concerning the contents of Polygons and Polyhedrons. 343 

 and consequently, if ABC, DEF be any two triangles, 



ABCxDEF=ADExFBC + AEFxDBC + AFDxBCE. 



This may be considered a theorem relating to two ternary 

 systems of points in a plane. The analogous and similarly ob- 

 tainable theorem for two binary systems of points in the same 

 right line is AB x CD = AC x DB - AD x CB. As in applying 

 this last theorem to obtain correct numerical results we must 

 give the same algebraical sign to any two lengths denoted by the 

 two airangements XY, ZT, according as the direction from X to 

 Y is the same as that from Z to T, or contrai-y to it, so in the 

 theorem for the products of triangles, the areas denoted by any 

 two ternary arrangements XYZ, TUV must be taken with the 

 like or the contrary sign, according as the direction of the rota- 

 tion XYZ is consentient with or contrary to that of TUVj so 

 that three of the six possible arrangements of XYZ may be used 

 indifferently for one another, but the other three would imply a 

 change of sign. If we analyse what we mean by fixing the direc- 

 tion of the rotation of XYZ, and reduce this form of speech to its 

 simplest terms, we easily see that it amounts to ascertaining on 

 which side of B, C lies, i. e. whether to its right or left, to a spec- 

 tator stationed at A on a given side of the plane ABC. 



Let us now pass to the corresponding theorems for two tetra- 

 hedrons put respectively under the forms 



a?, x^ x^ x^ fj 1^2 1^3 ^4 



Vi Vi Vs 1/4 Vi Vi Va Vi 



Zl Z^ Zq Z^ fj ^2 ^3 ?4 



1111 1111 



We may represent this product in either of two ways by the 

 application of our fundamental theorem, viz. as 



=^1 ^\ fa h ^4 -^2 *'3 ^4 



V\ Vi V^ Vs X V4 ya 2/3 2/4 +&C. 



•*! bl C2 b^3 b4 ~2 *3 *4 



1111 1111 



or as 



^1 *'2 fel ?2 ?3 ?4 % ^4 



Vy Vi Vi V2 X V:i V4 Vs 2/4 +&C. 



-1 *^2 bl ?2 Ci M ^a ~4 



1111 1111 



there being four products to be added together in the first ex- 

 pression and six in the latter; and the rule, if we wish that all 

 the products may be additive, being that on removing the sign 

 of multiplication the determinant to the square matrix formed 



