66 THE THEORY OF SCREWS. [76, 



the amplitude p of a twist about p which will produce the same effect as the 

 four given twists. We have seen ( 37) that the twist about any screw a 

 may be resolved in one way into six twists of amplitudes a a,, ... a 6 , on the 

 six screws of reference ; we must therefore have 



p pe = a a,, + /3 @ + 77 fi + 8 8 6 , 

 whence p and p l} ... p K can be found ( 35). 



A similar process will determine the co-ordinates of the resultant of any 

 number of twists, and it follows from 12 that the resultant of any number 

 of wrenches is to be found by equations of the same form. In ordinary 

 mechanics, the conditions of equilibrium of any number of forces are six, 

 viz. that each of the three forces, and each of the three couples, to which the 

 system is equivalent shall vanish. In the present theory the conditions are 

 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 system of the ?ith order if it be reciprocal to 

 6 n independent screws ; it follows that 6 n conditions must be fulfilled 

 when n + 1 screws belong to a screw system of the nib. order. 



To determine these conditions we take the case of n 3, though the 

 process is obviously general. Let a, /3, 7, & be the four screws, then since 

 twists of amplitudes a, (3 , 7 , & neutralise, we must have p zero and hence 

 the six equations 



a a, + /3 A + 77, + S S, = 0, 

 &c. 



from any four of these equations the quantities a , /3 , 7 , S can be eliminated, 

 and the result will be one of the three 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 system 

 of the (n + l)th order, twists of appropriate amplitudes about the screws 

 neutralise. The amplitude of the twist about any one screw must be pro 

 portional to a function of the co-ordinates of all the other screws. We thus 

 see that the evanescence of one function must afford all that is necessary for 

 n + 1 screws to belong to a screw system of the nth order.* 



* Philosophical Transactions, 1874, p. 23. 



