Section III, 1888. [ 3 ] Trans. Eoy. Soc. Canada. 



I. — A Table of the Cubical Expansions of Solids. 



By Prof. J. G. MacGregor, D.Sc. 



(Read May 23, 1888.) 



It has been fouud by experiment that in general the vohinie of any body at temper- 

 ature f may be expressed in terms of its vohime at any other temperature 7°, and of the 

 diifereuce of temperatures, by means of the expression, 



V,= V^(l + A(f-r)+Bit-rr), (1) 



where FJ and V^ are the volumes at f and r° respectively and A and B are constants for 

 the substance of which the body consists. In some cases an additional term C{t — t)' is 

 necessary. But in general the value of C is so small that it may be neglected. 



As the constants A, B, C have different valu.es for different values of r, it is con- 

 venient to choose some one temperature as a temperature of reference, and for this purpose 

 the temx^erature o'C is now universally chosen, the above exi^ression becoming therefore, 



r,= V„(l+at+bf). (2) 



The constants a and b (and c also, the coefBcient of f', in cases in which a term in f is found 

 necessary) having been determined for any substance, the volume of any body of that 

 substance, whose volume at o°C is known, may be determined at any other temperature 

 within the temperature limits of the experiments by which the A^alues of «, b and c were 

 found. 



Density may be substituted for volume in the above formula, provided the signs of a 

 and b be changed. For if p, and P^ are the densities of a substance at f and o° respectively, 

 we have, in general, 



p,/ P»= n/ V,= \/il + at + bf) = \-at-hf, 



since a and b are, in general, small quantities. 



In the case of isotropic solids, length may be substituted for volume in the above 

 formula, provided the constants a and b be divided by 3, change of length in any direc- 

 tion in such cases being numerically equal to one third of the corresponding change of 

 volume. The formula in the case of the linear expansion of such solids becomes there- 

 fore : — 



A = i„(H-|< + |f). (3) 



In the case of seolotropic solids, the linear expansion is different for different directions, 

 and must therefore be specially determined. 



