EQUILIBRIUM OF A RIGID BODY. 43, 



are reciprocal to a single screw must form the most 

 general type of a screw complex of the fifth order. 



50. Applications of Co-ordinates. If the co-ordinates 

 of a screw satisfy n linear equations, the screw must 

 belong to a screw complex of the order 6 - n. Let t\ be 

 the screw, and let one of the equations be 



AM +, &c., + A MS = o, 



whence ?? must be reciprocal to the screw whose co-ordi- 

 nates are proportional to 



It follows that TJ must be reciprocal to n screws, and 

 therefore belong to a screw complex of order 6 - n. 



Let cr, j3, 7, be four screws about which a body re- 

 ceives twists of amplitudes a, /3 X , y', S'. It is required to 

 find the co-ordinates of the screw p and the amplitude p' 

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

 four given twists. Wehaveseen (39) that the twist about 

 any screw a, may be resolved in one way into six 

 twists of amplitudes a'cti, . . . a'ae, on the six screws of 

 reference ; we must therefore have 



p'pi = of en + |3' )3i + y ji + ' Si 

 &c., &c. 



p'.pe = a a 6 + j3' j3 6 + y' 7e + 8' S 6 



whence p' and p l9 . . . p B can be found (37). 



A similar process will determine the co-ordinates of 

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

 1 5 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 



