A treatise on infinitesimal calculus Volume 4; containing differential and integral calculus, calculus of variations, applications to algebra and geometry, and analytical mechanics

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1862 Excerpt: ... (123) involves exponential expressions; and to, and o2 will increase indefinitely with the time. Whence we infer that if a body, free from the action of forces producing rotation, rotates at any time about an axis nearly coinciding with the principal axis of greatest or least moment, the rotation-axis will always nearly coincide with that principal axis. But if the principal axis, with which the rotation-axis nearly coincides is the principal axis of mean moment, the rotation-axis will deviate more and more from that axis. Hereby we have another conception of the stability and instability of principal axes; those of greatest and least moment are stable; that of mean moment is unstable. 165. We now return to the general case; although the invariable axis is fixed in space, yet in the body it describes a cone of the second degree, the equation to which is thus found. In reference to the principal axes of the body the equations of the invariable axis are but from (37) and (44) we have A(AA2-G2)a)12 + B(BA2-G2)»22 + C(C/fc2-G2)(03!! = 0; (jfc2_)+(2_!)y2+2_2 = 0. (126) which is the equation to a cone of the second degree coaxal with the momental ellipsoid; and is a circular cone if two principal moments are equal; and becomes two planes passing through the axis of y, if G2 = B k2, of which the equations are c(B-A)ff + a(c-b)z = 0. (127) Hence also we have the following image of the body's motion. Let the instantaneous angular velocity a at the time t be resolved into two components, the axis of one of which is the invariable axis, and the axis of the other is the line perpendicular to it in the invariable plane. Now if f is the angle between the instantaneous and the invariable axes, to cos j is the former component, and is, by reason of (59), Art. 153, co...