A Treatise on the Theory of Screws

42 THE THEORY Öl’ SCREWS. [47- Introducing the values just given for to this equation becomes («5 + O y - (a3 + a4) z = a (a, - a3) - + a2), as of course it ought to do, for the pitch is immaterial when the question is only as to the situation of the screw. 48. Transformation of Screw-co-ordinates. Let «i...a6 be the co-ordinates of a screw which we shall call to, with reference to a canonical system of screws of reference with pitches + a and — a on an axis OX; + b and — b on an intersecting perpendicular axis OY, and + c and - c on the intersecting axis OZ which is perpendicular to both OX and OY. Let x„, y„, zn be the co-ordinates of any point O' with reference to the associated system of Cartesians. Draw through O' a system of rectangular axes O'X', O'Y', O'Z' parallel to the original system OX, 0 Y, OZ. Let a new system of canonical screws of reference be arranged with pitches + a and - a on O'X', + b and - b on O'Y', and + c and - c on O'Z’. Let 03, 0.,... 06 be the co-ordinates of the screw to with regard to these new screws of reference. It is required to tind these quantities in terms of an ... «6, Let x, y, z’ be the current co-ordinates of a point on to referred to the new axes, the co-ordinates of this point with respect to the old axes being y, z, then » = »' + «„; y=y' + y9-, z = z' + z„. The equations of co with respect to the new axes are (§ 43), ( ^5 + 67) y' —(0.1+ 07) Y — a (0, — 0.7) — (0! + 0j) j ( 01 + 03)Y - (0S + ØJ x = b(03-0i)-pa(0i + 0i)- ..... (i). ( ^3 + 67) x — (0i + 0-7) y' = c (05- 07) - pa (0,. + øe)i We have also ( a5 + a6) y - (a3 + a4) z = a (a^ - a2) -jpw (a, + a2)) (otj + a.,) z — (a5 + a6) x = b (a3 — a,) — pm (a3 + a4); . (ii). (a3 + a4) x - (a, + a3) y = c (a5 - a6) - pu (a5 + a6)J Remembering that the new axes are parallel to the original axes we have — “i + a3; #3 + 04 = a3 + «4; Ö5 + ö6 = aä + a6.(iii).