383] THE THEORY OF PERMANENT SCREWS. 419 



will not depart from the original instantaneous screw except by an accelera 

 tion. This must be on a screw which stands to the restraining screw in the 

 relation of instantaneous to impulsive, but in the case supposed these two 

 screws are reciprocal, therefore they cannot be so related, and therefore there 

 is no acceleration. 



There is little to be said as to the restraining wrench when the freedom 

 is of the first order. Of course, in this case, as every movement of the body 

 can only be a twist about the screw which prescribes its freedom, the 

 restraining wrench is provided by the reactions of the constraints. It is 

 only where the body has liberty to abandon its original instantaneous screw 

 that the theory of the restraining wrench becomes significant. 



383. Two Degrees of Freedom. 



If a rigid body has two degrees of freedom, then it is free to twist about 

 every screw on a certain cylindroid. If the body be set initially in motion 

 by a twist velocity about some one screw on the surface, then, in general, 

 it will not remain twisting about this screw. A movement will take place 

 by which the instantaneous axis gradually comes into coincidence with 

 other screws on the cylindroid. If we impose a suitable restraining wrench 

 i), then of course can be maintained as the instantaneous screw; 77 is 

 reciprocal to 6. It may be compounded with any reactions of the constraints 

 of the system. Thus, given 6, there is an entire screw system of the fifth 

 order, consisting of all possible screws reciprocal to 0, any one screw of which 

 may be taken as the restrainer. Of this system there is one, but only one, 

 which lies on the cylindroid itself. There are many advantages in taking it 

 as the restraining wrench, and it entails no sacrifice of generality ; we there 

 fore have the following statement : To each screw on the cylindroid, regarded 

 as an instantaneous screw, will correspond one screw, also on the cylindroid, 

 as a restraining screw. 



The position of this restraining screw is at 

 once indicated by the property that it must be 

 reciprocal to the instantaneous screw. If we 

 employ the circular representation for the 

 screws on the cylindroid (fig. 42), and if be 

 the pole of the axis of pitch, then it is known 

 that the extremities of any chord, such as IR 

 drawn through 0, will correspond to two re 

 ciprocal screws ( 58). If therefore / be the 

 instantaneous screw, R must be the restraining 



screw. If a body free to twist about all the screws on the cylindroid be set 

 in motion by a twist velocity about /. it will be possible, by a suitable wrench 



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