1. Considering the equations

ax + by + cz = d, a'x + b'y + c'z = d', a"x + b"y + c"z = d", and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = 0, and the whole coefficient of z becomes = 0; the factors in question are b'c" - b"c', b"c - bc", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') + a'(b"c - bc") + a"(bc' - b'c), and this function, represented in the form |a, b, c |, |a', b', c'| |a", b", c"| is said to be a determinant; or, the number of elements being 3², it is called a determinant of the third order. It is to be noticed that the resulting equation is |a, b, c | x = |d, b, c | |a', b', c'| |d', b', c'| |a", b", c"| |d", b", c"| where the expression on the right-hand side is the like function with d, d', d" in place of a, a', a" respectively, and is of course also a determinant. Moreover, the functions b'c" - b"c', b"c - bc", bc' - b'c used in the process are themselves the determinants of the second order |b', c'|, |b", c"|, |b, c |. |b", c"| |b, c | |b', c'| We have herein the suggestion of the rule for the derivation of the determinants of the orders 1, 2, 3, 4, &c., each from the preceding one, viz. we have |a| = a, |a, b | = a|b'| - a'|b|. |a', b'| |a, b, c | = a|b', c'| + a'|b", c"| + a"|b, c |, |a', b', c'| |b", c"| |b , c | |b', c'| |a", b", c"| |a, b , c , d | = a|b', c', d' | - a'|b" , c" , d" | + |a', b' , c' , d' | |b", c", d" | |b"', c"', d"'| |a", b" , c" , d" | |b"', c"', d"'| |b , c , d | |a"', b"', c"', d"'| + a"|b"', c"', d"'| - a"'|b , c, d |, |b , c , d | |b', c', d'| |b' , c' , d' | |b", c", d"| and so on, the terms being all + for a determinant of an odd order, but alternately + and - for a determinant of an even order.