50 



Proceedings of the Royal Irish Academy. 



A pail- of screws (^i, Ai) and {in, Xj) will of course completely determine 

 the cylindi'oid which passes through theni. "We now 

 propose to determine p the vector from the origin to 

 C, the centre of the cylindroid. 



Let i, j be vectors along the two principal screws 

 of the cylindroid through C Let a, b be the pitches 

 of these screws, and pi, 5, the intensities of the two 

 wrenches upon them which are equivalent to A,, ni. 



We liave now to express that the force Ai at 0, 

 and the couple //, are equivalent to wrenches of inten- 

 sities 2h and q, respectively on i and j. 



Draw CL (fig. 15) equal, parallel, and in the same 



direction as OA, and di-aw CM equal and opposite to 



CL. Then OA is equivalent to CZ, and the couple whose vector is VX^p. 



We thus have 



IXi + V\,p = ;7,rti + Qibj. (68) 



X,=p,i + q,j. (69) 



If in like manner wrenches of intensities p^, qi on the two principal screws of 

 the cylindroid are equivalent to /i,Xj, we must have 



/i, + VX2P =p,cii + qjjj. (70) 



A, = /j,i + y,/. (71) 



From (69) and (71) wo see that r,/, A,,Aj are coplanar; whence multiplying 

 (68) and (70) by F"AiXa, and taking the scalars, we have 



SkiX^i + ,S'( TA.A, . A, . /)) = 0. (72) 



.S^.X,^, + S{ V\X .\,.p) = 0. (73) 



A third equation is obtained thus. By multiplying (68) and (71) 



A,;i, + AjFAip = {p,i + q-,j) {p,ai + qfij), 



whence taking scalars 



SkiH\ - SXi\ip = - apxjH - hqiqt. (74) 



By multiplying (69) and (70) 



A, //J 4^ X, FAap = (;)ii' + q\j) {pifii + qjyj) 

 whence as before 



S\\)ii + S\i\tp = - ap^p, - hqiqi; (75) 



and by subtracting (75) from (74), 



i (-SAvi, - <SA,^,) = Sk\'p. (76) 



