216 Mr. J. J. Sylvester on a new Class of Theorems 



or 



or 



or 



lx Q + my% + nz 2 = 0, 

 lx 3 + my 3 + nz 3 = 0, 

 lx 4 -{-my 4 + nz 4 =0, 



whence the theorem. 



Now suppose U and V to be each functions of four letters* 

 x, y, z, t; when 



IZZl {^u+i*v+(to+»ij/+w«+^)tt}=o, 



xyztu 



the conoid AU + ^V touches the plane 



Ix + my + nz +pt = ; 



and I | =0 being a cubic equation, in general three such 

 conoids can be drawn. 



Considerations of analogy make it obvious to the intuition, 

 that in the particular case of two of these becoming coinci- 

 dent, the given plane Ix -\-my + nz -\-jpt must be a tangent plane 

 to those two coincident conoids at one of the points where it 

 meets the intersections of U=0 V=0; i.e. 



lx + my-\- nz +pt—0 



will pass through a tangent line to, or in other words, may be 

 termed a tangent plane to the intersections. Hence the fol- 

 lowing analytical theorem, derived from supposing q, r, s, t 

 to be proportional to the areas of the triangular faces of the 

 pyramid cut out of space by the four coordinate planes to 

 which x, y 9 z, t refer. As these planes are left indefinite, 

 q, r, 5, t are perfectly arbitrary. 

 Theorem. — The resultant of 



4. 



U=0 1 where U and V are functions of 

 V=0J x } y, #, t. 



lx + my + nz -fp£= 

 'dU dU dU dU" 



= 0; 



dx 



dy 



dz 



dt 



dV 



dV 



dV 



dV 



dx 



dy 



dz 



dt 



I 



m 



n 



P 



9 



r 



s 



t 



which system, it will be observed, consists of three quadratic 



