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



axes, and is the most general which can be used for the case considered in the 

 present section. The direction of the molecular vibration will not be either 

 normal or transversal, but will be determined by the axes of an ellipsoid first 

 noticed by M. Cauchy in treating of bodies whose molecules attract and repel 

 each other. It may be worth while to examine whether there are any subordi- 

 nate functions which possess the property of not being changed in form by the 

 chancre of co-ordinates ; as such functions, if they exist, will contain the laws 

 of classes of bodies contained under the general function (22). I have already 

 discussed one such function (20), which is evidently a particular case of (22). 

 I shall now determine another remarkable subordinate function, which, while it 

 is much less general than (22), yet contains most of the bodies whose proper- 

 ties are at all interesting. The following formulas of transformation may be 

 easily demonstrated.* 



a = a-a + h"-^ ■{■ c-y' + bcii' + acv' + abw'; 



p = a"a' + h'-0 + c'Y + i'c'u' -t- a!c'v' + a'b'w'; 



y = a"- a' + b"-'^' + c"V + b"c"u' + a"c"v' + a"b"w'; 



u^2a'a"a'+^.h'b"^'+2c'c"y'+{b'c"+b"c')u' + {a'c"^-a"c')v'Ha'b" + a"b')w'; 



V = 2a«"a + 2M"/3' + 2cc"y'+ {bc"+b"c)u'+ {ac"+ a"cy+ {ab" + a"b)w' ; 



w= 2aa'a' + 2bb'^ + 2cc'y' + (be' + b'c)u' -(- {ac' + a'c)v' + {ab' + a'b) w'. 



If we now assume six functions of (a, /3, 7, u, v, w), defined by the following 



equations : 



\ — U- — ipy, n = v^ — 4a7, V = ur — ia^ ; 



= 2a?* — inc, X — ^^ ~ ■""^'' ^ = 27Z<7 — uv ; 

 it may be shown without much difiiculty, that these new functions are trans- 

 formed by the following equations : 



X = a-\' + b-fj! -t- cV -f 2bc(l)' + 2acy^ + 2abi^'; 



f, = a"\' + 6'y + c'V+2b'c'<i,' + 2a'c'x' + 2a'b'^'; 



V = a"^\' -f ?»'V + c"-v' -I- 26"e"0' -I- 2a'V'x' + 2a"6"f '; (31 ) 



<^=da"7^\b'h"ij:^c'c"v'^{b'c"^b"c')<^^{a'd'^a"c')y:^{n'b"^a"h')<f'; 



X = an"W 4- bb" f,: \ cc"v' ^ (be" + b"c)(})' + (nc" + a"e)x' + (<ib" + a"b)^'; 



^ = aa'\' -I- bb',x' -I- ee'u' -f (be' + b'c) 0' -I- (ac' -f a'c) x' + (ab' + a'b) ^'. 



* Transactions of the Eoyal Irish Academy, vol. xxi. p. 160. 



