equal respectively to cotr (0, x ), 

 tres ($, x ), tres ( x , ^) 5 it will be convenient to denote 



cotr (- 0, - x ), tres (- 0, - x ), tres (- x , - <p) by x, y, z. 



Then, as 



x 2 - yz = x, y 2 - zx - z, z 2 - xy = y, 



the equation of the tangent plane at the point (V, y, z') will 

 be 



x x + z'y + y z = 1. 



Hence dS, the element of the surface, is expressed by 



(x 2 + y + zy dydz 



