240 PKOFESSOR A. G. GKEENHILL ON THE 



To identify with the preceding results in 13, put 



, = "++ i.c_i = &_-;)-, 3c + l = ('' + ')\ 3c _4 = '' ! -f+' (u), 



and then the associated octahedron irrationality o \. 



15. For /A = 17, the quartic for q in terms of c is given in L.M.S., 25, p. 204, 

 obtained by the substitution in y, 7 = of 



, . 



This irreducible quartic is made reducible by putting q = < (e +1), uud with 

 t; = /> 1 becomes 



e (,, + L) 6* + c (:> + * + 1 ) If' + (<'' - * ;! - 3e* - 3c - l)l>- 



b 2(e+l) 



The alternating function 



,(Kr) - .s(2r) _ (/> _])(& + c + 1) - x /r + x '(^ + 9'' + 4) 

 ,,( 4r )_. s .( ,,)- ,/, -J(e+ 1) C 3 



and the division values are associated with elliptic functions of an argument 



1 



<<<' 





= ' , e (io>) = 1 , e (a)' + io>) = - - 1, o = -^^ (G), 



)8 + e/=0 (7). 

 By the transformation 



4e + 9 + 4 = ^, *-(^ = -!, 4 ^'- 1 ) e = ^-17 (8), 



and a comparison can he made with the equations of KIEPERT (' Math. Ann.,' 37, p. 386) 

 1G. The next case of /j. = 19 presents difficulties not yet surmounted, although it 



was hoped that the analogy with /n = 11 would give the clue required. 

 In the case of ^ = 11, the substitution of 



(1) 



