292 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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ten "' = zjtw (13) 



and if the first root be used, the amplitude at the first elongation is 



'4- ff 

 e~ at ' [ff • cos pt' + — sin />*']. (14) 



For the motion defined by (5), (6), (9), and (10), therefore, the first 

 amplitude can be found by substituting for 6' and w' in (13) and (14) 

 the values given by (11) and (12). The computation is, however, not 

 very simple, and we shall do well to treat the matter graphically, using 

 equation (9) as the basis of our work. 



If we define the function F(t) by the equation 



F(t) = e~ a( sin pt (15) 



and denote the constants — , — by p and q, (9) may be written in the 



P 9 

 form 



6=p.F(t) + q-F{t-h). . (16) 



For any given galvanometer with a given resistance of the coil circuit 

 a andp are definite, easily determined constants, and F(t) is therefore 

 determined. For the galvanometer represented by Figure 1, Plate 1, 

 for instance, p is twice a for a coil circuit resistance of about 150 ohms. 

 If we represent pt by x, pt x by X\, and the ratio of a to p by p., then 



6 =p ■ e~* x sin x + q ■ e~^ Jr ~ x ^ sin (x — Xi)=p-f(x) + q-f(x — Xi). (17) 



If then we draw the curves y = p • f(x), y = q '/(a), the ordinates 

 of which are in the constant ratio p/q, and displace the second curve 

 bodily to the right through the distance %i, the sum of the ordinates of 

 the first curve and the displaced curve will represent 0. For most 

 purposes only the ratio (r) of q to p is important, and in plotting the 

 curves we may make p = 1 and q, r. 



To illustrate the process just described, let us suppose that when the 

 galvanometer coil is at rest in its position of equilibrium, an impul- 

 sive current is sent through it, and after the coil, in response to this 

 impulse, has had about half time enough to reach its elongation, a 

 second impulse is given it half as strong as the first. The general 

 form of the diagram will be much the same whether the damping be 



