the Product of a Line mid a Force. 397 



In fig. 3, a. and 7 are repre- 

 sented drawn from the origin 0, 

 A and C act at the extremities of 

 the lines a and <y, and — A and 

 — C act at the origin. Now these 

 four forces are equal in magnitude, 

 and act along the sides of a square 

 in opposing directions; they there- 

 fore balance each other. But by 

 article II., 7A represents the couple 

 A and —A, and aC the couple C 

 and — C: therefore 7A + aC=!0(. ,, ^i 

 Hence it is obvious that > tiourui • .,t 



7A=-«C, «B=s-/9A, i8C=-7B. 



Also from article III. it follows that 



«A=:0, /9B = 0, 7C = 0. 



(VII.) By articles V. and VI. the symbol of the combined 

 effect of the forces U, U', U", &c., acting the points u, u\ u", &c. 

 oj^ a, r^d body, is 



. xo l) lu:X{{l+iCcc + yfi + sy){XA + YB + ZC)}, 'p 



^HicvlBy l!he results in article VI., immediately becomes 

 M.V,...., f ■■. A2X + BSY + C2Z 



;Now here A, B, C are units of force acting at the origin 

 parallel to the three axes ; also (by article II.) aB, /3C, 7A are 

 couples, each a unit of moment, acting in the three coordinate 

 planes icij, y::, zx respectively. Hence we obtain the well-known 

 statical result, that the set of forces U, U', U", &c. are equivalent 

 to the three forces 2X, 2Y, 2Z acting at the origin along the 

 axes J?, y, z respectively, together with the tliree couples 2 {xY — yX), 

 ]S(yZ — ^-Y), 'Z{z\—xZ) acting in the planes of xy, yz, s'a? re- 

 spectively. 



These results" are 'sufficient to show the meaning and use of 

 the projjosed interpretation of the product of a line and a force. 

 In a paper just presented to the Royal Society I have explained 

 the method of symbolizing the effect produced by the trmislation 

 of any directed magnitude, such as a force, velocity, traced line, or 

 any of those maguitudcs which wo represent on ])aper by arrows. 

 If U denote any magnitude of this kind, and u any line, I have 

 shown that the effect produced by the translation of U along u 

 may, under ccitain restrictions, be denoted by the product of u 

 and U ; and I have pointed out the statical applications of the 

 method, of which the result just given in art. VII. is an ex^p;iple. 

 [To be continued.'] 



