494 Prof. O'Brien on Symbolical Statics. 



(VI.) It is important to observe^ that the sign + between two 

 lines has quite a diiFereut signification from the same sign be- 

 tween two forces, u 4- u' repi'esents the change of position pro- 

 duced in a tracing point by causing it to describe the lines u and 

 u' in succession, while U + U' means the mechanical effect pro- 

 duced by the two forces U and U' acting simultaneously at the 

 same point, u + u' may be called a geometrical sum, and U + U' 

 a mechanical sum. It is easy to see that the parallelogram of 

 forces amounts to this, viz. that, if u and w' represent U and U' 

 respectively, u + u' will represent U -t- U'. Hence the parallelo- 

 gram of forces is a theorem which identifies the results of geo- 

 metrical and mechanical addition. This suggests the necessity 

 of caution in attempting to prove the parallelogram of forces 

 symbolically ; for it is clear that the proof must be vitiated by 

 any tacit assumption of the equivalence of geometrical and me- 

 chanical addition. If I mistake not, an error of this kind appears 

 to exist in some symbolical proofs which have been given of this 

 theorem. 



(VII.) To find the resultant of two parallel forces symbolically. 

 Let XA be one of the forces acting at 0, and X'A the other 

 acting at P, OP being denoted by u ; then the mechanical efi"ect 

 of the two forces is 



XA-h(l+M)X'A, 

 or 



{\+ ^^,uy^K+vA). 



Now this represents a force X A 4- X'A acting at the point 



X' 



-, u ; that is, a parallel force equal to the sum of the two 



X + X' 



forces acting at a point of the line u which divides it in the in- 

 verse ratio of X to X'. 



(VIII.) To find the resultant of any set of parallel forces symbo- 

 lically. 



Let XA, X'A, X"A, &c. be the parallel forces, and u, u', u", 

 &c. the distances of their respective points of application from 

 the origin ; then the mechanical efifect of these forces is 



(1 + m)XA -f (1 -f m')X'A -Kl -f- m")X"A -I- &c., 

 or 



Hence the resultant is a force SXA acting at the point whose 

 distance from the origin is 



^Xu 



"sx- 



