722 PROCEEDINGS OP THE AMERICAN ACADEMY. 



and when p is even, the form is similar. Whenever n is equal to a mul- 

 tiple of q, both A n and B n are zero, the first because each of the sine 

 terms is zero, and the second because each of the cosine terms is one and 

 the sum of the #'s is zero. Young's law is, therefore, always satisfied. 

 And finally, as has been said, it can be proved that Young's and Krigar- 

 Menzel's laws determine the motion of a fiuite string uniquely. 22 It is, 

 therefore, evident that the method proposed must lead in every case to 

 the desired solution. 



A number of solutions, obtained by this method from Krigar-Menzel's 

 law, are shown on the first and second of the accompanying plates. 

 The third plate shows some typical u t curves (a) as read off from the 

 various surfaces, and (b) as reproduced experimentally by the methods 

 of section 4. 



The surfaces of the first two plates show certain serial relations 

 which are interesting in the light of Young's and Krigar-Menzel's work 

 on the excitation of individual partials. The aliquot cases (l/k) in the 

 first line are obviously related, and are approaching Helmhotz's major 

 solution as a limit as k increases; 23 the amplitude for a given bowing 

 speed is, however, becoming infinite. The mode of vibration which one 

 would actually get in the limiting case, that is, if one were to bow at the 

 end of the string, would be absolute rest, something quite different from 

 the form toward which the surfaces are tending. 



A similar serial relation is to be found among the surfaces of the 

 second line. They are the cases which correspond to the various values 

 of the expression m/(2 m + 1) for integral values of m, just as those of 

 the first line are, essentially, cases which can be grouped under the 

 formula m/(m+ l). 24 The series in the second line begins with J and 

 approaches ^ as a limit as m increases. The surfaces show a tendency 

 toward a mode of vibration in which the middle of the string would be 

 nearly at rest and each half would be moving in accordance with Helm- 

 holtz's major solution ; but the amplitude of the motion is, as before, 

 increasing indefinitely. In other words, all the partials which have 



22 At least within a multiplicative constant which can be determined by means 

 of the velocity law. 



23 The integral surface corresponding to Helmlioltz's major solution alone is 

 built up of parts of hyperbolic paraboloids. Its contour lines are parts of rectan- 

 gular hyperbolae with two adjacent sides of a fundamental square as asymptotes. 

 Its appearance would be much like that of the surface for the case \. 



24 Any surface (1 — p/q) differs from the related surface [p/q) only in the signs 

 of its displacements. To get the surface f from the surface I one has only to 

 interpret each hill as a hollow and vice versa. 



