AND EXTENSION IN VELOCITY. 59 



and 



/Cf d (Ql, ... Qn)\% JT> Jp 



' ' J \d(P^ ~^P}) ' * 



"'<%>!... <jp. 



= c. 



J 



>!,-.. W 



The multiple integral 

 



>! . . . dp n dqi . . . rf^, (151) 



which may also be written 



1 . . . dq n dqi . . . dq n , (152) 



and which, when taken within any given limits of phase, has 

 been shown to have a value independent of the coordinates 

 employed, expresses what we have called an extension-in- 

 phase.* In like manner we may say that the multiple integral 

 (148) expresses an extension-in-configuration, and that the 

 multiple integrals (149) and (150) express an extensionrin- 

 velocity. We have called 



dpi . . . <Zp.<fyi . . . dq n , (153) 



which is equivalent to 



A-^! . . . dq n dq t . . . dq n , (154) 



an element of extension-in-phase. We may call 



A^ ...dq n (155) 



an element of extension-in-configuration, and 



. . . dp n , (156) 



See Chapter I, p. 10. 



