70 Proceedhif/s of the Roijal Irish Acaa<o/i;/. 



vector to any point a on the axis of the screiv [/j.i, A]). The invariant 

 reduces at once to 



8\aXiYX.X^+ 8Y(SXo\X,X, + SYyA,TAA2 



g^-^-^- +Pi^P2^pz = a^o^o; 



and on expansion of the terms in the numerator it becomes 



S(iS-y)\iSX,A.3 + S(y- a)X,SX,X, + S(a -/3)X,^X,X, 



Let A, B, c be any points on the axis of three screws (the extre- 

 mities of the vectors a, /S, and y) ; let BiCi denote the projection of the 

 line BC on the axis of the screw {fii, Aj) ; let (23) be the angle between 

 the axes of the screws (/x,, X2) and {1x3, A3) ; and let sin (123) denote 

 the sine of the solid angle determined by lines parallel to the three 

 axes (rotation fi-om Aj to A2 to A3 being supposed positive) ; then the 

 invariant is 



B,c, cos (23) + C0A2 cos (31) -r A3B3 cos (12) 



^^^1^3^ +P^^P^ + P^ = ^^i^o. 



In particular if three axes intersect, and if the three points are taken 

 to be at the point of intersection, 



Pi + P2 + p3 = a + i + c ; 



while for any points on the axes, 



BiCi cos (23) + C2A2 cos (31) + A3B3 cos (12) = ; 



and from this single theorem many other consequences may be de- 

 duced. 



