340 Mr. J. J. Sylvester on Staudt's Theoi-ems 



aud of the other p, q, r, s,8x 36, i. e. 288 times, their product 

 is representable by — 1 x the determinant 



and of course if p, q, r, s coincide respectively with a, b, c, d, 

 576 times the square of the tetrahedron abed will be represented 

 under Mr. Cayley's form, 



four out of the sixteen distances vanishing, and the remaining 

 twelve reducing to six pairs of equal distances. The demonstra- 

 tion of Staudt's theorem for triangles is obtained in precisely the 

 same way by throwing the product of the two determinants 



a-g y^ 1 and ^2 V^2 1 



under the form of — \\h of 



5;(^,-?,)^; Mx,-^.y; ^[^\-^^Y> 1 



S(^2-li)'; S(^2-f2)^ t[x,-^^Y> 1 



S(^3-^i)^; 2(^3-y^ S(a'3-f3)'; 1 



1; 1; 1; 



When the two triangles coincide, calling their angular points 

 a, b, c, the abo\'e written determinant becomes 



0; {abY; {(^oY; 1 



{baY; 0; {bcY; 1 



{caY; {cbY; 0; 1 



1; 1; 1; 



* The corresponding quantity to the above determinant for the case of 

 the triangle (hereafter given) is identical with the Nonn to the sum of the 

 sides. I have succeeded in finding the Factor (often dimensions in respect 

 of the edges), which, multiplied by the above Determinant itself, expresses 

 the Norm to the sum of the Faces, i, e. the superficial area of the Tetra- 

 hedron. 



