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Book: Bowditch's Laplace: Celestial Mechanics

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Bowditch-cover.png

First Part. General Theory of the Motions and Figures of the Heavenly Bodies.

First Book. On the General Laws of Equilibrium and Motion.

CHAPITRE PREMIER.

Je vais établir dans ce livre, les principes généraux de l’équilibre et du mouvement des corps, et résoudre les problêmes de mécanique, dont la solution est indispensable dans la théorie du système du monde.
It is my intention to give in this book the general principles of the equilibrium and motion of bodies, and to solve those problems of mechanics which are indispensable in the theory of the system of the world.

Chapter I. On the Equilibrium and Composition of Forces Which Act on a Material Point.

De l’équilibre et de la composition des forces qui agissent sur un point matériel.

1. Un corps nous paroît se mouvoir, lorsqu’il change de situation par rapport à un système de corps que nous jugeons en repos; mais comme tous les corps, ceux même qui nous semblent jouir du repos le plus absolu, peuvent être en mouvement; on imagine un espace sans bornes, immobile et pénétrable à la matière: c’est aux parties de cet espace réel ou idéal, que nous rapportons par la pensée, la position des corps, et nous les concevons en mouvement, lorsqu’ils répondent successivement à divers lieux de l’espace.
1. A body appears to us to be in motion, when it changes its relative situation with respect to a system of bodies supposed to be at rest ; but as all bodies, even those which appear in the most perfect repose, may be in motion; a space is conceived of, without bounds, immoveable, and penetrable by the particles of matter; and we refer in our minds the position of bodies to the parts of this real, or ideal space, supposing the bodies to be in motion, when they correspond, in successive moments, to different parts of this space.

La nature de cette modification singulière, en vertu de laquelle un corps est transporté d’un lieu dans un autre, est et sera toujours inconnue ; on l’a désignée sous le nom de force; on ne peut déterminer que ses effets et les loix de son action. L’effet d’une force agissante sur un point matériel, est de le mettxe en mouvement, si rien ne s’y oppose; la direction de la force est la droite qu’elle tend à lui faire décrire. Il est visible que si deux forces agissent dans le même sens, elles s’ajoutent l’une à l’autre, et que si elles agissent en sens contraire, le point ne se meut qu’en vertu de leur différence. Si leurs directions forment un angle entre elles, il en résulte une force dont la direction est moyenne entre celles des forces composantes. Voyons quelle est cette résultante et sa direction.
The nature of that singular modification, by means of which a body is transported from one place to another, is now, and always will be, unknown; it is denoted by the name of Force. We can only ascertain its effects, and the laws of its action. The effect of a force acting upon a material point, or particle, is to put it in motion, if no obstacle is opposed; the direction of the force is the right line which it tends to make the point describe. It is evident, that if two forces act in the same direction, the resultant is the sum of the two forces; but if they act in contrary directions, the point is affected by the 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.

Pour cela, considérons deux forces \(x\) et \(y\) agissantes à-la-fois sur un point matériel \(M\), et formant entre elles un angle droit. Soit \(z\) leur résultante, et \(\theta\) l’angle qu’elle fait avec la direction de la force \(x\); les deux forces \(x\) et \(y\) étant données, l’angle \(\theta\) sera déterminé, ainsi que la résultante \(z\), en sorte qu’il existe entre les trois quantités \(x\), \(z\) et \(\theta\), une relation qu’il s’agit de connoître.
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.

Supposons d’abord les forces \(x\) et \(y\) infiniment petites, et égales aux différentielles \(dx\) et \(dy\); supposons ensuite que \(x\) devenant successivement \(dx\), \(2dx\), \(3dx\), &c. \(y\) devienne \(dy\), \(2dy\), \(3dy\), &c., il est clair que l’angle \(\theta\) sera toujours le même, et que la résultante \(z\) deviendra successivement \(dz\), \(2dz\), \(3dz\), &c.; ainsi dans les accroissemens successifs des trois forces \(x\), \(y\) et \(z\), le rapport de \(x\) à \(z\) sera constant, et pourra être exprimé par une fonction de \(\theta\), que nous désignerons par \(\varphi\left(\theta\right)\); on aura donc \(x=z\mathord.\varphi\left(\theta\right)\), équation dans laquelle on peut changer \(x\) en \(y\), pourvu que l’on y change semblablement l’angle \(\theta\) dans \(\frac{\pi}{2}-\theta\), \(\pi\) étant la demi-circonférence dont le rayon est l’unité.
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 \(dx\), \(2\,dx\), \(3\,dx\), &c., and \(y\) becomes \(dy\), \(2\,dy\), \(3\,dy\), &c., it is evident that the angle \(\theta\) will remain constant, and the resultant \(z\) will become successively \(dz\), \(2\,dz\), \(3\,dz\), &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 \(x = z\mathord{.}\varphi(\theta)\), in which equation we may change \(x\) into \(y\), provided we also change the angle \(\theta\) into \(\frac{\pi}{2}-\theta\),  \(\theta\)  \(\pi\) being the semi-circumference of a circle whose radius is unity.

* Bowditch footnote: (1) A quantity \(z\) is said to be a function of another quantity \(x\), when it depends on it in any manner. Thus, if \(z\),  \(y\)  \(x\) be variable, \(a\), \(b\), \(c\), &c. constant, and we have either of the following expressions, \(z=ax+b\), \(z=ax^2+bx+c\); \(z=a^x\), \(z=\sin ax\), &c. \(z\) will be a function 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.

Maintenant, on peut considérer la force \(x\) comme la résultante de deux forces \(x’\) et \(x''\) dont la première \(x'\) est dirigée suivant la résultante \(z\), et dont la seconde \(x''\) est perpendiculaire à cette résultante.
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.

Bowditch footnote: (2) For illustration, suppose the forces \(x\) and \(y\) to act at the point \(A\), in the directions \(A\,X\), \(A\,Y\), respectively, and that the resultant \(z\) is in the direction \(A\,Z\), forming with \(A\,X\), \(A\,Y\), the angles \(Z\,A\,X=\theta\), \(Z\,A\,Y=\frac{\pi}{2}-\theta\).

Bowditch-1.1.1-image-01.png

Then as above, we have \(x=z\mathord{.}\varphi(\theta)\), \(y=z\,\varphi\left(\frac{\pi}{2}-\theta\right)\). Draw \(E\,A\,F\) perpendicular to \(A\,Z\), and suppose the force \(x\) in the direction \(A\,X\) to be resolved into two forces, \(x'\), \(x''\) in the directions \(A\,Z\), \(A\,E\), respectively, so that the angle \(Z\,A\,X=\theta\), and \(X\,A\,E = \frac{\pi}{2}-\theta\). Then, in the same manner in which the above values of \(x\), \(y\), are obtained from \(z\), we may get \(x'= x.\varphi\left(\theta\right)\); \(x''=x.\varphi\left(\frac{\pi}{2}-\theta\right)\). If in these we substitute the values \(\varphi\left(\theta\right)=\frac{x}{z}\); \(\varphi\left(\frac{\pi}{2}-\theta\right) = \frac{y}{z}\), deduced from the above equations, we obtain \(x'= \frac{x^2}{z}\); \(x''=\frac{xy}{x}\). In like manner, if the force \(y\), in the direction \(A\,Y\), be resolved into the two forces \(y'\), \(y''\), in the directions \(A\,Z\), \(A\,F\), making the angle \(Y\,A\,Z=\frac{\pi}{2}-\theta\), \(Y\,A\,F=0\), we shall have \(y'=y.\varphi\left(\frac{\pi}{2}-\theta\right)\); \(y''=y.\varphi\left(\theta\right)\); which, by substituting the above values of \(\varphi\left(\frac{\pi}{2}-\theta\right)\), \(\varphi\left(\theta\right)\) become \(y'=\frac{y^2}{z}\), \(y''=\frac{xy}{z}\), as above.

La force \(x\) qui résulte de ces deux nouvelles forces, formant l’angle \(\theta\) avec la force \(x'\), et l’angle \(\frac{\pi}{2}-\theta\) avec la force \(x''\), on aura \[x'=x\mathord.\varphi\left(\theta\right)=\frac{x^2}{z};\quad x''=x\mathord.\varphi\left(\frac{\pi}{2}-\theta\right) = \frac{xy}{z};\] on peut donc substituer ces deux forces, à la force \(x\). On peut substituer pareillement à la force \(y\), deux nouvelles forces \(y'\) et \(y''\) dont la première est égale à \(\frac{y^2}{z}\) et dirigée suivant \(z\), et dont la seconde est égale à \(\frac{xy}{z}\), et perpendiculaire à \(z\); on aura ainsi, au lieu des deux forces \(x\) et \(y\), les quatre suivantes: \[\frac{x^2}{z}, \frac{y^2}{z}, \frac{xy}{z}, \frac{xy}{z};\] les deux dernières agissant en sens contraire, se détruisent; les deux premières agissant dans le même sens, s’ajoutent et forment la résultante \(z\); on aura donc \[x^2+y^2=z^2;\] d’où il suit que la résultante des deux forces \(x\) et \(y\) est représentée pour la quantité, par la diagonale du rectangle dont les côtés représentent ces forces.
The force \(x\), which results from these two new forces, forms the angle \(\theta\) with the force \(x'\), and the angle \(\frac{\pi}{2}-\theta\) with the force \(x''\); we shall therefore have \[x'=x\,\varphi\left(\theta\right)=\frac{x^2}{z};\quad x''=x\,\varphi\left(\frac{\pi}{2}-\theta\right) = \frac{xy}{z};\] and we may substitute these two forces instead of the force \(x\). We may likewise substitute for the force \(y\) two new forces, \(y'\) and \(y''\), of which the first is equal to \(\frac{y^2}{z}\) in the direction \(z\), and the second equal to \(\frac{xy}{z}\) perpendicular to \(z\); we shall thus have, instead of the two forces \(x\) and \(y\), the four following: \[\frac{x^2}{z}, \frac{y^2}{z}, \frac{xy}{z}, \frac{xy}{z};\] the two last, acting in contrary directions, destroy each other; * the two first, acting in the same direction, are to be added together, and produce the resultant \(z\); we shall therefore have \[x^2+y^2=z^2;\] whence it follows, that the resultant of the two forces \(x\) and \(y\) is represented in magnitude, by the diagonal of the rectangle whose sides represent those forces.

* Bowditch footnote: (3) For, by the preceding note, the force \(x''=\frac{xy}{z}\), is in the direction \(A\,E\), and the force \(y''=\frac{xy}{z}\), is in the opposite direction \(A\,F\), and as they are equal they must destroy each other.

Bowditch footnote: (4) The sum of the two forces \(x'=\frac{x^2}{z}\), \(y'=\frac{y^2}{z}\) in the direction \(A\,Z\), being put equal to the resultant \(z\), gives \(\frac{x^2}{z}+\frac{y^2}{z}=z\), which multiplied by z becomes \(x^2+y^2=z^2\).

Let us now determine the angle \(\theta\). If we increase the force \(x\) by the differential \(dx\), without varying the force \(y\), that angle will be diminished by the infinitely small quantity \(d\theta\);=*= now we may conceive the force \(dx\) to be resolved into two other forces, the one \(dx'\) in the direction \(z\), and the other \(dx''\) perpendicular to \(z\); the point \(M\) will then be acted upon by the two forces \(z+dx'\) and \(dx''\), perpendicular to each other, and the resultant of these two forces, which we shall call \(z'\) will make with \(dx''\) the angle \(\frac{}{}-d\theta\); we shall thus have, by what precedes, \[dx''=z'\mathord.\varphi\left(\frac{\pi}{2}-d\theta\right);\] consequently the function \(\varphi\left(\frac{\pi}{2}-d\theta\right)\) is infinitely small, and of the form \(-kd\theta\), \(k\) being a constant quantity, independent of the angle \(\theta\); we shall therefore have

TODO test

\(A\!E\): $A\!E$

\(AE\): $AE$

\(A\,E\): $A\,E$

\(A\>E\): $A\>E$

\(A\;E\): $A\;E$

\(x=z\mathord.\varphi(\theta)\): $x=z\mathord.\varphi(\theta)$

\(x=z\mathop.\varphi(\theta)\): $x=z\mathop.\varphi(\theta)$

\(x=z\mathbin.\varphi(\theta)\): $x=z\mathbin.\varphi(\theta)$

\(x=z\mathrel.\varphi(\theta)\): $x=z\mathrel.\varphi(\theta)$

\(x=z\mathopen.\varphi(\theta)\): $x=z\mathopen.\varphi(\theta)$

\(x=z\mathclose.\varphi(\theta)\): $x=z\mathclose.\varphi(\theta)$

\(x=z\mathpunct.\varphi(\theta)\): $x=z\mathpunct.\varphi(\theta)$

Bill White (billw@wolfram.com) · Emacs 30.0.50 (Org mode 9.6.11)