338 Mr. Kelland on the Laws of Transmission 



of values of dy equal in magnitude, and opposite in sign: the 

 same is true of values of 8 : the sum of all such expressions 

 is consequently equal to zero. 

 • Making all the reductions, the equation (1.) is finally re- 

 duced to 



di^ \ rdr J 2 ' 



and by exactly the same process the foUovv^ing equations result 

 for the other directions : 



jt = -2r^|^M +7;^ 5^7sm^ ^-. 



These equations it must be observed have resulted from 

 the hypothesis that their solution has the form 



a = a cos {nt—Jcx) 

 &c. 



—7-^ = -^ a n^ cos hit— k x) 

 dt^ ^ ^ 



= — w^ a ; y 



hence n^ = 2:S l^r + -^^ Sjc^ Vsm^ -^--j 



&c. &c. 



and if v^ ty', t/', be the velocities of transmission of disturbances, 

 parallel respectively to x, y and si, v/e evidently have 



k 

 .;-^=2:r(^r+^^8.^)( sin^- ) 



I forbear to mention in this place, the facilities which are 

 afforded by these equations to the explanation of dispersion, 

 my object being the especial one of pointing out the reasoning 

 by which we are led to the assumption of the inverse square 

 (yftlie distance as one law of force. 



It will be evident to any one who attentively studies the 

 above equations, that the expression for the square of the ve- 

 locity of transmission is a series of terms beginning from 



