THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 239 



Pl = - 26 + 3 + 3 + 1-9-9-2, Vl = 3 (- fc 5 + - 2 + 2 + + 3) (35), 



^4= -2 - 9-9 + 1 + 3 + 3-2, ^=3(3 + + 2-2 + 0-1) (36), 



p., = - 2 +15-21 + 1 3 - 9 + 3 - 2, < h = 3 (3 - 4 + 2 - 2 + 4 -- I) (37), 



p g = - 2 + 3 - 9 + 1 3 - 21 + 1 5 - 2, </ s = 3 (- 1 + 4 - 2 + 2 - 4 + 3) (38), 



Pi + Pi + Pi + ft = - 4 (26- - b + 2) (// - U< + 3/r -6+1) (39). 



?i + Sf4+fc4 &== l^(/> 5 + I) (40); 



^ 8 = -2 + 8 + 3-11+ 3 + 3 - 2, </, = 3 (/, + I) (- /,' + I + I + 1 - I ) (41), 



p fi = -2+3 + 3 + 13 + 3 + 3- 2, 7i; = 3 (6 + 1) (- 6* + 1 + 1 + 1 -- 1) (42), 



so that 



P(k- V'3r = M' (43); 



p a =-2 + 8 + 3 + 1 + 3 + 3 - 2, 7 , = 3 (/, + 1) (- // 4- 1 - 3 + 1 - - 1) (44), 



14. The case of /x = 15 can also be derived hy a trisectiou of \L = 5, and so 

 generally when /u. = 3/6 is any multiple of 3. 



For when 



S = 4.s(.s + ./)' - !(1 +//)< + r^/;- (1) 



is expressed in the form 



S = 4(* + :! -(A.s-+B)^ (2), 



this implies that 



s (| )=-*, w'(|o,) = A-B (3); 



and now 



A^ = 12 + (1 + //)- - 8,r = - 12V1 w (4), 



so 



that 



2AB = l-2t- -- 4x- 

 B ; = 4 :l + .r/y 



reducing to 



3* + {(1 + y y - 8x] t* + 3x(2x-y- //-) f : + 3. 



a Jacobian quartic for ^, which can be resolved. 



For 



n = 5, ;?/ = a. 1 , 



(9), 



and putting t = ex, 



( c _ i)3 tf + 3c - (c - 1)- x + c :i = (10). 



