128 Prof. Kelland's Reply to some Objections against the 



forces which depend on the excess of the action due to the 

 right-hand direction above that due to the left can produce 

 any sensible effect. Let me repeat that it is not geometrical 

 symmetry which we assumed ; a cubical arrangement which 

 we sometimes speak of by way of illustration is not an arrange- 

 ment of geometric, symmetry. But what we do assume is a 

 medium of mechanical symmetry; an arrangement of such a 

 nature that all forces are independent of direction either 

 throughout or on either side of a particle. Perhaps the word 

 isotropc, which M. Cauchy uses, or isodynamical, might ex- 

 press the condition better than the word symmetrical, but 

 further than the employment of a term which is incorrect, and 

 of illustrations which are unsatisfactory, nothing can be urged 

 against the introduction of the hypothesis of perfect sym- 

 metry. 



d 2 V 

 To return to our argument. The value of , ^ is 



2(*-/)*-(y-g)*-.(3-a) 2 *' 



Zf m c • 



1* 



Now in a medium of symmetry 



%m- f-i- = 2 m w * J = 2 m± r^-. 



d* V d* V d 2 V 



Hence -™ = °- Similarly -j^- = 0, -j^~ = 0. 



Nor is it otherwise with an isotrope or isodynamical medium, 

 whatever be its constitution. In such a medium the value of 

 the square of the velocity of transmission of a vibration de- 

 pends on that of the function 



(r 3 



T> 9 X 



3{z-hf\ 9 n(y-g) 



c - /l 3(z-/*f\ 

 or of 2^^--^- jsin 9 



for the velocity is independent of the direction of vibration. 

 The equality of these two expressions gives us 



(x-ff . 9 *(y—g) v * {z-hf . a*{y—g) 



Now this equality is true whatever be the position of the 

 vibrating particle; that is, it is perfectly independent ofy—g. 

 Consequently the portions which depend on each particular 

 value of y— g must be separately equal to one another. This 



C r _n2 (z—hY 



gives us 2 m - — j~- = £ m v , . In exactly the same 



way does it appear that 



5 m (£z££ = 5 m k~g)l. Hence -5X = 0, &c. 



