460 MOSSOTTI ON THE FORCES WHICH REGULATE 



of the formula (5) from 8*^= o to 9' = V, from xj/' = o to 4/' = 2 ir, and 

 from r =: o to r = 00. 



Let us begin with performing the integrations of the formula (V). 

 In consequence of the quantity ^ct +/q being considered as constant, 

 and as the spherical form of the molecules renders p independent of w 

 and f, all the terms of the second and third line of this formula will 

 vanish, and it being observed that we always have 



fV 



n sin w c? w rf a = o, 



unless in the case of 11° = 1, which gives 



// 



Uo s\n u d uj d (p =: 4! If ; 



the expression for G will become G = -^ — ^ , 6 repre- 



senting the semidiameter of the molecule. 



This integral has been obtained under the supposition that the origin 

 of the coordinates is in the centre of the molecule ; but the origin may 

 be transferred to any point whatever, by restoring, instead of r, its 

 general expression, and writing 



4>ir (gm + fq)S^ 



(vy G = -x; 



3 { (x - xy + (y — y)^ +(z — zy} ^ 

 where x, y, z represent the coordinates of the centre of the molecule. 



Before we proceed to the expression for F, we had better clearly define 

 the signification of the term q which it contains. We must consider this 

 quantity (q) such as it is given by the equation (III), not as the entire 

 value of the density of the aether, but as the value only of its excess or 

 deficiency above or below the sensibly uniform density which the aether 

 diffused in equilibrium is supposed to have in that part of space. If we 

 represent the latter density by qo, the equations (III) and (VI), while 

 we suppress the terms due to the quantities G, G„ G^, &c., must be 

 satisfied by the substitution of y = ^-q : and that, in order that the 

 aether may remain in equilibrium spontaneously, or in consequence of 

 the action of the forces not expressed, whose centres must be supposed 

 to be at a very great distance. If, therefore, we take the difference be- 

 tween the equations resulting from this substitution and the original 

 equations themselves, we shall have 

 (III)' k(q - qo) = - F + G + G, + G„^ + G,+ &c. 



provided that, in F, we substitute for q the value of q — qo resulting 



