274 Prof. R. Clauslus on a new Mechanical Theorem 



dcrstand by i 2 the mean rotation-period of the connecting line. 

 For the mean angular velocity 6 1 we accordingly obtain the 

 equation 



&=™- ....... (25) 



h 



Further, on account -of the proposition that the radii vectores of 

 the points describe equal areas in equal times, we have the 

 equation 



rW = a, ....... (26) 



where a is a constant; and we can therefore put 



#' = # — and 6 ! =a— c> - 



rr r l 



By employing these equations the identical equation 



can be brought into the following form, 



a 2tt (1 T\ 



and multiplying this equation by dt, and integrating from to 

 t. we get 



which, on account of £=z 2 <p 2 passes into 



0=@ + 2^ 2 + «£(I--L)& . . . (27) 



This expression for 9 may now be variated, <j> 2 being regarded 

 as constant ; we thereby obtain 



8 f2 0=m + S f2 [aC(±-±)dt]. . . . (28) 

 The expression 



Jo Vf" r*J 



dt 



is a function of t which changes periodically; and its periods 

 have the same duration as the quantity r, namely i v We must 

 therefore endeavour to introduce for the variation of this expres- 

 sion denoted by S$ 2 that variation which is denoted by 8$ . 

 According to equation (17), for any function Z we can put 



8 fl Z = 8,Z + %, 

 ^ 3 Z = 6^Z + Z'^ 



