﻿702 Dr. F. L. Hitchcock on the Operator V in 



We now obtain \/Fu from dFu by changing dp into p', at 

 the same time writing V' to the left of the whole. That is 



VFw = V'<K/o' +uSup')r~\ .... (7) 



where the accents indicate that V acts only on the accented p. 

 The expression ^jFu therefore stands always for a function 

 homogeneous of degree — 1. This holds when Fp is either 

 scalar or vector, since the foregoing identities depend only on 

 the linear character of <£ and not upon its dimensionality. 



Finally, if Fp is homogeneous of degree n in p, \/Fp is 

 homogeneous of degree n — 1. For 



VF/3 = V(^Fw), by definition, 



= nr n ~ 1 Vr . Fic + r n \/Fu, by distributing V; 



but, in the first term on the right, \7r = u(hy Tait, Art. 145), 

 and in the second term, VFu, as has been shown, is homo- 

 geneous of degree — 1. Hence the right side may be factored 

 into r n ~ l and a function of u alone. It is therefore, by defi- 

 nition, homogeneous of degree n — 1. This, also, holds for 

 scalar or for vector. 



3. I shall now prove the following theorem in regard to 

 solenoidal vectors: — 



Any homogeneous vector may be rendered solenoidal by 

 adding to it a term of the form pt, where t is a properly 

 chosen scalar ; exception must be made of vectors of 

 degree — 2. 



For consider the effect of V upon the vector pSVFp, 

 where Fp is a vector homogeneous of degree n in p. We 

 have 



VOSVF/o) = Vp • SVFyo + VSVF/3 . p, by distributing V. 



But, in the first term on the right, \7p= — 3. Furthermore, 

 scalars are commutative, so that if we take the scalar part of 

 both sides we may write 



S V (/°S VFp) = - 3S VFyo + S/o V . S \/F P . 



Now the scalar SVF/3, as already pointed out, is of 

 degree n—1. We may therefore apply Euler's theorem to 

 the right-hand term, and have, (by (1)), 



S / oV.SVF /3 =-02-l)SVF / o. 

 By combining terms, therefore, 



SV(/oSVF / o) = -(w + 2)SVF / o. ... (8) 



