Mr Dixon, Note on Plane Unicursal Curves. 457 



Eliminating A, B, G we have a condition linear in F'a, F'j3, 

 F'y, Fa, which may be written thus 



F'a/ Fa, F'fi/Ffi, F'j/Fy, m = 0. 



X'a/Xa, X'/3/X/3, X'y/Xy, 1 



Y'a/Ya, Y'/3/Y/3, Y'y/Yy, 1 



Z'a/Za, Z'P/Z/3, Z'y/Zy, 1 



In the proof of the converse theorem it is to be remembered 

 that all but three of the curves <f> 1 = 0, <£ 2 = . . . may be made to 

 have a double point at the triple point of 11. Let <p lf (f> 2 , <£ 3 be 

 the three exceptions. Then \ 1} \ 2 , X 3 must be so chosen that 



shall have this double point, and the rest of the proof is as before. 



Other multiple points may be similarly treated. 



The results that have been found might have been deduced 

 from Abel's theorem. Take the case when U has p nodes. We 

 have 



Sf^ = const., (r=l,2...p) 



dx 



where the upper limits of the integrals summed are the inter- 

 sections of U with a variable curve such as f. In the integrals 

 change the variable from y to 0. Then <£,. becomes a fraction, 

 whose numerator is 



n{(0- a )(0-/3)} + (0-a r )(0-/3 r ), 



and denominator (Z0) n ~ 3 . The denominator of -~ is (Z0) 2 , and 

 its numerator vanishes at the 2n — 2 points where the tangents 

 are parallel to the axis of x. At these points ^— vanishes, and 



also at the nodes : the denominator of — is (Z6) n ~\ 



Hence ll' ■ is a constant multiple of ly^ r-r-= — 7r r, and 



J dU r J(0-a r )(0-(3 r ) 



dx 

 Abel's theorem becomes 



