January 2, 1903.] 



SCIENCE. 



ments 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 so- 

 lution of Laplace for the orbit of a comet 

 is general, but demands more labor of com- 

 puting than the method of Olbers, as ar- 

 ranged by Gauss. It is said by some 

 writers that the method of Laplace is to 

 be preferred because more than three ob- 

 servations can be used. In fact this is 

 necessary in order to get good values of 

 the derivatives of the longitudes and lati- 

 tudes 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 derivatives it gives an interesting 

 rule for judging of the distance of a comet 

 from the earth by the curvature of its ap- 

 parent 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 

 conditions 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 cor- 

 rect. But the interactions of planets be- 

 come important with the lapse of time, 

 and the labor of computing these perturba- 



tions 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 ques- 

 tion appears to be well toward a final solu- 

 tion. But the whole story has not been 

 told. 



The planets, on account of their relative 

 distances being great and because their 

 figures are neai-ly spherical, can be consid- 

 ered 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 

 3w differential equations of motion, and 6w 

 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 investigatoi-s soon 

 obtained this result, and it is clearly stated 

 by Lagrange and Laplace. The astron- 

 omer, therefore, is forced to have recourse 

 to approximate methods. He begins with 

 the problem of two bodies, the sun and a 

 planet, and neglects the actions of the other 

 planets. In this problem of two bodies 

 the motions take place in a plane, and the 

 integrations can all be made. Two con- 

 stants are needed to fix the position of the 

 plane of motion, and the four other con- 

 stants pertaining to the equations in this 

 plane are easily found. This solution is 

 the starting point for finding the orbits of 

 all the planets and comets. The mass of 

 the sun is so overpowering that the solu- 

 tion of the problem of two bodies gives a 

 good idea of the real orbits. Then the 

 theory of the variation of the elements is 

 introduced, an idea completely worked out 

 into a practical form by Lagrange. The 

 elements of the orbits are supposed to be 

 continually changed by the attractions of 



