Prof. O'Brien on Symbolical Mechanics. 123 



there will no longer be any occasion to distinguish between units 

 of force and units of length. I shall employ the Greek letters 

 a, ^, and 7 to represent both,, and substitute them everywhere ^ 

 in place of A, B and C. Hence^ instead of the relations aA=0^. 

 &c., ^A= — aB, &c., we have the following, viz. ;,^- 



/5a=— aft 7/3=— /37, uy=—yu.J 



And generally, supposing U and V to be forces parallel and 

 equal to the lines u and v respectively, I shall substitute the 

 latter for the former. Now, since ?;U= — wV*, this substitution 

 leads to the important result, that , 



vu=s--uv; (2*)i> 



that is, the factors in the symbolical product uv may be inter- 

 changed if we change the sign of the product. ' ' 

 If w and V be parallel, uV=0', hence we have another result' 

 of importance, namely, that the condition of parallelism of two 

 lines u and v is 



uv = . (3.) 



It will be remembered that uv denotes the effect produced by 

 the translation of the line v along the line u, that is, by the 

 parallel removal of the force represented by v from A to B, AB 

 being the line u. It will also be remembered that A and B are 

 supposed to be points in the same rigid body, and that uv is, in 

 fact, the couple consisting of the forces —v and v acting at A 

 and B respectively. 



All that is here said respecting forces applies equally well to' 

 impressed velocities. 



Units of Translation, — Representation by perpendicular lines. • 



If 6 be the angle between u and v, cc and y8 two units drawiir 

 at right angles to each other anywhere in the plane of u and v ; 

 and if x and y denote the numerical magnitudes of u and v ; then 

 it may be shown, that %avM} }i baR 



uv = [xy sin 6) up. • ' -• ■ ^^ • " «<-' . ^ T 



For, let <^ and -^Ir be the angles which u and v respectively 

 make with the direction a ; then, by geometrical addition, 



u={x cos (f>)u + [x sin </>)/3 

 ?; = (y cos '>^)cx, + {y sin '^)^. ! ' 



Hence, observing that uu^^^^O, and /5a = —aft we find 



uv=xy(cos (j> sin '\/r— sin </> cos yjr)oi^ n .(|t|^ ji 



= (^ysin^)aft ^Hibwgji 



* For t?U+wV=X'O«+«6p)=0. Ilidgli. 



