VECTOR DIFFERENTIATION. 
89 
Now, 
p o — (— cos <p .j — sin <p . k) (— sin ip . j -j- cos <p . k) — — 
and 
[) o f> = (— sin ^ . j + cos ip . k) (cos 6 .i- f- sin d cos ip .j -J- sin 0 sin p . k) 
— cos 0 sin p .k cos 6 cos <p . j — sin 0 . i , 
and 
p Q ^~ — — sin 0 sin ip .k — sin 0 cos ip .j — cos 0 . i ; 
00 
consequently the coefficient of reduces to — i -f i, that is, 0. 
When the function is scalar, the space-coefficient for is 
1 +W-=0; 
dp dp 
p —— —-— p 
r 00 00 r 
and so are the space-coefficients for the other two cross-terms. 
^2 
But when the function involves p. the term in -r-^r will not 
7 dr 60 
in general vanish, because one of its terms involves the modi¬ 
fied differentiation, while the other does not. For the same 
reason the term in --- - - will not in general vanish. If the 
ordip ° 
function involve the term in . S 
do 6<pd0 
will not in general 
vanish. 
The general operator / 7 2 so reduced for a scalar function 
becomes 
, _1_ , _1_, 2_ , cote _d _. 
dr* r 2 do 2 (r sin 0) 2 dtp 2 r dr r 2 00 ’ 
that is, Laplace’s operator for polar coordinates. 
From this deduction we see the great generality of the 
expression given for p 2 compared with the expression for 
Laplace’s operator, and the deduction also serves as a verifi¬ 
cation of the principles of differentiation employed. 
13—Bull. Phil. Soc., Wash., Vol. 14. 
