Page:Calculus Made Easy.pdf/25

 be a small fraction of the third order of smallness, being $$1$$ per cent. of $$1$$ per cent. of $$1$$ per cent.

Lastly, suppose that for some very precise purpose we should regard $$\tfrac{1}{1,000,000}$$ as “small.” Thus, if a first-rate chronometer is not to lose or gain more than half a minute in a year, it must keep time with an accuracy of $$1$$ part in $$1,051,200$$. Now if, for such a purpose, we regard $$\tfrac{1}{1,000,000}$$ (or one millionth) as a small quantity, then $$\tfrac{1}{1,000,000}$$ of $$\tfrac{1}{1,000,000}$$, that is $$\tfrac{1}{1,000,000,000,000}$$ (or one billionth) will be a small quantity of the second order of smallness, and may be utterly disregarded, by comparison.

Then we see that the smaller a small quantity itself is, the more negligible does the corresponding small quantity of the second order become. Hence we know that in all cases we are justified in neglecting the small quantities of the second—or third (or higher)—orders, if only we take the small quantity of the first order small enough in itself.

But, it must be remembered, that small quantities if they occur in our expressions as factors multiplied by some other factor, may become important if the other factor is itself large. Even a farthing becomes important if only it is multiplied by a few hundred.

Now in the calculus we write $$dx$$ for a little bit of $$x$$. These things such as $$dx$$, and $$du$$, and $$dy$$, are called “differentials,” the differential of $$x$$, or of $$u$$, or of $$y$$, as the case may be. [You read them as dee-eks, or dee-you, or dee-wy.] If $$dx$$ be a small bit of $$x$$, and relatively small of itself, it does not follow