[ 780 ] 



LXXXIII. A Method of Finding the Coefficients of Absorption 

 of the Different Constituents of a Beam of Heterogeneous 

 Rontgen Rays, or the Periods and Coefficients of Damping 

 of a Vibrating Dynamical System. By Sir J. J . Thomson, 

 O.M.,F.M.S* 



I HAVE found the following method useful in finding 

 the absorption coefficients of the various constituents 

 of a beam of heterogeneous Rontgen rays. In this case 

 the intensity of the rays after passing through a thick- 

 ness d of aluminium can be represented by the expression 

 A 1 6~ X ^ + A 2 6" M +...A, l €- x ^ ; the problem is to find 

 from the 2n observations which are necessary for this 

 purpose the values of A l5 A 2 , . . . A» ; X 1? 7u, . . . \ n . Though 



1 arrived at the method from the consideration of the 

 absorption problem, it is of much more general application 

 and applies to any quantity which can be represented as the 

 sum of a series of exponentials of some variable, whether 

 the coefficients of the exponent are real, imaginary, or 

 complex. Since the amplitude of the vibrations, damped 

 or undamped, of a dynamical system about a position of 

 equilibrium can be represented by a series of the form 



2 Ae^ a+l ^> the method can be employed to find the periods 

 and damping coefficients of a dynamical system, and is 

 especially useful when we have a graph representing the 

 variation of the displacement of the point in the system 

 with the time. In the case of undamped vibrations it 

 gives a simple method of finding the simple periodic terms 

 into which the motion can be resolved. 



The method depends upon the following theorem. Suppose 

 ? /o> Vi) 1/2-, - ' - y%n-\ 'ire 2n observed values, the observations 

 being made at equal intervals #, of the variable : in the case 

 of the X-ray problem this variable is the thickness of the 

 aluminium leaf through which the radiation has passed; 

 in the dynamical problem the variable is the time. We 

 shall denote this variable by t. Then y Q is the observation 

 when t = 0, yi when t — 0,y 2 when t = 20, and so on. 



y = A^^ + A^ + Ase^+A.eH 

 yo = Ax +A 2 +A 3 +A„, 

 yi = Aif +A 2 77 +A 3 ? +.. ., 

 ^ 2 = A 1 f + A 2 7 7 2 -fA 3 ? 2 + ..., \>, («) 



* Communicated by the Author. 



