702 PROCEEDINGS OF THE AMERICAN ACADEMY. 



brought under one simple, but as yet apparently arbitrary, geometric law 

 which has been called Figure 1. In this section it will be proved that 

 this law is a necessary consequence of Young's law, Helmholtz's law, and 

 the velocity law. 



The essential characteristics of the geometry of Figure 1 are, first, 

 that the points B l5 C 1? D l5 etc., and B 2 , C 3 , D 4 , etc., lie respectively on the 

 lines O' A x and OA : produced ; and second, that the other vertices of the 

 figure are the intersections of two pencils of rays through O and O' deter- 

 mined by these sets of points. These two statements will be considered 

 in turn. 



Let a string of length I be bowed downward at the point 1/k. Ac- 

 cording to Helmholtz's law, the motion of the point bowed would be an 

 ascent with a constant velocity /J,, followed by a descent with a constant 

 velocity g , and 



9o _ 1 

 fo * - 1 ' 



Let the amplitude of the point bowed be m . Then the time of one oscil- 

 lation would be 



1 2 m n 2 m n 2 m n k 



F Jo ffo go & — i ' 



where F is the frequency. Therefore 



m °- k 2F' 



According to the velocity law, cf would be equal to the velocity, V, of 

 the bow. Therefore 



£^J._F _ _T k- 1 

 k 2F~2Fl' k 



tn = — -; c^, = ^ „. • — ; — - /. 



Now for any of the points, A x , B 1? Ci, etc., of Figure 1, say C 1? (k — 1) Ijk 

 is the distance OC, and m is the distance CCi. Therefore, if Fis in- 

 dependent of k, 



AA X _ BB, _CC 1 _ 



WI~O r B~ 0'C~ etC '' 



and O'AjBjdDi, etc., is a straight line. The relative magnitude of the 

 envelopes is, therefore, that which the figure indicates. 



There remains to be proved only the fact that the shape of the kih 

 envelope (a) as given by Figure 1, and (b) as calculated theoretically, is 

 k straight lines inscribed in a parabola. 



