456 



municated to me a result at which he has lately arrived, and 

 on referring to my note book I found that Mr. Jellett's theo- 

 rem gave a physical reason for the hypothesis 1 have alluded 

 to. So far as Mr. Jellett's investigation relates to this subject, 

 it may be thus stated : 



" ' If in a system of molecules the forces developed by the 

 displacement of any two molecules be functions oi their rela- 

 tive displacements only, and tend to restore them to their ori- 

 ginal positions ; the function Ffor such a system will contain 

 thirty-six coefficients, the coefficients of ^(ai/Bs) being equal 

 in pairs.' 



" This theorem evidently supplies the link which was 

 wanting in my equations, which, perhaps, may not now be 

 deemed unworthy of notice, as they may be shown to rest on 

 a definite physical hypothesis. 



" The note from which the following abstract is taken is 

 dated December 26, 1848. I have slightly altered the no- 

 tation, and prefixed two theorems which facilitate the under- 

 standing of what follows. 



Theorem I. 



" Let (?/], U2, &c.) be functions of {x, y, z), defined by the 

 following equations : 



«i = i^aya - jSaya V\ = -yaas - 7302 Wi = a.^/Bs - 03/82 



"a = ^371 - i3i7.3 v-i = 7301 - 7103 w-z = os/Bi - ai/33 

 Us = /3i72 - (izy\ Vj = yia-i - jzai M'3 = ai/Bg - ag/Bi 



(ai,a2,/3i,&c.) denoting ^-,-,-,&c.j 



♦' If the co-ordinates be changed into x, y\ 2, by changing 

 the direction, without changing the origin, then the functions 

 (mi, 172> «^3> Vz + Wg, Wi + Ms, Ui + «,) wiU Tcproduce them- 

 selves by means of the following equations : 



