(H)U^ 



NEWSON: GEOMETRY OK ONE DIMENSION. 107 



The second polar of each point on the Steinerian has a point 

 common with the Hessian. Denoting by Gj the group of points 

 which are on second polars of the Steinerian and do not belong to the 

 Hessian, we can find the equation pf G, as before by eliminating be- 

 tween (,S)U and (Pj)-U and dividing out the Hessian factor. The 

 third, fourth, etc. to the (« — i)th polars of the points of the Steinerian 

 form successive groups which may be designated respectively by Gg, 

 G.^, . . . Gii_2. The group Gr (r-rr2, 3, 4, ... «— 2) consists of 

 2(« — 2){n — r — i) points. The last group consists of the 2(« — 2) 

 polar points of the Steinerian. 



We have heretofore considered U to be a non-singular (^uantic, i. e. 

 to represent a group of points having no double or multiple points. 

 We now proceed to examine the theory of a singular quantic U. 



Let the //ic have a multiple point of order k, and let this multiple 

 point be taken as one of the ground points of the system of binary 

 coordinates. The quantic may then be written in the form U=x'^V. 

 Substituting this value of U in the form for the Hessian, 

 d-U d2U 

 dx- dxdy 



d^U d^U 

 dxdy dy- 



it is easily shown that (H)x'^V contains x~''^~" as a factor. Hence 

 we infer 



THEOREM v.--// U have a multiple point of order k, this point 

 appears as a multiple point of order 2{Ji — z) 07i (^H)U. 



We come now to the consideration of the Steinerian of a singular 

 quantic U. It follows from Theorem IV that (H)U and (S)U are 

 two groups of points having a one to one correspondence; hence we 

 may infer from the "correspondence principle" that the Steinerian of 

 U has the same singularities as the Hessian of U. 



However this may be proved directly as follows: Let U be written 

 in the form x'^V, where V is of order M; then n--k-\ m. But the 

 first polar of any point (x,, yj) is of order (// — i) and contains x as 

 a multiple point of order {k — i); to find the Steinerian we have there- 

 fore to form the discriminant of a quantic of order (//—/'), or m. 

 This discriminant, which is the Steinerian less a factor x to a certain 

 power, will contain Xj and y^ to the degree 2(/// — i); but the Stein- 

 erian is of degree 2{n — 2), therefore x must be contained as a factor 

 2(// — 2) — 2{m — irr=2(« — ui — 1) = 2(/' — i) timcs. Wc Can put this 

 fact into the form of 



THEOREM VI . If U have a multiple point of order k, this point 

 appears as a multiple point of order 2{k — /) on {S ) CI. 



