30 Proceedings of the Royal Irish Academy. 



We thus obtain the ^dl•tual coefficient of r\ and a by the following equation, 

 which is succeeded by the two similar equations for ?) and /3, and »; and y: — 



2z3,„ = {pa. +-1\) cos ((^, - 0,) - (s, - Za) sin ((/„ - «„ 



2.^,3 = {p^ f 2;,) cos (03 - e,) - (t, - 2p) sin («3 - 0,) • (8) 



^''w = (Fy + Pi) <^os (9y - 0,) - (~~,, - ~y) sin (ff^ - 0„) . 



As the three twists are to neutralize, we must have from (4) 



a'TO,a + /3'w,p + y'jn^ = 0, 



and by substituting in this tho values just found for 



"'r\a' "JilP- ''lY' 



we see that 



= + aXpa cos 0„ + c. sin 0„) cos ft,, 



+ /3'(i5p cos 6^ + Zfi sin 6^) cos tf,, 



+ >'( Py cos Oy + 2y sin tf^) cos 6,, (9) 



+ a'(jp„ sin 0„ - s„ cos 0.) sin 0„ 



+ /SXp/s sin 63 - >^ cos flp) sin 0,, 



+ -/'( Py sin 0^ - ^v cos Oy) sin 0,, 



+ (a' cos 0, + /3'cos 03 + 7'cos 0y) ( 2', cos 0, + s, sin 0,), 



+ («' sin 0, + /3'sin 0^ + y'sin 0^) (2:*, sin 0, - 3, cos 0,). 



This equation must be satisfied for all values of 2^,, for all values of 0,, and 

 for all values of r, . These conditions will be fulfilled if, but only if, the 

 following four equations are true: — 



+ a'( Pa cos 0. + 2a sin 0,), 



+ {i'ipfi cos 03 + Zfi sin 03), 



+ y'[ Py cos By + Zy SlU 0^) = 0. (10) 



+ a'( p. sin 0a - 2a cos Wa), 



+ ^'{p^ sin 03 - Zp cos 03), 



4 7'(/>y sin -y^ - 2^ cos 0^) = 0. (Uj 



a'cos 0. + /3'cos 03 + y'cos »y = 0. (12) 



a'sin 0a + /3'sin 63 + y'sin 0^ = 0. (13) 



The foiTuulffi (10), (11), (12), (13) express the necessary relations of the 

 positions and pitches of the three vector-screws, and provide a determination 

 of the amplitude-ratios of the neutralizing twists. 



From (12) and (13) we obtain 



a' : /S' : 7' : : sin (^3 - By) : sin (By - 6.) : sin (6, - ^3). (14) 



