208 Mr. J. J. Sylvester on Poncelet's approximate 



we have now a geometrical solution of the question, which it is 

 important to observe is in general, but, as will be presently seen, 

 not universally applicable to the case when the limiting relations 

 of x, y, z are defined by means of the position of a variable point 

 limited to lie within a triangular area upon the surface 11 = 1, 

 whose sides are determined by the traces upon that surface ot 

 three planes drawn through the origin ; the plane drawn through 

 the angular points of this triangle will then take the place of 

 the plane Atf + By-f 0^—1=0 in the preceding investigation. 



The next thing to be done is to obtain the quantities L, M, N 

 in terms of A, B, C, and the coefficients of R, which is an easy 

 matter to accomplish. Let 



R = ax 2 + by 2 + cz 2 + 2fyz + 2gzx + 2hxy = (f> (x, y, z), 



and call f , rj, f the coordinates at the summit of the segment ; 

 the equation to the tangent plane at that point, which is of the 

 form Ax + By + Cz=0, will be identical with 



(«f+fc»+j'J)X+(«f+&»+/?)Y + (srf+/i, + eOZ=l. 

 Hence a% + hrj +g% = — , 



and 



and therefore 



<r <7 a 



1 Pfl(A,B,C) , 

 cr 2 A</>(A, B, C) ' 



where A<£ is the discriminant, and P(</>) the polar reciprocal of 

 <j> (A, B, C). Hence 



~v?. 



law of equality with all the logical rigour and precision of which the subject 

 might admit, as this would be beside my present object, which is not to call 

 in question the grounds of admitted truth applicable to the question in hand, 

 but to advance it one step further in the direction of practical application. 

 * We see from the above, that if A#+lfy=l, or Aa?-f By+C^r=l be 

 the equation to the chordal line or plane of a segment of a line or surface 

 of the second degree, the ratio of the perpendiculars to such line or plane 

 from the centre of the line or surface and the vertex of the segment re- 

 spectively, or, which is the same thing, of a ray to any point in the segment 

 to the portion of this ray produced, intercepted between the line or sur- 

 face and the tangent at the vertex, is expressed by Va ; VP. It may at 



