362 REFORTS ON THE STATE OF SCIENCE. 



Beltrami's work has recently been extended by Cesaro. 1 

 P. Hertz has shown 2 that the theory of Einthoven's string galvano- 

 meter depends upon an integral equation of Poisson's type. The 

 differential equation for the motion of the string, when the electro- 

 magnetic damping only is considered, is of the form 



where a and b are constant. The initial conditions are of the type 



**-?• (aL =0 - 



The equation is solved according to Eayleigh's method by putting 



v = 2r, (0 sm 

 i l 



where y,.(o) = 0. y,(o) = 0. 



Expanding F in a Fourier's series 



■m «^ 4F • VTTX 

 F = 2 — sin 



l'7T /. 



(.' = 1,3,5 . . . ) 

 and putting 



•(0 = F(0 - ^ (*' +1 + ) = F-6 ( d gdx 



we obtain the integral equation 



F(0 = <!>(/) + f N(* -'t))l(ri)dd. 



for the determination of <I> (/). The function N(<) is defined by the 

 equation 



(.' = 1,3,5 . . . ) 



and is such that ^ = + 2ab or — 2ab according as t lies between 

 ot 



n „ I -, 2ml i 2ml -.(Im + Vjl 



2m — 1) - and — or between and v — >-. 



a a a a 



Hertz also considers the more general equation 



— S-'R + ^-»ft* 



which corresponds to the case when the resistance of the air is taken 

 into account. 



1 Bull, tie Hoy. Acad. Belgique, 1902, p. 387 ; Cumptes renchts, Naples, ser. 3, t. 7, 

 pp. 284-289. 



2 ZeiUchrift fur Math, in Phys., 1909, Bd. 58. 



