and the Laws of Plane Waves propagated through them. 127 



(30) 



These are equivalent to equations (24) of my former paper, and are an imme- 

 diate consequence of (29), which express the relations among the resultants of 

 the molecular forces consequent upon the restricted form of the function. Equa- 

 tions (29), or the corresponding equations (9), 



are analogous to a statical theorem given by Fresnel in his Memoir on double 

 Refraction. If no external forces act upon the system, or if their influence may 

 be assumed to be constant at all points of the body, then the coefficients of V 

 will become constants, and the linear part of V will introduce no terms into 

 the differential equations, which will in this case depend upon a homogeneous 

 function of the second order. There will be, therefore, in general, twenty-one 

 constants, viz., the coefficients of the function 



2 F= (aOa- + (^'W + (r)r + (u')u' + {v'-y- + (w"-)w' 



+ 2K^)^7 + ("7)"7 + («^)«i3j + 2\(vw)vw + (uw)uw + (uv)uv\ 

 + 2u\(au)a + {pu)p + {yu)y\ + 2v\{av)a + (fiv)^ + (yv)y\ 

 + 2w\ (aw)a + (j3w)^ + (yw)y\. 



The equations of motion deduced from this form of the function T" are 

 identical with the equations used by M. Caught in his theory of light* They 

 are deduced by M. Cauchy directly from the consideration of attractive and re- 

 pulsive forces between the molecules. These equations have been used also 

 by Mr. Green, who seems to have considered them as more general than M. 

 Cauchy's equations, f 



The function (22) is not altered in form by an alteration of the co-ordinate 



* Exercices de Mathematiques, torn, v., p. 19. 



t Cambridge Pliilosophical Society's Transactions, vol. vii. p. 121. 



