A Treatise on the Theory of Screws

288 THE THEORY OF SCREWS. [273, this expression by their sum; and, consequently, if pa be changed into pa + x, and pß be changed into pß — x, the virtual coefficient will remain unaltered whatever x may be. We have found, however (§ 37), that the virtual coefficient admits of representation in the form PiCtißi + ■•• + pea6ß<l. To augment the pitch of a by x, we substitute for a1( a2> ... the several values (§ 265), aj + ~—cos«!, a2 + ~ cosa2, ... 2/>i 2^2 where alt a2, ... are the angles made by the screw a with the screws of reference. Similarly, to diminish the pitch of ß by x, we substitute for ßlt ß2, ... the several values ßi — ~~ cos blt ß2 — ~ cos L, &c. 2pi zpn With this change the virtual coefficient, as above expressed, becomes x 2pi cos bi or, tp^ßi + ~ få cos at + ß2 cos a2 + ... — cos — a2 cos b2 — ...) x2 /cos a2 cos Sj cos a2 cos b2 cos ae cos 6(i\ jpi p2 ”■ ps )' We have already shown that such a change must be void of effect upon the virtual coefficient for all values of x. It therefore follows that the coefficients of both x and x2 in the expressions just written must be zero. Hence we obtain the two following properties: 0 = (ß1 cos Oi + ... + ße cos æ6) — (ax cos b2+ ... + a6 cos 6(l), „ cos a, cos b, cos a6 cos bs Pi Pt The second of the two formulæ is the important one for our present purpose. It will be noted that though the two screws, a and ß, are com- pletely arbitrary, yet the six direction cosines of a with regard to the screws of reference, and the six direction cosines of ß with regard to the same screws of reference, must be connected by this relation. Of course the equation in this form is only true when the six screws of reference are co-reciprocal. In the more general case the equivalent identity would be of a much more complicated type.