44 EQUILIBRIUM OF A RIGID BODY. 



likewise six, viz., that the intensity of each of the six 

 wrenches on the screws of reference to which the given 

 system is equivalent shall be zero. 



Any screw will belong to a complex of the n th order 

 if it be reciprocal to 6 - n independent screws ; it follows 

 that 6 - n conditions must be fulfilled when n -t- i screws 

 belong to a screw complex of the n th order. 



To determine these conditions we take the case of 

 n = 3, though the process is obviously general. Let a, /3, 

 7, S be the four screws, then since twists of amplitudes 

 '> j3'> 7', $' neutralise, we must have p' zero and hence 

 the six equations 



7 / 7l + S'& = o, 

 &c. 



from any four of these equations the quantities a' y /3', 7', 

 $ can be eliminated, and the result will be one of the 

 required conditions. 



It is noticeable that the 6 - n conditions are often 

 presented in the evanescence of a single function, just as 

 the evanescence of the sine of an angle between a pair 

 of straight lines embodies the two conditions necessary 

 that the direction cosines of the lines coincide. The 

 function is suggested by the following considerations : 

 If n + 2 screws belong to a screw complex of the (n + i) th 

 order, twists of appropriate amplitudes about the screws 

 neutralise. The amplitude of the twist about any one 

 screw must be proportional to a function of the co-ordi- 

 nates of all the other screws ; this is evident, because if 

 one amplitude were ascertained to be zero, the remaining 

 screws must belong to a complex of the n th order. We 

 thus see that the evanescence of one function must afford 

 all that is necessary for n + i screws to belong to a screw 

 complex of the n 1h order. 



