518 



Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



1 rf'^ 5 . 



11 '^'^ S •■ 

 and -k a -r-r oxr~, 



^ ax- 



are of a different nature, though they consist of displacements in the same direction a ; for the 

 former disarrangement consists in the rotation (i. e the curvature) of physical lines parallel to (i 

 about the axis 7, and the latter of physical lines parallel to 7 about the axis j3. The same re- 

 marks apply to B., and jB..', ^3 and 5/. Fresnel virtually assumed that B^ = B', B. = J?,', Bj ■■= B3'. 



16. By reasoning as in article (6), we might easily shew, that the force brought into play by 

 the disarrangement, 



dxdy 



dwdy 



d'r 



dxdy ' dudy 



But we shall shew this somewhat differently, in order to find out what relation subsists (if any) 

 between C C and the constants already introduced. 



The disarrangement a — , ^x^y, is of the following nature. 

 dxdy 



Let OX and OY represent the directions a and /3, and O the point (ryji); take OM = ^x, 

 draw SMS' parallel to J'F', and PMP' making the tangent of the 



angle PMS equal to ^ 5.f. Then it is clear that the dis- 

 Qiiij ay 



d'P 



arrangement a ^- Sx^y causes the physical line SMS to 



° dxdy 



assume the position PMP'. In like manner, if ON = - Sx, and 

 TNT' is parallel to YY', the physical line TNT' will, in con- 

 sequence of the disarrangement, assume the position QNQ', the 

 angles QNT and PMS being equal. The physical lines (taken 

 parallel to YY') between SS' and TT' will suffer similar deviations, the tangent of the angle of 

 deviation being proportional to Sx. 



17- The effect of a disarrangement of this kind is obvious ; for it produces a uniformly 

 increasing expansion of the medium as we go along the line OY, and a uniformly increasing con- 

 densation as we go along the line OY , the rate of increase both of the expansion and condensation 

 being, as it is easy to see, 



dxdy 



The effect of this will be to bring into play upon the particle O a force in the direction Oi' 

 proportional to this rate of increase, i.e. a force whose symbol is, 



d'^ 



fiC 



dxdy' 



C being some constant. 



d'r 



In like manner we may shew that the disarrangement 3 - — — Sx^y brings into play upon 

 ■' dxdy 



a force whose symbol is. 



aC 



d' rj 

 dxdy 



, C' being some constant. 



