The London, Edinburgh and Dublin philosophical magazine and journal of science . . . (l2) Epochs in Vibrating Systems. provided . Ti , . To , . To arc tan T = ^ arc tan r,f = A arc tan ~ ==. 515 (13) the angles being taken as before. The times for the undisturbed epochs o£ a given harmonicare obviously in quadrature with those for the rest epochs;but the set of conditions (10) which make the general restepochs possible are incompatible with those which make thegeneral undisturbed epochs possible, and vice versa. Theconditions (10) and (13) evidently agree with those of thespecial cases 1. and
The London, Edinburgh and Dublin philosophical magazine and journal of science . . . (l2) Epochs in Vibrating Systems. provided . Ti , . To , . To arc tan T = ^ arc tan r,f = A arc tan ~ ==. 515 (13) the angles being taken as before. The times for the undisturbed epochs o£ a given harmonicare obviously in quadrature with those for the rest epochs;but the set of conditions (10) which make the general restepochs possible are incompatible with those which make thegeneral undisturbed epochs possible, and vice versa. Theconditions (10) and (13) evidently agree with those of thespecial cases 1. and II. above. In I. all the angles of thecondition (10) become zero, and of the condition (13) 90°or multiples thereof. The reverse holds in II., and thenecessity for the absence of the even harmonics in the casesremarked upon is shown by (10) and (13). These results are all very clearly illustrated by the usualgraphical representation for simple harmonic motion. Take a line XX to represent the rest position, YY theundisturbed position. Represent each harmonic by a vector. from 0 rotating in the positive direction with a uniformangular velocity m times that of the fundamental. Thegeneral motion will then be represented by the group o\vectors (which are infinite in number though in any actualsystem only the earlier ones in the harmonic series will beappreciable) rotating simultaneously and starting all at oncefrom positions scattered round the circle. These initial posi-tions will be determined by the initial conditions. A general 516 Special Epochs in Vibrating Systems. rest epoch will be indicated by all the vectors being foundsimultaneously along XX7; and a general undisturbed epochby their being found along YY7. If OP⢠bo the position, atthe beginning of the motion, of the vector corresponding tothe harmonic m, the angle XOP⢠is arc cot (rm/Tm). Take, for example, the conditions (10). These indicatethat in the diagram the points P1? P2, P3, &c, must haveangles (reck
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