138 THE LAWS OF 



If now, for abridgement, we make 

 ,. ,. i.il ,; i.i- 1.^ 



_ 4 p 



we shall obtain, by substituting the value of the integral just 

 found in that of V before given, 



which proves the truth of our assertion. 



From what has been advanced in the preceding article, it is 

 likewise very easy to see that if the density of the fluid within a 

 sphere of any radius be every where represented by 



(f) being the characteristic of any function whatever; and we 

 afterwards form the quantity 



where dv represents an element of the sphere's volume, and g the 

 distance between dv and any particle p under consideration, the 

 resulting value of V will always be of the form 



Y (i] being what Y' w becomes by changing /} OT', the polar 

 co-ordinates of the element dv into 0, OT, the co-ordinates of the 

 point p ; and E being a function of r, the remaining co-ordinate 

 of p, only. 



4. Having thus demonstrated a very general property of 

 functions of the form Y (i \ let us now endeavour to determine the 

 value of V for a sphere whose radius is unity, and containing 

 fluid of which the density is every where represented by 



a?', y t z being the rectangular co-ordinates of dv, an element of 

 the sphere's volume, and/, the characteristic of any rational and 

 entire function whatever. 



