46G MB. A. K. FORSYTH ON INVARIANTS, CO.VARIANTS, AND QUOTIENT- 

 117. By somewhat similar work it may be proved that 



. _ <'"' I , 2l / 49 \ 



~ +<f)*' lS ' Z]>l 



and the combination of (48) and (49) gives 



. _ (fld-bey> ( 







/l . _ (fld-bey> (ffz_+ XT r -, , , 



' ffz + 4 ~ (eh -/,,) (a, + d)- L 



which is the general formula of transformation for the simultaneous homographic 

 transformation of the dependent and the independent variables. 



118. The following simple case will sufficiently serve to illustrate the kind of 

 limitation, which prevents the converse of the proposition of 112 from being, in 

 general, true. From the general proposition it follows that if 



[<r, s] s = and [s, x]% = 0, 



then 



[o-, x\ = 0. 



The question then arises : What are the conditions to be satisfied in order that 



[>, x\ - 

 may be a necessary consequence of 



\cr, s] 3 = and [cr, x] 3 = ? 



Taking the two latter as given, we may replace them by an integral algebraical 

 equation : 



as 8 + 2bs + c Aa? + 2Ba; + C 



2B'a; + C' 



(50), 



the two fractions being the values of er, corresponding to the two derivative equations. 

 And, if it is to be necessary that 



|>, a:] 2 = 0, 



then this algebraical equation (50) must be equivalent to one or more equations of the 

 form 



aa; 



8 ' 



(51) 



Hence, when the value of s given by (51) is substituted in (50), it must become an 

 identity ; the conditions for which are 



