1(30 V- OSCILLATIONS AND CYCLIC MOTIONS. 



If we multiply the r ih equation 64), which may be written 



64) tf^?a rs A s + l.XrtA, +c rs A s = 



5 = 1 S = l 5 = 1 



by A r and take the sum for all the r's, we obtain 



r==n s=n r=^n s = n r=n s = n 



66) tf? a rs A r A s + *r,ArA, + c rs A r A s = 0. 



The double sum by which tf is multiplied is the value of the 

 function 2T when for every q' r is substituted A r . We shall denote 

 this by 2T(A). Similarly the coefficient of A is 2F(A), and the 

 constant term or that independent of I' is 2W(A). But by the 

 fundamentaF property of the three functions each must be positive 

 for every set of values of its variables. The equation 66) thus written, 



67) )?T(A) '+ IF (A) + W(A) = 



shows at once that A cannot be real and positive, for that would 

 involve the sum of three positive terms being zero. 



Secondly if jP=0, that is if there is no dissipation, 



2 _ W(A) 

 ' T(A) 



which is negative and I is a pure imaginary. In this case e lt and 

 e~ lt are replaced as above by trigonometric functions representing 

 an undamped harmonic oscillation of the same period for all the 

 parameters q. 



Thirdly, if F is large enough I can be real and negative. In 

 this case each parameter q gradually dies away to zero, the rate of 

 dying away being the same for all. This corresponds to case I of 

 the -preceding section/^ . 



Fourthly, in the limiting case of a system devoid either of 

 inertia or of stiffness, so that T or W is zero, F not zero, instead 

 of a pair of roots we have a single one which is real and negative, 

 so that the motion dies away. 



Fifthly, in other cases, that is when neither T, F, nor W vanish 

 and F is not too large, I is complex. This is the most frequent 

 case in practice. 



We shall prove that the real part of ^ is negative. When the 

 value of any root J, is determined, the equations 64) determine the 

 quantities A r except for a common factor. If complex values enter, 

 since any equation which involves i will also hold good if i be 



