148 On a Dynamical Theory of 



ferently either the element of volume or of mass. Then the 

 equation of motion will be of the form 



JJJ dxdydz ( S5 + -^ Srj + ? d% } = JjJ dxdydzSV, (1) 



where Y is some function depending on the mutual actions of 

 the particles. The integrals are to be extended over the whole 

 volume of the vibrating medium, or over all the media, if there 

 be more than one. 



Setting out from this equation, which is the general formula 

 of dynamics applied to the case that we are considering, we per- 

 ceive that our chief difficulty will consist in the right determina- 

 tion of the function Y ; for if that function were known, little 

 more would be necessary, in order to arrive at all the laws which 

 we are in search of, than to follow the rules of analytical me- 

 chanics, as they have been given by Lagrange. The determina- 

 tion of Y will, of course, depend on the assumptions above stated 

 respecting the nature of the ethereal vibrations ; but, before we 

 proceed further, it seems advisable to introduce certain lemmas, 

 for the purpose of abridging this and the subsequent investiga- 

 tions. 



SECT. II. LEMMAS. 



Lemma I. Let a right line making with three' rectangular 

 axes the angles a, /3, 7, be perpendicular to two other right lines 

 which make with the same axes the angles a', /3', 7' and a", /3", 7" 

 respectively, and which are inclined to each other at an angle 

 denoted by ; then it is easy to prove that 



sin cos a = cos ft' cos 7" - cos ft" cos 7', 



sin 6 cos j3 = cos 7' cos a" - cos 7" cos a', (A) 



sin cos 7 = cos a' cos j3" - cos a" cos ft' ; 



supposing the first right line to be prolonged in the proper direc- 

 tion from the origin, in order that the opposite members of any 



