516 



Mr. O'BRIEN, ON THE SYMBOLICAL EQUATION OF 



a, being thus assumed, we have d^U = 0, d^ U = 0, and therefore by (5), 

 a,d^U = ad,U + fid^U + yd^U=WU. 



Now a, is the direction of propagation of the disturbance U, therefore d^ U is the rate of 

 variation of U in the direction of propagation, and ad^U is that rate affected with its proper 

 sign of direction. 



Hence, the symbol 'W U expresses, in magnitude and direction, the complete rate of variation 

 of the quantity T7, that is to say, the direction of 39 (7 is that direction perpendicular to which there 

 is wo variation of U at the point xyz, and the magnitude of 13 t/' is the rate of variation of U in 

 that direction. 



It is manifest, therefore, that the symbol 33 has a very important signification, especially in 

 investigations relating to the propagation of waves. 



1 1 . Returning now to the equation (4) we shall, in the first place, interpret the meaning of 

 the symbols AH) . v, and Z>19 . n. 



Let ^^ >;^ ^, be the resolved parts of the displacement w in the directions a, (i^ y^ respectively; 

 then, choosing (as we may do)* the direction y^ so that ^^ = 0, we have 



V = ^,a, + >;,j8,, 

 and therefore, since W = ad^, we find 



Let OX and OY be the directions a, and /3,, y^ being perpendicular to the plane of the 

 paper ; let be the point {xyz) of the medium, POQ, the 

 line of particles which, in a state of equilibrium, lie in the 

 direction a^, and OT the tangent to POQ at O. Then since 

 OX is the direction of propagation, and since the disturbance 

 (w) consists of two parts, namely f, in the direction OX, and 

 t]^ in the direction OY, it is evident, that rf„f^ is the expansion 

 (i.e. the degree of expansion, or, what is called the rarefaction) 

 of the medium at the point O ; also d^ri^ is the tangent of the 

 angle TOX, and therefore measures the degree of lateral displacement of the medium at the 

 point O. 



This lateral displacement consists in the rotation of the line OT about the axis -y , and a 

 corresponding rotation of all the other lines of particles which constitute the medium in the im- 

 mediate vicinity of the point O, these lines being supposed to be parallel to OX in a state of 

 equilibrium. Hence it follows that the symbol Z)39 . v represents, in direction, the axis round 

 which the lateral displacement takes place, and in magnitude, the degree of lateral displacement. 



Thus it appears tjiat the symbols AiB.w and aJOjE).« have a very important signification in 

 investigations relating to the propagation of waves, the former expressing the degree of expansion 

 of the medium at the point xyx, and the latter representing, in magnitude, the degree of lateral 

 displacement at the point xyz, and, in direction, the axis about which that displacement takes 

 place. 



12. Hence it is evident that the symbol A39 . v defines the kind of disturbance which constitutes 

 normal waves, and DlB . v that which constitutes transverse waves. 



• a, is the direction of propagation, as in the preceding article, and y, is chosen at right angles, not only to o, , but also to the 

 direction of viliralion at the point xyx. In this case 5, is clearly zero. 



