428 
dcotr(psx)  _ tres (ys $) —— = tres (, x) 
do Xx “eat 
d ine “e eee (6 y). a = tres (x, ¢); 
dt : dt ) 
A ?) = tres ($, x)s ee = cotr (95 x). 
It appears, then, that the symbol 5 operating upon any one 
of the functions cotr (¢, x), tres (¢, x), tres (x, ¢), changes 
it into the preceding function in that cycle; whilst the symbol 
d + laa 2 : 
dp changes any one of these functions into the succeeding one 
in the same cycle. It follows, therefore, that ——, is inope- 
ad? 
dod 
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 
e+y+2—-32yz=1, 
as the co-ordinates x, y, z are equal respectively to cotr (¢, x), 
tres ($, x), tres (x; @), it will be convenient to denote 
cotr (- p - x), tres (— ps — x)s tres (- x, — $) by 2, yz. 
Then, as 
- yz =2, y? - 28 = 2, 2-ay=y, 
the equation of the tangent plane at the point (2’, y’, 2’) will 
be 2 CaP 
ce+Zyt+yz=l. 
Hence dS, the element of the surface, is expressed by 
(x + y t+ 2°) dydz - 
x 
