148 Mr. Ivory on the figure requisite to maintain the equilibrium 
provided we make s = 1, after the integration. Now for ■— 
substitute its developement,* and make e = 1 : then, 
4 ^ y —ffy' dfid™ 
+ s/f/.C^dp'd*' 
+ s//y.c (2) dv-'dw' 
+ ^i + i)ffy'.& ) d l dd^ 
"H &c. 
This expression is identical when y and / are functions of 
three rectangular co-ordinates. It is analytically true of 
every function that can be algebraically transformed into an 
expression of three rectangular co-ordinates ; and thus it 
may be said to comprehend every function of two variable 
angles. 
We have now obtained two expressions of y in the same 
quantities. But it is easy to prove that the same function 
can be so expressed only one way. The two values of y 
must therefore be identical ; and all the terms that cannot be 
made to coincide, must be evanescent. Hence we obtain 
o—JJy'. C {,) dp'd 7 
o=ffy.C (i) d\dd*s’ 
&c. 
* Section 2. 
