# I Help to understand second order differential definition

1. Oct 22, 2016

### jonjacson

Hi folks, I am reading Poisson's Teatrise on Mechanics. In the introduction he talks about the infinitesimals.

Let's say A is a first order infinitely small quantity, a differential of the first order, if the ratio of A to B is infinitely small too it means B is an infinitesimal of the second order.

Then he shows the meaning with an example I don't understand:

"For example, the chord of an arc of a circle being supposed infinitely small, the versed sine of the same arc is an infinitely small quantity of the second order, because the ratio of the versed sine to the chord is always the same as that of the chord to the diameter, and consequently becomes infinitely small at the same time as the second ratio."

As far as I know the relation between the chord and the sine is:

chord = diameter * sin (angle)

I can't see how his defition of ratios fits in the formula to get a second order differential.

Do you see what is he meaning in this example?

THanks!

2. Oct 22, 2016

### andrewkirk

The reference 'versed sine' is to the versine, which is 1 minus the cosine of the angle.
The chord length of a sector subtending angle $2x$ in a unit circle is $2\sin x=s\times (x-x^3/3!+x^5/5!-...)$, which is of order $x$.
The versine is $1-\cos 2x$ = $1-(1-(2x)^2/2!+(2x)^4/4!-....)=(2x)^2/2!-(2x)^4/4!+...)$ which is of order $x^2$.