436 HERMANN VON HELMHOLTZ 



scale, though never wholly free from external disturbances, 

 upon every rapidly rotating top that stands firm upon its point 

 with a vertical axis so long as it is in rapid rotation, and falls 

 over as soon as it loses the motion. 



'Accordingly these quantities, the velocity of the centre of 

 gravity of the planetary system and its direction, the direction 

 of the invariable plane and the value of the greatest rotational 

 momentum of the system, are, under the given conditions of our 

 universe, as much unalterable magnitudes as its supply of 

 energy. If perturbations from the fixed stars are present their 

 cumulative effect may conceivably, after a very great lapse of 

 time, become perceptible to posterity; and they perhaps may 

 have to take into account the motions of the centre of gravity 

 of these fixed stars and their rotary momenta. So long as we 

 reckon by millenniums only, the calculation may be confined 

 to the planetary system. 



1 And now we may say that we have come to the end of our 

 knowledge of these immutable motor magnitudes. To the end 

 of our knowledge, to the end of the list of things whose values 

 we can exactly reckon and determine, but not to the end of 

 the tale of all existing magnitudes of this kind. The number 

 of these, on the contrary, is so great that we can scarcely think 

 that the human race can ever succeed in recognizing and 

 enumerating all of them. 



'Let us now take counsel with the mathematician, who is 

 engaged upon mechanical problems. All that I have been 

 describing to you are known to him as Integration Constants ; 

 " Constants " because they are unalterable. And he terms his 

 method of resolving the equations that express the laws of 

 motion, and of discovering their final result for any later point 

 in time, " Integration." For him accordingly there are constants 

 which he meets in the integration of the equations of motion, 

 and the value of which he must find some means of determining, 

 in order to adapt his solution of the universal law of motion 

 to the special case of the given system of bodies, with which 

 he is occupied at the moment, the parts of which had all, at the 

 commencement of the time embraced in the calculation, their 

 definite position in space, and their velocities of definite value 

 and direction. 



' Now ask the mathematician how many integration constants 



