336 Mr. J. J. Sylvester on Stauclt's Theorems 



the area of a triangle in terms of the sides (in which, when ex- 

 panded, only even powers of the lengths of the sides appear), 

 is but a particular case of Staudt's theorem for polygons, 

 for it may be considered as the case of two equal and similar 

 triangles whose angular points coincide. So in like manner, 

 as observed by Staudt, a similar expression in terms of its sides 

 may be found for the square of a pyramid. This expression had, 

 however, been previously gi\-en (although, by a strange negli- 

 gence, not named for what it was) by !Mr. Cayley in the Cam- 

 bridge MathematicalJournal for the year 1841*, in his paper on 

 the relations between the mutual distances to one another of 

 four points in a plane and five points in space; the singularly 

 ingenious (and as singularly undisclosed) principle of that paper 

 consisting in obtaining an expression for the volume of a pyramid 

 in terms of its sides, and equating this, or rather its square, to 

 zero as the conditions of the four angular points lying in the 

 same plane. 



The analogous condition for five points in space is virtually 

 deduced by going out into rational space of four dimensions, 

 and equating to zero the expression obtained for the volume of 

 a plupyramid ; meaning thereby the figure which stands in the 

 same relation to space of four as a pyramid to space of three 

 dimensions. Mr. Cayley's method, if it had been pursued a step 

 further, would have led him to a complete anticipation of the 

 principal part of Staudt's discovery. The method here given 

 is not substantially different from Mr. Cayley' s, but is made to 

 rest upon a more general principle of transformation than that 

 which he has employed. As to Standt's o-mi method, it is as 

 clmnsy and circuitous as his results are simjile and beautiful. 

 Geometiy, trigonometry and statics, are laid under contribution 

 to demonstrate relations which will be seen to flow as immediate 

 and obvious consequences frojn the most elementary principles in 

 the algorithm of determinants. Perhaps, however, M. Staudt's 

 method is as good as could be found in the absence of the appli- 

 cation of the method of determinants, the powers of which, even 

 so recently as ten years ago, were not so well understood or so 

 freely applied as at the ]3resent daJ^ 



The following new but sim])le theorem, of which I shall have 

 occasion to make use, will be found to be a very useful addition 

 to the ordinary method for the multiplication of determinants. 

 " If the determinants represented by two square matrices are to 

 be multiplied together, any number of columns may he cut off 

 from the one matrix, and a corresponding number of columns 



* Quaere, Is not this expression for the volume of a pjTamid in teims of 

 its sides to be found in some previous wTiter ? It can hardly have escaped 

 inquiry. 



