APPLICATIONS OF ELLIPTIC FUNCTIONS TO PROBLEMS OF CLOSURE 127 



Fig H. 



h 

 As an example take A'j= 1, mi = 3, then — ^= i-, hence a conju- 



gate series has one revolution and six circles. In Figs. 10, 11, the 

 circles of the original series are drawn in solid lines, while those of 

 the conjugate series are represented by dotted lines. 



To find two conjugate series, each having the same number of 



revolutions and circles, put — ^ = ^-, which requires each of these 



ratios to be \. This case is illustrated in Fig. 11. 



IV. LOXODROMICS ON THE TORUS. 



1. Writers on the theory of surfaces frequently point to the 

 torus as an instructive example. Dr. Repetto in a monograph 

 "Sulle Geodetiche del Toro,'" has made a thorough investigation of 

 the geodesies of the torus and states the condition for closed curves 



(1) University of Sassari, Italy. See also article by Dr. G. A. Bliss, in Annals of Mathematics, 

 Vol. IV, No. 1. 



