,'nm^oH'i^'>J^^^n'|^^^ Classification, and their Uses) .>! J 525 



three points, which are necessary to determine the three coeffi- 

 cients a, bj c, will nearly coincide between those points. 



We may also now anticipate what the ultimate form of coeffi- 

 cients of expansion must be, if ever the knowledge of molecular 

 physics shall be so far advanced, that this form may be investi- 

 gated a priori from the laws of the elementary molecules, with- 

 out the necessity of having recourse to experiment except for 

 verification. 



It is well known that most solids expand uniformly by increase 

 of temperature until they reach the point of liquefaction. At 

 this point an abrupt change takes place, and the expansion of 

 the body commences to follow a different law. This new law 

 holds until the body reaches the temperature corresponding to 

 the point of vaporization, when a second abrupt change takes 

 place, and the expansion obeys a third law differing in character 

 from both the preceding. 



In the present state of our knowledge we are obliged to em- 

 ploy thx'ee distinct coefficients of expansion, expressed by empi- 

 rical formulae, for these three several stages. *It would seem, 

 however, that in general coefficients of expansion should be repre- 

 sented by discontinuous functions, with breaks corresponding to 

 the points of liquefaction and vaporization respectively, and the 

 temperatures corresponding to the points of maximum and 

 minimum condensation, if there be such, being given by the 

 roots of the respective derived functions. Any discrimination 

 of substances into sohds, liquids and gases, may be regarded, so 

 far as the previous remark is concerned, as unnecessary, since 

 modern skill has shown that they may be all reduced to the class 

 solid. 



Of the two classes to which we have proposed to reduce all 

 physical singularities, the one comprehending those deduced as 

 consequences of known laws, and the other those not yet so 

 deduced, the former has subserved the most important practical 

 purposes. Thus, in the science of optics, we avail ourselves of 

 our knowledge of the position of minimum prismatic deviation 

 in order to determine the indices of refraction and the dispersive 

 powers of various substances, accurate values of which are indis- 

 pensable to the perfection of all optical instruments and many 

 of the useful arts. The other instance by which this first class 

 was illustrated was borrowed from the science of terrestrial mag- 

 netism, and it is unnecessary to dwell upon the advantages which 

 have resulted from the reaction of the determination of the 

 magnetic poles upon a great branch of physical science. 



But while such valuable ends have been thus attained, how 

 comes it that the other class of these physical singularities has 

 as yet contributed little or nothing for the practical good of 



