214 Trans. Acad. Sci. of St. Louis. 



vious position of parallelism, — is the angle between two 



m' 



adjacent tangents at A', and m' is equal to the number of 



tangents at A', which is also equal to the number of loops. 



The last three paragraphs show that when m and m' are 



expressed in terms of their least common factor, 



the number of tangents at A is m, 

 " " " " at A, equal to the number of 



loops, is m , 

 the number of asymptotes is m — m' . 



In Fig. 5 



the number of tangents at A is m = 8, 

 " " " loops is m' = 3, 



" asymptotes is m — m' = 5, 



i< a 



angle between adjacent tangents at A is — = _ , 



" " asymptotes is 



.i i i 



m 6 



IT IT 



m' 3 



7T 7T 



?/< 972 



In Fig. 6 



the number of tangents at A is m = 4, 

 " " «< " «« yl' is m' = 1, 



" " " asymptotes ism — m' = 3. 



The dashed loop is the limiting line, and separates the 

 region having loops from that having infinite branches. The 

 limiting line is a portion of a curve which, just before it 

 became critical, also had m tangents at A, m tangents at A, 

 and m — m' asymptotes. Fig. 6 shows the whole curve 

 dashed, of which the limiting line is a part. In Fig. 2, 

 where m = 2 and — m = — 1, a complete curve gives place 

 to a circle and a straight line on reaching the critical 

 position. 



