36 M. H. Fizeau on the Expansion of Solids by Heat, 



heat, and their mode of expansion, which appears to depend on 

 their symmetrical structure about a principal crystallographical 

 axis. The expansions are here different in different directions ; 

 and reasons deduced from the symmetry of the structure show 

 that one of the axes of expansion must coincide with the principal 

 crystallographical axis, the two others being perpendicular to it. 

 If we assume, by the same reasons of symmetry, equality in the 

 coefficients corresponding to these two latter axes of expansion, 

 where a! = u", equation (1) will become 



D = ol cos 2 8 + «'(cos 2 8' + cos 2 8") ; 

 but equation (2) gives 



cos 2 8' + cos 2 8" = 1 — cos 2 8 = sin 2 8 ; 

 we shall then have (a being the expansion for the former axis) 



D = « cos 2 8 + a' sin 2 8 (3) 



Taking any direction at right angles to the first axis, which gives 



8 = 90°, cos 2 8=0, sin 2 8 = 1, 



the equation is reduced to 



D = a'; 



that is to say that, in any direction at right angles to the first 

 axis, which coincides with the axis of symmetry, the expansion is 

 constant, and that it is impossible to distinguish the axes of ex- 

 pansion. 



In any direction making with the first axis the same angle 

 8=54° 44', we have 



cos 2 8 = ^ and sin 2 8 = §, 

 and equation (3) becomes 



which is precisely the expression for the mean linear expansion 

 of the crystal. 



Varied experiments have been made on various kinds of crys- 

 tals with a view of verifying the accuracy of the following remark- 

 able property, which is deduced from theory : — All crystals which 

 have the form of right prisms, of rhombohedra, of regular hexa- 

 gons, or forms derived from these, present a certain angular 

 direction (the same for all crystals) which makes with the prin- 

 cipal crystallographical axis an angle of 54° 44' (an angle for which 

 cos 2 8= J); and along this direction we ought to find exactly 

 the third of the cubical expansion, or the mean linear expansion 

 for each crystal. 



It has been seen before that in the most general case this con- 

 dition must be satisfied by the normals to the faces of a regular 



