320 G. H. KNIBBS. 



dimensional homaloidal quantity in a direction parallel to the t 

 axis of the former, can be expressed by the equation 



j(x,y,z,w, etc.) = A t = A + Bt^ + Ct (i + Dt T + etc (1) 



in which the coefficients A, B, C, etc., have any real, finite values, 

 positive or negative including zero, and the indices p, q, r, etc., 

 are real and greater than - 1, but may be either fractional or 

 integral; then this n-dimensional quantity, for the limits to t, 

 will be 

 r,-/.Mi»-«(^+--J I f»+ ? -^+ ^-e + eto.) (2) 



The condition that V t is to be finite for all values of t from 

 up to but not inclusive of + go obviously requires that the indices 

 lie between, but shall not include, - 1 and + oo, for supposing 

 A t to have a term It' 1 , its integral, being I \og e t, will be co for 

 ^ = 0/ and this marks the inferior limit at which the function V t 

 becomes infinite for a finite index. 



I propose to investigate the range and generality of the functions 

 (1) and (2), and to develope certain theorems concerning their 

 relations, when p, q, r, etc., are subject to the one restriction that 

 they shall be greater than — 1, and the axis t is not necessarily 

 rectilinear. 



2. Form of graph of generating function immaterial. — Except 

 in so far as the interpretation of the integral is concerned, the 

 form of the generating (n - l)-dimensional function is immaterial : 

 so also is the angle of its inclination with the t axis, if this last 

 be rectilinear. That this is so, will be evident from the following 

 considerations. If the function be essentially one-dimensional, f{x) 

 say, its graph may be not only a straight line, but also a plane- 

 curved, or plane-spiral line, a closed curve, or a series of any or all 

 of these. So also it may be the intercept, parallel to the #-axis, 

 between two curves, or the intercepts between any series of curves 

 extending in the direction of the ((-axis assumed to be rectilinear; 



1 A negative index - m say, makes the graph of the function Kt m an 

 m ic hyperbola, and consequently infinite for t = 0, while if m be positive 

 the graph will be an m ic parabola, and finite for zero or finite values of t. 



