Approximately Simple Systems. 259 



Now let the system be slightly varied, so that T and V become 



T + 3T=i([l]+S[l])(/)?+...+S[12]^ 2 +..., 



V+8V=i({l} + 8{l})«+...+8{12}^ 8 +..., 

 giving for the equations of vibration of the altered system 

 ([l]D« + S[l]D a + {l}+8{l}> l + (8L12]D« + 8{12})0 ii 



'2 



f ...=o, 



(2) 



(8[12]D2 + 8{12})c/) 1 + ([2]D 2 + 8[2]D 2 + {2}H-8{2})^ 



+ ... =0, j 

 &c. 

 In the original system one of the natural vibrations is that 

 denoted by the sole variation of $,.. In the altered system this 

 will be accompanied by simultaneous small variations of the 

 other coordinates. If the whole motion vary as cos p r t, we get 

 from the sth equation, as was proved in the paper referred to, 



8[r*K-8{r*} (3) 



an equation which may be regarded as determining approximately 

 the character of the altered types of vibration. 

 Now the rth equation of (2) gives 

 <K-^M -P$\r\ + {r} +8{r}) + ... + &(-p 2 r 8[rs] 



+ 8{^}) + ...=0. (4) 



Using in (4) the values of (£> s : <f> r given in (3), we get for the 

 value of p 2 ri 



• (r) + 8{r} Oft[„] -8{r«}) » > , , (5) 

 ' H+8[r] HHW-« 



in which the summation extends to all values of s other than r. 

 The first term in (5) gives the value of p* calculated without 

 allowance for the change of type, and is sufficient when the square 

 of the alteration in the system may be neglected. If p r refer to 

 the gravest tone of the system, p\—p" r is always positive, and the 

 term of the second order in (5) is negative, showing that the 

 calculation founded on the unaltered type gives in this case a 

 result which is necessarily too high. 



If only the kinetic energy undergo variation, 



„,_ M_„..-jLs a Wl 2 



W+S[r] [r] H(^-tf) 





/>, 



%] *£r].L.](j^-, 



+ w 



S3 



a> (G) 



