Ball — Contributions to the Theorij of Screws. 51 



In the fundamental quaternion formula 



pSafiy = aSpjiy + jiSapy I ySaft/i, 

 we now write 



a = A„ ft = A=, J = KA.X,, 

 and 



pSkxX^VXiX, = \,S (pX.VXiXi) + XiSihpVXA.) + VX,\,SXXp, 



whence from (72), (73), (76) we obtain 



P (FAiX,)^ = A,SA.A,ju= - Aj.SA.A:^. + h {8X2/11 - SX^ft^) FA.X, ; (77) 



and thus p, the required vector from the origin to the centre of the cylindroid, 



has been determined. 



In the deduction of the equations (72) and (73) it wiU be noticed that no 

 use has been made of the fact that the screws on i and / are at right angles. 

 So far as these two equations are concerned, p might be the vector to any 

 point of intersection of two screws, i.e. to any point on the axis of the 

 cylindroid. 



If, therefore, t he a, variable scalar, 



p iVXiX,y- = A,.S'A,A.M2 - X.S\,X,ix, + tVXX. (78) 



is the equation to the axis of the cylindroid defined by (fiiXi) and (/UjA;), and 

 the origin will lie on the axis if SXX.Hi = and SXX/i, = 0. 



To complete the account of the cylindroid defined by (niAi) and {112X2) it 

 remains to find the values of the pitches of the principal screws. These are 

 obtained as follows : — 



If X be a variable scalar, a screw on the cylindroid wiU be represented 

 by {/ii + Xfii), (Ai + xXi), and its pitch ^j will be S (jii + .r/uj) (Ai + .cAj)"' ; 

 from which we easily find 



_ SfiiXi + X (SfijXi + SjuLiXi) + x-Sfi^Xt 

 ^~ Ar + 2a;^,A, + ,7;%/ ' ^'^' 



but the pitches of the two principal screws are a maximum and a minimum, 

 and accordingly we find for 2' the two values j)^ + m and p^ - m, where 



Po = mkxy ^ ^^'^' ^'^'"'^' ^ '^'"''^'^ " xrSfiX - x^SfiX), (so) 



™' = ,^ .,.,Vx ^ M {SXX (Ar5>,A= + A:^5;u,A,) - Ai%'-(S^iA, + 8fiX)y- 



■iAi'Au" ( y AiAjJ 



- iXrX.^iVXM ^'^^^"^^""^' ~ ^■''^''=^')'- (^1) 



The length of the a.xis of the cylindroid is 2m,; and the condition that 



the cylindroid shall be canonical, i.e. that the screws of zero-pitch shall be 



the bounding-screws of the surface, is found by equating the value just found 



for Pf to zero. 



