132 UNIVERSITY OF COLORADO STUDIES 



floses and the corresponding numhers iriy^ and n^ of revolutions 

 about the same axes are connected to m and n hy the relation 



n n, ^ — 7^ 



4. To discuss the nature of the curves on the torus, assume a 

 closed loxodromic, characterized by the ratio — In the z ' ■ plane 

 the equation of the corresponding straight line is 



n r 



y, = . u., 



but according to (3), 



r . r-j-R sin v 



■y, = : 8in~' , 



yR^ — r^ R-f r sin v 



, . n r+R sin v 



hence sin — u = ( 10) 



m R-j-^ sin v ^ ^ 



In Cartesian coordinates sin u = —pM=^ sin o= ' ^ ~ 



l/a?^+y- r 



Substituting these expressions in (10), the equation of the pro- 

 jection of the loxodromic upon the a?y-plane is obtained. Now 



sin — u may be expressed algebraically in terms of sin ti, hence in 



terms of . ^ . The proiection of the loxodromic is therefore an 



algebraic curve. As the torus is an algebraic surface, the theorem 

 follows :- — 



All closed loxodromics of a torus are algebraic curves. 



As an example take ;? =1, m = l, then the equation of the pro- 

 jection becomes 



which represents an ellipse. The corresponding loxodromic itself is 

 a circle as may be easily concluded. Hence the theorem: — 



