516 Special Epochs in Vibrating Systems. 



rest epoch will be indicated by all the vectors being found 

 simultaneously along XX 7 ; and a general undisturbed epoch 

 by their being found along YY 7 . If OP™ bo the position, at 

 the beginning of the motion, of the vector corresponding to 

 the harmonic m, the angle XOP™ is arc cot (r m /T m ). 



Take, for example, the conditions (10). These indicate 

 that in the diagram the points P 1? P 2 , P 3 , &c, must have 

 angles (reckoned backwards from OX) which are in the ratio 

 of the natural numbers. Then if all the vectors start off at 

 once with velocities proportional to their places in the 

 harmonic series, all will arrive at OX simultaneously, and 

 half a period (fundamental) later, all the odd ones will lie 

 along OX 7 and all the even ones along OX ; i. e. two rest 

 epochs (of different configuration) have occurred. But all 

 the vectors (starting off in this way) will never be found all 

 in the line YY 7 unless in the special case in which the even 

 harmonics are absent. 



It is clear that there are three types of possible configu- 

 ration at the rest epochs, and three types of velocity distri- 

 bution at the undisturbed epochs. This can be seen most 

 easily from the graphical representation. At a rest epoch, for 

 example, all the vectors may lie along OX, or they may all 

 lie along OX 7 , or they may lie some along OX and the others 

 along OX 7 . The first case occurs if condition (10) is satisfied 

 and the configuration recurs at intervals of the fundamental 

 period. The third case also occurs if (10) is satisfied and all 

 the rest configurations of this type are arrived at half a 

 period (fundamental) later than those of the previous type. 

 All the vectors corresponding to the odd harmonics lie 

 along OX', all those to the even ones along OX. The second 

 case will only occur as a result of (10) if the even harmonics 

 are absent. 



Similar results may be arrived at for the times and natures 

 of the undisturbed configurations. If condition (13) is 

 satisfied, the undisturbed epochs occur, and this is indicated 

 by all the vectors "being found simultaneously along YY 7 . 

 The velocity distribution of the system when these epochs 

 occur is of three types, corresponding to all vectors along OY, 

 all along OY 7 , some in OY and others in OY 7 . It is also 

 seen from the graphical method that in general conditions (10) 

 and (13) are incompatible. But that if the even harmonics 

 are absent, either condition gives both rest and undisturbed 

 epochs. 



Similar results to all these may be worked out for a 

 vibrating system free at both ends instead of fixed, as in this 

 case. 



