338 THE THEORY OF SCREWS. [312- 



312. Freedom of the First or Second Order. 



If the rigid body have only a single degree of freedom, then the only 

 movements of which it is capable are those of twisting to and fro on a 

 single screw a. If the impulsive wrench 77 which acted upon the body 

 happened to be reciprocal to a, then no movement would result. The forces 

 would be destroyed by the reactions of the constraints. In general, of course, 

 the impulsive screw 77 will not be reciprocal to a. A twisting motion about 

 a will therefore be the result. All that can be said of the instantaneous screw 

 is that it can be no possible screw but a. 



In the next case the body has two degrees of freedom which, as usual, we 

 consider to be of the most general type. It is required to obtain a con 

 struction for the instantaneous screw a about which a body will commence 

 to twist in consequence of an impulsive wrench 77. 



The peculiarity of the problem when the notion of constraint is introduced 

 depends on the circumstance that, though the impulsive screw may be 

 situated anywhere and be of any pitch, yet that as the body is restrained to 

 only two degrees of freedom, it can only move by twisting about one of the 

 screws on a certain cylindroid. We are, therefore, to search for the in 

 stantaneous screw on the cylindroid expressing the freedom. 



Let A be the given cylindroid. Let B be the system of screws of the 

 fourth order reciprocal to that cylindroid. If the body had been free it would 

 have been possible to determine, in the manner explained in the last section, 

 the impulsive screw corresponding to each screw on the cylindroid A. Let 

 us suppose that these impulsive screws are constructed. They will all lie on 

 a cylindroid which we denote as P. In fact, if any two of such screws had 

 been found, P would of course have been denned by drawing the cylindroid 

 through those two screws. 



Let Q be the system of screws of the fourth order which is reciprocal to 

 P. Select from Q the system of the third order Q t which is reciprocal to 77. 

 We can then find one screw ^ which is reciprocal to the system of the fifth 

 order formed from A and Qj. It is plain that -^ must belong to B, as this 

 contains every screw reciprocal to A. 



Take also the one screw on the cylindroid A which is reciprocal to 77, and 

 find the one screw rj 2 on the cylindroid P which is reciprocal to this screw 

 on A. 



Since 77, 77! and ?7 2 are all reciprocal to the system of the fourth order 

 formed by A l and Q 1} it follows that 77, 77^ and 773 must all lie on the same 

 cylindroid. We can therefore resolve the original wrench on 77 into two com 

 ponent wrenches on y^ and 772. 



