A Treatise on the Theory of Screws

254 THE THEORY OF SCREWS. [235, substitute p, = —, »2 = —, ®8= —, « = -; h q> y and suppose qlt q2, q3, qit qs, qe to be in descending order of magnitude. Thus —l- + - + ... + JSl_= 0. y - y - & y - q> That is X12(y-92)(2/-73)(y-74)(2/-76)(2/-26) + ... + V (y - ?i) (y - q2) (y - ?») (y - (y - ?6) = o. In the left-hand member of this equation substitute the values qi, q2, q3, qit q3, qe successively for y; five of the six terms vanish in each case, and the values of the remaining term (and therefore of the whole member) are alternately positive and negative. The five values of y must therefore lie in the intervals between the six quantities qt, q2,... q3, the roots are accordingly proved to be real and distinct (unless one of the quantities Xj, Å.2, X3, X4) X6, X6 = 0 and a further condition hold, or unless some of the quantities qlt be equal). The values of px, ... p3 are ± a, + b, + c; and we suppose a, b, c, positive and a > b > c. The values of y lie in the successive intervals between 1 1 1 _! _1. o’ b’ a' a’ b’ c’ and consequently of the roots of the equation in x. Two are positive and lie between a and b, and between b and c respectively. Two are negative and lie between — a and — b, and between — b and — c respectively. The last is either positive and > a or negative and < — a. 236. The Pectenoid. A surface of some interest in connection with the freedom of the fifth order may be investigated as follows. Let a be the pitch of the one screw &>, to which the five system is reciprocal. Take any point 0 on co and draw through 0 any two right lines OY, and OZ which are at right angles and which lie in the plane perpendicular to co.