86 
MACFARLANE. 
The symbol p o may be used to denote the unit-vector 
— sin (p .j -f- cos <p . k. The three unit axes p , p 0 form an 
orthogonal system. It follows that 
n > 1 ) i . *( ) i 
dr p dO dp d<p dp 
dO dip 
Hence 
f 2 = + 
d 1 
d 
dr p dO dp dip dp 
r w 
11 + 1 ^+ 5 1 
£_(*(. ) 1\ 1 
or \ 
dr p ) p 
dip ) f 
(i) + 
dr p SO Sp_ S</> up 
dO dtp 
+ 
+ 
+ 
*( ) 1 \ 1 
d( P r S P ) r d P ^ 
dtp J dip 
8<p 
» /*( ) 1 
dr [ dip 
dp ^ p 
dip J 
(5) 
oO 
+ -r— 
dtp 
' 8 ( ) 1 \ 1 
8f r d P ) r S _P 0) 
dtp J dO 
'd( ) 1 \ 1 
dO dp 
dO 
JjT (9) 
dip 
+ 
+ 
+ 
d 
r 8 ( 
> i 
\ 1 
Jo / 
so 
dp 
r 
dO 
-s ! 
©»| oj 
d 
r s ( 
) 1 
\ 1 
Jr / 
se 
r ' 
do J 
±i 
(K 
) M 
1 
dO ' 
\ dr 
p) r s p 
se 
d 
( H 
) 
1 
dip 
V dr 
p) r s _p 
dtp 
(4) 
16) 
( 8 ) 
From the above it is evident how j7 s , or in general p n , may 
be formed, and the function differentiated may be any func¬ 
tion, scalar, or vector of r, 0, <p. The simplest procedure is to 
apply these operators directly; but, in order to show the re¬ 
lation of this general operator to Laplace’s operator for polar 
coordinates, I shall expand the operators on the principle of 
differentiating a product. 
