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KANSAS UNIVERSITY QUARTERLY. 



points of the co-ordinate system; for the quantics may then be written 

 x-^V, and W. Forming the Jacobian we have 



J(VW)= 



The last determinant will not contain x as a factor unless V^ or W 

 contains it, a condition implicitly excluded by hypothesis. Whence 

 we infer the following theorem: 



THEOREM XIV.— A multiple point of order k on citlwr J' or 11' 

 7inU appear as a multiple point of order {k — /) on the Jaeobian op r 

 and IV. 



Next let us suppose that a given point P is a common multiple point 

 of both V and W; let us assume that it is a multiple point of order /' 

 on V and of order /'' on W. Taking P as one of the ground points 

 of the cordinate system, the two quantics may be written in the form 

 x^Vj and x'''VV,. Let it be assumed that V and W are of degrees // 

 and /// respectively, and that V^ and W^ are respectively of degrees 

 /;' and ;//. 



The Jacobian of these two quantics may be written: 



J(VW) 



/''x'^'-nVj+x'^' 



,dV 



dx 

 ,dW. 



,.dV, 

 f\x 



.dW 



dx dy I 



From this we can factor out x'^''^'"'! and have left the determinant 



dV, dV^ 

 "d7 



A--\-k'—l 



/'Vj+X 



dx 



/^'W,+x 



dx dy 



Since V, is if order n' and Wj of order ///', by Euler's theorem of 

 homogeneous functions ;.''Vj:=x 



d_V, 

 "dx 



•y 1 and w W , -^x-=-=r-"^ T 



•^ dv ' dx 



dW , 



V- — -. Substituting these values of \ , and \V,, we have 

 ^ dy ^ 11' 



