A Treatise on the Theory of Screws

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, ß, y, 8, e, Find the screw yjr which is reciprocal to y, 8, e, %, t). Let the seven wrenches act upon a body only free to twist about The reaction of the constraints which limit the motion of the body will neutralize every wrench on a screw reciprocal to •«/r (20). We may, therefore, so far as a body thus circumstanced is con- cerned, discard all the wrenches except those on a and ß. Draw the cylindroid (a, ß), and determine thereon the screw p which is reciprocal to i/r. Ihe body will not be in equilibrium unless the wi’enches about a and ß constitute a wrench on p, and hence the ratio of the intensities a" and ß" 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 <ot> &c. ®6, and let p be a given screw on which is a wrench of given intensity p". Let the intensities of the components be p/', &c. p(i", and let q be any screw. A twist about y 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 -a n be the virtual coefficient of y and the nth screw of reference, we have p"™v = p"^ + &c. pj'-ns^. By taking five other screws in place of r], five more equations are obtained, and from the six equations thus found p”, &c. p6" can be de- termined. This process will be greatly simplified by judicious choice of the six screws of which 7] is the type. Let r/ be reciprocal to w2> &c. a>6, then ct-,2 = 0, &c. = 0, and we have P'^v = p"™y- From this equation pL 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.