6 Me green, on THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



X = r' cos 0', y' = / sin 9' cos tst', %' = r' sin Q' sin tr', 

 it shall reduce itself to the form 



P = (l-/y./(0; 



f being the characteristic of any rational and entire function what- 

 ever: which is in fact equivalent to supposing 



p = (1 - /' - y" - %'f.f{x" + y" + z'^). 



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

 of the entire powers of the quantities x, y', %', 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, %, 

 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 -sr, 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 9, y = r sin 9 cos w, z = r sin 9 sin tst 



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

 powers of this variable. 



But since we have 



dv = r"dr d9' d-ur sin 0', and /> = (1 - ry.f{r'% 



the value of V becomes 



V= f-^, = jr'^dr'd9'd-w' sin 9' (1 - r''ff{r") .g'"", 

 J g" 



the integrals being taken from tst' = to tr' = 2 tt, from 9' = to 9' = w, 

 and from r' = to r' = l. 



* 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)"" 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, I have suppressed the demonstrations to save room. 



