485] 
PROBLEMS AND SOLUTIONS. 
553 
among the twenty-seven problems enumerated in my Note on this subject in the 
January Number of the Educational Times {Reprint, vol. v., p. 25); but two of these 
ten are the problems 25 and 26 which are in my Note stated to have been solved; 
there are, consequently, of the twenty-seven problems, in all twelve which are solved: 
viz. these are where it is to be observed that Zeuthen’s solutions apply to the case 
No. of Prob. 
1, 8, 10, 12, 14, 17, 19, 21, 23, 25, 26, 27 
Zeuthen’s No. 
-, 14, 13, 11, 8, 3, 12, 7, 2, 6, 1, — 
of a curve of a given order with given numbers of double points and cusps. The 
problems 25 and 26 had been previously solved only in the case of a curve without 
singularities. As to Prob. 27, the solution mentioned in my former Note is in fact 
applicable to the general case. The solution for Prob. 1 may also be extended to this 
general case, viz. for a curve of the order m with 8 double points and k cusps the 
required number is = m {12m - 27) - 24S — 27/e; or, if n be the class, then this number 
is = 12?i — 15m + 9/e; so that all the twelve problems are solved in the general case. 
The results obtained by M. de Jonquieres, as stated in my Note in the March 
Number {Reprint, vol. v., p. 57), seem to be all of them erroneous. In fact, for the 
number of conics passing through two given points and touching a curve of the order 
m in three distinct points (which is a particular case of Prob. 23), Zeuthen’s formula 
applied to a curve without singularities gives this 
instead of the value 
which is 
= |m(m-2) {m 4 + 5m 3 — 17 m 2 — 49 m + 108) 
^ m {m — 1) (m — 2) (m 3 + 6m 2 —19 m— 12) 
= J m {m — 2) (m 4 + 5m? — 25 m 2 + 7m + 12) ; 
and I have by my own investigation verified Zeuthen’s Number. So for the number 
of conics through a given point and touching a curve of the order m in four distinct 
points (which is a particular case of Prob. 17), Zeuthen’s formula applied to a curve 
without singularities gives this 
= m {m — 2) {m — 3) (m 5 4- 9m 4 — 15 m 3 — 225 m 2 + 140 m + 1050) 
instead of the value 
m {m — 1) (m — 2) (m — 3) (m 4 + 10 m 3 — 37 m 2 — 118 m + 282) 
which is 
= (m — 2)(m — 3) (m 5 4- 9m 4 — 47 m 3 — 81 m 2 + 400 m — 282), 
and it may I think be inferred that the expression obtained for the number of conics 
which touch a given curve in five distinct points (Prob. 7), containing as it does the 
factor (m — 1), is also erroneous. 
C. VII. 
70