240 THE THEORY OF SCREWS. [226, 



screw of the enclosing (n + l)-system whose co-ordinates are proportional 



to 



1 



P\ rj p n+1 rj n+1 



This we shall term the polar screw of rj with respect to the quadratic 

 w-system. It is supposed, of course, that the screws of reference are co- 

 reciprocals. 



If CL and ft be two screws of an enclosing (n + 1 )-system, and if i) and 

 be their respective polar screws with reference to a quadratic n-system, then 

 when a is reciprocal to % we shall have /3 reciprocal to 77. For we have, where 

 h is a common factor, 



1 dU a 1 dU a 



htji = -- r~ &amp;gt; ir )n+i i j 



P! da, p n+l da n+l 



whence 



h(p 1 1J 1 /3 1 + ... + Pn+i Vn+i Pn+l) = Uop- 



If therefore $ and rj are reciprocal the left-hand member of this equation is 

 zero and so must the right-hand member be zero. But the symmetry shows 

 that and a are in this case also reciprocal. We may in such a case regard 

 a and ft as two conjugate screws of the quadratic ?i-system. 



As a first illustration of the relation between a screw and its polar, we 

 shall take for U a = 0, the form 



Pitf + p*&amp;lt;tf + . . . + p 6 a&amp;lt;? - p (i 2 + 2 2 + . . . + 6 - + 2 1 2 cos (12) . . . ) = 0. 

 This means of course that U a = denotes every screw which has the pitch p. 



Take any screw a and draw a cylindroid through a. The two screws of 

 pitch p on this cylindroid belong to U and a fourth screw 9 may be taken 

 on this cylindroid so that a, 6, and the two screws of pitch p form an 

 harmonic pencil. 



By drawing another cylindroid through a. another screw of the 0-system 

 can be similarly constructed. If these five cylindroids be drawn through 

 we can construct five different screws of the ^-system. To these one screw 

 will be reciprocal, and this is the polar of a. We have thus the means of 

 constructing the polar of a. 



Seeing however that U a = includes nothing more or less than all the 

 p-pitch screws in the universe and that in the construction just given for 

 the polar of a there has been no reference to the screws of reference, sym 

 metry requires that the polar of a must be a screw which though different 

 from a must be symmetrically placed with reference thereto. The only 

 method of securing this is for the polar of a with respect to this particular 

 function to lie on the same straight line as a. 



