VECTOR DIFFERENTIATION. 
91 
The (2), (4), and (6) operators may be expanded, but in the 
terms marked with an * the operations require to be suitably 
modified when the function differentiated involves p or 
do 
- 
-*1 
d 2 ( 
) 
1 
, i n ) 
1 
1 
\*J 
r 2 
d0 2 
f 
0^0 > 
v 2 r 2 86 ( 
dp\ 
2 dO 2 dp' 
V 
Jo, 
1 V 
dO ) 
~dO 
(40 = 
1 
d 2 
i 
1 H ) 1 
VV 
r 
drdO 
—A p 
do r 
r 2 dO dp 
~dO P 
(6) = 
=*Jl 
d 2 
1 
4_ 
li( ) 1 dp 
1 
r 
do dr 
dp 
p 
r dr p 2 do 
*p ; 
do 
SO 
Hence 
„ 2 _< 5 2 ( ) 1 , 1 *( ) 1 , < 5 2 ( ) 1 
v dr 2 p 2 ^ r 2 do 2 (dp 2 
{%) 
8z 2 
, a a ) ii.ii 
r dr p 2 r 2 do 
1 J_ _ 
dp \ 2 P dp dp 
W ~do p 
m 
I !■ *( ) ( 1 ,1 
r dr dd < 
.f 1.L J | _L + i.. I 
drdz \ pk kp ) 
, 1 *(_) 
r dOdz 
1 + 1 
By introducing the principles of reduction, namely, 
b S P 
d P 
do 
d P 
do 
