﻿Combination with Homogeneous Functions. 703 



The following identity will now be evident, 



SV{F^ + ^Y|V| = 0, .... (9) 



Fp being a vector homogeneous in p of degree other 

 than — 2. This identity proves the theorem and shows 

 how to find the scalar t. 



The term pt is uniquely determined. For i£ there were 

 two values, their difference would be a scalar multiple o£ p 

 and would be solenoidal. Call this difference pt x . But by 

 the same order of reasoning as above, 8 V(^J = — (n + 2)^, 

 which cannot vanish unless ^ = or n=— 2*. 



4. As a simple, but important, extension of the foregoing 

 theorem, let us suppose (what is frequently the case) that a 

 non-homogeneous vector can be written as the sum of several 

 vector terms, each homogeneous in its own degree, e. g. let 



Fp = F 1 p + F 2 p+...+F n p + ..., 



where the subscripts denote the degrees of their terms. By 

 applying the theorem to the separate terms, we see that Fp 

 may be rendered solenoidal by adding the vector 



fBV { ¥+ se + ... + ^ + ... } . 



The series concerned may be infinite, provided they are 

 convergent. 



Conversely (as an example of integration w T ith V), if the 

 convergence of a vector, (SVF/)), be given, we can write 

 down a value for the vector itself, which shall be a scalar 

 multiple of p, provided we can expand the convergence as a 

 sum of homogeneous scalar functions of p lacking a term of 

 jeoTee —3. For example, if we have given 



convergence = S V Fp = t + ti -f t 2 + . . . + , 



where the subscripts denote the degrees of their terms, then 

 a possible value of Fp having this convergence is 



r ,.._,{£ + $. + * + ...} 



a flux directed toward the origin. 



5. These very simple results on the solenoidal character of 

 vector fields may naturally lead us to inquire whether there 



* In a similar manner we may show that a terra oi' the form pt, if t is 

 a scalar of degree —3 in p, is always solenoidal. 



