| lim.MN = 0 | MN · QR | = 0. |
| MN |
Thus the artifice by which differential elements of integrals are formed is in principle the same as that by which differentials are formed without first forming differential coefficients.
10. This principle is usually expressed by introducing the notion of orders of small quantities. If x, y are two variable numbers which are Orders of small quantities. connected together by any relation, and if when x tends to zero y also tends to zero, the fraction y/x may tend to a finite limit. In this case x and y are said to be “of the same order.” When this is not the case we may have either
| lim.x=0 | x | = 0, |
| y |
or
| lim.x=0 | y | = 0, |
| x |
In the former case y is said to be “of a lower order” than x; in the latter case y is said to be “of a higher order” than x. In accordance with this notion we may say that the fundamental artifice of the infinitesimal calculus consists in the rejection of small quantities of an unnecessarily high order. This artifice is now merely an incident in the conduct of a limiting process, but in the 17th century, when limiting processes other than the Greek methods for quadratures were new, the introduction of the artifice was a great advance.
11. By the aid of this artifice, or directly by carrying out the appropriate limiting processes, we may obtain the Rules of Differentiation. rules by which differential coefficients are formed. These rules may be classified as “formal rules” and “particular results.” The formal rules may be stated as follows:—
(i.) The differential coefficient of a constant is zero.
(ii.) For a sum u + v + ... + z, where u, v, ... are functions of x,