THIRD KI.UITtC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 238 



(L.M.S, 25, p. 241 ; ' Math. Ann.,' 52, p. 484), leads by the substitutions 



to the bicursal C. 



., 



o 



*0 2 ) + <(' + 1)~ = ( 2 ), 



1 2z = Y/C, C = 4, (c + 1)' + | /q\ . 



v / ^o; , 



^-~ic(l+ C )(l + 2 C + v /0) (4) , 



y = " ' ~ IV-M" " v/ ' L + y = 2 + c ~ ^t iV'-) ~ " v/( ' (5) - 



We now find 



_- c 

 - c 



" M S 2i" ' v//fJ ' ^ 7 ^' 



10t- 2 + 4c :i + (2 + 3c) ./C 



2(1+,) (8), 



/ V J \ /* 



2 



(t + r + iy + 22) + 5)./Cl (II) 



using detached coefficients. 



We find also 



where 



P 1= i4 + 19c+ - 2c 3 , Q,= 4+ c, 



P 2 = 6 + ^7c + 44c- + 18c 3 , Q, = 8 + I3c, 



P 3 =- 2 + 13C+ Gc\ Q 3 = 12+ 3c, 



P, = 12 + 43c + 44c- + 14c :i , Q 4 = - G - 9c, 



P.= 4+ 7c+ -10c 3 , Q.= - 2+ 5c (13) 



and the values of a r and b r are given in L.M.S., 27, p. 455. 



VOL. CCIII. A. 2 H 



