360 On the Vibrations of Approximately Simple Systems. 

 In order to find the variation of type, we have 



pi C l ■ r7rx • S7rx 7 l Po 



K-K-'o 



or, since p 2 r : ^=r 2 : $ 2 , 



(6) 



The normal component vibration is accordingly given by 



y=l<f) l sm -j- -f $ 2 sin —j- + . . .JcosQt?/ — e), 



where p has the value given by (5), and (j> s : <f> r the ratio just 

 determined. 



As an application of this theory let us calculate the displace- 

 ment of a node — for example, the node of the second component, 

 which would be in the middle of the string, were it not for the 



want of uniformity. In the neighbourhood of x = ^, the ap- 



proximate value of y is 



, . ir , . 2ir . . 37r 



^ = (/),sm^+</» 2 smy +<£ 3 sin— + ... 



-f bx < j <p 1 cos ^ + -j- 2 cos — -f . . . I , 



or 



7T 



y = 1 -0 3 + 6 -... + _g<^_20 2 + 4</> 4 - ...^ 



where ^ 



Hence, when 2/ = 0, 



^^{^-*3+.4-,.-j .... (7) 

 approximately^ where 



, 4 2 f ' P • 27T# . S7T# , 



& : *- s -^ 4 • 7 J o ^ S1Q — sin — & - 



The formula (7) gives the displacement of the node to the first 

 order of approximation. 



The generality of the method will perhaps be better brought 

 out by an application to a bar of nearly uniform density, where 

 the normal functions are more complicated. If these be denoted 

 by u v w s , &c, 



