Page:Mécanique céleste Vol 1.djvu/32

difference of the forces. If their directions form an angle with each other, the force which results will have an intermediate direction between the two proposed forces. We shall now investigate the quantity and direction of this resulting force.

For this purpose, let us consider two forces, $x$ and $y$, acting at the same moment upon a material point $M$, in directions forming a right angle with each other. Let $z$ be their resultant, and $\theta$ the angle which it makes with the direction of the force $x$. The two forces $x$ and $y$ being given, the angle $\theta$ and the quantity $z$ must have determinate values, so that there will exist, between the three quantities $x$, $z$ and $\theta$, a relation which is to be investigated.

Suppose in the first place that the two forces $x$ and $y$ are infinitely small, and equal to the differentials $dx$, $dy$. Then suppose that $x$ becomes successively $dx$, $2dx$, $3dx$, &c., and $y$ becomes $dy$, $2dy$, $3dy$, &c., it is evident that the angle $\theta$ will remain constant, and the resultant $z$ will become successively $dz$, $2dz$, $3dz$, &c., and in the successive increments of the three forces $x$, $y$ and $z$, the ratio of $x$ to $z$ will be constant, and may be expressed by a function of $\theta$, which we shall denote by $\varphi (\theta )$;^[1] we shall therefore have $x=z.\varphi (\theta )$, in which equation we may change $x$ into $y$, provided we also change the angle $\theta$ into ${\frac {\pi }{2}}-\theta$, $\pi$ being the semi-circumference of a circle whose radius is unity.

Now we may consider the force $x$ as the resultant of two forces $x'$ and $x''$, of which the first $x'$ is directed along the resultant $z$, and the second $x''$ is perpendicular to it.^[2] The force $x$, which results from these two new forces,

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1. A quantity $z$ is said to be a function of another quantity $x$, when it depends on it in any manner. Thus, if $z$, $y$ be variable, $a$, $b$, $c$, &c. constant, and we have either of the following expressions, $x=ax+b$, $z=a^{2}+bx+c$; $z=a^{x}$, $z=sin.ax$, &c. $z$ will be a function [1a] of $x$; and if the precise form of the function is known, as in these examples, it is called an explicit function. If the form is not known, but must be found by some algebraical process, it is called an implicit function.
2. (2) For illustration, suppose the forces $x$ and $y$ to act at the point A, in the directions AX, AY, respectively, and that the resultant $z$ is in the direction AZ, forming with AX, AT, the angles ZAX$=\theta$, ZAY$={\frac {\pi }{2}}-\theta$. Then as above, we have $x=z.\varphi (\theta )$, $y=z.\varphi \left({\frac {\pi }{2}}-\theta \right)$. Draw EAF perpendicular to AZ, and suppose the force $x$ in the direction AX to be resolved into two forces, $x'$, $x''$, in the