OF ARTS AND SCIENCES. 325 



9°. The equatioQ of the parabola may be written 



Q="{a^x^, (19) 



where a _L ^. Then 



q' =. xa '\- (3, 



and if Ta = a, and T(3 = h, 



Ti,/ = sjcrx' + U\ 



and 



X 



s — Sq = I sjd'x- -\- hi^ 





h'' Vox 

 2a L I 



Let V = Sh~ 1 — ; then the expression for the arc becomes 



*-^o = £-[Sh2^; + 2^,]^ (20) 



10°. The equation of tlie lielix on tlae elliptic cylinder (or, rather, 

 of a curve analogous to the helix) is 



p = « cos x -\- ^ s'm x -\- yx, (21) 



where «X(3_L7_La, a and ^ being semi-diameters of the elliptic 

 base. By differentiation, 



q' = — a sin aj -(- ^ cos x -\- y; 



Tq' = y/(a^ sin^ x -\- P cos'^ x -f- c^), 



if Ta = a, T/3 = b, Ty = c. 



X 



,•, s — Sq = j \J{d^ — b") sin^ a; -j_ 6- -j- c\ (22) 



This integration, as it should, involves elliptic functions. If the base 

 of the cylinder be circular, then a = b, and 



X 



s — s, =Js/¥J^' = s/¥^:7\x - x,) (23) 



