PROFESSOR CHRYSTAL ON THE HESSIAN. 



647 



Now, provided none of the linear factors of u 12 occur in xu 5> and none of 



those of u b in u 10 we have 



U^ 2 = 8xl2 



Um 5 = u 10 it 5 = 10x5 



whence finally UV=96 -50 = 46. 



We may consider the more general case. 



Ex. 2. 



U X m U k - m + U k +r m>fx 



VEZxH!^ +v lc+p r< P , 



of which Ex. 1 is a particular case ; it may be shown by the above method that 



UV = /* + ?>, 



This obviously agrees with the result of Ex. 1, and also with the following. 

 Ex. 3. 



U=x 2 -y 3 

 Y=x — y 3 



UV=2xl+lxl 

 = 3. 



The figure corespond- 

 ing to this case is 



s: 



Figl 



which may be looked 

 upon as the limiting 

 case of 



If in Ex. 2 r>p the application of the above method is not so simple, and 

 the result is not in all cases the same as will be shown directly. 



For the sake of comparison with the results of another process shortly to be 

 indicated, I work out two more examples by the present method. 



Ex. 4. 



we have 



U =x 3 u b + x 2 u 7 + u 10 

 "V =x\ +x 3 v z + v 7 , 



KEE^U — u s V=a^(v 1 u 1 — u b v 3 ) + xv^q — u 6 v 7 



z =LX 3 u i +« 12 say 

 L =?i 5 K — w 8 U=« l7 +w 18 say 



.-. 8xl7 = UL = UK + Uw 5 



= VY+fu 5 + Uu B 



136 =UV+90 

 UV= 46 



