296 PROCEEDINGS OF THE AMERICAN ACADEMY. 



similar to that which we have discussed above for the case where a is 

 less than ($. If </>(£) is defined by the equation 



<H0 =o.te~ at , 

 * = ~*<0 + ~* (*-*)■ (23) 



Figure F shows the form of the curve y = xe~ x . 



In considering the magnitude of the throw of a damped ballistic gal- 

 vanometer due to a given continuously varying current which flows 

 through the coil for a finite time interval, we shall do well to use Dorn's 

 results in nearly the forms into which they have been put by Diessel- 

 horst in his important paper on the subject. 



When the suspended system is at rest in its position of equilibrium, 

 a short-lived current shall flow through the coil and shall have the in- 

 tensity, I, which is a given function of the time. From the epoch 

 t=r,I shall have the value zero. The product of the strength of the 

 magnetic field between the poles of the permanent magnet, at the place 

 where the coil is, and the effective area of the turns of the coil shall be 

 denoted by q, so that while the current is flowing, the equation of 

 motion of the coil, for such small angles as are used in mirror instru- 

 ments, has the form 



or 



If, as before, m and n are the roots of the equation ,r 2 -f 2 ax + fi 2 = 0, 

 if Qt represents the whole flux of electricity through the coil from 

 t = to t = t, and if M t , Nt represent the ratios to Qt of the integrals 



f'l- e~ mt ■ dt, i I- e-** • dt, (26) 



respectively, then the solution of (25) is 



6 = — ^— \e mt ■ fl e~ mt -dt-e nt f I- e^ 1 ■ dt~\ (27) 

 m — n Jo Jo 



^ [l/r^-iWel. (28) 



m — n 



