342 The Rev. M. O'Brien's Additional Remarks 



sin 2 



n 2 = — % l3<pr+r~Fr\ 



i 



The quantity n is the coefficient of t in the expression of a 

 displacement, a. = a cos (nt— kg), 



1 3 



Now on the Newtonian law <t> r = — o- and F r = — -3-, 



and therefore n* = ; which being substituted, the Professor's 

 equations of motion assume the following forms : — 



d*a d 2 /3 ^7 



d* 2 ~ °' ~dW~ 0i IT _0 ' 

 which indicate that on the Newtonian hypothesis no forces are 

 called into play by the vibratory displacements of the particles. 

 Now the Professor, having treated the quantity n 2 as finite in 

 all his investigations on this subject, will see that all argu- 

 ments based on them against what I have written fall to the 

 ground, and that my arguments remain in full force. 

 Cambridge, August 19, 1842. 



LX. Some Additional Remarks upon a Communication of Pro- 

 fessor Kelland, published in the Philosophical Magazine for 

 May last. By the Rev. M. O'Brien, late Fellow of Caius 

 College, Cambridge*. 



N the Philosophical Magazine for June 1842, I asserted 

 that certain fundamental equations in Professor Kelland's 

 memoir on Dispersion (in the Cambridge Phil. Trans., vol. vi.) 

 were erroneous. A friend has suggested to me that I ought 

 to have proved more distinctly the existence of those errors. 

 This I will now do in the following manner: — 



(1.) With respect to the equations qf motion in page 159, vol. 

 vi. Camb. Trans. Professor Kelland has overlooked the terms 

 arising from the part of the equation in page 158, which is 

 multiplied by S /3 and 8 y : for instance, there is a term in the 



expansion of 8 /3 (viz. ^ — -r-lxly \ which gives rise to the 



d? a. 

 following term in the expression for -j-^ , viz. 



r J dxdy 



which term does not appear as it might in Professor Kel- 

 land's equation. 



And there is another similar term omitted, viz. 



r ax dz 



* Communicated by the Author. 



I 



