On the Natural Qi'oupings of the Elements, 423 



is probable that the general case might be treated in an analo- 

 gous manner by first grouping together the terms which corre- 

 spond to a given difference x~y, and ultimately summing the 

 sums of these partial series; but I have not examined this 

 question. 



2 Stone Buildings, W.C. 

 April 18, 1857. 



LVIII. Note on a Theorem relating to the Rectangular Hyperbola. 

 By A. CayleYj Esq.'^ 



THE following theorem is given in a slightly different form 

 by Brianchon and Poncelet, Gergonne, vol. xi. p. 205, viz. 

 Any conic whatever which passes through the three angles of a 

 triangle and the point of intersection of the perpendiculars let 

 fall from the angles of the triangle upon the opposite sides is a 

 rectangular hyperbola. And there is an elegant demonstration 

 depending on the properties of the inscribed hexagon. The 

 theorem is, however, a particular case of the following : viz. "Any 

 eonic whatever which passes through the four points of intersec- 

 tion of two rectangular hyperbolas is a rectangular hyperbola." 

 And this, again, is a particular case of the following : viz. If 

 there be a conic 12 and a line P, then considering any two conies 

 U, V such that the points of intersection of P, U are harmonics 

 in respect to the points of intersection of P, O, and the points 

 of P, V are also harmonics in respect to the points of intersection 

 of P, fl, then any conic whatever W which passes through the 

 four points of intersection of U, V will have the like property, 

 viz. the points of intersection of P, W will be harmonics in 

 respect of the points of intersection of P, fl ; a theorem which 

 is an immediate consequence of the theorem that three conies 

 which intersect in the same four points are intersected by any 

 line whatever in six points which are in involution. 

 2 Stone Buildings, W.C. 

 April 23, 1857. 



LIX. On the Natural Groupings of the Elements. By William 

 Odling, M.B. Lond., Professor of Practical Chemistry, Guy's 

 Hospital ; Secretary to the Chemical Society*. 



Part I. 



THAT certain elements have certain properties in common is 

 now a time-honoured doctrine in chemical science; but 

 the majority of chemists have been satisfied with a simple ad- 

 & mission of the fact : they have not investigated the extent of 



I 



