202 The Kev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 



II. — Hypothesis of Modified Action. 



14. Let m, Jii' be two molecules of the medium under consideration, m beiag 

 that whose equilibrium or motion is required. Then if, as before, we suppose 

 the force which m' exerts upon m to be composed of two parts, one depending 

 upon the relative displacement of these two particles, and the other existing pre- 

 viously to the displacement of either, we shall still have, as in p. 183, 



F=F, + A{^' - +B{n' - ,,) + C(r - ?). 



Now it is easily seen that the difference between this case and the preceding 

 will show itself in the nature of the quantity F„. In the former case, in which 

 the action of m' is independent of the other particles of the medium, i^„ must 

 be of the form 



f{x, y, z, p, e, 0), 



and may, as we have seen, be neglected in the case of a body whose original 

 position is one of free equilibrium. But in the present case,»in which it is sup- 

 posed that the displacement of the other particles has itself the power of deve- 

 loping a force between m' and m, the form of -Fo is completely changed. Our first 

 object, then, must be to determine the new form to be assigned to this quantity. 



Let m" be a third molecule of the given medium ; |", »/", f", its displace- 

 ments; and pi,6i,(pu or pi, "i, |3|, 71 , its polar co-ordinates with regard to m. 

 Then it will appear, by reasoning similar to that of p. 183, that the mathematical 

 expression for its effect in developing a force between m and ?«' will be 



f(a:,y,z, p,0,(p, pi, 61,^1, ^'-?, 17"-)/, ^"-f, €"-^, V - '/'. ^'-^)\ 



or, as it may be otherwise written, 



f{x,y,z, p,G,<p, p„e,,cp,, §"-?, >/'-'/, r-f' «'-^' '/-'7. C-n- 



Treating this expression as in p. 184, it becomes 



/ d^ f/? d^' 



fo + ap [co% aj^ + cos § ^^- + cos 7 ^^ 



, / dn „dv d)j 



+ Jp(^COSa^ + co8^^ + cos7^ 



