32 THE THEORY OF SCREWS. [28- 



not for this condition a distinct solution would be required for every variation 

 of the order in which the successive twists were imparted. 



If the number of screws were greater than seven, then both problems 

 would be indeterminate ; if the number 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, ft, 7, B, e, 77. Find the screw ty which is 

 reciprocal to 7, 8, e, , 77. Let the seven wrenches act upon a body only 

 free to twist about ty. The reaction of the constraints which limit the 

 motion of the body will neutralize every wrench on a screw reciprocal to 

 i/r (20). We may, therefore, so far as a body thus circumstanced is con 

 cerned, discard all the wrenches except those on a and ft. Draw the 

 cylindroid (a, ft), and determine thereon the screw p which is reciprocal to -^r. 

 The body will not be in equilibrium unless the wrenches about a and ft 

 constitute a wrench on p, and hence the ratio of the intensities a&quot; and ft&quot; 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. (See Appendix, note 1.) 



29. Intensities of the Components. 



Let the six screws of reference be w l , &c. &&amp;gt; 6 , and let p be a given screw 

 on which is a wrench of given intensity p&quot;. Let the intensities of the 

 components be p/ , &c. p 6 &quot;, and let 77 be any screw. A twist about 77 must 

 do the same quantity of work acting directly against the wrench on p as 

 the sum of the six quantities of work which would be done by the same 

 twist against each of the six components of the wrench on p. If TS^ be 

 the virtual coefficient of 7; and the nth screw of reference, we have 



P&quot;^r,p = P&quot;^ + &C. p G &quot;^rfl- 



By taking five other screws in place of 77, five more equations are 

 obtained, and from the six equations thus found p/ , &c. p 6 &quot; can be de 

 termined. This process will be greatly simplified by judicious choice of 

 the six screws of which 77 is the type. Let 77 be reciprocal to G&amp;gt; 2 , &c. &&amp;gt; 6 , then 

 oj-,,2 = 0, &c. -or^ = 0, and we have 



P&quot;^r,p ^ Pl ^rft- 



From this equation p&quot; is at once determined, and by five similar equations 

 the intensities of the five remaining components may be likewise found. 



Precisely similar is the investigation which determines the amplitudes of 

 the six twists about the six screws of reference into which any given twist 

 may be decomposed. 



