392 Proceedings of the Royal Irish Academy. 



19. The osculating planes of the quarfic envelop the quad/ric of 

 Art. 8. 



More especially for the quartic curve, it is easy to show that its 

 osculating planes envelop the quadric ISp6~^p = - 1 of Art. 8. Taking 

 the point x^yi three times over, and an arbitrary point Xojjo once, the 

 pole is (1112), and it lies on the tangent (111) to the quartic. iN'o'u-, 

 in the osculating plane [11] the points are x^yi taken three times over, 

 and the fourth point x^^y'i, in -which the plane meets the curve. The 

 pole (1111') of these four coplanar points lies in their plane, and con- 

 sequently lies on the quadric ISpO~^p = - 1, and the osculating plane is 

 the tangent plane thereat. It should be noticed also that, taking the 

 point (1111) twice, and two other points 0:33/3 and z^i, which lie in a 

 phme with (1111) taken twice, that is to say, the points which lie in 

 a plane through the tangent line (111), the locus of their poles (1134) 

 is a right line in the osculating plane. For, the points being coplanar, 

 the theorem of Art. 8 holds good, and the locus of poles of a system of 

 planes through a line with respect to a quadric surface is a right line. 

 This line meets the conic (11) in two points. Corresponding to these 

 poles, the variable plane touches the quartic in a second point, or it 

 contains two tangent lines, or every tangent to the quartic meets two 

 others, or the rank of the developable formed by these tangent lines 

 is 6,^ as will be otherwise proved later on. 



It wiU also be shown that there are four planes which pass through 

 four consecutive points on the curve. The theorem of Art. 8 holds with 

 respect to one of these points taken four times. These four points con- 

 sequently lie on the quadric ISp6~^p = - 1, and as the osculating planes 

 touch the quadric, the quartic touches it likewise at each of the four 

 points. 



20. Characteristics and reciprocal of unicursal curves. 



There is no difficulty in determining the characteristics of these 

 unicursal curves, using the principles laid down in Arts. 326 and 327 

 of Salmon's "Three Dimensions." In accordance with Dr. Salmon's 



' See " Three Dimensions," Ait. 330. The number of tangents which meet a 

 given tangent is r — 4, where r is the rank. 



