VECTOR DIFFERENTIATION. 
81 
Now this is the sum of two terms, an absolute 4 and a term 
which reduces to 2. The expression, therefore, is heterogene¬ 
ous. But carry on the differentiation a step further. Accord- 
24 
ing to the former principle, pV = — ; according to the 
latter, it is if the differentiation is made before reduc¬ 
tion, and it is 0 if the differentiation is made after reduction. 
Hence it is only V r ~ ~~ which leads to consistent results, 
and the reduction p 2 — 1 cannot be made when p is variable, 
excepting after all the p operations affecting it have been 
performed. 
In the harmonic analysis the following principle is re¬ 
quired : 
pV m — m (m T 1) r m 2 . 
On the above principles it is proved as follows: 
j 7 r m = mr m 1 pr = mr m 1 
1 
P 
5 
pV m — m (m — 1) r m 
J_ _|_ mr m i J A 
p p r 
On finishing the p operations, p 2 may be made 1. This 
formula is also true when m is negative or fractional; it will 
vanish when m = — 1. 
The fact that the differential coefficient of — does not have 
P 
a minus sign makes it necessary to investigate the rule for 
. differentiating by parts the expression p m ~ where m is a 
positive integer. 
Now, 
P m P m 1 ; therefore 
P 
(m + x) p m 1 —- + P m — ( m — 1) P m 2 ; whence 
m + x+1 — m — 1, or x~ 2. 
12-Bull. Phil. Soc„ Wash., Vol. 14. 
