Ball — Contributions to the Theory of Screws. 



35 







D 





X - 



\ 





\ 



c 







|0 1 

 1 J 



\ 



s 





\ 



M 1 ^' ~-, 



Fio, 9. 



It is instructive to prove tlio same thooreius geometrically as follows : — 



Let AA', BE', CC, DD' (fig. 9) be a cube of which is the centre, 

 and draw LM through ami || AA', BE', 

 CC, DD'. Draw XY through and || CB', 

 C'B, AD', A'D. Let AB and CD be the 

 bounding screws of the cylindroid, with 

 centre at and axis LM. We shall sup- 

 pose that the cylindroid has been made 

 canonical — i.e., that the pitches of the 

 bounding-screws are both .zero, and that a' 

 the pitches of the two principal screws are 

 equal in magnitude and opposite in sign. 

 Any cylindroid can of course be made can- 

 onical without any other alteration than the 

 addition of a certain magnitude, positive or negative, to the pitches of all the 

 screws it contains.'* 



From the fundamental property of the cylindroid we see that a twist 

 about any screw on the canonical cylindroid can be resolved into rotations 

 about AB and CD. A rotation about AB does not alter the position of any 

 point on AB, and consequently the effect of any twist on the cylindroid 

 upon the point L will be the same as if the twist were produced merely 

 by a rotation about CD. But remembering that the amplitude of the twist 

 is a small quantity, this is the same so far as L is concerned as a displace- 

 ment of L along AB. In like manner it is shown that the effect on M 

 produced by a twist about any screw on the canonical cylindroid can never 

 be anything but a displacement of M along CD. 



Consider now the effect on the point L produced by the combination of 

 a right-handed rotation round the vector XY, with a translation parallel 

 to XY. If a be the amplitude of the twist, and p^ be the pitch of the 

 screw on XY, and if m be the semiaxis of the cylindroid and equal to OL, 

 then LL. = ma is the distance through which L is moved by the rotation. 

 As, however, the total effect of the twist on Z is to move B along AB, we 

 see that the movement LL2 on one side of AB must be compensated by 

 another displacement to the other side of AB. This is of course ZLi = a'p^, 

 and consequently foi' a twist about the vector-screw a, the translation ZZ, 

 must be that of a right-handed screw — i.e., p„ is positive. 



But AB and XV form a left-handed pair, as do also XY and CD, and 

 thus the desired theorem has been proved. 



" "Treatise," p. 47. 



