178 



THE THEORY OF SCREWS. 



[176- 



We see from this that the sum of the reciprocals of the pitches of three 

 co-reciprocal screws is constant. This theorem will be subsequently 

 generalised. 



177. Locus of the feet of perpendiculars on the generators*. 



If p be the pitch of the screw of the three-system which makes angles 

 a, ft, 7 with the three principal screws, it is then easy to show that the 

 equation of the screw is 



(p a) cos a + z cos /3 y cos 7 = 0, 

 z cos a + (p b) cos ft + a; cos 7 = 0, 

 + y cos a x cos ft + (p c) cos 7 = 0. 



If perpendiculars be let fall from the origin on the several screws of the 

 system, then if x, y, z be the foot of one of the perpendiculars 



sc cos a + y cos ft + z cos 7 = 0. 



Eliminating cos a, cos /3, cos 7 from this equation and the two last of those 

 above, we have 



a; y z = 0, 



z p b x 



+ y x p c 



or (pV)(p-c)x + x (# 2 + y- + z 2 ) + yz (b - c) = ; 



from this and the two similar equations we have, by elimination of p 2 and p 

 and denoting x 2 + y z + z* by r 2 , 



x, (b + c) x, bcx + (b c) yz + xr* = ; 

 y, (c + a) y, cay + (c a) zx + yr n - 

 z, (a+b)z, abz + (a b) xy + zr 3 



multiplying the first column by r- and subtracting it from the last, we have 



x, (b + c) x, bcx + (b c) yz = 0, 

 y, (c + a) y, cay + (c a) zx 

 z, (a + b) z, abz + (a b) xy 

 which may be written 



(a - b) 2 #y + (b - cf y*z- + (c - a) 2 z-ac&quot; = (a -b)(b- c) (c - a) vyz. 



* This Article is due to Professor C. Joly, On the theory of linear vector functions,&quot; Transac 

 tions of the Royal Irish Academy, Vol. xxx. pp. 601 and 617 (1895), where a profound discussion 

 of Steiner s surface is given. See also by the same author Bishop Law s Mathematical Prize 

 Examination, Dublin University Examination Papers, 1898. 



