332 M. Edlund on the Thermal Phenomena which 



v the excess of temperature at the moment /. The differential 



equation 



d x ax 



-tx -f m {x — a? ) -|- 2n -=r — kv = 



is easily obtained. 



As, further, the excess v varies with the time independently 

 of x, in consequence of the cooling of the wire, in conformity 

 with Newton's law, 



dv-\- avdt=0, 

 whence 



v = v e~ at , 



and the differential equation becomes 



— + m(x— x ) +% n jj- —kv e- at = Q, 

 the general integral of which is 



fry e -at 



Jb wC\ — — 



+ e~ nt {C cost Vm— w 9 -f C x sin / \/ m — n*. 



a 2 — 2an + m 

 As, at the time / = 0, x—x and -=- are simultaneously zero, the 



two constants C and C, are easy to determine, and we get 



[ e -at + e -ntl a - U ■ f ^/^Z^2_ cos/i /^r^) 



kv 



a z — zan + mL \ \/ 



If, in this expression, the smallest of the values which reduce the 



dx 

 velocity -=-. to zero be placed for t, the corresponding value of 



x— x will be the amplitude of the first range of the needle; and 

 it will be evidently proportional to v , that is, to the initial varia- 

 tion of the temperature of the wire. 



The reasoning which leads to this conclusion supposes that 

 the calorific phenomenon, of which the wire is the seat, is of very 

 short duration, and that it is ended before the displacement of 

 the needle has become sensible. But experiment shows that it 

 is not necessary that this condition be rigorously satisfied, but 

 that the duration of the fall of the weight may be prolonged to 

 six seconds without the original range of the needle being 

 sensibly diminished. 



Experiment shows that, for the same wire, the amplitude of 

 the first range is really proportional to the quantity of heat ab- 

 sorbed or disengaged in the wire. If, in fact, these be deter- 

 mined by means of four series of successive experiments — (1) the 

 amplitude X, which corresponds to the total fall of the weight d 

 from the axis b to the end of the lever, (2) the amplitude Xj, 



