SECT. 1.] TO POSITION IN THE ORBIT. 3 



the sun, in order that these phenomena may be continually produced. In this 

 way it is found that the action of the sun upon the bodies moving about it is 

 exerted just as if an attractive force, the intensity of which is reciprocally 

 proportional to the square of the distance, should urge the bodies towards the 

 centre of the sun. If now, on the other hand, we set out with the assumption of 

 such an attractive force, the phenomena are deduced from it as necessary 

 consequences. It is sufficient here merely to have recited these laws, the con 

 nection of which with the principle of gravitation it will be the less necessary to 

 dwell upon in this place, since several authors subsequently to the eminent 

 NEWTON have treated this subject, and among them the illustrious LA PLACE, in 

 that most perfect work the Mecanique Celeste, in such a manner as to leave 

 nothing further to be desired. 



3. 



Inquiries into the motions of the heavenly bodies, so far as they take place in 

 conic sections, by no means demand a complete theory of this class of curves ; 

 but a single general equation rather, on which all others can be based, will answer 

 our purpose. And it appears to be particularly advantageous to select that one 

 to which, while investigating the curve described according to the law of attrac 

 tion, we are conducted as a characteristic equation. If we determine any place 

 of a body in its orbit by the distances x, y, from two right lines drawn in the 

 plane of the orbit intersecting each other at right angles in the centre of the 

 sun, that is, in one of the foci of the curve, and further, if we denote the distance 

 of the body from the sun by r (always positive), we shall have between r, x,y, 

 the linear equation r-\-ax-\-(iy = y, m which a, ft, y represent constant quan 

 tities, y being from the nature of the case always positive. By changing the 

 position of the right lines to which x,y, are referred, this position being essentially 

 arbitrary, provided only the lines continue to intersect each other at right angles, 

 the form of the equation and also the value of y will not be changed, but the 

 values of a and ft will vary, and it is plain that the position may be so determined 

 that ft shall become = 0, and a, at least, not negative. In this way by putting for 

 , y, respectively e. p, our equation takes the form r-\-ex=p. The right line to 



