232] FREEDOM OF THE FJFTH ORDER. 251 



while they are also reciprocal to the cylindroid because they cross two screws 

 thereon with pitches equal in magnitude but opposite in sign. They are 

 therefore reciprocal to X. In like manner it can be shown that two of the 

 other system of generators possess the same property. 



On every cylindroid there is as we know ( 26) one screw of a given five- 

 system. This important proposition may be otherwise proved as follows. 

 Let 6 be the co-ordinates of a screw on the cylindroid, then these co-ordinates 

 must satisfy four linear equations. There must be a fifth equation in the 

 six quantities l} ... 6 6 inasmuch as 6 is to lie on the given five-system. 

 Thus from these five equations one set of values of 8 lt ... 6 6 can be 

 determined. 



On a quadratic two-system ( 224) there will always be two screws 

 belonging to any given five-system. For the quadratic two-system is the 

 surface whose screws satisfy four homogeneous equations of which three are 

 linear and one is quadratic. If another linear equation be added two 

 screws on the surface can, in general, be found which will satisfy that 

 equation. 



232. Impulsive Screws and Instantaneous Screws. 



We can determine the instantaneous screw corresponding to a given 

 impulsive screw in the case of freedom of the fifth order by geometrical 

 considerations. Let X, as before, represent the screw reciprocal to the freedom, 

 and let p be the instantaneous screw which would correspond to X as an 

 impulsive screw, if the body were perfectly free ; let 77 be the screw on which 

 the body receives an impulsive wrench, and let be the screw about which 

 the body would commence to twist in consequence of this impulse if it had 

 been perfectly free. 



The body when limited to the screw system of the fifth order will 

 commence to move as if it had been free, but had been acted upon by a 

 certain unknown wrench on X, together with the given wrench on rj. The 

 movement which the body actually acquires is a twisting motion about a 

 screw 6 which must lie on the cylindroid (, p). We therefore determine 6 

 to be that one screw on the known cylindroid (, p) which is reciprocal to the 

 given screw X. The twist velocity of the initial twisting motion about 6, as 

 well as the intensity of the impulsive wrench on the screw X produced by 

 the reaction of the constraints, are also determined by the same construction. 

 For by 17 the relative twist velocities about 6, , and p are known; but 

 since the impulsive intensity rj &quot; is known, the twist velocity about is 

 known ( 90) ; and therefore, the twist velocity about is known ; finally, 

 from the twist velocity about p, the impulsive intensity X &quot; is determined. 



