Principles of Dynamics. 175 



§ 10. Now we do not know either N or n. For the 

 equations (3) and (4) claim perfect generality ; they must 

 be true for all the particles in the universe and all the 

 observations that have been or will be made. It is true that 

 for certain values of the Partial Accelerations certain of the 

 bodies can be left entirely out of consideration, but we cannot 

 tell which these bodies are till we know what the Partial 

 Accelerations are, and this is precisely what we are trying 

 to discover. Accordingly, in the first instance, we cannot 

 put for N or n any number less than that of all experiments 

 and all particles. Moreover, our measurements must include 

 the relative distances of all particles at each of the times 

 considered. In this form the problem is absolutely intract- 

 able ; in practice the matter is simplified by the introduction 

 of a very large number of assumptions which eliminate com- 

 pletely most of the particles and reduce greatly the number 

 of variables. The assumptions made are of three kinds : — 

 (1) It is assumed that the particles may be divided into two 

 groups, one of which includes only a small number of par- 

 ticles, and that all Partial Accelerations, the suffixes of which 

 are taken from different groups, are nil. Assumptions of 

 this kind are made in all estimations of masses. (2) It is 

 assumed that some of the Partial Accelerations are known 

 functions of the Absolute Coordinates : such assumptions are 

 made in astronomical estimates of Masses. (3) It is assumed 

 that the values of some of the Absolute Coordinates are 

 known : such assumptions are made in the measurement or 

 estimation of Masses by ballistic methods in the laboratory. 



The number of independent equations has doubtless not 

 been estimated quite accurately above, but it is certain 

 at least that, for very large numbers of particles such as 

 would be concerned if our observed system included the 

 whole physical universe, the number of independent variables 

 would be greater than the number of equations, and the 

 problem of determining the Masses and Partial Accelerations 

 would be indeterminate. But the assumptions introduced in 

 practice are always so numerous as to make the number of 

 equations greater than the number of variables. Some of 

 the assumptions are precarious and might be proved to be 

 inconsistent with the others or with the propositions of 

 dynamics : nor can we avoid tbis precariousness by making 

 exactly the right number of assumptions, for we do not know 

 how many assumptions to make. Theoretical mechanics is 

 useful in experimental science only because certain fortunate 

 people ppssess that remarkable quality called " scientific 

 intuition/' which enables them, in a very small number of 



