442 Mr. Earnshaw's Reply to Prof. Kelland's Defence 



position of the wave's front. Hence A r and sin 2 — de- 



2 



pend upon the direction of transmission ; but does A, i. e. 



(t h \ 

 A' r sin 2 -— J , also depend upon the direction of transmis- 

 sion ? This question, and a similar one for each of the other 

 coefficients, M. Cauchy has not answered, but I have an- 

 swered it for myself in the negative on experimental grounds, 

 as follows. My equations of motion (and they are M. Cauchy's 

 also) are, 



%*£= - A£-F)j~E£ 



dti.m _Ef-D>j-cr. 



The question is, are the coefficients dependent on the po- 

 sition of the wave's front? Multiply these equations respectively 

 by cos a, cos |3, cos y, and add the results, at the same time 



7 9 A . T* COS fi . T-« COS 7 T* T-v COS V 



assuming ft 2 = A + F — + E r - = B + D — % 



° cos a cos a cos /3 



-r, cos a ~ , -n cos a , ,-. cos Q „ ,.,,.. 



+ F ^ = C + E + D £ ; from which ejrauna- 



cos p cos y cos y 



ting cos a, cos /3, cos y, we find the following cubic in Jc\ 



(£ 2 -A) (F-B) (&*-C)-D 2 (& 2 -A)-E 2 (F-B) 

 -F 2 (F~C) = 2DEF. 

 Having from this found three roots k t % k 2 % k s % we can then 

 find three corresponding sets of values of cos «, cos /3, cos y ; 

 and our equations of motion by this process of mere algebra 

 take the following simple forms, 



d?v = --x% #i = - **% i*tv = - vfc 



where £' = £ cos «j + )j cos /3 2 + $ cos yj 



V ss £ cos « 2 + >j cos & + $ cosy 2 

 £' = | cos « 3 + )j cos /3 8 + £ cos y 3 , 

 that is, £' V £' are the displacements of the particle m estimated 

 parallel to a new set of rectangular axes. The forms of the 

 new equations of motion show that these axes are axes of 

 dynamical symmetry, — those in fact which are better known 

 as the axes of elasticity. Now from experiment we know that 

 for waves of a given length k^, & 2 2 , k 3 2 are constant quantities, 

 i. e. independent of the position of the waves' front (by the 

 above process I have only changed the axes of coordinates, 

 the waves' front remains unaltered in position). And not to oc- 

 cupy room unnecessarily, I now refer the Professor to the note 

 (July, p. 48) to my letter for the remainder of the proof that 

 " A, B, C, D, E, F are independent of the position of the 



