86 DYNAMICS OF A RIGID BODY. 



complex of the n th order consists of all the screws reci- 

 procal to 6 - n screws, it follows that the modified group 

 must still be a screw complex. 



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 17, of pitch p a + k, on the same line as a, then we 

 have just shown that all the screws /*, of zero pitch, 

 which can be drawn through the point A, so as to be 

 reciprocal to r/, lie in a plane. Since fj, and 7j are reci- 

 procal, the screws on the same straight lines as ^t and rj 

 will be reciprocal, provided the sum of their pitches is 

 the pitch of ij ; therefore, a screw X, of pitch , on the same 

 straight line as ju, will be reciprocal to the screw a, of 

 pitch / a ; but all the lines ju 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^ 2 , . . . . , will 

 correspond a point A ly A z . . . . ; and it is worthy of re- 

 mark, that all the points A ly A^ must lie on a right 

 line which intersects a at right angles ; for join A ly 

 A Zy then a screw on the line A^A^ which has for pitch 

 either k or / 2 , must be reciprocal to a ; but this is. 

 impossible unless A A 2 intersect a at a right angle. 



81. 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 complex 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. 



