VECTOR DIFFERENTIATION. 
83 
There ought to be a corresponding p operator in the case 
of spherical coordinates. It ought to be applicable to any 
scalar or vector function of the coordinates, and it ought to 
be such that p 2 or, in general, p n may be deduced by direct 
operations of the differential calculus, suitably generalized 
for space analysis. In works such as Love’s Theory of Elas¬ 
ticity and Webster’s Theory of Electricity and Magnetism there 
is given an expression for p 2 in terms of spherical coordinates 
which applies when the function differentiated is scalar, but 
I have not seen, anywhere, a general expression for' p from 
which an equally general expression for p 2 can be derived by 
direct operations. The nearest approach which I have found 
is in Whitehead’s Universal Algebra, where, however, the 
principle is adopted that pr = p, and p 2 cannot be formed by 
direct operations. 
When the function is a function only of r, the spherical 
modulus, 
( ) 
ti 
n _) 1 
dr p * 
.. = ^ 2 _( 
dr 
1 I * ( ) 1 
0 dr p 1 
d\ ) 1 , 9( ) 2 
dr 2 p 2 dr rp 2 ' 
Introduce the reduction p 2 — 1, then 
p 2 = > + 1 *( ) . 
dr 2 r dr 
But this reduction must not be introduced here, if we are to 
go on to derive pi 
Suppose that the function to be differentiated is a function 
of r and p, the spherical modulus and axis. The operator p 
is the sum of the partial differentiations, each multiplied by 
the p of the variable. In this case 
dp r * 
