386 



chords, PnAiPnj-i, . . . P2n-i A„P2„. Again, draw the n chords, 

 P2nA.iP3„+i, . . . P3„_, A„P3„. Draw tangent planes at p,i and P3„, 

 meeting the two new chords pPam and p^Psb in points R, r'; 

 and draw any rectilinear tangent bc at b. Then one or other 

 of the two following theorems will hold good, according as 

 n is an odd or an even number. When n is odd, the three 

 points brr' will be situated in one straight line. When n is 

 even, the three pyramids which have bc for a common edge, 

 and have for their edges respectively opposite thereto the three 

 chords PPgn, PgnPnj PmPsmj being divided respectively by the 

 squares of those three chords, and multiplied by the squares 

 of the three respectively parallel semidiameters of the surface, 

 and being also taken with algebraic signs which it is easy to 

 determine, have their sum equal to zero. Both theorems con- 

 duct to a form of Poncelet's construction (the present writer's 

 knowledge of which is derived chiefly from the valuable work 

 on Conic Sections, by the Rev. George Salmon, F. T. C. D.), 

 when applied to the problem of inscribing a polygon in a plane 

 conic : and the second theorem may easily be stated generally 

 under a graphic instead of a metric form. 



The analysis by which these results, and others connected 

 with them, have been obtained, appears to the author to be 

 sufficiently simple, at least if regard be had to the novelty and 

 difficulty of some of the questions to which it has been thus 

 applied ; but he conceives that it would occupy too large a 

 Space in the Proceedings, if he were to give any account of 

 it in them : and he proposes, with the permission of the Coun- 

 cil, to publish his calculations as an appendage to his Second 

 Series of Researches respecting Quaternions, in the Transac- 

 tions of the Academy. He would only further observe, on the 

 present occasion, that he has made, in these investigations, a 

 frequent use of expressions of the formQ+ -v/(- 1) Q> where 

 \/{- 1) is the ordinary imaginary of the older algebra, while 

 Q and q' are two different quaternions, of the kind introduced 



