694 PROCEEDINGS OF THE AMERICAN ACADEMY. 



In two respects the present paper goes beyond the published work on 

 the violin string. In the first place, the fact that a given state of vibration 

 can be maintained indefinitely and can be reproduced at will has made it 

 possible, for the first time, to measure amplitudes under conditions which 

 are accurately known. This has resulted in a good quantitative verifica- 

 tion of the established theory, and especially of the law connecting 

 amplitude with bowing speed. And in the second place, it is hoped that 

 the graphical method which is here presented for solving the differential 

 equation of vibrating strings, subject to the most general boundary condi- 

 tions which the problem of the rubbed string requires, will prove to be a 

 valuable addition to the theoretical side of the subject. It is possible 

 that this part of the paper will have some interest for the pure mathe- 

 matician, for it is rather unusual that graphics should be able to give, 

 not simply an approximate, but an exact and rigorous solution of a prob- 

 lem of such complexity. A striking instance of this kind is, of course, 

 Euler's solution of the problem of the plucked string. 



As has been indicated, all previous work on rubbed strings has been 

 concerned with transverse vibrations. In this form the problem is fully 

 as old as that of the plucked or of the struck string, but it differs essen- 

 tially from them in that it is not easy to set up, a priori, a satisfactory 

 analytical statement of the action of a bow, as was attempted by the ear- 

 lier mathematicians. 1 The first results of any importance are due to 

 Thomas Young, 2 who formulated for plucked strings, and less definitely 

 for bowed strings, 



Young's Law : No overtone is present ivkich would have a node at the 

 point of excitation. 



But if, he continues, the bowing be, not at a principal node, but near 

 one, " the corresponding harmonic is extremely loud," a fact which has 

 been very prettily verified by Krigar-Menzel, 3 who showed also that its 



1 For example, D. Bernoulli, in his " Memoir sur les sons produit par les tuyaux 

 d'orgue," likens the action of a bow to that of a toothed wheel, and J. Antoine 

 [Ann. de chim. et phys., Se'r. Ill, 27, 191 ; Pogg. Ann., 81, 544 (1850)] speaks of 

 " a series of gentle blows." Duhamel [C. R., 3, 646, (1836) ; C. R., 9, 567, (1839) ; 

 Mem. d. Savants e'trang., 8, 131 (1843)] properly assumes the action to be fric- 

 tional, but his treatment is not entirely satisfactory. That of F. Lippich [Mittheil. 

 d. deut. math. Ges. in Prag, 1, 118 (1892)] is much better, but it would scarcely 

 have been possible apart from the experimental work which preceded it. 



2 Phil. Trans, of the Roy. Soc. of London, 90^ 106 (1800). 



3 " Ueber d. Bevvegung gestrich. Saiten," Iuaug. Diss., Berlin, 1888. See also 

 J. Ritz, "Zusammensetzung d. Kliinge d. Streichinstr.," Inaug. Diss., Munchen, 

 1883. 



