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The first error is called a differential of the first order and denoted by d, the second a differential of the second order denoted by d₂. Thus if we call the base of our rectangle x and its height y, the area will be xy. Let us suppose x to receive the addition of a very small increment dx, and y the corresponding increment dy, what will be the corresponding increment of the area, or d.xy? Clearly the difference between the old area xy and the new area (x + dx) multiplied by (y + dy). This multiplication gives

x + dx y + dy xy + ydx xdy + dx.dy xy + xdy + ydx + dx.dy

The difference between this and xy is xdy + ydx + dx.dy. But dx.dy is, as we have seen, a differential of the second order and may be neglected. Therefore dxy = xdy + ydx. In like manner dx² = (x + dx)²-x² = 2xdx + dx², which last term may be neglected, and dx² = 2xdx. In this way the differentials of all manner of functions and equations of symbols representing dimensions and motions may be found. Conversely the wholes may be considered as made up of an infinite number of these infinitely small parts, and found from them by summing up or integrating the differentials. Thus if we had the equation