Page:American Journal of Mathematics Vol. 2 (1879).pdf/7



to propose a new notation for the lines and points connected with the Pascal Hexagram, to give a brief account of the discoveries of Veronese on the subject and to develop a few additional properties of the figure.

The vertices of the hexagon inscribed in the comic, $$S,$$ are $$A,$$ $$B,$$ $$C,$$ $$D,$$ $$E,$$ $$F;$$ the lines tangent to the conic at these vertices respectively are $$a,$$ $$b,$$ $$c,$$ $$d,$$ $$e,$$ $$f.$$ In general, a large letter will represent a point, a small letter a line. Lines joining vertices of the inscribed hexagon are called fundamental lines; intersections of sides of the circumscribed hexagon are called fundamental points. The intersection of the two fundamental lines $$AB,\,DE$$ is called $$P\,(AB\,.DE);$$ the line joining two fundamental points, $$ab,\,de,$$ is called $$p'\,(ab\,.de).$$ It is evident that $$p'\,(ab\,.de)$$ is the pole of $$P\,(AB\,.DE).$$ There are $$45$$ points $$P$$ and $$45$$ lines $$p'.$$ The Pascal line obtained by taking the vertices of the hexagon in the order $$ABCDEF$$ is called $$h\,(ABCDEF).$$ It passes through the points $$P\,(AB\,.DE),$$ $$P\,(BC\,.EF),$$ $$P\,(CD\,.FA).$$ Similarly, the intersection of the lines $$p'\,(ab\,.de),$$ $$p'\,(bc\,.ef),$$ $$p'\,(cd\,.fa)$$ is the Brianchon point $$H'\,(abcdef)$$ of the hexagon $$abcdef,$$ the pole of $$h\,(ABCDEF).$$

The three Pascal lines which meet in a Steiner point are (Salmon's Comic Sections, 5th ed., note, p. 361) $$h\,(ABCFED),$$ $$h\,(AFCDEB),$$ $$h\,(ADCBEF).$$ We shall call the Steiner point in which they meet $$G\,(ACE\,.BFD).$$ In this symbol, the relative cyclic order of the letters in each group of three is all that it is necessary to observe; for instance, $$G\,(AEC\,.FBD)$$ and $$G\,(ACE\,.BFD)$$ are the same as $$G\,(ACE\,.BFD).$$ Given a $$G$$ point, the $$h$$ lines through it are obtained by taking one group of three in a fixed order for the odd letter and permuting cyclically the other group of three for the oven letters. The Pascals which pass through the conjugate $$G$$ point are $$h\,(ABCDEF),$$ $$h\,(ADCFEB),$$ $$h\,(AFCBED),$$ and the symbol of that $$G$$ point is $$G\,(ACE\,.BDF);$$ hence two Rh