230 PfcOFESSOR A. G. GftEENHlLL ON THE 



8. This suggests that in the general case of /u, = 2n + 1 it is simpler to work with 



and to put ,s = t z ; and then with 



T! = 2* ;i + (1 + .//) t- + 2:rf + xy (2), 



T., = 2f :i - (1 + y) /, 2 + 2tf ary (3), 



we si i all have 



" ] "'" + ' ''"" ' " + 



(4), 



according as w is even or odd ; and the results are of one-quarter the degree that 

 would be given hy ABKL'S method of the periodic continued fraction ; and since 



(t"- 1 + V/'~ 3 + ... ) 2 T 1 + (t"- 1 - hj"-* + . . .) 2 T 2 = 4 2 '< +1 (5), 



the determination of h { , />.,, . . . can be carried out by a consideration of the reduites 

 (HALPHEN, ' F.E.,' 2, p. 576) in preference to continued fractions, once the coefficients 

 of / in 'I', and T., have been assigned. 

 9. Thus (L.M.S., 25, p. 222) for 



H = 7, v = , 7l 0, or xy x~ f = (1), 



is a unicursal C 3 , in which 



x = z(l-zy, y = z(l-z) (2), 



and 



TL T, = 2 3 (1 -f- 2 - 2 ) t 2 + 2z(l - zf t z* (1 - 2 ) (3) 



P (2v) = 3 - +^ 2 (4), 





(6). 



