APPENDIX. 331 



Hence 



. ( A 4 K \(du dv dw\* 



* = -( A -l B )(dx + Ty + d^) 



dw\* duo du\* du dv\* 



du\* (du dv 

 Tx) + S~ 



It thus appears that A - B y -B, B are each of them the 



o o 



coefficient of an essentially negative expression. Hence, in order 

 that < 2 may always be negative, it is necessary and sufficient 



that B should be positive, and A > - B. 



3 



Note to p. 253. 



Let P, Q be the positions of two particles of a medium in 

 equilibrium distant from each other by a small interval. Let 

 the medium receive a small displacement, in consequence of 

 which these particles assume the positions P, Q', respectively. 



Let the co-ordinates of P be #, y, z> 



Q x + Sx, y + Sy, z + Sz, 

 P x+u, y + v, z + w, 

 then those of Q r will be 



du du 



dw ~ dw ~ / dw 

 _ & 



Hence the co-ordinates of P relatively to Q' are 

 du\ ~ du 



= * > suppose. 



