84 
MACFARLANE. 
Hence 
' d 
1 
p 
+ ± 
°p 
4 )( 
d 
Hr 
1 
P 
+ 
dp 
2_> 
r ) 
) 
d 
(*<■ 
> 
1 
4- 
£ 
(t 
( 
) 2 
dr 
V dr 
P ) 
P 
r 
V 
dp 
r , 
+ 
d 
(H 
) 2\ 
1 
4- 
d 
p 
( 
) i 
dr 
V s P 
r ) 
P 
dp 
V 
dr 
) i 
■ + SK 
( 
) 
4 
d 
r 2 p 2 
dp 2 
r 2 
H 
+ 
+ 
<S 2 ( ) 2 
^ 2 ( 
rp 
) 2 * 
( ) 2 
r 2 p 
d P 
dpdr rp 
+ 
( ) 2 
dr rp 2 
Should the function involve p, then the differentiation in¬ 
dicated in the term marked with an asterisk must be modi¬ 
fied, because the other factor differentiated is —. If the 
p 
function involve p m , then the multiplier will not be m, but 
m — 2. The expression for p 2 when the function is a func¬ 
tion of r only may be deduced from the above more general 
formula by canceling all the terms in which appears. 
Example of the application of p 2 to a vector function of 
r and p : Let the function be R 2 = r 2 p 2 . By applying the 
four terms of p ' 2 directly we get 
nmm i o $ 2 (r 2 p 2 ) 4 _ 8 
dr 2 p~ ‘ dp 2 r 2 
± i\ J_ = 4. A ( *(** ^ Ml = 4. 
dr \ dp r ) p dp \ dr p ) r 
Hence 
p 2 R = 18. 
The result in this simple case may be easily verified, for 
V R = 2 R V R = 6 R and p 2 R 2 = 6 V R = 18. 
