assigning certain Portions of a Sphere . 155 
quantities, s and 9; and, since we have x — R = funct. (y, ? ), 
y == O = funct. (g, 0), x and y may be deduced functions of £ 
and 9. Hence, * = [f) f + (f) ' s 1 p.) |±) <j &c . b6 i n g 
y = a J partial fluxions of x andy, 
to determine (y=j and (y-j = y, and |?-j was made equal 
to 1 4- 1 on the supposition thaty was constant, or thaty = o; 
hence, (5-j must be determined from the equations * — | 4- J £ -f- [y] $ 
ando =!f)f + (f) 4 
ri \ 
eliminating 9, 
= — . =3 _J_ 
- (r) (f) 
r)(# 
is transformed into 
4 -) 14-) — 1 4-) (4-jj 
The preceding transformation is .general, whatever functions 
of £ and 0, a; and y are. To obtain a particular solution in the 
present case, let the co-ordinates x and y be transformed into 
a radius vector £, and an angle 9 , which £ makes with x ; 
.% j: = ^ cos. 0 
