184] 



FREEDOM OF THE THIRD ORDER. 



185 



!,...; ft l} ... /3; 71, ... 7 6 . Then if X, p, v be three variable parameters, 

 the co-ordinates of the other screws of the three-system will be 



Xotj + yttySj + vji , Xa. 2 + /i$j + v%, ... Xa 6 + /Jift 6 + i&amp;gt;7 6 . 



We shall denote the pitch of this screw by p, and from 43 we have for the 

 equations of this screw with reference to the associated Cartesian axes : 



= + (Xa 5 + pfa + vy, + Xa 6 + fj,j3 9 + vy 6 ) y 



(X 3 + fj,ft s + vj 3 + X 4 -1- pfti + v%) z 



(Xj + jjifii + vyi X 2 /A/3 a 1/72) a 



with two similar equations. 



From these we eliminate X, /*, v and the determinant thus arising admits 

 of an important reduction. 



To effect this we multiply it by the determinant 



4 , 



. 



I 7i + 7a . 73 + 74 

 For brevity we introduce the following notation : 



P = x [(ft, + &) (73 + 74) - 08, + A) (75 + 7.)] 



+ y [(& + &) ( 75 + 7) - (& + ft) (71 + 7.)] 



+ z [(& + &) ( 7l + 7.) - (^ + /3 2 ) (7, + 74 )] , 

 with similar values for Q and R by cyclical interchange. 

 We also make 



L aft = a(a 1 + a ) (A - &) + 6 ( 3 + 4 ) (/3 3 - /9 4 ) + c (a, + 6 ) (/3 5 - &), 

 ^ = a (A + A) (i - a,) + & (& + /3 4 ) (a 3 - 4 ) + c (/3 5 + &) (a 8 - 6 ), 

 with similar values for Z ay , Z ya , L fty , L^ by cyclical interchange. 



The equation to the family of pitch quadrics is then easily seen to be 

 0= 



If the three given screws a, ft, 7 had been co-reciprocal, then as 



L a p + Lp a = 2-57 a = 0, 

 it follows that L af} and L fta only differ in sign, so that if 



