﻿Combination with Homogeneous Functions. 705 



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

 two possible values their difference would be of the form 

 V/9T 1? and would be lamellar. Now the term Ypr x is of 

 degree n, hence t x is of degree n — 1. We therefore have 

 identically* 



VVV /0 T 1 =-(n + l)T 1 -. / oSVT 1 . . . (15) 



If n does not equal —1, Txdoes not have degree —2, and 

 may be rendered solenoidal by a term in p without altering 

 the value of V/jtv Hence we may suppose SVt"i = 0, and 

 the right side of (15) cannot vanish if t x does not vanish and 

 n is not —1; that is, the term Ypr is uniquely determined. 



6. From the identity (14) may be easily deduced a second 

 example of inverse operation (integration) with V. Suppose 

 a rotation vector, (Y\/Fp), to be known at all points of a 

 given region, and to be expressible as a sum of vectors each 

 homogeneous in p, lacking a term of degree — 2. For 

 example, let 



rotation vector = VVF/3 = t + t 1 + t 2 4- ..., 



where the subscripts denote the degrees of their terms. 

 A possible value of Fp then is 



^P=-^p\j + r i + l +-}' • • (16) 



a vector everywhere at right angles to p. The vector Fp is 

 often called the vector potential of its derived vector Y\/Fp. 

 Thus (16) shows how to write down a possible vector potential 

 for any assigned solenoidal vector whose components are either 

 polynomials or other sums of homogeneous terms (exception 

 noted) . (16) may be directly verified by expanding the right 

 side with the aid of identities like (15)f. 



* Phil. Mag. loc. cit. 



t It is well known that possible values for a required vector potential 

 can be found by partial integration with respect to the point-coordinates 

 #, y, and z. The above method illustrates how v may replace partial 

 integration, — a principle probably more far-reaching than any appli- 

 cation which has yet been made of it. As another illustration, let 

 Xifo'-r Ydy+'Zidz = dJ ) =0 be an exact differential equation. X, Y, and Z 

 are components of the vector VP. Suppose P = SpFp. If VP can be 

 written as a sum of homogeneous vectors 



vP = o- -f-o- 1 -j-o- 2 +. . . 



we may write down P by the formula 



P=-Sp|?»+i+|.+... }, 



proved by multiplying both sides of (13) by p and taking sealars. 



Phil Mag. S. 6. Vol. 29. No. 173. May 1915. 2 Z 



