576 Prof. C. V. Raman on the Partial 



the complete series of harmonics. When the bow is applied 

 at one o£ the points of aliquot division, e. g. 7/5, or 1/6 &c, 

 the subordinate series of harmonics ha vino- a node at that 

 point drop out, and it can easily be shown that the resulting 

 Helmholtzian type of motion is determined by one large 

 positive discontinuity, and a number of small equal negative 

 discontinuities. For example, with the bow applied at l/5 y 

 we have one large discontinuity equal to 4V, and four small 

 discontinuities each equal to — V, where V is the velocity of 

 the bow. Similarly, with the bow at 7/6, we have one dis- 

 continuity equal to 5V, and five discontinuities each equal 

 to — V, and so on. In the intermediate or " transitional 

 modes " also, the motion is o£ a generally similar character ; 

 the positions and magnitudes of the discontinuities must be 

 such that in the forward motion at the bowed point the 

 speed is uniform and equal to that of the bow. The speed 

 of the backward motion is necessarily non-uniform in greater 

 or less degree except when the bow exactly coincides with 

 one of the points of aliquot division. 



To illustrate the method of calculation of the amplitudes 

 and phases of the harmonics from the formula given above, 

 we may take a specific case in which the bow is applied at 

 a point a? = a, where 7/5>a>7/6. We have then six dis- 

 continuities of which five are necessarily equal and negative, 

 and the other is larger and positive. The magnitude of the 

 large discontinuity may be taken to be 5V, and of the small 

 discontinuities — V. We have then the following scheme: — 



d 1 = d 2 = d d = — V, d 4 =5Y, d b — d & = — V. 



C 1 = 0, G 2 = 2a, C 3 = 4a, C 4 =2/-6a, (J 5 = 27-4a, C, = 2l-2a. 



When a = l / 6, the initial position of the discontinuities is 

 identical with that in the Helmholtzian type obtained by 

 bowing at 7/6. It will also be seen that when a = 7/5, 

 3 and G 4 become identical, and the discontinuities d z and d^ 

 therefore merge into each other, reducing the number of 

 discontinuities to five, and the magnitude of the large dis- 

 continuity to 4V. Thus, both when a = 7/6 and when a = 7/ 5, 

 the transitional modes become identical with the Helmholtzian 

 types. The cases in which the bow is applied between 

 7/7 and 7/6, or be ween 7/8 and 7/7, may be similarly 

 worked out. 



