﻿Sir James Cockle on Fractional Criticoids. 365 



and, by (6), we must have 



dy\a) dy\b) ' 



e f 

 in other words the fraction - or t must be free from y. If this 



a o 



be the case, then 



and the criticoid is 



a_(da db \e de df 



~\dx dij ) a dx dy 



8. Next, of the group (3) let bf—ce = alone be satisfied. 

 Then 



f= 0, *=-{ — {; 

 and, by (6), 



dx\c) dx\bJ ' 



f e 

 that is to say, the fraction - or ■=■ must not contain y. If this 



be so, then the criticoid is 



~\dx dy J)c y~~ dx dy 

 and 



v 



In this and in the preceding case the criticoid may be written in 

 various forms, determined by the mode of elimination. 



8. Let all the equations of the group (3) be satisfied. In this 

 case multiply the given quotoid into a, and it becomes, after the 

 elimination of c and f, 



a?r + 2abs + bH + 2e(ap + bq) + agz ; 



and if we transform this last quotoid, and divide the result as 

 before, we have the transformed and divided quotoid, | 



o 2 r + 2abs + bH + 2\{ap + bq) 



+ \a*(p + ?) + 2ab(a + Z V ) + b*(T + V *)+2e(a£ + br ] )+ag\z=x(z)> 



wherein A has the value written opposite to it in the group (1); 

 that is to say, 



\ = aj; +br] + e. 

 Hence 



t=-Jj *=-£*■ 



