HI] FREEDOM OF THE FIRST ORDER. 103 



increased by a certain quantity, and all the pitches of the reciprocal system 

 be diminished by the same quantity, then all the first set of screws thus 

 modified are reciprocal to all the second group as modified. Hence, since a 

 screw-system of the nth order consists of all the screws reciprocal to 6 n 

 screws, it follows that the modified set must still be a screw system. 



We shall now apply this principle to prove that all the screws X of any 

 given pitch k, which can be drawn through A, to be reciprocal to a, lie in a 

 plane. Take a screw ij, of pitch p a + k, on the same line as or, then we have 

 just shown that all the screws p, of zero pitch, Avhich can be drawn through 

 the point A, so as to be reciprocal to ?;, lie in a plane. Since fj, and 77 are 

 reciprocal, the screws on the same straight lines as /u, and 77 will be reciprocal, 

 provided the sum of their pitches is the pitch of r) ; therefore, a screw X, of 

 pitch k, on the same straight line as p, will be reciprocal to the screw a, of 

 pitch p a ; but all the lines //. lie in a plane, therefore all the screws X lie in 

 the same plane. 



Conversely, given a plane and a pitch k, a point A can be determined 

 in that plane, such that all the screws drawn through A in the plane, and 

 possessing the pitch k, are reciprocal to a. To each pitch k 1} & 3 , ..., will 

 correspond a point A 1} A. ... ; and it is worthy of remark, that all the points 

 A l} A. 2 must lie on a right line which intersects a at right angles; for join 

 A l} A, then a screw on the line A^.^, which has for pitch either k : or k., 

 must be reciprocal to a; but this is impossible unless A^A^ intersect a at 

 a right angle. 



111. Equilibrium. 



If a body which has freedom of the first order be in equilibrium, then 

 the necessary and sufficient condition is, that the forces which act upon the 

 body shall constitute a wrench on a screw of the screw system of the fifth 

 order, which is reciprocal to the screw which defines the freedom. We thus 

 see that every straight line in space may be the residence of a screw, a 

 wrench on which is consistent with the equilibrium of the body. 



If two wrenches act upon the body, then the condition of equilibrium is, 

 that, when the two wrenches are compounded by the aid of a cylindroid, 

 the single wrench which replaces them shall lie upon that one screw of the 

 cylindroid, which is reciprocal to a ( 2G). 



We can express with great facility, by the aid of screw co-ordinates, the 

 condition that wrenches of intensities 6&quot;, &amp;lt;&quot;, on two screws 6, &amp;lt;f&amp;gt;, shall 

 equilibrate, when applied to a body only free to twist about a. 



Adopting any six co-reciprocals as screws of reference, and resolving each 

 of the wrenches on 9 and &amp;lt;/&amp;gt; into its six components on the six screws of 



