GENERATION OF FIGURES ON STRAIGHT OR TORTUOUS AXES. 329 



Consider first an element 8s parallel to, but not necessarily on 

 the y axis, (i.e. the radius of the osculating circle and its prolonga- 

 tion): this will generate an element o'l surface 8s . 8t (X+Apjp) 

 whatever be its position. 1 An element 8s on the x axis lying in 

 the binomial axial surface and coincident therewith, will generate 

 a surface the magnitude of which, depending upon its distance Ao- 

 from the plane of osculation, is since d^^dt/o; easily seen to be 

 8s . dt v[l + (A<r/o-) 2 j. If however it is parallel to the x axis, 

 and distant Ap therefrom, the generated element will become 

 8s. 8t v{(l + Ao/p) 2 + (Ao-/o-) 2 j 2 For brevity let Ap or y, divided 

 by p, be denoted by A, and similarly Act or x, divided by cr, be 

 denoted by /x; so that A and jx are ratios merely; then for any 

 element 8s, making the angle y with the x axis, and whose rational 

 coordinates are A and //., we shall have for the generated surface 8A t , 



8A t = 8s.8t v{(l + A) 2 + ^ 2 } sin rj (13) 



7] being the angle of inclination between 8s and the element of 

 surface it generates. Let £ denote the angle made by any point 

 in 8s with the radius of osculation as it moves subject to the 

 torsional as well as the onward motion : then tan {"= (1 + A)//x and 

 since 



cos 7] = sin y cos f =/x sin y/V(l + 2A + \ 2 + p?) (14) 



equation (13) may be written 



8A t = K.8s.8t = ^{(l + Xy + p, 2 cos 2 y}8s.8t (15) 



which resolves into the quantities previously mentioned for y = 

 and y = 90°. It is evident from this last equation, seeing that 

 (1 + A) 2 , fi 2 , and cos 2 y are necessarily always positive, that the 

 value of k cannot be made unity by the imposition of any condition 

 whatsoever in the case of a closed curve ; for any variation with 

 t would involve changes in A and /jl. For a line, curved or straight, 

 it is necessary that 2|(1 + A 2 ) + /x 2 cos 2 y) ds should conform to (1) 

 throughout. We may therefore say that there is no general con- 

 dition by means of which the generated quantum can be made to 



1 That is to say, the element of surface is independent of its distance 

 from the y axis, 



2 It may be noted that the torsional component depends upon Act alone, 

 not at all on Ap. 



