6 TRANS. ST. LOUIS ACAD. SCIENCE 



When f {yznd) = 0^ then a 



Therefore, f {yzn ) = ar' ( i — 



c = 



a', and f^ = cJia', or 

 7? a 1 



i?' 



i?«' 



If a = 6* , f{yzn x ) = r', or r' is the initial value of 

 / (^yzn a). These values in the general equations give 



X = {^R -{- r\\— -^-7 cos no] sin a 

 y = L-^ ~f" ^•(i ^^ cos na\ cos a 



i?a' 



z = r' {\ 



Ra ] 



(19) 



(20) 

 (31) 



y = r (I-- 



(22) 

 (23) 



Transferring the origin, as w^as done in the case of the helix, 

 and making R = cc . 



X = Ra = S __. 



r' 

 cos 6 =-~ {S'— x) cos - - 



z = r' {\ — ~^,\s,m d = -^, {S' — x) sin - - (24) 



These are the equations of a spiral on a cone. 



These examples will suffice to show how simple is this method 

 of generating curves in space. The generating curve, or the curve 

 traced by the axis of the point, or both, may be an hyperbola, an 

 ellipse, a parabola, an Archimedean spiral, or, in fact, any curve 

 whose equation is known. 



