This may be more concisely written in the form
a_1 b_1 c_1 d_1 a_2 b_2 c_2 d_2 a_3 b_3 c_3 d_3 a_4 b_4 c_4 d_4
the expression on the left being a determinant of the fourth order.
Also we see that the coefficients of a_1, b_1, c_1, d_1 taken with their proper signs are the minors obtained by omitting the row and column which respectively contain these constituents.
554. More generally, if we have n homogeneous linear equations
a_1 x_1+ b_1 x_2 + c_1 x_3 + \ldots + k_1 x_n = 0, a_2 x_1+ b_2 x_2 + c_2 x_3 + \ldots + k_2 x_n = 0,
a_n x_1+ b_n x_2 + c_n x_3 + \ldots + k_n x_n = 0,
involving x unknown quantities x_1, x_2, x_3, \ldots x_n, these quantities can be eliminated and the result expressed in the form
a_1 b_1 c_1 \ldots k_1 a_2 b_2 c_2 \ldots k_2
a_3 b_3 c_3 \ldots k_3
The left-hand member of this equation is a determinant which consists of n rows and n columns, and is called a determinant of the nth order.
The discussion of this more general form of determinant is beyond the scope of the present work; it will be sufficient here to remark that the properties which have been established in the case of determinants of the second and third orders are quite general, and are capable of being extended to determinants of any order.
