A Treatise on the Theory of Screws

381] THE THEORY OF PERMANENT SCREWS. 417 screw. We ought, therefore, to find that the expressions for the co-ordinates of 7? remained unaltered if we substituted for ... 0e, the co-ordinates of any other screw on the same straight line as 6. These are (§ 47) 01 + -(01 + 02), 02-^(01 + 0a), d d 03 + y(03+^ 04-y(03+04), 05 + ^(05 + 06)) 06-^(06 + 06), c v where H is arbitrary. Introducing these into the values for i?/', it becomes / 27f \ 7 / 21/ \ ,, . — «c&>2 f ps + — ®3 I + ab<i)3 ( p2 4—a>2 J + (P" — C“) m2ms, from which H disappears, and the required result is proved. The restraining screw is always reciprocal to the instantaneous screw, and, consequently, if e be the angle between the two screws, and d their distance apart, (Pi + Pe) cos e — sin e = 0. We have seen that this must be true for every value of pe, whence cos e — 0 ; d = 0 ; i.e. the two screws must intersect at right angles as we have otherwise shown in § 37 6. This also appears from the formulae rjx' + 1/2" = 2bp2co3 — 2cp3a>2 > Vs" + v” = 2cp3a>! - 2apxa>3, Vs" + v" ~ ^Pi^i ~ multiplying respectively by &>!, w2, a>3, and adding, we get (''Zi + V2) (A + ^2) + (% + ’h) + "*■ (^5 + ~ 0’ which proves that ?? and 6 are rectangular; but we already know that they are reciprocal, and therefore they intersect at right angles. 381. A Particular Case. The expressions for the restraining wrenches can be illustrated by taking as a particular case an instantaneous screw which passes through the centre of gravity. b.