[ 289 ] 



XXXIII. On a Geometrical Theorem. 

 %William Spottiswoode, M.A.> ofBalliol College, Oxford*. 



IF three cones of the second order, having a common ver- 

 tex, cut one another two and two in at least three straight 

 Jines, and if in each cone there be inscribed a hexahedral 

 angle, such that each of the nine straight Jines of intersection 

 shall be a common edge of two hexahedral angles; then, adopt- 

 ing a former notation, the equations to the three cones may 

 be written thus: — 



S(V.VA 1 A 9 .V Wl .V.V f t 1 |x 2 .Vv 2 x 1 .V.Vv 1 v 2 .VA 9f t 1 ) = 0* 

 S( V. Va 2 a . V^v 2 . V. V^ • VvA 2 . V. V v . V A/* 2 ) = o !> i 

 S(V.VavV^v .V/lty^ .Vv^. V.VwiV. A^) = o 

 or 



SAjA^g . S[^ ] ^ 2 v l . SvjVgAg . SA 1 ^, 1 v 2 = SAj A 2 v 2 . S^ 1 ^t 2 A 1 . Sy 1 v 2 /x 1 . Sa^Vj 

 SA 2 A/x . S/x- 2 ^v 2 . 8v 2 vA . SA 2j w, 2 v = SA 2 Av . S|«, 2 ju,A 2 . Sv 2 vj& 2 . SAj«,v 2 

 SAAjju-j . Sp^v . SfVjAj . Sa^Vj zsSAA^i . Sju-jw^A . SvVjjx . Sa^v 



and if these be written thus, 



U=W, U.^W,, U 2 =W 2 , . . . (3.) 

 the equation 



UU 1 U 2 =WW 1 W 2 (4.) 



will be of the fourth order in each of the vectors A, A 15 A 2 , . . , 

 and will be satisfied when any of them [e. g. v) is made to co- 

 incide successively with each of the others ; (4.) is consequently 

 the equation of a cone of the fourth order, on which all the 

 nine edges lie. Hence the following 



Theorem. If three cones of the second order, having a com- 

 mon vertex, intersect one another two and two, the nine lines of 

 intersection (three being selected from each pair of cones) will 

 lie on a cone of the fourth order, 



Raigmore, Aug. 27, 1850. 



*■ 



XXXIV. On the Chemical Formida of the Nitroprussides. 

 By John Kyd1\ 



AT the suggestion of Professor Will, I have made some 

 experiments upon the nitroprussides recently described 

 by Dr. Playfair, mainly with the view of testing the simple 

 and elegant formula which Dr. Playfair considers as the pro- 

 bable one, although his experiments do not permit him to 



* Communicated by the Author. 



f From Liebig's Annalen, vol. Ixxiv. p. 340. 



Phil. Mag. S. 3. Vol. 37. No. 250. Oct. 1 850. U 



>5 



(2.) 



