VIBRATORY MOTION OF AN ELASTIC MEDIUM. 513 



is a uniformly increasing expansion of the medium in the direction a, the rate of increase of the 

 expansion being — . Hence, according to well-known principles, this disarrangement brings 



into play on P, a force proportional to -— in the direction a, that is, a force whose symbol is 



Aa — - , A being some constant. 



Again, 1 /3 -— ; So," represents a small line, proportional to Sx", drawn in the direction /3 ; 

 therefore the disarrangement indicated by 



is a ciirvdture of the physical line MJV (see Art. 3.), and a similar curvature of all the other 

 physical lines, the symbol of the index of curvature 

 {i.e. a line equal to the reciprocal of the radius of curva- 

 ture drawn towards the centre of curvature) being 



Hence, according to well-known principles, this disarrangement brings into play upon P a 

 d^r] 

 force proportional to -— in the direction /3, that is, a force whose symbol is 



fi/3 -r-T, , B being some constant. 

 ax- 



In the same manner we may shew that the force brought into play by the disarrangement 



dx^ 



^" = ^7 3^.-^ 



is represented by the symbol 





Hence the whole force brought into play by the disarrangement 



is represented in magnitude and direction by the symbol 



d \ ■-■ , 





(5. To find the force brought into play by the disarrangement represented by 



Sua 



