219] FREEDOM OF THE FOURTH OKDEK. 229 



already seen that two of the roots of this must be infinite, whence this 

 equation reduces to a quadratic, and its roots are as we have seen equal 

 but opposite in sign to the pitches of the principal screws of the reciprocal 

 cylindroid. 



After a few reductions and replacing p e by p we obtain the following 

 equation 



P*(P!P* sin 2 (34) + p,p a sin 2 (24) + . . .) 



We thus deduce from any four co-reciprocal screws the quadratic equation 

 which gives the pitches of the two principal screws of the cylindroid to 

 which the given four-system is reciprocal. 



218. Equations to the screw in a four-system &amp;lt;/ 



The screws of the four-system are defined by the equations 



(p e + a) cos X + z cos yu, y cos v = 0, 

 z cos X + ( p e + b) cos fj, + x cos v = 0, 



where p g is the pitch where cos X, cos p, cos v are the direction cosines and 

 where oc, y, z is a point on the screw. By these equations the properties of 

 the various screws of the system can be easily investigated. 



If p e be eliminated we obtain 



x cos X cos v + y cos p cos v z (cos 2 X + cos 2 p,) (a 6) cos X cos //., 



whence we obtain for the equation to the cone of screws which belongs to the 

 four-system, and has its vertex at #, y 0) Z Q 



x (as - x,} (z - z,} + 7/0 (y - y ) (z - z,) 



- z, {(x - a? ) 3 + (y - y ) 2 } ~(a-b) (x - x ) (y - // ) = 0. 

 This is of course the cone which has been referred to in 123. 



219. Impulsive Screws and Instantaneous Screws. 



A body which is free to twist about all the screws of a screw system of 

 the fourth order receives an impulsive wrench on the screw 77, the impulsive 

 intensity being 77 &quot;. It is required to calculate the co-ordinates of the screw 

 6 about which the body will commence to twist, and also the initial re 

 actions of the constraints. 



Let X and /j, be any two screws on the reciprocal cylindroid, then the 

 impulsive reaction of the constraints may be considered to consist of 

 impulsive wrenches on X, p of respective intensities X &quot;, //,&quot; . If we adopt 



