512 Mb. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



Now, by a well-known principle, the force on P resulting from the disarrangement 



iv =-- CIV + —- 6y + &c., 

 dx ay 



is the resultant (or symbolical sum) of the forces due to the separate disarrangements 



^« = — Iw, 6v=--~6y, 6v = --dx, &c. 

 dx dy dz 



Hence, if we find the forces due to the several terms of the expression (1), and add them 

 together, the resulting sura will express, in magnitude and direction, the whole force brought into 

 play upon P by the disarrangement (l). This we now proceed to do. 



3. To find the force brought into play on P by the disarrangement, 



Sv = — Sx = a-r^ Sa; + ^ — dx + y -f- da:. 

 dm dx dx dx 



a —Sx represents a small line, proportional to Sx, drawn in the direction a ; therefore the 

 dx 



disarrangement indicated by 



Sv = a — — Sx 

 ax 



is a uniform expansion of the medium in the direction a. This brings no force into play upon P. 



Q JL^x represents a small line, proportional to Lv, drawn in the direction /3 ; therefore the 

 dx 

 disarrangement indicated by 



dx 

 takes place as follows: Suppose the medium when at 



rest to be divided into physical lines parallel to the , 



direction „, let MN be any one of these lines, M being the point when it meets the plane per- 

 pendicular to a containing >, and let MN' be a line parallel to the plane of xy, makmg an 



angle tan -> (^) with MiV. Then the disarrangement consists in the displacement of the line 



MN into the position MN\ and a similar displacement of all the other physical lines. This 

 disarrangement evidently brings no force into play upon P. 



The same reasoning applies to the remammg term 7^'''''' 



4. Reasoning therefore in this way it is clear, that the disarrangement represented by the 

 first three terms of the expression (l) brings no force into play upon P. 



5. To find the force brought into play on P by the disarrangement represented by 



a„ = X j!^J,.= = 1 ^4 (a e + /B. + 7D ^^=- 

 •^ dx- dx'' 



la^Sw- represents a small line, proportional to ^x\ drawn in the direction a ; therefore 

 ^ dx- 

 the disarrangements indicated by 



Sv = :^a^Jx' 

 ^ dx' 



