DAVIS. — LONGITUDINAL VIBRATIONS OP A RUBBED STRING. 719 



Figure 8. The parallels are laid out as before by means of the points 

 A, B, and C ; the height of A is taken arbitrarily as 4 ; and Krigar- 

 Menzel's law gives D = 2. Then since G = 0, F = -f 4 = 4. Also 

 F and H give I = J -f- 4 = 4, and E and I give K = D — 4 = — 2 and 

 M = L + 4 = 4, which brings us back via N to the starting point A. 

 One such round can always be made, and it will always cross the known 

 section A E in such a way as to start a second round. In this case 

 either of the pairs D I or D M enables one to make this second round 

 and to complete the figure. In a more complicated case a third round, 

 a fourth round, etc., would follow ; and as new points are filled in, former 



E 



H 



E 



Figure 7. Showing the construc- 

 tion of the integral surface for the ali- 

 quot case \. 



B o c o 



Figure 8. Showing the construc- 

 tion of the integral surface for the non- 

 aliquot case f. 



rounds may be repeated, until the surface is completed. The method will 

 always yield one, and only one, surface. 



No attempt will be made in this paper to deduce the two rules mathe- 

 matically from general principles. 19 As is so often necessary in the theory 

 of La Place's equation, they are to be regarded simply as an arbitrary 

 device for producing something that can subsequently be proved to be 

 (a) a solution, and (b) the only possible solution, satisfying the given con- 

 ditions. Thus, since such a solution is built up of planes, it is always true 

 that 



19 It is hoped that the mathematical questions which are suggested by this work 

 can he treated more fully in a later paper. In particular, this graphical method 

 can be so generalized as to give a solution of any kind, whether periodic or not, 

 when its values are known along any three incommeasurably spaced, non-charac- 

 teristic, parallel lines. 



