2 TRANS. ST. LOUIS ACAD. SCIENCE. 



The axis first mentioned will be called the axis of the generat- 

 ing curve ; and the last, the axis of the point. 



It is evident, that, if the equation of the curve traced by the 

 ^xis of the point is known, and, also, the equation of the generat- 

 ing curve, referred to the axis of the point as an origin, the equa- 

 tion of the spiral can at once be determined. 



In Fig. I. the plane o^ yx is the plane of the paper, and the axis 

 of z is taken perpendicular to ^^. 



OT is the trace of the moving plane on yx at a given time. It 

 revolves about the axis of ^ with an uniform angular velocity. 



The initial position of the plane OT" is zy. The projection of 

 the generating curve on yx is ace. 



In order to simplify the equation it is supposed that the curve 

 traced by the axis c of the point is a plane curve {bed) onyx. 



P' is the projection ox\ yx of the point at the given instant. 



The point was in its initial position on yx when OT coincided 

 with Oy. 



Revolve the moving plane about OT as an axis until it falls 

 on yx. 



Let aPe be the revolved position of the generating curve, and 

 P that of the point. 



Draw P'ni perpendicular to Oy ; also draw Pc and P P' . 



Let o. be the angular velocity with which the plane moves 

 about Oz, and «' the angular velocity with which the point moves 

 about c. 



If t is the time the plane has been moving, 



< TOy — a — t.o.. 

 <:^Pce = d = t.a'. 



a d a a . 



t =z — — ox u zn — a zr. n a^ where « is a 



a a' ^ a 



a' 

 constant equal to — , 

 a 



X =z P' m z=. OP' sin a = [(9c -|- cP cos d ] sin a 

 y = mO =. OP cos a =: \_0c -\- cP cos d ] cos a 

 z = PP' — cP sin d. 



Oc depends upon the curve bed., and cP upon the curve aPe. 



