396 Prof. O'Brien on the Interpretation of 



factors in the product wU, and we are at liberty to assume that 

 mU = Um if Ave please. It will not be found necessary, however, 

 to settle anything as regards this point. 



(II.) If AP" and BF (fig. 2) be ^-~ ■^^^ ^V "<•'• ^ ^-'ini 'iii 

 two forces each parallel and equaf ^ ^'^ .ni:Ei^vi2.JR ioB 3- 

 to P, it is immediately obvious, ■'"SJ»«f"'i«^pai>'"'«2^i^''^'<''i 

 that, the couple P'P" being applied ■ '^ '' '^'> ^-^i^-ijlx^of^ ion bnu 

 as in the figure, the force AP is ' \^''''' -,i"'^,^'M'!«^' 



thereby translated to BP' : the \iK^^)OMiiii;^«i 



couple' P'P" is therefore that which ' ' -* jfiJ j;Jrr>^a.u|«^^^lll ohira; 

 translates the point of application '^''P^ ^' * '"^ ''''^ ' ' '.' ■'■ 

 of the force U from A to B. Hence '^"^' 



we have the following important symbolical result, viz. f^ 

 symbol of a couple is the product uU, where u denotes the line AB, 

 U one of the forces of the couple supposed to act at B, and — U 

 the other force supposed to act at A. 



(III.) If we suppose that all the forces we may be concerned 

 with act on the same rigid body, and this I shall always do ex- 

 cept the contrary be specified, then, since a force may be sup- 

 posed to act at any point of its line of direction, nothing is re- 

 quired to translate the point of application when u and U coin- 

 cide in direction. In this case therefore we have uV = 0; that 

 is, the product of u and U vanishes when u and U coincide in 

 direction. 



(IV.) Using the sign -(- in full generality, and assuming that 

 the force U acts at the origin (which we shall suppose to be A), 

 it follows that U + wU is the symbol for the force U supposed to 

 act at the point B, u being the symbol of AB. Hence we obtain 

 another important symbolical result, namely, the symbol which 

 denotes a force U, when its point of application is at a distance u 

 from the origin, is U + uU, or ive may write it (l+u)U, '■ - 



(V.) Hence, if U, U', U", &c. be a set of forces which act 'oft 

 a rigid body at points whose distances from the origin arfe re- 

 spectively u, u', u", &c., it follows that the combined effect of 

 these forces is represented by the symb(Sl' '.,"'•. 



S(U + «U). ' ,., .. 



(VI.) Let «, /3, 7 denote three lines, eachlai'uiitit'of length, 

 drawn parallel to three rectangular axes ; also let A, B, C denote 

 three forces, each a unit of magnitude, acting parallel to the 

 same axes respectively. Let x, y, z be the projections (expressed 

 numerically) of the line u on the three axes, and X, Y, Z the 

 components (expressed numerically) of the force U parallel to the 

 three axes. Then we have by the first principles of symbolical 

 aleebra ** '^ IT< .J<jt, ii. . i,. -./sj-i aUi flyidv' iy ,L«l'. 



U = XA-^YB + ZC. 



