14 THE CYLINDROID. 



choosing different screws for rj. From each group of 

 three equations the amplitudes can be eliminated, and 

 four of the equations thus obtained will involve all the 

 purely geometrical conditions as to direction, situation, 

 and pitch, which must be fulfilled by the screws when 

 three twists can neutralize each other. 



But now suppose that three wrenches equilibrate on 

 the three screws a, |3, 7. Then ( 10) the sum of the 

 works done in a twist about any screw T\ against the 

 three wrenches must be zero, whence 



o"w all + $"*fr + 7"w = O, 



and an indefinite number of similar equations must be 

 satisfied. 



By comparing this system of equations with that pre- 

 viously obtained, it is obvious that the geometrical con- 

 ditions imposed on the screws a, /3, 7, in the two cases 

 are identical, and that the amplitudes of the three twists 

 which neutralise are, respectively, proportional to the in- 

 tensities of the three wrenches which equilibrate. 



When three twists (or wrenches) neutralise, then a 

 twist (or wrench) equal and opposite to one of them must 

 be the resultant of the other two, and hence it follows 

 that the laws for the composition of twists and of wrenches 

 must be identical. 



1 6. The Cylindroid. We now proceed to study the 

 composition of twists and wrenches, and we select twists 

 for this purpose, though wrenches would have been 

 equally convenient. 



A body receives twists about three screws ; under 

 what conditions will the body, after the last twist, oc- 

 cupy the same position which it had before the first. 



The problem may also be stated thus : It is required 

 to ascertain the single screw, a twist about which would 

 produce the same effect as any two given twists. We 



