Approximately Simple Si/stems. 261 



By a suitable transformation any two of the functions T, F, V 

 can be reduced to a sum of squares, but not in general all three. 

 When all three occur, the types of vibration are more complicated 

 than those of a conservative system, or of that of a dissipative 

 system with one degree of freedom. When, however, the fric- 

 tional forces are small, as in many important applications they 

 are, it is advantageous to proceed as if the system were conser- 

 vative, and reduce T and V to sums of squares, leaving F to take 

 its chance. In this way we obtain equations of the form 





(13) 



in which the coefficients (11), (22), (12), &c. are to be treated 

 as small. 



Let the type of vibration considered be that which differs little 

 from the sole variation of cf> r , and let all the coordinates vary as 

 (Pr t i where p r will be complex, as also the ratios of the coor- 

 dinates. From the 5th equation, 



0?W + {*}>.+ (")pA+ • • • =°> 



we get, by neglecting the terms of the second order, 



which determines the alteration of type. Although p is com- 

 plex, the real part is small compared with the imaginary part ; 

 and therefore (14) indicates that the coordinates (j> s have approxi- 

 mately the same phase, and that phase a quarter period different 

 from that of </> r . The rth equation gives, by use of (14), 



PW+{r} + MP-Z p ^\ s} =0, . . (15) 



from which it appears that p r may be calculated approximately 

 from the equation 



l>]#+Wiv+H=0j .... (i6) 



that is, as if there were no change in the type of vibration. The 

 rate at which the motion subsides will not be altered, even though 

 the terms of the second order in (15) be retained. 



The reader may apply these formulae to the case of a uniform 

 string whose middle point is subject to a small retarding force 

 proportional to the velocity. 



It is scarcely necessary to point out that these methods apply 

 to other physical problems than those relating to the vibrations 



