34 



(where a /3 y represent three lines, each a unit in length, drawn at 

 right angles to each other, and x y z are any arbitrary numerical 

 coefficients,) and supposes that the differentiation denoted by d affects 

 a jS y, but not x y z. This secures the condition that u is invariable 

 in length, and leads to the following expression for Xdu, viz. 

 Xdu = (zy ' — z'y) a, + (xz' — x'z) jS + (yx ' — y 'x) y , 



x' y' z' being arbitrary coefficients. 



Assuming n'=x'a,+y'fi + z'y, it appears from this expression for 

 Xdu, that du=0 when m=m', and therefore that d denotes a differen- 

 tial taken on the supposition that u' is constant. 



On this account the author substitutes the symbol T> u ' in place of 

 Xd ; he then shows that the operation D M ' is distributive with respect 

 to u' (i. e. that D M '+ M "=D M ' + D K ")> and to indicate this he elevates 

 the subscript index u', and writes Dm'.m instead of D U 'U. Thus he 

 obtains the expression 



Du'.u= (zy' — z'y)cc + (xz' — x'z)(d + (yx'— 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 Du'.u is rr' sin 0, where 

 r and r' are the numerical magnitudes of u and u' , and 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, Am'.m, 

 which represents the expression xx' +yy' ' -\-zz s ', or rr' cos 0. 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, andM, m', m", &c. the symbols f of their respective points 

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

 the two equations 



SU=0 and 2D«.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', U'', &c. do not balance each 

 other, and putting SU=V, 2Dw.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 



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

 X Y Z are the three components of the force. 



f By the symbol of a point is meant the expression xa-\-yfi-\-zy, where 

 x y z are the coordinates of the point. 



