concerning the contents of Polygons and Polyhedrons. 339 



series of quantities. This simple transformation is of coui'se 

 derived by adding to the respective cokmnis in the first matrix 

 the last column (consisting of units) multiplied respectively by 

 h^, /<2, . . . h^^, ; and to the respective lines, the last line (con- 

 sisting of units) multiplied respectively by k^, k^, . . . k^^, 0. 



SujDpose, now, that we have two tetrahedrons whose volumes 

 are represented respectively by one-sixth of the respective deter- 

 minants 



*'i y\ ~i 1 li Vi ^1 1 



*2 y% ^■g 1 ^2 Vc, ?2 1 



% Vi -3 1 ^3 ^3 ^3 1 



^4 Vi 5-4 1 h V4 ^4 1 



^r' !/r> ^r representing the orthogonal coordinates of the point r 

 in one tetrahedron, and |^,., 77,., ^^. the same for any point (r) in 

 the other. 



By the first theorem their product may be represented 

 (striking ofi" the last column only from each matrix) by the 

 matrix 



^^i^\; Sa^i^s; 2*'i?3; S^il^4J 1 



Sa-gliJ 2a^2^2; Sa's^s; ^as^^^^; 1 



•^*'3?i i -'^3?'2 > Sa^a^s ; 2.^3^4 ; 1 



2a'4?,; Zt^^^; 2^-4^3; 2a?4f4; 1 



li 1; 1; 1; 



where, in general, any such term as 2*'^ • ^g represents 

 a; .^+v .7] +z . t. 



r ^s ' Jr 's ' r *s 



Again, by virtue of the second theorem, adding 



-l^"""' -\-^'' -l^^^'> -^2^4^ 



to the respective lines, and 



-l^V; -l^^,'; -\^U; -|sr/ 



to the respective colunms, the above matrix becomes (after a 

 change of signs not affecting the result) the — ^th of 

 r2(a',-|.)^ X{x,-^,Y; 2(^,-^3)2; 2(^,-^4)^; 

 2(.r,-e,)^ 2(.r,-^,)^ 2Cr,-^3)^; M^^,-^,f; 



2(^3-|l)^ 2(^3-^2)'^; 2(^3-13)2; 2(^3-f4)^• 

 .2(a.4-|,)2; 2(^^4-0,)^; ^i^.-hY', ^{^.-^f ', 



1; 1; 1; 1; . 



or calhng the angular points of the one tetrahedron a, b, c, d, 



Z2 > ' ' > 



