;eometry of one dimension. 



t^5 



J(VVV); 



/I'X , ' -f /'V ~ --^// X— — - 



dx cly dx 



dV 

 ^dx 



dW, 



k x-^ ' -^ k y— — ' -^m X —' 

 dx dy dx 



(/'^;/)x 



dV 



(k' 



dx 

 dW. 



-ky 



dV, 



dy 



^ dx ^^ dy 



// 



dy 



,dW, 



" ^r ! 



,dV, 



,dVV. 

 ^7 





(^'' + ;//)- 



dW 



dV, dVV, 

 dx dy 



dy 

 dW^ 

 dx dy 



From this last result we see that generally x is containeil in 



J(V\V) as a factor (X'-L/t'— i) times; we also see that, when the 

 I /, „' I 



o, X is contained once more in JO'^^)- 



X' _ ;/' k-r-n' 



determinant 

 Puttirm 



/y m' 



o, we have k/ii' ^^k' ii' . 



-"-. Making these substitutions in the last e(|ualion and omitting 

 in 



censtant factors, we have 



J(VW)^x''-'^' i 



|dV, 

 I dx 



dV, I 



dy I _^ 

 dW, dW, \'" 

 I dx dy I 

 These important results may be formulated as follows: 



THEOREM XW— If a point P be a multiple point of order k on V 

 and of order k' on IV, wliere V and IV are two g nan ties of de^i^recs n 

 and ni respeetivelx; then, 7C'lien km — k' n±0, P is a multiple point vf 

 order {k-\-k' — /) on Ji^VW); but when km — k'n^^O, P is a multiple 

 point of order {k-\ k') on /{III'). 



When the two quantics V and \V are of the same degree // and at 

 the same time the condition km — k' n^^o is satisfied, then must k=k' . 

 The Jacobian then contains x-^' as a common factor. 



THEOREM XVI.— When tiuo groups of points V and W of the 

 the same degree have a common inultiple point of order k, this appears 

 as a multiple point of order 2k on their Jocobian. 



(See Salmon's Higher Algebra, Art. 178.) 



Since the two groups of points (J(VW) and M(VW) have a one to 

 one correspondence, it follows from the '• correspondence principle "' 



