0/ 



> Y 



Ball — Con fribtif ions to the Theory of Screws. 3& 



locus of the screws of a 2-system, and the law of distributiou of tlie pitch in 

 the following simple maiiuer : — 



Let OX, F be vectors along the two rectangular A 



screws intersecting at with pitches p\ and pa i;espec- / 



tively (p, > p.). / 



Let OA be the projection in the plane of XY of / 



the vector-screw a, which intersects the axis OZ /" 



normal to the plane of the paper. As OX has the '. 

 greatest pitch ^j,, it must, as just shown, form a left- 

 handed pair with a ; and consequently a is ahovc the ^l 

 plane of the paper at the distance z. 



Let us imagine another screw on OX to which o" \^ 



the pitch - pi is attributed. Then this is reciprocal Fia. 12. 



to the screw on OX with pitch + p^ ; and it is also 



reciprocal to the screw on OY, inasmuch as OJTintersects OF at right angles. 

 Thus the screw of pitch - pi on OX must be reciprocal to a ; because, when- 

 ever a screw is reciprocal to two screws on a cylindroid, it is reciprocal to 

 every screw. 



Observing that a is above the plane of the paper, the right-handed angle 

 between OX and a is 360° - Q ; and hence we have, from the condition of 

 reciprocity, 



(jy - p,) cos (360° -B) -z sin (360° - 0) = 0. 



In lilve manner, observing that a must be reciprocal to a screw of pitch -pi 

 lying on OY, and that 90° - is the right-handed angle between a and OY, 



(p - ih) cos (90° -Q)-z sin (90° - 0) = 0. 



But if X, y, z be the coordinates of a point on a, we have tan = yjx. 



Wlience 



a; (2? - pC) + yz = 0, y Qj - 2h) -xz = 0; 



and, eliminating jj, 



z(x^ + y-) = (pi -2h)xy 



is the equation of the cylindroid ; and 



2^ = 2h cos'6 + 2h sin'-0. 



The figure is drawn so as to keep in view the suggestive measurement 

 of d by the watch dial, OX points to XII, and OY to III. As p^ > p,, 

 we see that when x and y have the same signs z is positive. Thus the 

 surface rises above the paper at OJ^ to meet the paper again in Y. — 



R. I. A. PROC, VOL. XXVIir., SECT. A. [6] 



