268 
SECOND MEMOIR ON THE 
[407 
the point P is at a cusp, then the n — 2 tangents become the n— 3 tangents from 
the cusp, and the tangent at the cusp; hence the curve 0 meets the given curve in 
2 (n — 3) + 3, =2n — S points, that is, (n — 2) + (n — 1) points; this does not prove (ante, 
No. 96), but the fact is, that the cusp counts in the Supplement (n — 1) times, and the 
expression of the Supplement is =(n — 1) k. It is clear that we have /3 = /3' = (n — 2)(m — 3), 
so that the equation is 
8D — 2k + 2t — 2 (n — 2) (in — 3) + (n — 1) k = (n — 2) 2D, 
that is 
2r = 2 (n — 2) (m — 3) + (n — 6) 2D + (— n + 3) k ; 
or substituting for 2D its value = n — 2m + 2 + k and reducing, this is 
2t = v? + 8 in — lOn — 3/e, 
which is right. 
104. As another example, suppose that the point P on a given curve of the order 
m and the point Q on a given curve of the order in have an (a, a!) correspondence, 
and let it be required to find the class of the curve enveloped by the line PQ. Take 
an arbitrary point 0, join OQ, and let this meet the curve m in P'; then (P, P') 
are points on the curve in having a (m'a, mu') correspondence; in fact to a given 
position of P there correspond a! positions of Q, and to each of these in positions of 
P'; that is, to each position of P there correspond mu 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 0, hence the 
curve 0 does not pass through P, and we have k = 0. Therefore the number of the 
united points (P, P'), that is, the number of the lines PQ which pass through the 
point 0, 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 0 is = ma' + m'a, that is, the order of the scroll generated by the line 
PQ is =ma'+m'a. 
Article Nos. 105 to 111.—Application to the Conics which satisfy given conditions, one at 
least arbitrary. 
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, (3-£T)(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.