70 KANSAS UNIVERSITY QUARTERLY. 



NOTE A. 



The theorem concerning the three points on a conic A, B, and C, 

 whose osculating circles pass through a fourth point O on the conic, 

 is due to Steiner. From the properties of the harmonic polars of the 

 points of inflection on a nodal cubic we may infer many other theor- 

 ems concerning the points A, B, and C on a conic. Let the cubic be 

 projected into a circular cubic and then inverted from the node. Its 

 points of inflection Aj, Bj, Cj invert into the points A, B, and C. 

 The harmonic polar of A^ inverts into the common chord O P of the 

 circles osculating the conic at B and C; and similarly for the other 

 harmonic polars. 



The pencil 0-|ABPC [- is harmonic. Any circle through A and O 

 meets the conic in S and T so that the pencil 0-jASPT[- is har- 

 monic. The two circles through O and tangent to the conic at S and T 

 intersecton O P. If two circles be drawn through O and A intersecting the 

 conic one in S and T and the other in and U V, the circles O S U and 

 O T V intersect on O P; so also the cirles O S V and O T U. But one 

 circle can be drawn through O and A and tangent to the conic; its point 

 of contact is on O P. Let 1, m, and n be three points on the conic on 

 a circle through O. Draw the circles O A 1, O A m, 



and OA n intersecting the conic again in 1^, m^, n^; 1^, m^, n^, are 

 also on a circle through O, and the. circles through 1, m, n and 

 Ij, m^, n^ intersect on O P. 



NOTE B. 



From the fundamental property of the Cissoid of Diodes we can ob- 

 tain by inversion an interesting theorem concerning the parabola. In 

 the figure of the Cissoid given in Salmon's H. P. C. Art. 214, 

 AMj=MR, whence AMj=A R — AM;orAR = A M+A Mj. In- 

 verting from the cusp and representing the inverse points by the same 

 letters, we have for the parabola 



III 



A R ~ A M ' AMj 



This result is interpreted as follows: — draw the circle of curvature 

 at the vertex of a parabola; this circle is tangent to the ordinate B D 

 which is equal to the abscissa A D; draw a line through A cutting the 

 circle in R, the ordinate B D in M, and the parabola in M^; then 



III 



+ 



AR AM ' AMj 



Draw the circle with centre at D and radius A D; any chord of the 

 parabola through the vertex is cut harmonically by the parabola, the 

 circle, and the double ordinate through D. 



