APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 129 



1_L ^ • 



1-j- 8in V 



+ sin V 



r 



The values of u and v can each vary from to 27r. In this 

 interval u^ varies from to 27r and according to the expression (3), 

 V, from 



r r r ^ n 



/ ^ sin-'— to:;7^ , [sin"' HStt]. 



■r<x 

 The sides of the primitive rectangle are therefore 



r27r 



w, ■=■ 27r and w» 



t/r^ 



(4) 



To the lines ?^j = const., Vi = const, which are respectively parallel to 

 the sides of the rectangle {w{w^^ correspond the meridians and par- 

 allels of the torus. Conversely to the meridians u — const., v = const., 

 correspond in the rectangle lines parallel to the sides %o^ and n\ 

 respectively. 



Evidently these lines form orthogonal systems. Putting 

 z=u^-\-iv^ and designating by a and /3 two complex quantities 

 a=p—iq, y8 = r(l-|-^), the function 



represents also an orthogonal system in the (w, ^i) —plane. There is 



^ = au^-\-hVy^-\-c, 



y{r= —hti^-\-av^-\-c, 



(5) 



Which represent two perpendicular pencils of parallel rays. On the 

 torus they correspond to a system of orthogonal loxodromics. 



3. Among the loxodromics of the torus I shall consider those 

 that close after a certain number of revolutions about the axis and 

 the axial circle of the torus. For this purpose consider the elliptic 

 integral 



