OF THE RECIPROCAL OF A DETERMINANT. 



627 



which are all presentable in the form x 1 ' F 1 , where F x is a complete cubic in 

 x : , x 2 , . . . , x 5 , consisting of 125 positive terms, viz., in the case of the determinant 

 specified, five terms of the form b 1 c 1 d 1 e 1 ' x x 3 , twenty compound terms of the form 



X \ X 2 " (&l c l^i e 2 + ^l C 1^2 e l + ^l C 2^l e l)> 



and ten compound terms of the form 



X l X 2 X 3 ' (&l C 1^2 e 8 + ^l C l C ^3 e 2 + ^l c 2^i e » + &l C 2*Vl + ^Y'l^i e 2 + ^3 C 1^2 e i) • 



By adding all the other columns to the first column we see at once that the deter- 

 minant is equal to 



- h x z 



61 



cl 



d 3 x 3 



l>r<- 4 



dj^j + d^c. 2 + d 3 x s + d y r- : 



~h X 5 



- d b x 6 



6i»£i "T €cyi',y *T" G0X0 *t* &Ap*-± 



and by treating the cofactor of 6 } here in the same way we can separate it into two 

 parts of the same kind as the cofactors of b 3 , b i , b b . F x is thus seen to be the sum of 

 five similar parts, viz., 



Vi 



c? 1 d^ + d 2 x% + d 3 x 3 + d 5 x { 



6-t &aX. 



- c 5-h 

 -diTt 



^4*4 



"5" 5 



&-\ x \ *r* &i)Xi) ~f- @$X.> ~r ^4*^4 



Taking the first of these and dealing with it after the manner followed in the preceding 

 paragraph we find it expressible as the sum of twenty-five positive terms, viz., 



.'•., 



h \ x i 



C-iCt-iC-i C-i6t-i6rt Cit'iCo ^i^'i^i ^111 



C 1^3 6 l C 1^3^2 C 1^3 e 3 C l"3 6 4 C \':/'Z 



C^Cl-^6-^ C^.1^62 ^4*^1^3 ^4^i^4 ^'o^l^4 



c i^o e i c 5^2 e i c b'h e \ c 4 c h e i c 'A e \ 



*4 



*5> 



and the like is true of the four others. Removing the b at the beginning of each of 

 these five expressions and attaching it to every element of the square array to which it 

 belongs, we obtain a variant form consisting of x 1 multiplied by a quadric in x^ , x 2 , 

 . . . , x b with coefficients of the form b r c s d t e u , x 2 multiplied by a similar quadric. x 3 

 multiplied by a similar quadric, and so on. It follows therefore that the natural way to 

 represent the result is to use space of three dimensions, each square array being made 

 to overlie the one before it. We should thus have x 1 ,x 2 , x 3 , x 4 , x 5 placed at equal 

 distances along each of three mutually perpendicular straight lines, and the 125 

 coefficients disposed within the cube of which these lines are concurrent edges. As 

 x 1 x. 2 would arise from x^x x x 2 or x^x^x-^ or x 2 x 1 x 1 to each of which a coefficient attaches, 

 the coefficient of x-^ x 2 would clearly be three-termed : and for a similar reason the 

 coefficient of x 1 x 2 x 3 would be six-termed. Our theorem is thus established. 



(21) The first set of twenty-five positive terms specified in the preceding paragraph 

 may be put in the form 



