SUSCEPTIBLE OF AN EQUILIBRIUM. 63 



Let V stand for the integral in the equation (7.) ; and supposing that p and t^ vary 

 so as always to satisfy that equation, we shall have 



Now, r2 representing any positive quantity, we may conceive it to increase from zero 

 to be infinitely great ; in which case it follows from the nature of the function V, 



dV 

 that during the whole increase ;^r^ r d r will be negative : wherefore the other term 



dY 



^ dp will be positive; which requires that p decrease continually. Since p de- 

 creases when r^ increases, the greatest value oi p will answer to the least value of r^, 

 that is, to zero; and hence, by making r^ = in the formula (7.)? we shall obtain 

 this equation, viz. 



^ -Jo (l+px*)^ 



for finding the greatest value of jo. 



It is obvious that there is only one value of p that will verify the equation just 

 found ; for the integral can pass only once from being positive to be negative while 

 p increases from 1 to be infinitely great. Let p = l^, Ix = z ; and the equation will 

 be changed into this which follows, 



Jo (I + z^)^ 



of which the integral is, 



n _ 72 I (^ + ^^f 3g+ 5^ _ 3 + \4P + 31^ rl dz 

 _ / s; + g ' {\ ^ z^f 8 Jo \ + z^' 



and hence, by making z = /, we deduce 



_ 3Z+ 13/^ 

 arc tan / — 3 _^ j4 ^2 _(_ 3 ^4- 



The only solution of this equation is / = r3934 ; and 1'9414 is therefore the greatest 

 value of /? = /2 Thus, in all the ellipsoids susceptible of an equilibrium by revolving 

 about the least axis, XX' = p is contained between the limits 1*9414 and 1, while 

 (X — ?i')2 = r^, increases from zero to be infinitely great. 



An elliptical spheroid formed of a homogeneous fluid, can be in equilibrium by the 

 action of a centrifugal force, only when it revolves about the least axis. What has 

 been said determines completely the series of ellipsoids with which an equilibrium is 

 possible, when the three»a?:es are unequal. Representing these axes by 



k, Jc^/i +X^ k>/i -\-X''\ 

 it has been shown that X X' must be contained between the limits 1*9414 and 1, while 

 (X — X'y varies from zero to be infinitely great. One limit is when X = X', being a 

 spheroid of revolution of which the axes are 



k and k ^/2*9414 = k X 1*7150. 



