550 
PROBLEMS AND SOLUTIONS. 
[485 
corresponding points), then AC, A'G will meet the curve in the corresponding points 
B', B; and AB, A'B' will meet on the curve in a point C' corresponding to C, giving 
the inscribed quadrilateral (A, B, C, A', B', C"); the triangle ABC is therefore constructed. 
It is to be remarked that the equation fgh — ijk being satisfied, we may without 
any real loss of generality write f=j,g=k, h — i, and therefore a = /3 = r y; hence 
changing the constants we have the theorem : the inverse points (x, y, z), (ar 1 , y~ x , z~ r ) 
are corresponding points on the curve 
ax (y- + z 2 ) + by (z 2 + x 2 ) + cx (x 2 + y 2 ) + 2Ixyz = 0. 
[Yol. Y. pp. 57, 58.] 
Addition to the Note on the Problems in regard to a Conic defined by five Conditions of 
Intersection. 
Since writing the Note in question, I have found that a solution of Problem 7 
has been given by M. De Jonquières in the paper “ Du Contact des Courbes Planes, 
&c.,” Nouvelles Annales de Mathématiques, vol. ni. (1864), pp. 218—222: viz. the number 
of conics which touch a curve of the order n in five distinct points is stated to be 
n (n — 1 )(n — 2) (n — 3) (n — 4) 
1.2.3.4.5 
(n s + 15?i 4 — 55n 3 — 4>9o?i 2 + 1584?i + 15). 
There are given also the following results; the number of conics which pass 
through two given points and touch a curve of the order n in three distinct points is 
n (n — 1) (n — 2) 
(n 3 + 6n 2 — 19?i — 12), 
and the number of conics which pass through a given point and touch a curve of 
the order n in four distinct points is 
n S?L 3 )( W 4 + ion 3 - 37n 2 - 118n + 282). 
These formulae are given without demonstration, and with an expression of doubt as 
regards their exactness—(“ elles sont exactes, je crois ”); they apply, of course, to a 
curve of the order n without singularities; but assuming them to be accurate, the 
means exist for adapting them to the case of a curve with singularities. 
[There is also a paper on the same subject in the Annales for January, 1866 
(pp. 17—20), from the Editors Note to which we have introduced a correction (+ 15 
instead of — 35) in the formula given above.]