Io8 KANSAS UNlVERSnv C)UARTERLY. 



There yet remain to be examined the singularities of the group G, 

 when the group U has a multiple point of order k. Let A be a 

 multiple point of order k on U. The Steinerian then consists of the 

 point A counted 2{k — i) times, and 2(/; — /' — i) single points. The 

 first polar of each of these single points consists of the point A 

 counted (/' — i) times, and (;/ — /') other points; so that the first polars 

 of all the single points on the Steinerian consist of the point A 

 counted 2(/' — 1)(''/ — k — t) times, and 2{ii — k'){n — k — i) other points. 

 The first polar of the point A on the Steinerian consists of A counted 

 /' times, and (// — /' — i) other points. But A occurs 2{k — i) times on 

 the Steinerian; hence all the first polars of A consist of A counted 

 2\{k — i) times, and 2{k — \){it — k — i) other points. Therefore all 

 the first polars of the Steinerian together contain the point A 2{k — i) 

 (// — i) times. But the Hessian contains the point A 2{k — i) times, 

 and the square of the Hessian contains it \{k — i) times. Subtract- 

 ing this number from the above, we have 2(/(' — 1)(;; — i) — \{k — i)=r 

 2{Ji — 1)(;/ — 3). Therefore the group G contains the point A 

 2(/&— i)-f (;;— 3) times. 



Besides the multiple point at A, the group G also contains {n — k 

 — i) other multiple points, each of order 2{k — -i ). For, as we pointed 

 out above, the first polar of the point A on the Steinerian consists of 

 the point A counted k times and (// — k — i) other points. But A 

 occurs 2(X' — i) times on the Steinerian; hence each of these (// — k 

 — i) other points occurs 2{Ji — i) times on the group G. 



By subtracting from the order of G the sum of the multiplicities of 

 the multiple points on G, we find the number of single points on G to 

 be 2(;; — /' — 2)(« — k — i). We can now sum up the singularities of 

 the group G in the following: 



THEOREM VII.— If U have a miilliple point A of order k, this 



point appears as a multiple point A of order 2{k — /)(// — J) on (^G)U; 



^(^G)U also has {n — k — /) other multiple points eae/i of order 2{k — /) 



at the sin_s;le first polar points of A; eonseguentlv, in t/iis ease, {G)U 



has only 2{n — /' — 2){ii — k — /) single points. 



By a process of reasoning similar to that above, it can be shown 

 that the group Gj consists of a multiple point at A of order 2\_n{k — 

 2) — {k — 3)], (// — k — 2) multiple points each of order 2(/(' — i) at the 

 single second polar points of A, and 2{k — ;/ — i) single points. Sim- 

 ilar formulae for the groups Gg, G3, . . . etc. might be developed, but 

 they are not especially important. 



