A Treatise on the Theory of Screws

äSg&M 380 THE THEORY OF SCREWS. [348 the outlines of this theory, with the view of generalizing it into one adequate for our purpose. We can easily conceive of two systems of corresponding forces in two planes. To each force in one plane will correspond one force in the other plane, and vice versa. To any system of forces in one plane will correspond a system of forces in the other plane. We are also to add the condition that if one force x vanishes, the corresponding force x will also vanish. The fundamental theorem which renders this correspondence of im- portance is thus stated :— If a group of forces in one of the planes would equilibrate when applied to a rigid body, then the corresponding group of forces in the other plane would also equilibrate when applied to a rigid body. Draw any triangle in each of these planes, then any force can be de- composed into three components on the three sides of the triangle. Let x, y, z be the components of such a force in the first plane, and let x', yz' be the components of the corresponding force in the second plane; we must then have equations of the form x' = ax + by + cz, y' = a'x + b'y + c'z, z' = d'x + b"y + d'z, where a, b, c, &c., are constants. These equations do not contain any terms independent of the forces, because x', y', z must vanish when x, y, z vanish. They are linear in the components of the forces, because otherwise one force in one plane will not correspond uniquely with one force in the other. Let xlt ylt zx; x.2, y2, z2, ... xn, yn, zn be the components of forces in the first plane. Let a?/, y{, z(\ x2, y2, z2\ ... x^, yü, z^ be the components of the corre- sponding forces in the second plane. Then we must have xk=axk +byk + czk , yk=a'®k +b'yk +c'zk, Zk = + b"yk + c"zk, where k has every value from 1 to n. If therefore we write 2« = + x2 + ... + xn, and Sx' = x2 + x2 + ... + xn',