﻿708 Operator V in Combination with Homogeneous Functions. 



point, hence consists of two straight lines. We may then put 



8 P Fp = $xp8/3p, 



where a and /3 are constant vectors. Operating by V gives 



VS/oFp=-aS/3/>-/3S«/o. 



The vector WF/), being now of degree zero, is constant, 

 and must be parallel to Y*j3. Hence the most general 

 orthogyral linear vector is of the form 



aYpYuj3 + 6(aS/3/o + £S*/o), 



where a and b are constant scalars. 



If we let n equal 2, the associated scalar defines a cubic 

 with two double points, hence degenerate. 



If n = 3, the most general orthogyral vector has for its 

 associated scalar a quartic of deficiency zero. 



In a similar manner, if we start with any two homogeneous 

 scalars we may write down orthogyral vectors in the form (13) . 

 For the vector YS/u\/v is solenoidal, whatever scalars u and 

 v may be. Hence 



aYpYVuVv + bV {uv) 



is orthogyral if u and v are homogeneous, a and b being 

 constants. That is, to any pair of algebraic plane curves 

 corresponds a two-parameter family of orthogyral vectors. 



9. In conclusion it may be said that the differential and 

 integral relations of this paper are extensions to space of the 



one-dimensional formulas for <—= — -' and I x n dx. In fact, most 



of the preceding results reduce to these, or to identities, if we 



put p — ix and S?z=i—. That a calculus with V is worthy 



of systematic and extensive development there can be no 

 doubt. We should naturally expect greater variety and 

 complexity in proportion as the geometry of space is many- 

 sided in comparison with that of one dimension. It would 

 be essential to consider next the values of n treated above 

 as exceptional cases — not a difficult matter, but leading 

 to logarithms and other non-homogeneous functions, beyond 

 the special domain of the present paper. 



