Cambridge Philosophical Society, 186 



the subacript index m', and writes Dm'.m instead of Dm/m, Thus he 

 obtains the expression 



Du' .u=i{zy'~z'y)a,-\-ixz' —x'z)^-\-iyx' —y'x)y. 



From this it follows that Dm'.m is a line perpendicular both to u' 

 and u, and that the numerical magnitude of Dm'.m is rr' sin ^, where 

 r and r' are the numerical magnitudes of u and m', and fl the angle 

 made by u and u' . 



Having investigated the principal properties of the operation Dm'., 

 the author, by a similar method, obtains another notation, Au'.u, 

 which represents the expression xx' -{-yi/'+zz', or rr' cos Q. He then 

 gives some instances of the application of these two notations to 

 mechanics, which may be briefly stated as follows : — 



1st. If U, U', U", &c. be the symbols* of any forces acting upon 

 a rigid body, and m, m', m", &c. the symbolsf of their respective points 

 of application, then the six equations of equilibrium are included in 

 the two equations 



2U=0 and I:Dm.U=0. 



2nd. That these two equations are the necessary and sufficient 

 conditions of equilibrium, may be proved very simply from first prin- 

 ciples by the use of the notation Dm. 



3rd. The theory of couples is included in the equation 2Dm.U=0. 

 In fact the symbol Dm.U expresses, in magnitude and direction, the 

 axis of the couple by which the force U is transferred from its point 

 of application U to the origin. 



4th. Supposing that the forces U, U', TJ", &c. do not balance each 

 other, and putting 2U=V, 2;Dm.U=W, we may show immediately, 

 by the use of the notation Am, that the condition of there being a 

 single resultant is 



AV.W = 0; 



and when there is not a single resultant, the axis of the couple of 

 minimum moment is 



AV.W y 



AV.V. ■ * 



5th. The three equations of motion of a rigid body about its 

 centre of gravity are included in the equation 



-('2Du. — Sm\='LDu.VSm; (1.) 



dt\ dt J 



u being the symbol of the position of any particle hm of the body, 

 and U the symbol of the accelerating force acting on Jm. 



6th. If u) be assumed to represent the expression cWia + cwojS + Wjy, 

 where Wy, w.^, Wj are the angular velocities of the planes of yz, zx, 

 xy about the axes of x, y, z respectively, then the symbol of the 



* By the symbol of a force is meant the expression X«+Y/3+Zy, where 

 X Y Z are the three components of the force. 



t By the symbol of a point is meant the expression jf«-t-^/3+2y> where 

 X y z are the coordinates of the point. 



