492 



Prof. O'Brien on Sijnibolical Btutks. 



^(uU) a parallel and equal force acting at a point whose distance 

 from the origin is n, it is required to determine the form of the 

 functio7i (p. In what follows I attach the usual signification to 

 u + k', and by 0(mU) -f- (f){u'\J') I mean the mechanical effect which 

 is produced upon the rigid body by the simultaneous action of 

 the forces U and U' acting at the „ ^ 



points M and?/ respectively. U + U' ,., V.,„„„,„ , ,r r k% i „,. 

 means the resultant of U and U',.. ^v, \bJI08oi 9f# uri^ift 

 or the mechanical effect produced^, v i, ,„ \,i/ -i V' '» ■. 

 by U and U' acting at the sato^^,^ •;^-, ,^,,A»^^^^V_.,„Ai> 



P«"it- . ' \ I W-X' %h 



'"' A/ hn, 



V 



It is immediately obvious that 



</)(mU) + <^(mU') = «/)(«; u + U'). 



Hence (J3 is distiibutive as regards U, and we may therefore put 



(^(Mtl) = (^(«))U. 



Again, suppose the two parallel and equal forces UU to act at 

 B and C respectively^ O being the origin, OB = Wj OC = m' : then 

 it is clear that these two foi-ccs produce the same effect as U at 

 D and U at (BD and CD being parallel to OC and (D^).: 

 hence, expressing this by the assumed notation, ,TV;'SiiiJ,^^j(^ngl£t; 



(</)(h))U + (0(m'))U = (<!>(« + t<'))'ir.-H<U>5>mo sniad 



therefore :?)'»•! sb oo-iiii oaorf? 



(/)(m)+<^(m')=^(m+V);+1*}''--" ' '^'^^r-^ SYX 



{</)(m) -1 } + {<^(m') -1 } '= {«^(« + w') -1 li'-'^'t^'^'l'^^'' 



Wlience it appears that the function <p{u) — \ is distribittiva axc^ 

 therefore, by properly assuming the imits, we may put v, ,^,. 



^(m) — !=?<, or <f){u) = 1 + M. 

 Hence 



(f>{uV)-{l + u)\] ; 



or the mechanical effect produced by a force U aQt^lg at the 

 point u is represented by the symbol (1 +m)U: * ' '• 



(IV.) Hence, if « and /3 be two units of : . , <; . 



length, and A and B two units of force 

 acting at the origin (O) in the same di- 

 rections as those of « and /3, we may show 

 that 



«A=0, /3B = 0; 

 and 



/3A=-aB. 



For, let OQ = a, OP = /S, and complete the parallelogram 

 OPRQ : then^ since A may be supposed to act at Q instead of 0, 



