THIRD ELLIPTIC INTEGRAL AND T1IK KLLII'SOTOMIC 1'KOBLKM. 269 



32. ,.= 18, /srlOLl^J. 



The relation D y = becomes by tbe substitutions 



a; + 1 



y + 1 2 V (g^ + 7 



Now the p employed in (2), 10, distinguished here as /;, is given by 



^ = ^/( 1 - 2 ). Z -V = ~ P 



returning for a moment to the original x and y of 10, so that 



the same as those employed with D H in 22, a new relation 



representing a C 5 in (p, e) ; and putting 



1 1 



this (J 5 reduces to a (J.j. in (,r, ?/) with deficiency p = 2, 



,,f __ (tf + -2x - 2) //' - (x + 4) .nj + x~ (.,: + 2) = (4), 



and this with y = (q + 1 ) ,, as in L.1N1.S., 27, p. 4U3, I)ecoiues 



= 7 v 



2( V + 1)^ 

 = ,/ + 2,/' + 5,y' + I0<f' + 10 7 - -H 4 V + I (8), 



- 



v - y 3 + v 3 - 2 9 - - 1 + v^ do) 



2(3 + 1) 



