A Treatise on the Theory of Screws

_______________ ______ 40 THE THEORY OF SCREWS. | 44- If pa be indefinitely great with respect to a and dal, then _7>«cosf(ai) joacos(ai) “i-----------1 a., = — - „ - —-; 2a 2a pa cos (as). pa cos (as) 2b ’ tti~~ 2b ; _______ a, = cos ■ a, = _P«co^(a5) If the co-ordinates of a screw not itself at infinity satisfy a, + a2 = 0; a3 + a4 = 0; a5 + aB = 0; then we must have Pa= OC for the equations a _ (£« + a) cos (ai) — dai sin (ai) 1 ' 2a ’ b, = ~ a) cos (M1) ~ sin (gl) — 2a ’ and two similar pairs could not be otherwise satisfied. We are not however entitled to assume the converse, i.e. that if the pitch is infinite then the three equations a, + a., = 0, &c. must be satisfied. It will however be true that —1 = — 1; -=-l; ^ = -1 a= «4 ’ a„ but some at least of the co-ordinates being infinite, we are in general prevented from replacing these equations by the ordinary linear form. 45. Indeterminate Screw. It may however be instructive to investigate otherwise the circumstances nt a sciew a possessing the pi’operty that its six co-ordinates an a___a« are submitted to the three conditions + a2 = 0; a3 + a4 = 0; a5 + «„ = 0. Two distinct cases must be considered. Either the screw a must have some finite points, or it must lie altogether at infinity. The first alternative is now supposed. The second will be discussed in the next article. If there be any finite points on a then for such points the three left- hand members of the equations in § 43 are all zero. The three right-hand members must also reduce to zero. The only way in which this can be accomplished (for we need not consider the case in which all the co-ordinates are zero) is by making pa infinite.