THE SCIENCE OF ASTRONOMY. 293 



among people. It was therefore a happy duty of the astronomers to 

 show that the comets also move in orbits around the Sun, and are sub- 

 ject to the same laws as the planets. This work was easier because the 

 comets move nearly in parabolas, which are the simplest of the conic 

 sections. Still the general problem of finding the six elements of an 

 orbit from the six data given by three observations remained to be 

 solved. The solution was given by Gauss a century ago in a very elegant 

 manner. His book is a model, and one of the best ever written on 

 theoretical astronomy. No better experience can be had for a student 

 than to come in contact with such a book and with such an author. The 

 solution of Laplace for the orbit of a comet is general, but demands 

 more labor of computing than the method of Olbers, as arranged by 

 Gauss. It is said by some writers that the method of Laplace is to be 

 preferred because more than three observations can be used. In fact 

 this is necessary in order to get good values of the derivatives of the 

 longitudes and latitudes with respect to the time, but it leads to long and 

 rather uncertain computations. Moreover it employs more data than 

 are necessary, and thus is a departure from the mathematical theory of 

 the problem. This method is ingenious, and by means of the deriva- 

 tives it gives an interesting rule for judging of the distance of a comet 

 from the earth by the curvature of its apparent path, but a trial shows 

 that the method of Olbers is much shorter. Good preliminary orbits 

 can now be computed for comets and planets without much labor. This, 

 however, is only a beginning of the work of determining their actual 

 motions. The planets act on each other and on the comets, and it is 

 necessary to compute the result of these forces. Here again the condi- 

 tions of our solar system furnish peculiar advantages. The great mass 

 of the sun exerts such a superior force that the attractions of the planets 

 are relatively small, so that the first orbits, computed by neglecting this 

 interaction, are nearly correct. But the interactions of planets become 

 important with the lapse of time, and the labor of computing these 

 perturbations is very great. This work has been done repeatedly, and 

 we now have good numerical values of the theories of the principal 

 planets, from which tables can be made. Practically, therefore, this 

 question appears to be well toward a final solution. But the whole story 

 has not been told. 



The planets, on account of their relative distances being great and 

 because their figures are nearly spherical, can be considered as material 

 particles and then the equations of motion are readily formed. In the 

 case of n material particles acting on each other by the Newtonian law, 

 and free from external action, we shall have ?>n differential equations 

 of motion, and 6n integrations are necessary for the complete solution. 

 Of these only ten can be made, so that in the case of only three bodies 

 there remain eight integrations that cannot be found. The early 



