274 



PROFESSOR A. G. GREENHILL ON THE 



reducing to 



2 ) ( x - 



(y 



y - x 



(13), 



where x and y are given in terms of s by equation (64). 



Substituting these values of c and ^/C in terms of a; and y in the expression for 

 I (v) in (42), 11, for p. = 11, we obtain a result applicable to fj. = 22, and associated 

 functions of the Second Stage. 



34. The next two cases of //, = 13, 26 and //, = 15, 30 still present analytical 

 difficulties not yet surmounted, although /JL = 30 could be treated by the trisection 

 method of G, applied to /j. = 10. 



35. When /j. is of the form 4n, so that 



2r + 1 TT i 2r + \ v ,. f 2r + 1 



T = o), -4- -Wo, or K + Ki, / = 



2/i 2*i 2n 



then J/j,v = 2?ty is congruent to j, as in Case TIL, and T (/.') is given by 



T ( v ) - x - 1 r 1 p i s "" 1 + P 2 S> " 2 + ' ' ' + P /fs ,s 1 



2n (a- isY V( ' ' 3> 



?i 



( . 



J ]vS .-2 _|_ _ 

 ((T - S)" 







cr .s j 



Here the degree is halved by putting 



so that 



and 



f * -f 



/s 



dx 



X = 



j/u, odd 



C + C^ - x 2 



(Xj X) (X 2 + 



C CjO; x 2 



also 



and putting 



= (ajj + as) (x 2 x) 



even 



= C + (> + a; 2 

 = (a?! + a) (asg + x) 



-^ WQ ^* \^^OC ~p X 

 = (Xj x) (X 2 X) 



S = a- - 3 = (, Sl - a s ) dn 2 /K' = a^ 2 dn 2 /K' 



(1), 



(2), 

 (3). 



(4), 



(6), 



(8), 

 (9). 



