696 PROCEEDINGS OF THE AMERICAN ACADEMY. 



any aliquot bowing point completely, and has been ingeniously verified 

 by a number of observers. 6 



Non-aliquot, rational points were considered for the first time by 

 Krigar-Menzel and Raps,? who showed that Helmholtz's law must, in 

 the general case, be replaced by 



Krigar-Menzel's Law: Whe?i a string is bowed at any rational 

 point p/q where p and q are prime to each other, the part of the string 

 immediately under the bow moves to and fro with constant i^elocities, whose 

 ratio depends only on q and is l/(q — 1). 



This law, which includes Helmholtz's law as a special case, they veri- 

 fied for values of q up to 10, at which point it seemed to break down. 

 In general, this breaking-point would be determined in some obscure 

 fashion by such circumstances as the pressure and speed of the bowing, 

 the material and size of the strings used, etc. It can easily be proved 

 that in those cases in which Krigar-Menzel's law does hold, this law is 

 enough, with Young's law and Helmholtz's velocity law, to determine 

 the motion of the string uniquely. The theory is therefore complete for 

 all simple rational cases, whether aliquot or not. It must be remem- 

 bered, however, that it rests wholly on an empiric foundation, and that 

 the foregoing laws, which are that foundation, were enunciated solely for 

 the case of a string vibrating transversely under the action of a violin 

 bow. This paper will show that they are equally applicable to the case 

 of a string vibrating longitudinally under the action of a rough wheel. 



The string upon which the following experiments were performed was 

 a steel wire about 0.4 mm. in diameter (density 8.6; Young's modulus, 

 as calculated from the observed frequency, 23,430 kg. wt. per mm' 2 ). It 

 was stretched horizontally, one end passing over a pulley and support- 

 ing 10 kgs., rather less than one third the breaking weight. Effective 

 ends for the vibrating portion were obtained by clamping the wire firmly 

 to the faces of two cast-iron blocks, each about 5 kg. in weight. The 

 length of the vibrating portion was 676 cms. During most of the work 

 a node was formed in the middle of the wire by means of a brass knife 

 edge, the pressure on which was that due to a displacement of the wire 

 of about a centimeter from its position of equilibrium. The wire was 

 then rubbed on one side of the bridge and observed on the other, which 



6 Mach, Pogg. Ann., 134, 311 (1868); Clem. Neumann, Sitzber. I, and K. 

 Acad. d. Wissenschaften, Wien, Math. CI., 41 2 , 89 (1870) ; Clifton, cited by Donkin, 

 Acoustics, p. 138. See also Mach, Zeitsch. f. d. phys. u. chem. Unterr., 1, p. 264 

 (1888) ; Fernbach, ibid., 9, 239 (1896) ; Kuhfahl, ibid., 10, 92 (1897). 



i Wied. Ann., 44, 623 (1891). 



