169-172] FREEDOM OF THE THIRD ORDER. 171 



six independent displacements. Its position is, therefore, to be specified by 

 six co-ordinates. If, however, the body be so constrained that its six co 

 ordinates must always satisfy three equations of condition, there are then 

 only three really independent co-ordinates, and any position possible for a 

 body so circumstanced may be attained by twists about three fixed screws, 

 provided that twists about these screws are permitted by the constraints. 



Let A be an initial position of a rigid body M. Let M be moved from 

 A to a closely adjacent position, and let x be the screw by twisting about 

 which this movement has been effected ; similarly let y and z be the two 

 screws, twists about which would have brought the body from A to two 

 other independent positions. We thus have three screws, x, y, z, which com 

 pletely specify the circumstances of the body so far as its capacity for 

 movement is considered. 



Since M can be twisted about each and all of x, y, z, it must be capable 

 of twisting about a doubly infinite number of other screws. For suppose 

 that by twists of amplitude x , y , z , the final position V is attained. This 

 position could have been reached by twisting about some screw v, so as to 

 come from A to V by a single twist. As the ratios of x to y , and z , are 

 arbitrary, and as a change in either of these ratios changes v, the number 

 of v screws is doubly infinite. 



All the screws of which v is a type form what we call a screw system of 

 the third order. We may denote this screw system by the symbol S. 



171. The Reciprocal Screw System. 



A wrench which acts on a screw 77 will not be able to disturb the equili 

 brium of M, provided t] be reciprocal to x, y, z. If rj be reciprocal to three 

 independent screws of the system S, it will be reciprocal to every screw of S. 

 Since rj has thus only three conditions to satisfy in order that it may be 

 reciprocal to S, and since five quantities determine a screw, it follows that tj 

 may be any one of a doubly infinite number of screws which we may term 

 the reciprocal screw system S . Remembering the property of reciprocal 

 screws ( 20) we have the following theorem ( 73). 



A body only free to twist about all the screws of S cannot be disturbed 

 by a wrench on any screw of S ; and, conversely, a body only free to tAvist 

 about the screws of S cannot be disturbed by a wrench on any screw of S. 



The reaction of the constraints by which the freedom is prescribed 

 constitutes a wrench on a screw of S . 



172. Distribution of the Screws. 



To present a clear picture of all the movements which the body is 



