TRIED ELLIPTIC INTEGKAL AND THE ELLIPSOTOMIC PROBLEM. 273 



33. n = 22, f= - 

 11 



The relation D n = in (21), 22, gives a C s in (p, e) ; but putting 



e = ^Ti' ' > = ?/ +i 0). 



it becomes 



( x + 1 )2 y& + ( 4x 2 + 9a . + 4 ) ^ _ (_ r + 2 ) ^ _ 3a . _ 2 ) ^ 



+ a: (a- + 2) (a* - 7.r - 2) f + 4.r'> (x + 2) y - ^ (.r + 2) = (2), 



a C 7 in (x, ij) witli triple point at the origin ; so putting y = sx, 



. S .5 X 4 + (2.,' 1 - 4.s^ - 3* - 1)*V + (.s 3 - 9.s' 4 - S.s" + 5,s 2 + 4.v + l).x- 2 



- 2 (2,s' - 4.s ;i - 8* 8 - 4.s - 1) x + 4,s" (.s + 1) = (3), 



a quartic for x, which can be resolved 



[2^ 2 + (2.s ;! - 4** - 3x - 1) x - 2 (2.s- + 2x + 1 )]' = {(-' - 1 ) a' - 2 j 3 8 (4), 



S = 4*(s+l) 3 +l (5). 



Writing it 



[2**(Aj-2- l) + (2s 2 + 2s+ L)K-- l)x--2|.] 2 



= {( s _l) a! _2} 2 S (G), 



and putting 



(8), 



zj.> 



so that, with 



r x ,1 . 



(9), 



= (tt.-t) (10). 



This is the elliptic integral of which the ikosahedron irrationality 77 of KLEIN is 

 unity ; it is curious also that this integral occurs us one of ABEL'S numerical exercises 

 ('CEuvres,' 1, p. 142). 



Then to connect up with ^ = 11, as in L.M.S., 27, p. 469, 



, , s ('Zv) s (V) (x iir ,, N 



l+c n =_p u = Y\ / \ = / w~~ "o~TH \ ' '' 



after reduction ; and 



_ s (3v) s (2v) 



VOL. COIII. A. 2 N 



