THIRD ELLIPTIC INTEGRAL AND THE ELLIPSOTOMIC PROBLEM. 277 



38. p. = 8, v = K + - 1 - 3 K'i, 



8 - s 3 



where the octahedron irrationality o = ^/K is given in terms of a by 



The result is obtained by putting 



/ \ / 1 \ '" ( I '") 



.< (o>) - ,s (4v) = 0, a = ; , /3 = I 



I ZWl 



and the values of .s-,, ,s-.,, .s'. { are rationalised by putting 



Ii 

 + a 



(I), 



o = A- ( a 



\a 



unchanged by the substitution (a, j, which interchanges /= } and |. 

 Also, in the region v /2 > a > v /2 1 , f }, 



4a 



P (v) = * = xn | K' (4) 



Q(r) = K/-S = (1 +a) (l 



16a s 



(5). 



a 



\ / - 

 ,.181 2w/) a = 

 <t - n- 1 + 2(t - <i~ 



To normalise the results, a homogeneity factor M is introduced, such that 



. - h = M^,, M = v /(,, - ,,) = (I + o) ( f; 2a _ a)) . (8). 



The division-values are shown in tabular form, and the numerical test of 

 = \ (v/5 1) is required for yu, = 15. 



