4 TRANS. ST. LOUIS ACAD. SCIENCE. 



\{?iaz=/n " and m is an odd integer^ then sin na r= :£; i,. 

 and cos na z= 0, then 

 </j' /? sin a ± rn cos a i? tan a ± f'fi 



dx R cos a =F r« sin a R ^ rn tan a 



(12). 



Equation (i i) gives the direction of the curve at such points as 

 a or (5, Fig. II., and shows tliat it is here perpendicvilar to the 

 radius, drawn to the points, of the projection of the circular axis 

 of the torus. 



Equation (12) gives direction of the curve at such points 

 as c or d. 



Let dS be the tangent to the curve at d ; the tangent of the 

 angle ,3 that dS makes with Ox is given by (12). 



Draw od^ and dm perpendicular to od. dm is tangent to the 

 projection of the circular axis of the torus. 

 <^ doy ^z a z=L dmo. 



<^ Idm z=. \i -\- a is the angle that the curve at d makes with 

 the projection of the circular axis of the torus. 

 <^ Idin = (f. 



tan B + tan a . . . . 



tan c =■ ; but from (12) 



I —tan p tan a 



R tan a -h m 

 R ^ rn tan a 



tan y5 = _ -y,— - . Therefore, 



R tan a -\- rn , 



— —^ \- tan a 



R ^ rn tan a 



tan (f = — 



I R tan^ a 4= ^'^ tan a 



I -j- == 



R ^ rn tan a 



— R tan a dr r?i -\- R tan a-^rn tan^a 



~ R'^ rn tan a -\- R tan^a ± ^^^ tan a 



± r« (tan^ a -\- i) , rn 



~ R (tan2 ^+1) ~ ~R 



This means that the curve always crosses the circle cdIC at a 

 constant angle. 



