124 ON THE LAWS OF 



Let us therefore suppose that the body A is a sphere, whose 

 centre is at the origin of the co-ordinates, the radius being 1 ; 

 and p is such a function of x, y\ z r , that where we substitute for 

 x, y' 9 z their values in polar co-ordinates 



x = / cos 0', y r sin ff cos OT', z' = r' sin & sin OT', 

 it shall reduce itself to the form 



f being the characteristic of any rational and entire function 

 whatever : which is in fact equivalent to supposing 



* Now, when as in the present case, p can be expanded in a 

 series of the entire powers of the quantities a?', y, z', and of the 

 various products of these powers, the function V will always 

 admit of a similar expansion in the entire powers and products 

 of the quantities x, y, z, provided the point p continues within 

 the body A*, and as moreover V evidently depends on the 

 distance Op = r and is independent of 6 and w, the two other 

 polar co-ordinates of p, it is easy to see that the quantity V, when 

 we substitute for x, y, z these values 



x = r cos 0, y = r sin 6 cos OT, z = r sin 6 sin -or, 



will become a function of r, only containing none but the even 

 powers of this variable. 



But since we have 



dv^r'dr'dffdvsuiO', and p = (1 -r *)*./(/), 

 the value of F becomes 



= f P^ = /r' 2 dr' d& d*' sin & 



* The truth of this assertion will become tolerably clear, if we recollect that V 

 may be regarded as the sum of every element pdv of the body's mass divided by 

 the (n - l) th power of the distance of each element from the point p, supposing the 

 density of the body A to be expressed by p, a continuous function of x, y , z. For 

 then the quantity V is represented by a continuous function, so long as p remains 

 within A ; but there is in general a violation of the law of continuity whenever the 

 point p passes from the interior to the exterior space. This truth, however, as 

 enunciated in the text, is demonstrable, but since the present paper is a long one, 

 J have suppressed the demonstrations to save room. 



