710 X. ON THE STRUCTURE OF 'MATTER' 

Only practice can show which definition is more workable or more fruitful in results. In the matrix calculus the definition of Cayley is generally accepted, as it has led to the most workable results. It was based on considerations of the composition of linear tranformations.
The definition is approximately as follows: The product ab of two square matrices of the »th order gives a square matrix of the nth order in which the element which lies in therow andcolumn is obtained by multiplying
each element of therow of a by the corresponding element of the jth
column of b and adding the results.
If we denote by a,, and &<, the elements in the f'th row and jth column of a and b respectively, then by definition the element (ij) of our product ab would give,
(5) and the (i,j) element in the matrix ba would be
(6)
In general, the quantities (5) and (6) are not equal and therefore we see that the multiplication of matrices is, in general, not commutative. The order in which we perform our multiplication is of importance and ab is not generally equal to
It should be noticed that the vector calculus has made us familiar with new operations which differ from arithmetical operations. For instance, the sum of two vectors differs in general from the arithmetical sum and is defined by the law of the parallelogram (see Chapter XXXIII). This definition is more general, and the arithmetical definition expresses only the particular case in which the vectors have one direction. Similarly, the noncommutative law of multiplication corresponds more closely to vector multiplication than to arithmetical multiplication.
We will not go further into the details of the matrix calculus, which is a welldeveloped mathematical discipline with a large literature, but will emphasize some methodological points of importance.
One of the main applications of the theory of matrices is found in the subject of linear transformations.
In mathematics, instead of using the given variables, we very often introduce new variables which are functions of the old. Such transformations, or change of variables, are particularly simple and important when the functions in question are homogeneous and linear.
If xi, Xi.....x_{n} represent the original variables, and X\, . . . , x_{n}' the
new variables, we have, by definition, the formulae of transformations: 







The square matrix made up of the coefficients is called the matrix of the transformation and the determinant is called the determinant of the transformation and is completely determined by the matrix. We have already seen the importance of linear equations and linear transformations in physics and therefore in 
