ON THE FUNDAMENTAL FORMULAE OF DYNAMICS. 15 



If the variations Sco lt <Jo) 2 , etc. are capable both of positive and of 

 negative values, we must have 



dUi ~ dU 



3ZT Ql ' ^ = Q2 ' ^ (28) 



dU dV dU dV 

 r > ;7^~ = ;r~> 77^ = j> etc - ( 29 > 



To illustrate the use of these equations in a case in which dw 1 , 

 etc. are not exact differentials, we may apply them to the problem of 

 the rotation of a rigid body of which one point is fixed. If dw^ c?o> 2 , 

 dw 3 denote infinitesimal rotations about the principal axes which pass 

 through the fixed point, Q 1 , 2 2 , 2 3 will denote the moments of the 

 impressed forces about these axes, and the value of U will be given 

 by the formula 



(a + b + c)(<*\ + <% + wW-(<l+<*l + ^ 

 + 2(6 c)(b 2 w 3 cl}i + 2(c 



where a, b, and c are constants, a + b, b + c, c+a being the moments 

 of inertia about the three axes. Hence, 



dU ,-, . -, . dU . 



-r = (a b] w^ + (a + 6) o> 3 ; 



and the equations of motion are 



a+c 

 (b 



