252 PROFESSOR A. U. GREENHILL ON THE 



Denoting by a' 2 , b' 2 the values of y z corresponding to a 2 , 6 3 of s, 



reducing to 



/7 7 /! " 



+ 4a ^/{i _ 8 (1 - 2m) a} 

 so that 



a' = en/K', V = dn/K', in Region I. (11); 



a' = sn/K', 6' = dn ^ R/ , in Region II. (12). 



Changing the variable to y in the integral 



i^)==r P (') r - Q ("') 

 )>,* y 8 ' 



where 



Y = 4(r- !)(?/- a' 2 ) (r-// e ) (14), 



P (,') = P (^~ V "> , W> = m * a ( ~ ^ + (15), 



Q(,/)=i^^ = a'6' = G 2 (16); 



/"( ^X' 1 ?,/-' / 1 >7 V 



: " 



20., = '' + 6'- = a 



4a(a m + 1) 



1 - a'- ,, V- I 



- 



A/1 8 (] 2y/i)a 



25. To connect up ABEL'S results for even values of ^ we take n = 1 in (9), 5, so 

 that 



2a; y y~ 1 x (u x ii~\ 

 a- , & = - _A2_ 2 _ - ^ ' , p = 4a;^/ (1), 



Z = - (2 2 - 2 + Vf + _p 2 



= - (z- - az + b + 2s) 3 + 4 (Az + B) 2 (2), 



where 



A- = s, 2AB = a,s> + ajy, B 2 = s (s + 6) (3), 



aud therefore the resolvin cubic becomes 





(4), 



