5. Reverting to the system of linear equations written down at the

beginning of this article, consider the determinant |ax + by + cz - d , b , c |; |a'x + b'y + c'z - d', b', c'| |a"x + b"y + c"z - d", b", c"| it appears that this is = x|a , b , c | + y|b , b , c | + z|c , b , c | - |d , b , c |; |a', b', c'| |b', b', c'| |c', b', c'| |d', b', c'| |a", b", c"| |b", b", c"| |c", b", c"| |d", b", c"| viz. the second and third terms each vanishing, it is = x|a , b , c | - |d , b , c |. |a', b', c'| |d', b', c'| |a", b", c"| |d", b", c"| But if the linear equations hold good, then the first column of the original determinant is = 0, and therefore the determinant itself is = 0; that is, the linear equations give x|a , b , c | - |d , b , c | = 0; |a', b', c'| |d', b', c'| |a", b", c"| |d", b", c"| which is the result obtained above. We might in a similar way find the values of y and z, but there is a more symmetrical process. Join to the original equations the new equation [alpha]x + [beta]y + [gamma]z = [delta]; a like process shows that, the equations being satisfied, we have |[alpha], [beta], [gamma], [delta]| = 0; | a , b , c , d | | a' , b' , c' , d' | | a" , b" , c" , d" | or, as this may be written, |[alpha], [beta], [gamma] | - [delta]| a , b , c | = 0: | a , b , c , d | | a', b', c'| | a' , b' , c' , d'| | a", b", c"| | a" , b" , c" , d"| | | which, considering [delta] as standing herein for its value [alpha]x + [beta]y + [gamma]z, is a consequence of the original equations only: we have thus an expression for [alpha]x + [beta]y + [gamma]z, an arbitrary linear function of the unknown quantities x, y, z; and by comparing the coefficients of [alpha], [beta], [gamma] on the two sides respectively, we have the values of x, y, z; in fact, these quantities, each multiplied by |a , b , c |, |a', b', c'| |a", b", c"| are in the first instance obtained in the forms |1 |, | 1 |, | 1 |; |a , b , c , d | |a , b , c , d | |a , b , c , d | |a', b', c', d'| |a', b', c', d'| |a', b', c', d'| |a", b", c", d"| |a", b", c", d"| |a", b", c", d"| but these are = |b , c , d |, - |c , d , a |, |d , a , b |, |b', c', d'| |c', d', a'| |d', a', b'| |b", c", d"| |c", d", a"| |d", a", b"| or, what is the same thing, = |b , c , d |, |c , a , d |, |a , b , d | |b', c', d'| |c', a', d'| |a', b', d'| |b", c", d"| |c", a", d"| |a", b", d"| respectively.