150 
PEOFESSOE CAYLEY’S SECOND MEMOIE ON THE 
it be required to find the class of the curve enveloped by the line PQ. Take an arbi- 
trary point O, join OQ, and let this meet the curve m in P' ; then (P, P') are points on 
the curve m having a (m'a, ma') correspondence ; in fact to a given position of P there 
correspond a' positions of Q, and to each of these m positions of P' ; that is, to each posi- 
tion of P there correspond ma! positions of P' ; and similarly to each position of P there 
correspond m'a positions of P. The curve 0 is the system of the lines drawn from each 
of the a' positions of Q to the point O, hence the curve 0 does not pass through P, and 
we have Jc=0. Therefore the number of the united points (P, P), that is, the number 
of the lines PQ which pass through the point O, is =ma' +m'a, or this is the class of 
the curve enveloped by PQ. 
It is to be noticed that if the two curves are curves in space (plane, or of double 
curvature), then the like reasoning shows that the number of the lines PQ which meet 
a given line O is =ma' -{-m'a, that is, the order of the scroll generated by the line PQ is 
—ma'-\-m'a. 
Application to the Conics which satisfy given conditions , one at least arbitrary . — 
Article Nos. 105 to 111. 
105. Passing next to the equations which relate to a conic, we seek for (4Z)(1), the 
number of the conics which satisfy any four conditions 4Z and besides touch a given 
curve, (3Z)(2) and (3Z)(1, 1), the number of the conics which satisfy three conditions, 
and besides have with the given curve a contact of the second order, or (as the case may 
be) two contacts of the first order; and so on with the conditions 2Z, Z, and then finally 
(5), (4, 1), . . . (1, 1, 1, 1, 1), the numbers of the conics which have with the given curve 
a contact of the fifth order, or a contact of the fourth and also of the first order . . ., or 
five contacts of the first order. 
106. As regards the case (4Z)(1), taking P an arbitrary point of the given curve m, 
and for the curve © the system of the conics (4Z)(1) which pass through the given 
point P and besides satisfy the four conditions, then the curve © has with the given curve 
(4Z)(I) intersections at P, and the points P are the remaining (2m— 1)(4Z)(1) intersec- 
tions ; in the case of a united point (P, P'), some one of the system of conics becomes a 
conic (4Z)(1) ; and the number of the united points is consequently equal to that of 
the conics (4Z)(1) ; we have thus the equation 
{(4Z)(l)-2(2m-l)(4Z)(l)} + Supp. (4Z)(I)=(4Z)(1). 2D. 
107. It is in the present case easy to find a priori the expression for the Supplement. 
1. The system of conics (4Z) contains 2(4Z • ) — (4Z/) point-pairs*; each of these, re- 
garded as a line, meets the given curve in m points, and each of these points is (specially) 
a united point (P, P') ; this gives in the Supplement the term m{2(4Z • ) — (4Z/)}. 2. The 
number of the conics (4Z) which can be drawn through a cusp of the given curve is 
* The expression a point-pair is regarded as equivalent to and standing for that of a coincident line-pair : 
see First Memoir, No. 30. 
