GENERATION OF FIGURES ON STRAIGHT OR TORTUOUS AXES. 331 



rigorously applies to solids bounded by what may be called 

 "circularly warped surfaces," in which the centre of inertia of any 

 section (formed by a plane rotating about an axis and cutting the 

 solid) changes its distance, from the rotation axis, linearly 1 in 

 relation to the arc through which the section turns. It will now 

 be shewn that this is an elementary case of a much more general 

 theorem. 



If the terminal values of a generating function of the form, 



A t = A+Bt + Ct 2 + Dt 3 + etc (21) 



together with a series of equidistant intermediate values, are 

 given, the generated quantum, between the limits t x and t 2 , is 

 absolutely determined, when the highest index in the generatrix, 

 (1), is the same as the number of given values inclusive of the 

 terminal ones, if the number is odd, or one less than the number, 

 if it is even 2 Consequently if the indices p, q, r, etc. and a, 6, c> 

 etc., are positive integers, that is to say, if the quantum of the 

 generatrix and of the departure of its centre of inertia (in the 

 direction of the radius of curvature) from the axis, are both 

 positive integral functions of the position thereupon, then it is 

 easy to determine the number of equidistant values of the gener- 

 ating function necessary to fix the quantum of the generated 

 figure. For let m denote the sum of highest indices, s + dsay, 

 then if an odd number be taken, if must be ra, but if an even, 

 m+ 1. 



10. Conclusion. — Since an expression of the form (1), can be 

 designed to represent, even with a few terms in most cases, almost 

 any spatial quanta related to a variable, it is evident from the 

 foregoing consideration of the curvature of axes, that the generality 



1 Such a figure for example, as a railway cutting on a circular curve, 

 when the slope of the natural surface changes at a uniform rate with 

 respect to a line rotating about the rotation axis at the centre of the 

 curve. 



2 See my paper "On the relation, in determining the volumes of solids, 

 whose parallel transverse sections are n ic functions of their position on 

 the axis thereof, between the number, position, and coefficients of the 

 sections, and the (positive) indices of the functions." — Journ. Roy. Soc, 

 N. S. Wales, Vol. xxxiv., pp. 36 - 71 1900. § 18. 



