464 Royal Society : — 



And if we assume k properly, z + kp may be made to touch the 

 multiplier of ^;, i. c. the multiplier may be made to take the form 



-fj,{z + kpf+2)(j^z + l<p)v + m''). 



From the symmetry which reigns between x and y, it is obvious 

 a priori that any value of k which is rightly assumed for the 

 object in view will make co (when z is eliminated from it by means 

 of the equation z-\-kp-=Qi) a multiple either oi cc—y or x-\-y; 

 the latter obviously cannot be true, since such values would make 

 the given cubic a function of a.' + ;/ and z ; the proper values of 

 k will therefore make .r— 7/ = 0, from which, combined with the 

 equation 2a^ + z^ + Qmx^z = 0, the values oix-.y.z may be deter- 

 mined. These will be three in number ; and as we may write, 

 instead of x and y, px, p^y, or py, p^x, we obtain three sets of 

 three points, corresponding to p being taken x + y + 2mz ; and 

 consequently, by interchanging z with x and with y successively, 

 we obtain altogether three systems of three sets of three points 

 each; any such factor as x + y + 2mz is a tangent to a point of 

 inflexion, and it is clear a priori that if the cubic is put under 

 the form u^ -^ v[uiv + o>^) , since v=-0 make i«^ = 0, v can only be 

 a tangent at an inflexion. Hence the nine sets of three points 

 just assigned ai'e all that can be found enjoying the property in 

 question, and it is readily seen that x—y is the straight line 

 containing the three points of intersection in which the second 

 emanant, 



(*' Tx ^y' Ty +"' i)' (f' + f + ^'-^r>ixyz), 



at the point of inflexion \_x + y = 0, ^=0] cuts the given cubic 

 over and above the three coincident points x + y=0, ^=0. In 

 other words, each ternary group of the twenty-seven points in 

 question consists of the three points in which the curve is met 

 by the tangents drawn from a point of inflexion, which agrees 

 with the geometrical construction given by Pliicker in Crelle's 

 Journal. 



"Woolwich, May 3, 1856. 



LXII. Proceedings of Learned Societies. 



ROYAL SOCIETY. 

 [Continued from p. 393.] 

 June 21, 1855. — The Lord Wrottesley, President, in the Chair. 



r|"^HE following communication was read : — 



-*- " On the ultiinate arrangement of the Biliary Ducts, and on 

 some other points in the Anatomy of the Liver of Vertebrate Animals." 

 By Lionel S. Beale, M.B. 



In his valuable communication to the Royal Society in 1833, 



