Il6 KANSAS UNIVERSnV OUARTERLV. 



that they must have the same singularities For this reason it does 

 not seem necessary to give a formal discussion of the singularities of 

 the function M(V\V). It may be readily verified in each particular 

 case that a singularity on J(VW) appears also on M(V\V). When 

 either V or \V has a multiple point, or when the two have a common 

 multiple point, the results may be readily obtained from Theorems 

 XIII, XIV, and XV by simply changing J(VVV) into M(VW). But it 

 is better, however, to write out the results for future reference. 



THEOREM XVII.— A multiple point of order k on cither V or IF 

 loill appear as a multiple point of order (/('—/) on M{V]V). 



THEOREM XVIII— If a point P />e a multiple point of 

 order k on V and of order k' on JV, rcliere V and IV are tiuo qnantics 

 of degrees n and m respectively; then, when km — k' n±o, P is a multi- 

 ple point of order (/--[ k' — i) on M(VIV); hut when k'm—kn----0, P is 

 a multiple point of order {kA-k') on M(VIV). 



THEOREM XIX. — When two groups cf points V and JV of the 

 same degree have a common multiple point of order k, this appears as a 

 multiple point of order 2k on M(VJV). 



It yet remains to point out the obvious truth that, when V and W 

 are first polars of a third quantic U, the Jacobian of V and W is the 

 same as the Hessian of U; and the function M(VW) then becomes the 

 Steinerian of U. It will prove to be an interesting exercise for the 

 reader to compare the propositions concerning the Hessian of U with 

 those for the Jacobian of V and W; also the Steinerian of U with 

 M(VW), and thus verify in detail the correctness of the whole. 



I hope to be able in the near future to prepare a paper discussing 

 in detail the application of these theorems to the linear geometry of 

 Cubic, Quartic, and Quintic. 



