Discontinuous Mare-Motion. 353 



are concerned only with the cases in which two discontinuous 

 changes having the same sign define the form of vibration. 

 These cases may be considered in two broad divisions, the 

 fir.-t comprising the limiting types in which the two discon- 

 tinuities are of the same magnitude and sign ; and the second, 

 those in which they are unequal. We may discuss these 

 separately. 



Two Equal Discontinuities. 



In this case the velocity-diagram of the string must 

 evidently consist of three parallel straight lines, each passing: 

 through one of the three nodes of the second harmonic, 

 namely, the two ends of the string and its centre. At par- 

 ticular epochs, however — that is, when the discontinuities 

 pass each other or reach an end of the string, — one or even 

 two of the straight lines may contract and vanish, leaving 

 only two lines or one line on the diagram. For instance, if 

 the two discontinuities are at the two ends of the string 

 simultaneously, the diagram reduces to one line passing- 

 through the centre of the string. As the discontinuities 

 move in towards the centre, two new lines appear, each 

 passing through an end of the string, and the third line 

 passing through the centre gradually contracts and finally 

 vanishes when the discontinuities both arrive and pass each 

 other at the centre. The line then reappears and the form 

 of the velocity-diagram then goes back to its initial state, 

 passing through the same stages in the reverse order. It is 

 obvious that in this case the centre of the string remains 

 completely at rest, and that the string vibrates in two seg- 

 ments with double the ordinary frequency, the fundamental 

 component being entirely absent. In general, however, the 

 discontinuities do not cross at the centre of the string, but at 

 some other point ; if one such crossing takes place at a point 

 distant 1/2 + b from an end (I being the length of the string), 

 a second crossing would take place half a period later at the 

 point 1/2 — b. The vibration-curve at either of the points 

 1/2 + b is readily seen to be a simple two-step zigzag. For, 

 the velocity of either point remains unaltered when the 

 two discontinuities pass over it simultaneously in opposite 

 directions, and changes only at the two epochs at which either 

 discontinuity passes over it singly. If tana be the slope of 

 the lines of the velocity-diagram, the velocity at either point 

 changes at the first epoch from b tan a to (1/2 — b) tan a, and 

 changes back again to its original value at the second epoch. 

 The fraction of the total period of vibration during which 

 the larger velocity subsists is given by the ratio b/2l, or by 



