the Product of a Line and a Force. 395 



Cambridge Philosophical Society, Nov. 1846, and published in 

 vol. viii. part 4. of the Transactions. Other results were pub- 

 lished in subsequent papers. But I employed a new notation 

 to express these results, and so far obscured their meaning. I 

 am now able to put them all into the ordinaiy notation of algebra 

 without introducing anything novel in principle, or assuming 

 any but the simplest symbolical laws. 



The nature of the proposed inteiiDretation may be thus 

 briedy stated. If u denote the magnitude and direction of a 

 right line AB drawn from a given origin A, and U the magni- 

 tude and direction of a force supposed to act at A, then (1 4-m)U 

 denotes the force U acting at B, and ?<U denotes the couple com- 

 posed of the forces U and — U acting at B and A respectively. 



A similar interpretation holds with reference to velocities, or 

 mere lines instead of forces. 



By this mode of interpretation, the various statical propositions 

 which constitute the theory of couples and the conditions of 

 equilibrium of a rigid body may be proved with extraordinary 

 facility. The same may be said of the equations of motion of a 

 rigid body about a point. 



In what follows I shall very briefly explain the intei-pretation, 

 and exhibit its application to Statics. 



(I.) If U be the symbol of the force AP (fig. 1), and u the 

 symbol of the line AB, it may be showTi, as follows, that the 

 product mU represents the translation of the point of application 

 of the force from A to B, the magnitude and direction being 

 unchanged. 



Let f{u, U) denote that, whatever Fig. ] . 



it be, which effects this translation ; ^:.jr 



then the translation from A to any 

 third point C is the same thing as 

 the translation from A to B, together 

 with the translation from B to C j 

 also, if we assume u' to be the symbol 

 of BC, M + m' will be the symbol of 



/{ (w + m'), u } =/KU) +/(«', u). 



Again, it may be easily proved that 



/{«,(U+u')}=/KU)+/(«,u'). 



Hence it follows, according to the first principles of symbolical 

 algebra, that 



/KU)=mU. 



It appears therefore that the product mU denotes that, what- 

 ever it be, which translates the point of application of the force 

 U from one extremity of the line u to the other. 



N.B. Nothing here is detennined as regards the order of the 



