( uMVVfcfcSITY 



Of 



37,38] ENERGY INTEGRAL 'CGE'S EQUATION. 125 







and our differential equations are 



d , 72Q , N dT d 



di( ml )-d& = ~~ 



*(mvwt*. 9 ')-. 



dt^ ^ ^ Off dtp 



Now since m and I are constant the equation for # becomes 



64) ^ sin#cos# qp' 2 == -- ~- sin#, 



in which the centrifugal force -component according to & is 



The equation for qp (cp has no centrifugal part), 

 65) |^(7 2 sin 2 # V) = 



may at once be integrated, giving 



66) 



which is the integral equation 50), 21. 



Substituting in 64) the value of qp' derived from the integral 

 equation 66), we obtain the differential equation for #, which is the 

 same as the derivative of equation 51), 21. The remainder of the 

 solution is accordingly the same as in 21. 



38. Equation of Activity. Integral of Energy. Let us 



multiply each of Lagrange's equations by the corresponding velocity ql, 

 and add the results for all values of r, obtaining 



The expression on the right, otherwise written 



^^ d q r d A. 

 ^J rl>r ~d^ == ~dt' 



represents the time -rate at which the applied forces do work on the 

 system. The equation 67) is accordingly the equation of activity, 

 27, 20), in generalized coordinates. 



By means of the property of T expressed in equation 38), 36, 

 we may transform the left-hand side of the equation, for, since T 

 depends upon both the g's and q n s, both of which in an a'ctual 

 motion depend upon t, differentiating totally, 



dT x^i/dTdtf dT d 



