...only thus proved for curves of the third class, if they do not involve any mention of the cusp or of the number of tangents which can be drawn to the curve from any point, we may see will be true for all curves of the third degree. Ex. 1. A chord is drawn through...

...through these several pairs ; and as these lines arc not tangents in the ordinary meaning of the word, the number of tangents which can be drawn to the curve from a given point is to be diminished by three for each cusp. Hence if a curve has K cusps, the number...

...substitute the co-ordinates of that point in the equation u = 0, and determine a so as to satisfythat equation. This problem will have a definite number...of the curve. For example, the envelope of the line act* 4 3 W + 3oa + d = 0, where a, J, c, rf, are linear functions of the co-ordinates, is plainly a...

...number of inflexions, and number of double tangents, — first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents. The...

...number of inflexions, and number of double tangents, — first, as regards the class, this is equal to the number of tangents which can be drawn to the curve from an arbitrary point, or what is the same thing, it is equal to the number of the points of contact of these tangents. The...

...the curve (other than at the multiple points), no one of which is common to all of them. Also that the number of tangents which can be drawn to the curve from an arbitrary point is 2m + 2p — 2. We have also shewn that the theory, for a curve whose multiple points have any complexity,...