mh green, on the laws of the equilibrium of fluids. 19 

 and consequently, there will result 



where ni = cos 9^ and ^ is a quantity independent of 6/ and tr/, but 

 which may contain the co-ordinates 9, -ar, that serve to define the 

 position of the axis JCi passing through the point p. 



It now only remains to find the value of the quantity k, which may 

 be done by making 0i' = O, for then the line r coincides with the axis 

 JTi, and K*'' during the integration remains constantly equal to Y^\ 

 the value of the density at this axis. Thus we have 



^ ^rin ^ 7 [-. ii—l i.i — l.i—2.i — 3 „ \ 

 V 2.2?— 1 2.4.2«— 1.2^ — 3 I 



or, by summing the series within the parenthesis, and supplying the 

 common factor 2 7r, - 



•jr(i) _ ^-^-^ ^ J, 



1.3.5 2«-l ' 



and, by substituting the value of k, draAvn from this equation in the 

 value of the required, integral given above, we ultimately obtain 



If now, for abridgement, we make 



^^> = ^' - 2:27:11^' + 2.4.2i-1.2i-3 ^' -^^- 



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

 that of V before given, 



r= r(^27rr'%i44^^^-^^^^^/_}r?^/(^H(^-2rr'M/ + r'^); 



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 



c 2 



