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Mr. M. Gardiner on Undevelopable 



[May 14, 



the direct, general, and complete solution of which is claimed to be given 

 in it for the first time. 



Chapter I., after some preliminary general properties, resulting imme- 

 diately from the known properties of homographic systems of points on 

 plane conies, treats more particularly of the simplest case of such sys- 

 tems on undevelopable quadrics, viz. of the case of systems in perspective, 

 the several pairs of whose corresponding constituents possess manifestly 

 the property of interchangeability ; shows that systems having three double 

 points not in the same tangent plane to the quadric on which they lie are 

 necessarily of that class, except only when they have a fourth double point 

 not in the plane of the other three, in which case they altogether coincide ; 

 and gives simple instances in which the problem, whose solution is the 

 principal object of the memoir, is manifestly either " wholly or partially 

 parismatic," as he terms it. 



Chapter II. treats of systems whose several pairs of corresponding 

 points are interchangeable but which are not in perspective ; shows that 

 their several chords of connexion intersect the same two reciprocal lines 

 with respect to the quadric on which they lie ; shows the relation between 

 either system and the perspective of the other to any point on either of 

 those lines, also the relation between either of two systems in perspective 

 and the perspective of either to any point conjugate to their centre of per- 

 spective with respect to the surface ; and shows how to construct the two 

 reciprocal lines from two pairs of corresponding points of the systems. 



Chapter III. treats of systems whose several pairs of corresponding 

 points connect through a single common line, which have therefore an 

 infinite number of double planes passing through that line ; shows that 

 their several chords of connexion, besides intersecting the line, all touch 

 a second quadric having double contact with the original, both at its two 

 points of intersection with the line, and also at its two points of intersec- 

 tion with the reciprocal line ; proves a property of the cone enveloping 

 either surface from any vertex taken arbitrarily on either line , shows the 

 relation between either system and the perspective of the other to any 

 point on either line ; and shows how to construct the two reciprocal lines 

 from two pairs of corresponding points of the systems. 



Chapter IV. treats of systems having two of their four double planes 

 non-tangential to the quadric on which they lie ; shows that the chords 

 connecting their several pairs of corresponding points touch two cones en- 

 veloping the quadric along two planes colinear with and harmonically con- 

 jugate to each other with respect to the two non- tangential double planes ; 

 proves that those touching along a plane section of either cone generate a 

 skew quadric ; shows that the two homographic systems determined by the 

 two correspondents in the two systems of a variable point on the quadric 

 are of the class considered in the preceding chapter, having an infinite 

 number of double planes passing through the line of intersection of the 

 two non-tangential double planes of the original systems ; and shows the 



