SUSCEPTIBLE OF AN EQUILIBRIUM. 61 



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The inspection of the same equation is sufficient to show that q is positive for all 

 values of t^ ; and as it vanishes both when /- is zero and infinitely great, it must pass 

 at least once from increasing to decreasing, or it will admit of at least one maximum 

 value. By differentiating with regard to / we obtain 



^.tdl^Jo {i+l^xy ' \^') 



from which formula we learn that ^-^ is positive between the limits /^ = and 



/2 = 1 ; that it will consist of a positive and a negative part when l^ is greater 

 than 1 ; and the positive part decreasing while the negative part increases, that it 



will ultimately be negative when /^ is infinitely great. It follows therefore that oT§/ 



can be only once equal to zero, and consequently that q can have only one maximum 

 value, while l^ increases from to go. Applying to the equations (4.) and (5.) the 

 known method of integration, we get ^ 



9 "= — 2P — ^^^ ^^^ ^ "~ 27i' 



dq 3 (9 + l^) ^ J 3l{9 + 7l^) 

 arc tan I — 



of which expressions the first will verify the other. To determine the maximum .of q, 

 we have 



Q.ldl~^ 



9l-)-7P 



arc tan / = 



(1 +/2)(y^;2), 



and the only value of / in this last equation is 



/ = 2-5293. 



By substituting this value of I we obtain 0*3370 for the maximum of q. With respect 



to spheroids of revolution it thus appears that an equilibrium is impossible when q 



f 

 or ^^ IS greater than 0*3370 : in the extreme case, when q is equal to 0*3370, there 



is only one form of equilibrium, the axes of the spheroid being 



li and k x/l + (2*5293)2 = 2*7197 ^ ; 

 but when q is less than 0*3370 there are two different forms of equilibrium, the equa- 

 torial radius of one being less, and of the other greater, than 2*7197 h, k being the 

 semi-axis of rotation. 



The number of the forms of equilibrium in spheroids of revolution is purely a ma- 

 thematical deduction from the expression of q ; and as this has been known since the 

 time of Maclaurin, the discussion of it was all that was wanted for perfecting this 

 part of the theory. 



