8 J. LOUDON ON 



sum of the moments rouiicl C eAddeutly vauishes. Hence the sum also vanishes for 

 the original forces when each is turned through any angle. 



Eetaiuing C as origin, the sum of the moments G of the given forces round the 

 point P{a, b) is evidently bR, whilst the virial V is — aR. If therefore each force be 

 turned through an angle C, these quantities become 



G' = Rb' — R[h cos 6 — a sin 0) = V «n 0+ G cos d, 

 V — —Ra' = —R(a cos d+h siu 6) — V cos 6— G sin 6, 



where {a', b',) are the coordinates of P referred to the lines of zero moments and virial 

 for the new forces. 



Hence also (?'-+ V" = G'+ V- R%(r + b') varies as CP\ 



II. 



When any set of forces are reduced to a single force R ai and a couple G, they 

 may, as is well known, be still further reduced to two forces acting along Hues which 

 are perpendicular to each other. This reduction can be readily effected by referring jR 

 and G to ordinary polar coordinates, as in the accompanying figure, and by resolving 

 the force and couple into two forces and two couples in such a way that each component 

 force shall be perpendicular to the axis of a component couple. 



Thus R at O and G are equivalent to R cos cp, R siu cp, at 0, and the couples 



Jj z= G sin d cos qi, L' = \/cos- é'+sia'' 6 sin- tpi 



the axes of which are perpendicular to the forces. 



Now R sin cp and L are equivalent to R sin ç), parallel to Oi/, at a distance from LOi/, 



equal to 



L G sin 



R sin <p R tan tp 



whilst R cos (p and L' are equivalent to R cos (p, parallel to Ox, at a distance from L'xO, 



equal to 



G am 8 



Rcot cp 

 Hence the distance between these reciprocal lines of force is 



G siu H 2G sin 



(cot ç/+tan qj) = . 



R R sin 2(p 



