TRA NSACTION S 



A Spiral on a Torus. 



By J. H. KiNEALY. 



Last September I showed to the Academy a model of a stove- 

 pipe elbow that had been made of a single strip of tin wound so 

 that there was only one joint. This joint was a continuous spiral 

 around the elbow. The spiral joint had not been made accord- 

 ing to any law, but had simply been guessed at. This elbow had 

 been made by Mr. E. F. O'Toole, a tinner living in the northern 

 part of this city, who, when he gave it to me, asked if it would 

 be possible to make a spiral that would be the same in all parts 

 similarly situated. 



In order to show me the spiral he wished, he took a helical 

 wire-spring and bent it so as to form a quadrant. I saw at once 

 that the spiral wanted was a spiral on a torus, which, when the 

 radius of the circular axis of the torus was equal to infinity, be- 

 came a helix. 



I then attempted to generate a spiral on a torus by moving a 

 point on the surface of the torus so that the tangent to the curve 

 would always make a given angle with the circular axis of the 

 torus. This, however, involved a very difficult differential equa- 

 tion. After some further study I found that a spiral could be 

 generated upon any surface by the following method : 



Let the surface upon which the spiral is to be drawn be gener- 

 ated by a curve in a plane that revolves about a given axis with 

 an uniform angular velocity. Let the point by which the spiral 

 is generated always remain on the generating curve and move 

 about an axis, perpendicular to the moving plane, with an uni- 

 form angular velocity. 



,. ,_ , (June 28, 18S7. 



