KINEALY A SPIRAL ON A TORUS. 3 



Therefore, 



Oc z=zf{xya) and cP ^ f{yzd). 



Putting- these values in the equations above, and also putting 

 na for 6, the general equations for the spiral become 



X =z [f{xj'a) -j- y{yzna) cos na'] sin a - - - (l) 



y z= {^yixfa) -f- fi^yzna) cos na\ cos a - - - (2) 



z ■=. f {yznd) %\\\ na - - - - - - - (3) 



If y(A'ya) = R^ 2i constant, and fi^yzna') =r r, also a con- 

 stant, the curve traced by c is a circle whose center is at <9 ; and 

 the generating curve is also a circle with its center at c, — the 

 general equations become 



X = \^jR-{-r cos na] sin a - - - - - - (4) 



y =. [^R -\~ r cos na] cos a ------ (5) 



z :=. r sin na - - - - . - - - - (6) 



These are the equations of the spiral on a torus. 



The projection of such a spiral on the plane yx is shown in 

 Fig. II. The equation for this projection is obtained by squaring 

 (4) and (5)1 and adding. It is 



x"- -\- y-^ — \_R -\- r co% aY (7) 



The direction of this curve at any point is obtained by finding 



dv 

 the value of -- at that point. From (4) and (5) is obtained 



[7? -|- r cos 7ia] cos — rn sin a sin na - .. (8) 



da 



-J— =z — [^R -\- r cos 7ia] sin a — rn cos a sin na - - (9) 



da 



Divide (9) by (8) and obtain 



dy \_R -\- r cos na] sin a-\- rn cos a sin na 



dx \_R -f- r cos na] cos a — rn sin a sin na 



(10) 



7Z 



If in (10) na =1 m - where m is an even integer^ then 



2 



sin na =z 0, and 



f =^ — tan a (11) 



dx 



