32 Proceedings of the Royal Irish Academy. 



and thus we have the following remarkable expression for the pitch of the 

 screw corresponding to 6, viz. 



^ = cos' Op^ + sin' Opp. (22) 



This is of course an elementary result in the Theory of Screws * but tliis 

 method of obtaining it from the virtual coellicient has not been given in the 

 pre\'ious papers. 



We can introduce much simplification into the formula (10) and (11) 

 by taking as origin the point wliich is so obviousl}' suggested by being 

 the intei-section of the two screws of stationary pitcli. We then have 

 2a = 2/3 = 0. and as 5. = and 6^ = 90°, we find (14) 

 a ■.^' -.i : : COS ^ : sin Q : - \. 

 Thus formulie (10) and (11) become respectively 

 p^ cos d - p cos 6 - z sin = 0, 

 j)f^ sin Q - p sin Q ^ z cos 5 = 0, 

 whence c = (/>„ - p^ sin cos 0. (23) 



If the line wliich all llio screws intei-sect is the axis of z, the surface on 

 which all tiie screws lie, so well known as llie cyliuihnid,! has as its e(iuation 



2 (a,-* 4 y') = (;>, - p^) xy. (24) 



This is perhaps the most satisfactory method i4 investigating the 

 equation of the cyliudroid so far as the Theory of Screws is concerned. 

 In the deduction of the equation of the surface previously giveuj I assumed 

 tliat tlic two principal screws of the cyliudroid intei-sected at light angles. 

 No doubt the legality of this assumption was subsequently justified, but the 

 method here followed seems not open to objection. 



As a further illustration of the formuhe connected with the virtual 

 coefficients, we may prove tiie following tiieorem : — 



If in a 2-sy8tem ;j,, p,, y, are the pitches of three vector-screws 1, 2, 3, 

 which make right-handed angles fl,, 0,, 0, respectively with a standard vector- 

 screw also perpendicular to the axis of the 2-system, show that we have the 

 following three equations : — 



px sin (0, - e.) + a,, sin (fl, - 0,) + ra„ sin (0, - fl.) = 0, 



w„ sin (0, - 0,) + p, sin (0, - «,) + S3„ sin (0, - 0,) = 0, . (25) 



Wis sin (0, - 0,) + -On sin (0, - 0,) + p^ sin (0, - 0,) = 0, 



* "Trcilise," p. 19. 



tThc relation of this surface to the Theory of Screws wm first given in Trans. K.I. A., 

 vol. XXV., p. 161 (I87I). The discovery of the siirraco is, himevcr, due to Hamilton (1830) : see 

 "Treatise," pp. 510-11. 



I " Treatise." p. 10. 



