APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 131 



Pig. 13. 



then the corresponding lines on the torus are closed loxodromics, and 

 the trigonometric tangent included by them is easily found as 



«^ in. 



n n. 



r" 



The condition for a right angle is 



n IK R^ — ^ 



in m^ 



(9) 



This must be a rational fraction. 



Hence, — must be rational. 

 R 



In this case the loxodromics on the torus are orthogonal. Evi- 

 dently, m gives the number of revolutions of the loxodromic about 

 the axis (perpendicular to the planes of parallel circles) and n the 

 number of revolutions about the axial circle of the torus. Hence 

 the theorem: — 

 "P 



If — is a rational number and if a loxodromic on the corre- 

 sponding torus closes after m revolutions ahout the axis and n revo- 

 lutions about the axial circle^ then every orthogonal loxodromic 



