438 On a Porismatic Property of two Conies. 



other boulders. At Boyndie^ further west, the flint boulders 

 cover the shore ; and at Delgaty, ten miles inland, they occur 

 in great abundance, along with boulders of quartz rock, but 

 no fossils except their own. It would therefore appear, that 

 we owe the flint boulders and the lias boulders to different 

 periods ; and as the chalk overlies the lias, it may be that its 

 denudation was completed, and its fossils thrown upon the 

 high grounds of the interior, previous to the formation of the 

 boulder clay containing the fossils of the lias. Although appa- 

 rently not here, the boulder clay has in other places (as on 

 the banks of the Thorsa in Caithness) been found to contain 

 " fragments of chalk flints, and also a characteristic conglome- 

 rate of the oolite," as well as comminuted fragments of exist- 

 ing shells. (H. Miller.) These facts seem also to favour this 

 hypothesis. 



The subject altogether is one involved in considerable dark- 

 ness, and it is perhaps vain to attempt any generalization upon 

 it till the local geology has been far more accurately examined 

 and determined. 



LVII. On a Porismatic Property of two Conies having with one 

 a?iother a contact of the Third Order. By J. J. Sylvester, 

 M.A., F.R.S* 



IF two conies have with one another a contact of the third 

 order, i. e. if they intersect in four consecutive points, it 

 will easily be seen that their characteristics referred to coor- 

 dinate axes in the plane containing them must be of the rela- 

 tive forms x 2 -\-yz, k(y 2 + x* +yz) respectively, y characterizing 

 their common tangent at the point of contact f« 



Hence if we take planes of reference in space, and call t 

 the characteristic of the plane of the conies, the equations to 

 any two conoids drawn through them respectively will be of 

 the relative forms 



U = a? + yz -f- tu = 



V =y 2 + x 2 -{-yz + tv=Q. 



* Communicated by the Author. 



t These relative or conjugate forms are taken from a table which I shall 

 publish in a future Number of this Magazine, exhibiting the conjugate 

 characteristics in their simplest forms, correspondent to all the various 

 species of contacts possible between lines and surfaces of the second degree. 

 This table is as important to the geometer as the fundamental trigono- 

 metrical formulas to the analyst, or the multiplication table to the arith- 

 metician ; and it is surprising that no one has hitherto thought of con- 

 structing such. 



