1132 Prof. O'Brien on Symbolical Mechanics. 



In ordinary mechanics, where addition is in all cases merely 

 numej'ical, it is immediately obvious that lines may be assumed 

 to represent forces in magnitude and direction ; but whether the 

 same mode of representation can be adopted in symbolical me- 

 chanicS; where -f is used in the two different senses just alluded 

 to, is a point to be determined. For, if we suppose AB and AC 

 to represent the forces U and V respectively, AB + AC ought to 

 represent the force U + V ; that is, AD ought to represent the 

 resultant of the two forces represented by AB and AC : otherwise 

 lines cannot be assumed as proper representatives of forces. 

 Now this immediately leads us to the parallelogram of forces^ 

 and shows that the general representation of forces by lines 

 assumes the truth of that theorem. In fact, the parallelogram 

 of forces is a principle which identifies geometrical and mecha- 

 nical addition, and shows, that, if the lines u and v represent 

 the forces U and V respectively, in magnitude and direction, then 

 the geometrical sum u + v will also represent the mechanical sum 

 or resultant U + V. That u-\-v repesents U + V admits of remark- 

 ably simple proof by means of the symbolization explained in 

 the former papers, as I shall now briefly show. 



Let a and /& denote units of length, and A and B units of 

 force parallel respectively to a and y8. Let U=XA, V=YB, 

 X and Y being the numerical magnitudes of the forces ; then, if 

 u and V represent U and V, we must have, msbX*, «=Y/3. 

 Hence 



(^,4.v)(UH-V) = X2aA + XY(«B-|-/3A)-f-Y2/9B; 



but we have shown that ak, aB + /8A, and /SB are each equal tor' 

 zero; consequently 



(M + t^)(U-HV)=0, 



and therefore the force U -f- V is parallel to the line w + v ; that 

 is, the latter represents the former in direction. 



Again, let € and E be imits of length and force in the common 

 direction oi u-\-v and U-}-V, and let w-f w=re, U + V = RE, r 

 and R being the magnitudes of w -f- v and U -f V respectively. 

 Then we have 



wU5;:(r€-«;)(RE-V), 

 or 



X««A=rR6E-ryeB-- YR)5E- Y2/3B ; 



but a A, eE, and ySB are each zero, and eBss — /8E ; hence 



0=rY-RY, orr=R. 



It appears, therefore, that u-{-v represents U + V m magnitude 

 as well as in direction. 



I shaU now always use lines to represent forces, and, therefore. 



