42 THE THEORY OF SCREWS. [47 



Introducing the values just given for eo this equation becomes 



- (a a + a 4 ) z = a fa - o 2 ) - p a fa + a,), 



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...a 6 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 X ; + b and b on an intersecting perpendicular axis Y, 

 and + c and c on the intersecting axis OZ which is perpendicular to both 

 OX and Y. 



Let x , 7/0, z be the co-ordinates of any point with reference to the 

 associated system of Cartesians. 



Draw through a system of rectangular axes O X , O Y , O Z parallel to 

 the original system OX, OF, 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 1} 0., ... # 6 be the co-ordinates of the screw o with regard to these 

 new screws of reference. It is required to find these quantities in terms of 



!, ... 6 . 



Let x, y , z be the current co-ordinates of a point on w referred to the 

 new axes, the co-ordinates of this point with respect to the old axes being 

 x, y, z, 



then x=x + x ; y = y + y - z = z + z&amp;lt; ) . 



The equations of &&amp;gt; with respect to the new axes are ( 43), 



(0, + 0) &amp;lt;/ - (&, + OJ z = &quot; (0i - 0.) - JP. (0i + 0,)) 



(0 1 + 0JS-(0 s + 0Jaf = b(0 &amp;gt; -0 4 )-p..(0 t + 0^ ......... (i). 



(0, + 4 ) x - (0, + 0,) y = c(0 5 - 6 ) - p M (0, + 6 }) 



We have also 



( 5 + 8 ) y ~ fa + 4&amp;gt; z = a (tfj - a,) - p u fa + 2 )j 



(! + ,) .2 -fa + a 6 )x = b(a 3 -a,)-p ta (a 3 +a 4 )[ ......... (ii). 



( 3 + ce 4 ) x - fa + a.,) y = c (a g - a fi ) - _p u (a 5 + or 6 )J 



Remembering that the new axes are parallel to the original axes we have 

 1 + 3 = a, + a a ; 3 + 4 = a 3 + a 4 ; 5 + 0, = a, + a ti ......... (iii). 



