A Treatise on the Theory of Screws

230 THE THEORY OF SCREWS. [219- the six absolute principal screws of inertia as screws of reference, (§ 79) then the body will commence to move as if it were free, but had been acted upon by a wrench of which the co-ordinates are proportional to p^, p636. It follows that the given impulsive wrench, when compounded with the i eactions of the constraints, must constitute the wrench of which the co- oidinates have been just written; whence if h be a certain quantity which is the same for each co-ordinate, we have the six equations — y yi + + p" pi, hp636 = p % + X/z X(i + p"'p6. Multiply the first of these equations by X1; the second by &c.: adding the six equations thus obtained, and observing that 6 is reciprocal to X, and that consequently 'Zpid^i = 0, we obtain + p"'^Xipi = 0, and similarly multiplying the original equations by plt ..., p6 and adding, we obtain y'"^yiPi + V'SMa + /"S/*? = 0. 1’rom these two equations the unknown quantities X'", p" can be found, and thus the initial reaction of the constraints is known. Substituting the values of X'", p" in the six original equations, the co-ordinates of the required screw 3 are determined. 220. Principal Screws of Inertia in the Four-System. We have already given in Chapter VII. the general methods of deter- mining the principal screws of inertia in an n-system. The following is a different process which though of general application is in this chapter set down for the case of the four-system. Choose four co-reciprocal screws a, ß, 7, 8 of the four-system and let their co-ordinates be as usual .,.,a0; ß,,...^; 8lt ...,8e; referred to the six absolute principal screws of inertia (§ 79). Let an impulsive wrench on one of the principal screws of inertia 3 in the four-system be decomposed into components on a, ß, y, 8, and let the impulsive intensities be a!", ß"', y’", 8"'. Let X, p be any two screws on the reciprocal cylindroid. Then the body will move as it it had been free and had received impulsive wrenches on the absolute principal screws of inertia, the impulsive intensities being a ai + ß ".ßi + 7/"y1 + S'"?, + X'"Xj + p"p1 « ct6 + ß /36 + y 'yt] + 8'”86 + X'"Xe + p"'p6.