M. II. Fizeau on the Expansion of Solids by Heat. 33 



of facts have been connected by very general geometrical consi- 

 derations, in like manner for the phenomena of expansion we 

 may, by analogous considerations, express the law of the varia- 

 tions met with in the numerical values of the expansions when 

 they are considered according to different directions. There are 

 then three distinct physical phenomena which may be connected 

 with analogous theoretical views ; these are the propagation of 

 light and of heat through crystals, and the expansion by heat 

 of the crystal itself; and these theoretical views are precisely 

 of the nature of those which geometricians use when they in- 

 vestigate ellipsoidal surfaces. In fact, one and the same prin- 

 ciple serves as a common starting-point in the theoretical ex- 

 planation of these three orders of phenomena — that is, the consi- 

 deration of the three principal directions or rectangular axes 

 endowed with well-defined physical and geometrical properties, 

 and around which are attached as rigorous consequences the 

 totality of these phenomena in their most varied manifesta- 

 tions. 



I must here simply attempt to define the action of these axes 

 relatively to the phenomena of expansion with which we are oc- 

 cupied; and I shall adduce a certain number of experiments 

 which clearly prove that these axes correspond to real and 

 distinct physical properties, which prevent us from regarding 

 them as a mere geometric fiction sufficient to group empirically 

 the data of observation. For the future they will be desig- 

 nated as axes of expansion. The expression axes of elasticity 

 used in my first paper having become inadequate owing to the re- 

 sults observed in oblique crystals, we shall see in the sequel 

 that in these crystals the three kinds of axes are no longer 

 superposed as in other crystalline systems, but are really sepa- 

 rated from each other by angular distances which are frequently 

 considerable. 



It has been shown in the first memoir that, if it be attempted 

 to express in a general manner the value of the expansion of 

 a crystal in any direction referred to three rectangular axes, 

 a yery simple formula is obtained, merely containing the squares 

 of the cosines of the angles made with the three axes, as well 

 as the three principal coefficients of expansion corresponding to 

 these axes. 



But it is important to remark that the reasoning which has 

 led to this result depends really on the following principle : — 



However complex be the crystalline form, however varied the 

 expansions observed in the various directions, be they even in 

 one case expansions, in another contractions, there are only three 

 primitive expansions distinct and independent of each other, and 

 solely manifested in three fixed directions at right angles to each 



Phil Mag. S. 4. Vol. 36. No. 240. July 1868. D 



