342 Mr. J. J. Sylvester on Staudt's Theorems 



method of doing so, wliicli is simple and ingenious, and does not 

 admit of material improvement, tlie reader is referred to the 

 memoir in CrcUe^s Journal or Terquem's Annates already adverted 

 to. It is, however, to be remarked (and this docs not appear to 

 be sufficiently noticed in the memoirs referred to), that whilst the 

 expression for the product of any two polygons in terms of the 

 distances given by Staudt's theorem is unique, that for the pro- 

 duct of two polyhedrons given by the same is not so, but will 

 admit of as many varieties of representation as there arc units in 

 the product of the numbers respectively expressing the number 

 of ways in which each polygonal face of each polyhedron admits of 

 being mapped out into triangles. I cannot help coujectming (and 

 it is to be mshed that Professor Staudt or some other geome- 

 trician M'ould consider this point) that in every case there exists, 

 linearly derivable from Staudt's optional formulae (but not co- 

 incident with any one of them), some unique and best, because 

 most symmetrical, formula for expressing the product of two 

 polyhedrons in terms of the distances of the angular points of 

 the one from those of the other. In conclusion I may observe, 

 that there is a theorem for distances measured on a given straight 

 line, which, although not mentioned by Staudt, belongs to pre- 

 ciselj'^ the same class as his theorems for areas in a plane and 

 volumes in space ; viz. a theorem which expresses twice the rect- 

 angle of any two such distances under the form of an aggregate 

 of four squares, two taken positively and two negatively ; that is 

 to say, if A, B, C, D be any four points on a right line 2AB x CD 

 =AD2 + BC2— AC^— BD2. I know not whether this theorem 

 be new, but it is one which evidently must be of considerable 

 utility to the practical geometer. 



Note on the above. 



The fundamental theorem in determinants, published by me 

 in the Philosophical ]\Iagazine in the course of last year, lead.s 

 immediately to a class of theorems strongly resembling, and 

 doubtless intimately connected with, those of Staudt. 



Thus for triangles we have by this fundamental theorem 



Xi a^2 ''^3 ?1 b2 63 



yi ^2 Vs X vi vo vs 

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*1 ^1 ^2 fs '^'2 '^3 '^1 ^2 ^d ^l '■^2 ^3 



=yi Vi Vi X % 2/2 1/3 + U\ Vo Vs X Vi Vi Vs 



111 111 11] 111 



•^1 63 tl 62 "^2 "^'a 



+ y\ V3 Vl X ^2 7/2 .V3 

 111 111 



