180 The Rev. J. H. Jellett on the Equilibrium and Motion of an Elastic Solid. 



2. Two general methods have been adopted by the various authors who have 

 treated of this problem. Of these, the one consists in forming expressions for 

 the forces which act upon each particle in the medium under consideration, 

 and then determining the laws of its equilibrium, or motion, by the general sta- 

 tical or dynamical equations. This method is followed by Poisson and Cau- 

 CHY. It is also adopted by Navier in the commencement of his Memoir, but 

 soon abandoned, as being less complete than the second method. This latter, 

 which is the method of Lagrange, and is followed by Mr. Green, Professor 

 Mac Cullagh, and INIr. Haughton,* takes as its basis the equation derived 

 from the combination of D'Alembert's principle with that of virtual velocities, 

 and is distinguished by the greater completeness of the solution which it affords; 

 the same analysis giving both the general equations of equilibrium, or motion, 

 and the particular conditions which must be satisfied at the bounding surface 

 of the body or medium under consideration.! This is the method which I pro- 

 pose to adopt in the present Memoir. The discussion of a problem like the 

 present must, of course, rest upon principles more or less hypothetical, inas- 

 much as the nature of molecular action cannot (at least in the present state of 

 physical knowledge) be ascertained by direct experiment. The classification, 

 however, with which the present investigation commences, cannot be considered 

 as other than positive, inasmuch as the two kinds of force, upon the distinction 

 between which it is founded, are known to exist in nature, and cannot, without 

 a hypothesis, be reduced to one. The principle of this classification I shall 

 now proceed to state. 



* All these writers commence witli the assumption that the sum of the internal moments of a 

 medium may be represented by the variation of a single function. To this method it may, per- 

 haps, be objected that it takes, as the foundation of a physical theory, a principle which is almost 

 purely mathematical, and to which it appears difficult to give a definite physical meaning. This 

 hypothesis, moreover, does not give to the equations of motion all the generality of which they 

 are susceptible. I have, therefore, preferred taking, as the basis of the present Memoir, a prin- 

 ciple essentially physical; more especially as the equations of motion derived from this principle 

 are, in the case of homogeneous bodies, possessed of the full number of constants, and have, there- 

 fore, the greatest amount of generality which their form admits. 



•j- The investigation of these conditions, according to the method ordinarily adopted, is, how- 

 ever, open to serious objections. These the reader will find noticed in a subsequent part of the 

 present Memoir. 



