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



IT 



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\^ 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 7n — m' = 5, 



(( (( 



" ansrle between adiacent tangents at yl i^^ — = — , 



it (I i( a a K ^' i jj _^ __ ]^ 



m 3 



TT TT 



" " " " asymptotes is = 



m — m' b 



In Fiof. 6 



B 



the number of tangents at A is m = 4, 

 <( «* «' " «< A is 111 = 1, 

 *' " " asyniiDtotes is wi — m' = '6. 



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, vi 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 — 7n' = — 1, a complete curve gives place 

 to a circle and a straight line on reaching the critical 

 position. 



