4 PROF. J. G. MACGEEGOR ON 



To determine the vohime at /'° when that at t" is given, we have : — 



F, = F„ (1 + af + hf\ 

 and F„=. F„(l + ar-h6n- 



T^ T^ 1 + at'+ bt" 

 Hence T'"= ^ ' -i +at + bf, 



= r,{l + a{t'-t)+bit''-f)), (4) 



since a and b are small quantities. 



The mean coefficient of thermal expansion between two -temperatures is by some 



writers defined as the change of volume per degree and per unit volume at the lower 



temperature, and by others as the change of vohime per degree and per unit volume at 



the temperature of reference (o°C). In terms of the symbols used above, it is in the 



y Y T V F 



former case the value of =~, -, and in the latter case the value of^'^ , ' . From (4) 



it follows at once that : — 



^j^^=.a + b(t' + t) (5) 



provided a and b are so small that their powers and product may be neglected. From (2) 

 and a similar equation with t' substituted for t, it follows that : — 



F„ - r, = v.. {a (r -t) + b (f ■' — f) ) 



exactly, and therefore that 



tÇ^;=«+k^'+o, (6) 



whatever the magnitudes of a and b may be. Provided a and b are sufficiently small, 

 therefore, the mean coefficient has the same value between given temperature limits 

 according to both modes of definition; and if a and b are known, this value may be de- 

 termined for any tetnperature range. 



The " true " coefficient of thermal expansion at any temperature is the rate at which 

 volume varies with temperature at that temperature per unit volume at zero. It is thus 



-, and bv differentiation of (2) is seen to have the value « +2 bt. The true coefficient 



V, dt ^ 



at a given temperature is clearly the mean coefficient (per unit volume at o°C) between 



two temperatirres indefinitely near one another and including the given temperature ; 



and the above value is also obtained from (6) by noting that ultimately t' -\- 1 =^2t. 



Sometimes, but rarely, the true coefficient at any temperature is defined as the rate 



at which volume varies with temperature at that temperature per unit volume at that 



1 dV 

 temperature — in symbols, — •— ^; and by (.5), its approximate value, provided a and b be 



y ( etc 



small, is rt + 2 bt, the same as the value of the true coefficient according to the former 



definition. 



The following table contains the values of a and b (and of c also in cases in which c 



is found to have an appreciable value) in the case of the more important and interesting 



solids. A table containing an exhaustive list of the determinations of these constants 



would be so long that in most cases I have thought it well to give only the most recent 



and most accurate determinations, though in some important cases, as in that of glass, a 



