42 



Proceedings of the Royal Irish Academy. 



If we uow substitute for the virtual coeflicients 



2^,3 = {J>. ^ Pi) cos (03 - f^O - f?23 siu [0, - 6,), 

 2ir3, = (pi + p,) COS (0, - fts) - ^31 sin {d, - e,), 

 2t-,2 = (2h +Pi) cos (0o - 0,) - (^12 sin (9. - 0i), 



where attention should be paid to the right measurement of the angles as 

 explained in connexion with fig. 6 ; it will be seen that the terms involving 

 Pi, Pi, Pt disappear ; and assuming that no two of the screws coincide, 



the equation reduces to 



f7,3 + f/,3 = (7,3. (36) 



This of course proves no more than the well-known property of the 2-system 

 that the common perpendicular to (1) and (3) also intersects (2). 



It is instructive to observe how the original formula (31), taken 

 in conjunction with the pitch-operator, gives at once the fundamental 

 characteristics of the cylindroid. 



Let (1) and (2) be the two principal screws of the cylindroid, then toi2 = 

 and also Aw,, = cos (12) = 0. Thus the formulae (31) and (32) become 



PilhPa - Pi -^""a - pi ^\> = 0, 



;>,/>, . 2>j/'i + 2''P' ~ =''31 - ^''» - '2piV3iCOsd - 2;j,irjj8in(? = 0, 



where 360° - 6 is the right-handed angle between (1) and (3). 



We have 



2xj, = (^jj + 2)i) cos + d sin 



-r„ = (j), + Pj) sin - d cos ; 

 substituting these values in (37) we obtain 



(Pi - Ih cos' - 2h sin' 0)' ■¥ \d - (p, - /?,) sin cos 0}' = 0, 

 whence the well-known fundamental equations 



(37) 



d = (j), - 2h) sin B cos ti, 

 Pi = pi cos' + />, sin' 0. 



"■} 



(38) 



In like manner if four screws belong to a S-system, we must have 



U 



/'I S^l; 



■r-3, ""jj 2'» 



M 



II 



II 



(39) 



as a general relation to be satisfied by the pitches and the virtual coefficients. 



If a screw belongs to a 3-system, it must fulfil three conditions; for 



example, it must be reciprocal to three screws of the reciprocal system. 



