of the Mechanical Theory of Heat, 165 



whence it follows as a rule that, if we wish to variate equation 

 (3), we must regard the magnitude (/> as constant. On the con- 

 trary, if we wish to differentiate the same equation, we must 

 regard % as constant, because the differentiation refers to the 

 course of a determinate motion, in which the time of a revolution 

 i is a given magnitude. We thus obtain 



dt=idcj) (5; 



4. These preliminaries being settled, we can now proceed to 



the proposed mathematical development. Taking the expression 



dx 



■j- hx, and differentiating it according to (/>, we obtain 



ai 



d (dx £ \ _ d 2 x ~ dx d(Sx) f „. 



^\W d *)-aTd$ dX+ ~di'~df' ' ' • () 



Now, as in variation the phase <j> is regarded as constant, we can, 

 when a magnitude varies and is to be differentiated according to 

 <£, change the order of these two operations and therefore put 



d(Sx) £ dx M . 



sf =8 # (7) 



Thereby the preceding equation changes into 



d (dx ~ \ d 2 x ~ dx ~dx fQ . 



d$\dt s *rdid4> s * + dt s d4, (8) 



This equation may be transformed in the following manner : — 



d /dx £ \_ drx. dt £ dx Jdx dt \ 



dj>\di n~dp' df + dt \dF'd$) 



_a n x dt ~ dx dt *dsc /dx\ z *dt_ 

 ~dF'didt'didi\dt) d6 



-~dt*'dcf> 6 ^ + 2# \di) :\dt) d$ 

 Putting herein, for the differential coefficient -jj 3 its value from 

 equation (5), there results 



4(£*)=<S^*(S'-«-(£)** • ^ 



This equation shall now be multiplied by d<j> and then integrated 

 from <£ = to </) = l ; that is, for an entire revolution. 



The integration on the left-hand side may proceed at once, 

 and we obtain 



