Page:Mécanique céleste Vol 1.djvu/32

2 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 $$\varphi(\theta)$$; 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. The force $$x$$, which results from these two new forces,