20 On the Constitution of Matter, 



vibration would be assimilated to, and fall in witb that corres- 

 ponding to some smaller value of \|/. As a increases, this effect 

 increases and extends to values of \I/ nearly equal, within a certain 

 range, to the particular value in question. 



The temperature of the ?•*'' atom in the case under considera- 

 tion is 



l^- a- K cos- (2 » ■ — 2 r + 1) \{/ 



A being some constant ; and the average temperature is 

 u?a' K cos- (2n — 2 r -|- 1) •], y/ a" a 



4 n cos- (2 ?i -f 1) v|/ 8 cos' (2 w + 1) \I/ 



nearly when n is large. The maximum temperature is 



fxr a" A 

 4 cos' (2 71 + 1) ^|/- 



There is thus a concentration of heat at the joints, so 

 that at those points a greater softening takes place than would 

 occur if the heat were uniformly distributed. This effect is in- 

 creased as the temperature increases, on account of the increasing 

 proportion of the heat, employed in softening the joints, which 

 becomes latent. The temperature therefore at which melting 

 occurs is much lower than what, if uniformly distributed, 

 would dissever the atoms. 



The effect of radiant heat upon a system of n atoms, such as 

 that under consideration, may perhaps be represented by suppo- 

 sing the additional atom at the beginning of the series to be of 

 feeble power, proportional to A w", when A is very small. Let x^ 

 be the displacement of this atom, and 



Xq ^^ a cos ju-if 

 The equation of motion for the first of the n atoms is 



' m" {Xi — cci) + X 7n- (xq — ajj) 



df 



1 d-x^ 



or — ~~y + (I + X) Xi = X Xq + X.-, 

 mr dt 



neglecting A in comparison \A'Ca. 1, and putting 



1 /(/\' 



*~M \'T~.\ + 2 = o = 2 cos 9, we have 

 m \dtj 



(q — 1) a^i = ic^ + A Xq 



q X2 = Xi + Xi 



