A Treatise on the Theory of Screws

304 THE THEORY OF SCREWS. [284- Multiplying by pp^, ...pea6, respectively, and adding, we have ««aa = y'"^, where uaa = pfa* + ... + pli2a62-, because, as p belongs to the reciprocal system, we must have Wpa = 0. Similarly, if we multiplied the six equations by pißlt... pA, respectively, and added, we should get, since p is reciprocal to ß also, ^■^aß R ^rjß> where uaß = pf CC& + ...+pe2 aeß6. Eliminating å and y'" we have the concise result*, Uaa/tt ß-q = Wa(g'STa1). In a similar way we can deal with the pair of screws, ß and £, and by eliminating a, the reaction of the constraints in this case, we obtain the result ’Ußß'STa.^ = Ußn^ßi- Finally, from these two equations we can eliminate uaß, and we obtain __ ■5r^'arj3J ^ßß This formula is a perfectly general relation, connecting any two pairs of impulsive and instantaneous screws p, a and £, ß. It holds whether the body be free or constrained in any way whatever. If the body be perfectly free, then it is easy to show that it reduces to the result already found, viz. P°- rT _ Pß cos (a?/) cos (30 285. Another Proof. From the theory of impulsive and instantaneous screws in an n-system we know (§ 97) that if a1;... an be the co-ordinates of an instantaneous screw, then the co-ordinates ylt ... pn of the reduced impulsive screw may be deter- mined as follows: Pl tt R'n MVn= — <*n- Pn Multiplying severally by p^, ... pnan> and adding, we have ^aa • * Trans. Roy. Irish Acad., Vol. xxx. p. 573 (1894).