217] FREEDOM OF THE FOURTH ORDER. 227 



If X, /j,, v be the direction angles of 6 we have therefore ( 44) 



fi _ (p e a) cos X d ei sin X _ 

 c/ 2 (j 



a 



a (Pd - b) cos /u, - d e3 sin /i 

 4 - -g- :0, 



/, _ (P - c) cos v - d 65 sin v 



t/G = U. 



c 

 The direction cosines of the common perpendicular to 6 and 1 arc 



cos v cos //, 



sin X sin X 



whence the cosine of the angle between this perpendicular and the radius 

 vector to a point ac, y, z on is 



y cos v z cos /j, d 6l 

 r sin X r sin X r 



or d ei sin X = y cos v z cos /x. 



We have thus the three conditions 



(p e a)cos\ +z cos //, ?/ cos z/ = (i), 



2 cos X + (p 9 b) cos /i + cos v = (ii), 



+ ycosX # cos //, -f ( p Q c) cos v = (iii), 



whence eliminating cos X, cos //,, cos v we obtain 



( p e a) (p e b) (p e c) + (p 6 a)x 1 -\-(p e H)y- + (p e c)z&quot; = 0. 



Thus we deduce the equation otherwise obtained in 174, for the family of 

 pitch-hyperboloids on which are arranged according to their pitches the 

 several screws of the three-system. 



217. Principal pitches of the Reciprocal Cylindroid. 



From a system of the fourth order a system of canonical co-reciprocals 

 can in general be selected which possesses exceptional facilities for the investi 

 gation of the properties of the screws which form that four-system. 



Let OA and OB be the axes of the two principal screws of the reciprocal 

 cylindroid. Let a and b be the pitches of these two principal screws and 

 let c be any third linear magnitude. Let 00 be the axis of the cylindroid. 



Then the canonical co-reciprocal system now under consideration consists of 

 Two screws on OA with pitches + a and - a. 

 Two screws on OB with pitches + b and - b. 

 Two screws on 00 with pitches + c and c. 



152 



