240 Prof. R. Clausius on a new Mechanical Theorem 



As the differential coefficients of T in this equation will fre- 

 quently recur in what follows, it will be advantageous to intro- 

 duce for them a simplified symbol. For this we will choose the 

 letter p, and accordingly, understanding by v any of the integral 

 numbers from 1 to n, put 



P *=S w 



The preceding equation then becomes 



2T=W (7) 



According to Lagrange, the differential equations of motion take 

 for the general variables q the following form — 



d_/dT\_dT _dV } 

 dt \dq* v ) dq v dq v 

 or, pursuant to (6), 



*v = ^T_rfU 



dt dq v dq v ' 



4. As regards the equations given by Hamilton in his me- 

 moirs of 1834 and 1835*, they are, if the initial values of the 

 quantities q v q 2 , • • • g^and p v p 2 > . . ./J„be denoted by k^k^,... k n 

 and h v h 2 , . . . h n , as follows : — 



•£ 



2Tdt = 2{p8q-h&k)+t8E; . . . (I.) 



; j> 



V)dt=2{pSq-m)^mt. . . (La) 



These two equations are not essentially different the one from 

 the other, because, presupposing the equation T + U = E, the 

 one immediately results from the other. Hence they can be 

 designated as one equation in two different forms. 

 In the first form of the equation the integral 



i 



is to be regarded as a function of the quantities q ly q& , . . q ni 

 k lf kz, . . . k nt and E, and the equation can be analyzed into as 

 many different equations as there are independent variations on 

 the right-hand side. As soon as the function which that inte- 

 gral represents is known, we can, from the equations resulting 

 from the analysis, deduce by mere elimination of E all the first 

 and second integrals of the differential equations of motion. 



* Philosophical Transactions, 1834 and 1835. 



