Ball — Contrihitions to the Theory of Scretos. 63 



equal to the distance botweeu their rellectioiis, a rij^ht-aiij^led triangle will 

 rellcct into a right-angled triangle, and accordingly, the right angle haa not 

 been altered by reflection. 



We may see this otherwise by observing that, as siiown in the paper 

 already referred to, any two reciprocal screws reflect into reciprocal screws. 

 But two screws intersecting at right angles are reciprocal whatever be their 

 pitches. 



It is hence plain that, if the origin lies in the plane of reflection, the 

 perpendicular from the origin on a screw will reflect into the perpendicular 

 from the origin on the reflected screw. 



We know that F/a'/A', F/i/A are respectively the perpendicular vectors 

 from the origin on the reflected screw and the original screw ; and accordingly 

 from the well-known quatei'uion relation of a vector and its reflection where 

 i is the unit vector 



A A 



= - VifiX-H-' = - Vifdi-^X-H-' = - f!^% 



As the pitch of a screw is equal and opposite to the pitch of its reflection, 



S^, = - S^ = -ST'/xA-' = &>X-i 

 A A 



whence 



= - SifiiiX-H = - Sifirn\-H-' = -,5^; 



1X1 



AAA, *A« ^A^ %Xi 



but A' = iXi, ) 



and consequently t^' = - '>fii- ' 



It is easy to verify that the virtual coeflicient of a pair of screws (/i,. A,) 

 and (/ij, X>) is equal in magnitude but opposite in sign to the virtual 

 coefficient of their reflections. 



The virtual coefficient of the two screws is ^6'(^,Ao + /(.A,), and the virtual 

 coeflicient of their two reflections is 



- ^S{inin\ni + ip,iiiXii) = - ^S(i.ti\i i /xoA,). 



The screw whose equation is 



p = F ^ + .rX 

 A 



pierces the plane Spi = at the point whose vector is 



^ A -S-Ai ' 



