of Pascal's Hexagramme Mystique 

 When I is in the arc EG and F in the arc 



335 



Class II. — 'When B is 'within the conic section; and 



When I is in the arc GD and F in the arc 



When I is in the arc DH and F in the arc 



When I is in the arc HE and F in the arc 



When I is in the arc EG and F in the arc 



. 17 

 . 18 



• 19 

 . 20 

 . 21 

 . 22 

 . 23 

 . 24 

 . 25 

 . 26 

 . 27 

 . 28 

 . 29 

 . 30 

 . 31 

 . 32 



These appear to be all the variations which can take place,' 

 though Mr. Drummond, in his Demonstration of the converse 

 property, states (Phil. Mag. No. 299.) the number of cases to 

 be forty. But as that gentleman has not given any specific 

 enumeration, it would become tedious to inquire into the 

 source of his oversight. 



This theorem (in the form of a property of the inscribed 

 hexagon) has been so often demonstrated, — and as the demon- 

 stration of that case applies, mutatis mutandis, to all the others, 

 — it is unnecessary to annex a new one here. A remark on one 

 or two of the cases will be sufficient. Case 9. is that property 

 of the inscribed hexagon so often mentioned: Case 21. is that 

 figured in Maclaurin's Properties of Curve Lines, fig. 27 ; 

 and Case 32. is that mentioned in my paper on the Trapezium, 

 scholium to Prop. VI. It may seem superfluous to remark 

 that from this theorem are also derived extremely simple 

 demonstrations of the well-known properties of the inscribed 

 and circumscribed trapezia found in many of our best treatises 

 on Conies : and further, that if the rectilineal angle be consi- 

 dered (as Pascal considered it) one of the conic sections, the 

 sixth proposition of my paper before alluded to, also becomes 

 a specimen of the 32nd case enumerated above. 



The first series of corollaries that follow from combining the 

 different cases of this theorem are concerning the trapezium. 



Cor. 



