﻿Combination with Homogeneous Functions. 701 



of V in the third of the above ways, — in combination with 

 homogeneous functional operations. A few facts are already 

 known : chief of which is Euler's theorem, written in qua- 

 ternion form as 



SpV.Fp=-nFp, (1) 



where Fp is any function of p (scalar or vector), homogeneous 

 of degree n in p. Aside from Euler's theorem, most of the 

 known results on homogeneous functions in connexion with 

 V are combinations of V with linear vector operators, and 

 are due to the writers above mentioned. For example, if 

 dFp = (f)dp, and if <£>' is the linear vector operator conjugate 

 to </>, then 



<£'a-0a = V«VVF/o, (2) 



where a is any vector not acted on by V*« 



2. Before proceeding to the proof of new theorems, it 

 will be necessary to enter briefly into a few elementary 

 considerations. First, with regard to notation, I shall write, 

 for brevity, Tp = r and Up = u, so that p = ru. 



Next, as to the definition of a homogeneous function, it is 

 most natural for a vector algebraist to write 



Fp = r n Fu, (3) 



as the definition of homogeneity, either for scalar or vector. 

 This is of course precisely equivalent to the usual definition, 

 and much more available. In words : A homogeneous function 

 of p is one that can be factored into a power of r (that is Tp), 

 and a function of u (that is Up), alone. 



The differentials of Tp and of Up are important, and may 

 be expanded in many forms (Tait, Art. 140). For the 

 present purpose we may take as most convenient for the 

 tensor of p, 



dr=—$udp, (4) 



and for the unit vector 



du=r~ 1 (dp + u$udp) (5) 



Again, we often need to apply V to a function of u alone. 

 This is achieved by writing 



dFp = <f>dp (6) 



We then have 



dFu = cf)du 



=<j>(dp+u$udp)r-\ by (5), 



* Tait. ' Quaternions,' 3rd Ed., Arts. 185, L86. 



