SUSCEPTIBLE OF AN EQUILIBRIUM. 65 



wherefore 



^ = (,p + -pj; .y — ^ — + py du (i _ ^2) 



^-^.Jdu. Q, ; 



in consequence of the formula (7.) the first term is zero, so that we have 



dq V Pj /, 9\ . /""^ Pj ^^* + (3 + 4»):r4 — »^^:6 _f. 2^2^ 

 ^ = py ^Z. (1 - ^2) + _.y ^,, . ^1L_ 



And because 3 + 4/? is always greater than ;?2, it follow^s that ^ is essentially po- 

 sitive. 



Again, by taking the fluxion relatively to r^, we have 



dq_ __ r d u f p^ ^^ P^ - M p^ .r^ Q^ - 3 x^ M 1 



that is. 



-/du 



d q p + p^ /* , 1 — p^ X 



Tdr— p3 y "". Q3 



-f/-^ 



Q^ 



Of this value the first term is zero by the formula (7.) ; and attending to the limits of 

 p and of the integral, the second term is essentially negative. 

 Now we have 



1 dq , dq , 



^^ = d-p'^P-^7d-r'''^'''- 



if we suppose r^ to increase, p will decrease ; and according to what has been shown 

 the two parts o£ d q will be negative. Wherefore, while r^ increases from zero to be 

 infinitely great, ^ will decrease continually from its first value to zero; and for every 

 possible value of q there will be only one value of r^, and consequently only one 

 ellipsoid susceptible of an equilibrium. 



It would be superfluous to pursue this investigation further, and a mere waste of 

 labour to seek the easiest formulas for solving a problem which, it appears from what 

 has been shown, can have no application in the theory of the figure of the planets. 

 It is extremely probable that no such figures as those required for the equilibrium 

 of ellipsoids with three unequal axes, will be found to exist in nature. It seems diffi- 

 cult to admit that any circumstances, or the action of any forces we are acquainted 

 with, could induce upon a mass of fluid a figure adjusted with such mathematical 

 nicety to the attraction of the mass and the centrifugal force. If the existence of 

 such a figure can be supposed, would it be permanent ? Would not the least action 

 of the other bodies of the system upon it be sufficient to destroy the exact confor- 



MDCCCXXXVIII. K 



