Approximately Simple Systems, 359 



Pr [r]+8[r] r l [r] + lyf J ' ' ' & 



where P r denotes the value of p r before the change. The inter- 

 pretation is that the altered period may be calculated from the 

 potential and kinetic energies without allowance for the varia- 

 tion of type which will in general accompany the change in the 

 system, provided we are content to neglect the square of the 

 change. This result, proved in a different manner, was given in 

 a paper read before the Mathematical Society in June last*, and 

 appears to be of considerable importance. It may be applied, 

 for example, to the calculation of the beats of a slightly unsym- 

 metrical circular plate, which are analogous to those often given 

 by bells. 



An example or two will probably clear up any point that may 

 be obscure (whether from defective exposition, or the extreme 

 generality of the method) in the general theory. 



Consider the case of a string whose longitudinal density p is 

 not quite constant. Since there is no change in the potential 

 energy of a given configuration, 8{r } =0. On the other hand, if 



. . Tra? , , . 2ttx , 

 £/ = (/>! sin y +0 2 sin-y-+ ... 



be the form of the string at any time, 



T =i 1/of^! sia -7- +02 sin -y-+ • • •) efo?=i#n/>sin s -y«fo' + . 



+ 9!9 2 1 P sin ~T~ sin ~~T u® .+ 



If p were constant the products would disappear, since <f> l &c. are 

 the normal coordinates for a uniform string. As it is, the inte- 

 gral coefficients, though not evanescent, are small quantities. 

 Let p=p + 8p; then, in our previous notation, 



H = 2^o> £[>]= | o>sin 2 ^d#; 

 and therefore, by (4), 



^{'-IO-t*}- • • • « 



a formula from which the pitch of an approximately uniform 

 string may be calculated. It should be noticed that no parti- 

 cular selection of p is necessary, since P£ contains p as a divisor. 



* " On some General Theorems relating to Vibrations," by the Hon. J. 

 W. Strutt. 



