■536 Proceedings of the Royal Irish Academy. 



"We may replace this by the three equations 

 Xi = Xo tan a, 

 Xz = Xi tan ji, 



Sin 2a = :; 7 s]n 2p. 



The pitch is then found to be expressed as follows : 

 a sin^/3 + h cos^yS 



P = 



cos'^a + ah sin' a 



An interesting test of this expression may be made ; for, if we 

 substitute for^, a, I, respectively, 



p + x a + X h + X 

 1+px' 1+ax' l+bx' 



we shall find that the formulae remains unaltered whatever x may be. 

 For example, if a; be infinite, we find 



sin^ a+ ah cos'^ B 



h sin*/3 + a cos^/S 



while we have also 



a sin*,^ + h cos^ (3 



cos* a + flJ sin* a^' 

 from which we can deduce that 



a-h . r> '' 



sin 2a = ± , 7 sin 2B. 



1 -ah 



By taking x small, we easily prove that p must always satisfy the 

 differential equation 



l-o» = (l-o*)^ + (l - h')db. 

 da 



These results are in form only a slight generalization on those of 

 the "Theory of Screws" in ordinary space; for we have, in both 

 cases, 



a sin*/3 + h cos-^ 



cos* a + ^sin*a 



If we make S= ah, we have the formulae of this paper ; if we make 

 JI= 1, we have the familiar law of pitch distribution on the cylin- 

 droid. 



