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″′ - a′″ b,c,d ,
a′,b′,c′,d′ b″,c″,d″ b′″,c′″,d′″ b,c,d b′,c′,d′
a″,b″,c″,d″ b′″,c′″,d′″ b,c,d; b′,c′,d′ b″,c″,d″
a′″,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.