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A PEOPEETY OF AXISYMMETEIC DETEEMINANTS, 

 CONNECTED WITH THE SIMULTANEOUS VANISH- 

 ING OF THE SUEFACE AND VOLUME OF A 

 TETEAHEDEON. 



By Thomas Muib, LL.D. 



1. In a striking and characteristic paper,* published by Sylvester 

 in 1853, he dealt with a consequence of the simultaneous vanishing 

 of the volume and surface of a tetrahedron. Denoting the squared 

 areas of the faces of the tetrahedron by F, G, H, K, and the volume 

 by V, he supposes one of the vertices to become a point in the 

 opposite face, with the result that 



x/F + Vg + n/H + s/K = 

 and V = 



and he thence concludes that if F, G, H, K, V be expressed in 

 terms of one and the same set of variables — say the edges a, b, c, 

 f, g, h of the tetrahedron — the norm of \/F + \/G + >/H + \JK 

 must contain V as a rational factor. Further, the said norm N 

 being 



( VF + JG + JR + JK) 



• (- VF + JG + JTL + n/K) ( n/F - jGc + n/H + JK) 



( s/F + JG~ JE + JK) ( JF + JG + x/H - x/K) 



• (- n/F- s/G + x/H + x/K) (- x/F + VG - x/H + x/K) 



(- x/F + x/G + x/H - x/K), 

 or 



SF 4 - 4SF 3 G + 62F 2 G 2 + 42F 2 GH - 40FGHK, 

 where 



16F = 2a 2 h 2 + 2h 2 g 2 + 2g 2 a 2 -a*- hA - g\ 



16G = 2b 2 f 2 + 2f 2 h 2 + 2h 2 b 2 - M -f* - li\ 

 16H — 2c 2 g 2 + % 2 / 2 + 2/ 2 c 2 - c* - g* -/*, 

 16K = 2b 2 c 2 + 2c 2 a 2 + 2a 2 b 2 -cH-fa-c*; 



* Sylvester, J. J., "On the Kelation between the Volume of a Tetra- 

 hedron. . . ." — Cambridge and Dub. Math. Journ., viii., pp. 171-178; or Col- 

 lected Math. Papers, i., pp. 404-410. 



