34 M. H. Fizeau on the Expansion of Solids by Heat. 



other ', these are the three axes of expansion. In any other di- 

 rection than those, there will be observed only the simultaneous 

 effects of these three primitive expansions, which will always 

 manifest themselves individually, according to their intensities 

 and their own directions, and constant for the unit of length. 



It may be added that, all the elementary parts of the crystal 

 being identical among themselves, these axes are not represented 

 by three lines having a determinate position in the interior of the 

 crystal, but really by three rectangular systems of parallel lines 

 taken at each of the internal points. 



To complete the enunciation of what appears essential in these 

 singular properties,, which in some sense seem to indicate a tri- 

 nary arrangement in the elements of the crystallized substance, 

 it remains to be stated in what each of the principal expansions 

 must differ from any other resultant expansion, or, in other words, 

 what is the distinctive character of an axis of expansion. 



Suppose a sphere isolated in the body of the crystal at a cer- 

 tain temperature ; if it be heated, the sphere will expand un- 

 equally in the direction of its different radii, and in the most ge- 

 neral case its form will be ellipsoidal ; but there will always be 

 three diameters rectangular to each other, at the ends of which 

 the displacement of a point on the surface of the sphere will be 

 in a radial direction— that is to say, in the prolongation of the 

 radius itself and without any lateral deviation. 



That is the true character of axes of expansion, and the 

 principle of the geometrical construction from which, in the first 

 memoir, we have deduced the formula which we here examine. 



I shall now give the results of experiments made with a view 

 of controlling the accuracy of the general formula in many of 

 its most important consequences, by trying to find in the various 

 crystalline systems the most decisive phenomena and those most 

 accessible to observation. 



Denoting by D the coefficient of expansion in any direction 

 whatever, determined by the angles S, h' } and &" which this direc- 

 tion makes with the three axes of expansion, and calling a, ct', 

 and a" the three coefficients of expansion along the three axes, we 

 have the following relation, 



D = acos 2 S + a'cos 2 S / + a"cos ? 3"; .... (1) 

 but we have at the same time the known relation which expresses 

 that the three angles 8, S ! , and 8" are referred to three rectan- 

 gular axes, CO s 2 S-fcos 2 S'+eos 2 ^ = l (2) 



Cubical System. 

 The general characters and the properties of this crystalline 

 system lead us to consider the three principal expansions equal 



