120 The Rev. S. Haughton on a Classification of Elastic Media, 



co-ordinates, and consequently the proper form of the function T', for bodies 

 which transmit only normal pressure in every direction, will be 



V= F(w). (20) 



There are two other forms of the function V (considered as containing the 

 nine differential coefficients) which possess this property of reproducing them- 

 selves by transformation of co-ordinates. Let (A', Y, Z) be determined by the 

 following equations, 



A' = j33— 70, r= 71 -eta, Z=a2 — /3i. 



It appears immediately from equations (19) that (A", Y, Z) are expressed by the 

 following relations in terms of (A'', Y', Z'), 



X=aX'+bY'+cZ', 

 Y=a'X' + b'r+c'Z', 



Z=a"X' + b"Y+c"Z'. 



These formultc are well known, and prove that another self-producing form of 



the function V will be 



V=F{X,Y,Z). (21) 



I have proved in my former memoir that if (a, /3, 7, u, v, iv) denote the co- 



*'- {!• !■ !■ CM), (24!) (I- g)}."-^"'"^— ■ 



by transformation of co-ordinates, linear functions of (a, ^, 7', u', v', w') ; and 

 that the formidte of tranformation are the same as for (.r), (j/-), {z-), {2yz), 

 (2.z-j), (2,r?/). Hence, a function of these six quantities will reproduce itself, 

 by transformation of axes of co-ordinates. The third form of F which possesses 

 this property is, therefore, 



V=F{a,^,^,u,v,w). (22) 



The forms of the function F, which have just been determined, are perfectly 

 general, and do not merely express properties of the body with reference to 

 particular axes, but general properties of its molecular structure, independent 

 of the directions of the co-ordinates. 



I shall consider in the present section the function (20),which is peculiar 



