A Treatise on the Theory of Screws

278 THE THEORY OF SCREWS. [265, If x be a variable parameter, then the co-ordinates x dR x dR 4^ cZaj 4p,; da6 must denote a screw of variable pitch x on the same screw as a. We are thus conducted to a more general form of the results previously obtained (§ 47). These expressions may be written OC CU + .y— cos , «2 + cos a2, ... where alt a2, ... are the angles which a makes with the screws of reference. 266. A general Expression for the Virtual Coefficient. We may also consider that function of the co-ordinates of a Dyname which, being always proportional to the pitch, becomes exactly equal to the pitch when the intensity is equal to unity. More generally, we may define the function to be equal to the pitch multiplied into the square of the intensity, and it is easy to assign a physical meaning to this function. It is half the work done in a twist .against a. wrench, on the same screw, where the amplitude of the twist is equal to the intensity of the wrench. Referred to any co-ordinates, we denote this function by V expressed in terms of Xj,... X(j. If we express the same function by reference to six co-reciprocal axes with co-ordinates a1;... a6, we have the result jMia+ ••• peat2 = V. Forming now the first emanant, we have n n dV dV Spießt + ... + Ip^ß, = . + M(. ; but the expression on the left-hand side denotes the product of the two intensities into double the virtual coefficient of the two screws; hence the right-hand member must denote the same. If, therefore, after the differentiations we make the intensities equal to unity, we have for the virtual coefficient between two screws X and p, referred to any screws of reference whatever one-half the expression dV dV ^dxf+^dxf Suppose, for instance, that X is reciprocal to the first screw of reference, then ^=o.