AND THEORY OF INTEGRATION. 29 



ing constant (a), is also a quadrature, since the equation to 

 be integrated may be expressed in the form 



Now, apart from any uch considerations as have been ad- 

 duced, if we limit ourselves to the changes which take place 

 in time, we have identically 



r 2 dr r^ dr z = 0, 



and r and r 2 are given in terms of r v . . . r 2n by the differential 

 equations of motion. When we have obtained 2 n 2 integral 

 equations, we may regard r 2 and r^ as known functions of r l 

 and r 2 . The only remaining difficulty is in integrating this 

 equation. If the case is so simple as to present no difficulty, 

 or if we have the skill or the good fortune to perceive that the 

 multiplier 



d(c,...h) ' (79) 



d(r.,...r fc ) 



or any other, will make the first member of the equation an 

 exact differential, we have no need of the rather lengthy con- 

 siderations which have been adduced. The utility of the 

 principle of conservation of extension-in-phase is that it sup- 

 plies a ' multiplier ' which renders the equation integrable, and 

 which it might be difficult or impossible to find otherwise. 



It will be observed that the function represented by b' is a 

 particular case of that represented by b. The system of arbi- 

 trary constants , 5', c . . . h has certain properties notable for 

 simplicity. If we write b' for b in (77), and compare the 

 result with (78), we get 



= 1. (80) 



d(a, b', c, . . . A) 

 Therefore the multiple integral 



da db f do . . . dh (81) 



