1864.] The Rev. J. Bayma on Molecular Mechanics, 131 



II. Mathematical Evolution of these Principles. 



After establishing principles, we must proceed to investigate the formulas 

 of motion and of equilibrium, first between the elements themselves, then 

 between the several systems of elements. The difficulties to be overcome 

 in establishing the principles were chiefly philosophical: the difficulties 

 which occur in the present part are mathematical, and can only be over- 

 come by labour and patience. 



As long as we confine ourselves to two elements, the mathematical for- 

 mula expressing their motion is easily found. Thus, if there are two attrac- 

 tive elements of equal intensity, and if v be the action of one for a unit of 

 distance in a unit of time, 2a the distance between them at the beginning 

 of motion, x the space passed through by one in the time ty the equation 

 of motion will be 



^^^-^(^^ x{a — x)-{-a .2iXctmg = 



And since it is clear, from other considerations, that these two elements 

 must vibrate together indefinitely in vibrations of equal times and constant 

 extent, the time of one oscillation will easily be found from the above 

 formula. 



If the two attractive elements have unequal forces, or if one be attractive 

 and the other repulsive, or both repulsive, the equation of motion may easily 

 be obtained. 



But when we have to do with a more complex system of elements, after 

 obtaining the differential equations corresponding to the nature of the 

 system, it is scarcely possible to obtain their integration, as will appear 

 from the examples which I shall give below. Consequently, if we wish to 

 deduce anything from such equations, we must proceed indirectly, and a 

 long labour must be undertaken, sometimes with but slender results. This 

 material difficulty will be diminished, or perhaps disappear, either by some 

 new method of integration (which I can scarcely dare to hope for, though 

 it is a great desideratum) or by certain tables exhibiting series of numerical 

 values belonging to different systems. 



But there occurs another difficulty in these systems. For since the 

 agglomerations of simple elements can be arranged in an infinite variety, 

 and it would be neither reasonable nor possible to treat of all such agglo- 

 merations, we must limit the number of them according to the scope we 

 have in view, i. e. according to the use they may be of in explaining natural 

 phenomena. Even this is a very difficult matter. How I have endeavoured 

 to overcome this difficulty I will briefly explain. 



First, I considered that the molecules of primitive bodies, such as oxygen, 

 hydrogen, nitrogen, &c., cannot reasonably be supposed to be irregular — 

 a conclusion which, though I cannot rigorously demonstrate, yet I can 

 render probable by good reasons. Consequently, while treating of primi- 

 tive systems I may confine myself to the examen of forms that are regular. 



