SCREW CO-ORDINATES. 29 



applied to a rigid body in succession, the body after the 

 last twist shall occupy the same position which it had 

 before the first. 



The problem w r e have last stated has been limited as 

 usual to the case where the amplitudes of the twists 

 are small quantities, so that the motion of each particle 

 produced by each twist is sensibly rectilinear. Were it 

 not for this condition a distinct solution would be re- 

 quired for every variation of the order in which the suc- 

 cessive twists were imparted. 



If the number of screws were greater than seven, 

 then both problems would be indeterminate ; if the num- 

 ber were less than seven, then both problems would be 

 impossible (unless the screws were specially related) ; 

 the number of screws being seven, the problem of the 

 determination of the ratios of the seven intensities (or 

 amplitudes) has, in general, one solution. We shall 

 solve this for the case of wrenches. 



Let the seven screws be a, /3, 7, S, e, , rj. Find the 

 screw \p which is reciprocal to y, 3, e, , rj. Let the seven 

 wrenches act upon a body only free to twist about i//. 

 The reaction of the constraints which limit the motion 

 of the body will neutralize every wrench on a screw re- 

 ciprocal to ^ ( 22). We may, therefore, so far as a body 

 thus circumstanced is concerned, discard all the wrenches 

 except those on a and /3. Draw the cylindroid (a, /3), 

 and determine thereon the screw p which is reciprocal to 

 i//. The body will not be in equilibrium unless the 

 wrenches about a and j3 constitute a wrench on /o, and 

 hence the ratio of the intensities a." and $" is determined. 

 By a similar process the ratio of the intensities of the 

 wrenches on any other pair of the seven screws may be 

 determined, and thus the problem has been solved. 



31. Intensities of the Components. Let the six screws 

 of reference be w,, &c. w 6 , and let p be a given screw 



