26-t CELESTIAL MECHANICS. I. vii. 29. 



vanish, and multiplying them respectively by j9', 5', and r', 

 /d^zi— --..y^rVd^; q'dq':=Z' q'rydt, and r'dr' = 



~^. rYq'dt;hutBC-AC-{-JC-AB + JB—BC= 0, 



and the sum of the three equations becomes Ozip' dp' +5'd^' 

 4-r'dr'; or taking the fluent, p'^ -\-q'^ +r'^=:k^, k being a 

 constant quantity, to be determined by tlie conditions of 

 the motion. 



Again, if we multiply the three equations by ABp\ 

 BCq', and ACr, and add them together, we obtain, by 

 taking the fluent, JBp'^-\-BCq^-{-ACr'^=:H ^, an equa- 

 tion which includes the condition of the preservation of 

 the impetus of the system, [being equivalent to ABC{p^ + 

 q^-\-r^)=:H^, which implies that the square of the angu- 

 lar velocity of rotation is constant ; and H^znABCk'^^ if 

 h' be the angular velocity.] 



Now since ^C(p'2+^'«4./2)zz^C^2, we have ACk^ 

 — H^ - AC {p'^ + q'^) — ABp'^ — BCq'^ and q'^ = 



ACk^—H^+iAB—AC) p'^ , . ,, 



jj^ — — — -^-—; and m the same manner we 



AC — x>C 



. , ,„ h^-BCk^+{BC—AB)p'^, , . , 



find r^zz ^ ^f— : whence we may tind 



q' and / from p' if H and k are known. Now the first 



of the equations (2>) gives in this case difzr ^ ^ ■■; 



and by substituting for q' and / we obtain the equation of 

 the theorem, which, however, can only be integrated when 

 to of the three quantities. A, B, and C, are equal. 



The determination of ja', q\ and / from t includes there- 

 fore that of three independent quantities H, k, and the 

 constant quantity to be introduced in the fluent of t. But 



