340 THE THEORY OF SCREWS. [314- 



had been free, would have been the impulsive screws, corresponding to the 

 instantaneous screws belonging to A. 



Let Q be the system of screw of the (6 - n)th order, which are reciprocal 

 to P. 



Take from A the system of the (n- l)th order, reciprocal to 77, and call 

 it A,. 



Take from Q the system of the (5 ?i)th order, reciprocal to 77, and call 

 it Q,. 



As A is of the nth order, and Q x of the (5 - w)th, they together define a 

 system of the fifth order. Let ^ be the single screw, reciprocal to this system 

 of the fifth order. 



As Jj is of the (n-l)th order, and Q is of the (6 - w)th order, they 

 together define a system of the fifth order. Let r} 2 be the single screw, 

 reciprocal to this system of the fifth order. 



77! is reciprocal to A l of the (n l)th order, because J.j forms part of A. 

 It is reciprocal to Qj of the (5 n)th order, because it was so made by con 

 struction. Thus 77! is reciprocal to both A l and Q lt that is to a system of the 

 fourth order. 



In like manner, it can also be shown that rj 2 is reciprocal to both A^ 

 and Q lt 



A l and Q 1 were originally chosen so as to be reciprocal to 77. It thus 

 appears that the three screws, rj, tj^, 772, are all reciprocal to the same system 

 of the fourth order. They are, therefore, co-cylindroidal. 



The initial wrench on 77 can therefore be adequately replaced by two 

 components on ^ and ?7 2 . The former of these is destroyed by the reaction 

 of the constraints. The latter gives rise to an initial movement on a 

 determined screw of A. Thus, the most general problem of the effect of 

 an impulsive wrench on a constrained rigid body has been solved geo 

 metrically. 



315. Freedom of the Fifth Order. 



The special case, where the rigid body has freedom of the fifth order, may 

 be viewed as follows. 



Let p be the screw reciprocal to the screw system of the fifth order, about 

 which the body is free to twist. 



Let A, be the instantaneous screw, determined in the way already explained, 

 about which the body, had it been free, would have commenced to twist in 

 consequence of an impulsive wrench on p. 



