DAVIS. LONGITUDINAL VIBRATIONS OF A RUBBED STRING. 703 



Take the line A O' of Figure 1 as an a: axis and a line perpendicular 

 to it at A as a y axis, and let the units of length along the two axes be 

 AO' and AA[ respectively. Call the vertices of the k\\\ envelope or 

 K , Kj, K 2 , . . . K k _y, and K k or O'. Let the coordinates of any 

 vertex, K p be x(j, k) and y(j,k). When j= 1 the corresponding 

 vertex lies on the line 0'A 1 , whose equation is 



y — — x + 1. 

 Also 



Therefore 



2/(1, k)=-x (1, ft) + 1 = - l=i + 1 = 2^ . 



Similarly when^' = k — 1, the vertex lies on the line A l5 whose equa- 

 tion is 



y = x + 1. 

 Also 



x(k-l,k)=-l +— — • 2 = —^-. 



Therefore 



y(k-l,k) = x(k-\,k) + 1 = ^^ + l = 2^i. 



Now the law by which Figure 1 was constructed may be expressed 

 analytically by the statement that the point [x (j, k), y (J, £)] is collinear 

 with the points [x(j — l,k — 1), y (j — l,k— 1)], and [— 1, 0], and also 

 with the points [x (j, k — 1), y(j\ k — 1)], and [+ 1, 0]. It follows, in the 

 first place, that [x (j, k),y(j, k)], [x(l, k — j+ l),y(l, k—j+ 1)], 

 and [ — 1, 0] are colliuear. Therefore 



y(j,k) x(j\k)+ 1 



y(l,k-j+ 1) x(l,k-j+ 1) + 1 



Substituting the values found above and simplifying gives 



(k —f)-x (j, k) - y (j, k) =j - k. 



Similarly [x(j, £), y (j, £)], [x(J,j + 1), y{j,j + 1)], and [1, 0] are 

 collinear. The resulting equation is 



j-x(j, k) + y(j, k) =j. 



