88- 
MACFARLANE. 
but for any spherical axis a we have * — = 1 absolutely ; 
2 
hence the above coefficient reduces to —, giving for the term 
11 . 1 . 
r dr p 1 
The space-coefficient of the term 
. d . 
m 
1^1 1 , 
1 d 2 p 1 
/ s py <v §? / 
V do) do do r \ 
( dp \ 2 dtpdO dp 
{ dO ) d(p 
111 
j_COS 0 1 
/ d py d p d p 
\ do) do do y 
sin 0 ( dp y' 
\~dOj 
When the function is scalar, the further final reductions 
are introduced that 
= 1j and p ~ 
\d 0 J ’ r do 
dp 
= -~M P ' 
for p and are at right angles to one another. By this 
means the coefficient is reduced to cot 0 and the term to 
A cot e A 
r 80 
■* 
The space-coefficient for is 
1 dp n 1 sin 0 
COS o 
p Q 2 d<p p o p Q p 
dp * 
r ° do 
For a scalar function the coefficient is reduced by the prin¬ 
ciple that for any spherical axis a we have <? = 1, and this 
allows all the terms to be placed in the denominator, giving 
