Inverse Problem of Criticoids. 195 



whence 



Vl-12L=-(N-L) + ^ 



«/l-12N=-(N--L)-»; 



and we can give (8) the form 



V1-12L- * / l-12N=2o> = -1 +s/T^fb. 



28. We need not rationalize this equation, which indeed 

 is, practically, rational. For, if we take the criticoid of the 

 biordinal 



(D-ft)(D-A)w + (D-« 1 )(D-« s yw=0, 

 we get a result of the form 



(1 + a?)W + (I + ma? + ra# 4 ) v = ; 

 where, taking 



%— «2, «i + «2— ft— fti ft— ft=J, U, I, 

 we have 



4Z=1-P, 4m=(U-2) 2 -(2 + I 2 + J 2 ), 4n=l-J 2 . 



29. In the last biordinal change the independent variable 

 from A' 2 to t d ; then write # in place of t and take the criticoid. 

 We get a result of the form 



and this, compared with the biordinal of art. 16, gives 



\=3L, /* = 3M, v=3N. 



30. But 



*=*'-*> p=4«Hb "=*»-Ai 



relations which yield 



X— ^-^=§(7 — m + ?i), X— v=f(J— n). 



31. Consequently 



12L=4X=9Z-f = 1-|P; 

 so 



12N=1-£J 2 , 

 and 



6 = L-M + N=i(X-/A + ^) = |(/-?w + M). 



32. Hence 



1-12L = £I 2 , 1-12N = £J 2 , 

 and 



l-|6 = l-(Z-m + n) = (U-2) 2 -4-4, 



