PROFESSOR KELLAND ON THE THEORY OF WAVES. 139 



2 1 2 . a 2 



LOR. 5. 



... bgta 



.: velocity =^7, 



which shews that the actual velocity of a particle after a small time t varies inverse- 

 ly as the product of its distances from the two extremities of the displaced mass. 

 Con. 6. If a be small, velocity a inversely as square of distance from centre 

 of displacement, which is M. POISSON'S result. 



74. Let us return to the value of -3* : 



dy 



d(b M/0 f" sin u. (a a') + sin uaf . / A u.(bz\, , 



It is -7-*-= I-* / "srnVgiJLte ^" Z) d p. nearly 



".'/ 7T Jo vM 



_ b*/g r m cos (fJLa-a'-V^jj. t) - cos Qu a -"<? + */ff p f) + fee. ^(6 *) , 



" iv J ~jjT '" 



To find the value of this integral, we will first determine that of the ex- 

 pression 



/ e f(* *) 



(cos (VfffJLt+rfjL) j d/ji ; which let us designate by V. 



Assume Vfffj.f-trfj.= 6; then V/x=- 2r + N 'T~2 + ~ 



rfu 



-- - , and 



a . u., /" 



V=/ cosDe ^ '-. ~AI 

 Jo Vfft 2 + r[6 Jo 



Now _ _ 



1 2r 2 2 ^ 



TJ 2 ^' ' 7 S3= ~3 nearl J- 



4 r 2 y| ^ J ) 2 ^ ^ 2 



Hence V= 



2n 



if 



