relative to Stationary Motions. 2G7 



paths, and with which also the individual variables by which the 

 positions of the points are determined do not simply change their 

 values periodically, a somewhat more general conception must 

 be employed, which can be comprehended as the phase in a 

 wider signification of the term. 



Employing again the quantities q v q q , . . . q n for the determi- 

 nation of the positions of the points, without presupposing that 

 every quantity repeats its variations regularly in periods of de- 

 termined length, we shall yet introduce for each quantity a cer- 

 tain interval of time, which may be denoted by i lf i 2 , . . . i n . With 

 the aid of these, we will define the phases belonging to the dif- 

 ferent quantities, and which may be called ^> x , 2 , . . . (j) n , by the 

 following equations : — 



' = ^i^Vf>2---=*A (13) 



Now let q x , q v . . . q n be variated, and with each variable the 

 phase belonging to it be regarded as the measuring quantity, 

 which in the variation remains constant, while the time-interval 

 may undergo an alteration. The variations so formed are, ac- 

 cording to the above, to be represented by the symbols 



^Pitfli %$&2> • • • V*2V 

 Employing such a variation, we will form for the variable q n , 

 the fraction 



p v 8{ V q v —h$k 



If the quantity q v accomplished its variations in a periodical 

 manner, and i v were the duration of its period, the variation 

 $9v Q v would also alter only periodically, and accordingly the 

 fraction, which has t in its denominator, would make continually 

 smaller fluctuations and so approximate to zero c The same 

 would hold for all the n variables if they changed in a periodical 

 way, in which each might have its special period-duration. But 

 now we will not make the definite assumption that the varia- 

 tions of the quantities q v q 2 , . . . q n are periodic, but only fix the 

 condition that the mean value of the sum 



* 1 



for great times shall become very little — a condition which, ac- 

 cording to the preceding, is at all events satisfied by periodic 

 variations,, but can also be fulfilled by other variations if they 

 take place in a stationary manner. 



After these preliminary remarks the following theorem can 

 now be stated : — 



If the variations , in the formation of which the quantities 



