CONTEMPORARY ADVANCES IN PHYSICS 669 



pendent of both / and x, and equal to a constant which (for the sake of 

 consistency with prior notation) I denote by — mr. Solutions of these 

 differential equations into which the underlying one was broken up 

 are these : 



f = A cos mx + B sin mx, g = C cos nnU + D sin mut. (116) 



So far, there is no limitation upon m. 



Now come the boundary-conditions, formulated thus: 



m =KL) = 0. (117) 



We have now encountered, in its simplest example, the peculiar and 

 characteristic problem of the Theory of Acoustics, which is also the 

 peculiar and characteristic problem of the type of Atomic Theory which 

 is inherent in wave-mechanics. This is not the question which we 

 meet in the theory of moving particles, where we are asked what path 

 a particle will follow through all future time if its position and velocity 

 at a single moment are given. A similar question will indeed presently 

 be asked and answered ; but this peculiar problem intrudes itself at the 

 beginning. 



To adjust the function /(x) to the boundary conditions, it is evident 

 that we must set A = and sin mL = 0; therefore we must assume 

 that m has one of the values : 



m = kir/L k = 1, 2, 3, 4 •••. (118) 



The boundary-conditions have compelled the coefficient m to choose 

 among a rigidly defined series of values. The wave-lengths, and conse- 

 quently the frequencies, of the permitted vibrations are strictly deter- 

 mined. 



The permitted values of m are known in German as the Eigenwerte 

 of the differential equation for the boundary-conditions in question. 

 The English term would be "characteristic values"; but it is long and 

 has many meanings, and I think it preferable to borrow the German 

 word as a foreshadowing of the application which Schroedinger has 

 made peculiarly his own. To each Eigenwert of m there corresponds a 

 value of the vibration-frequency mu/lir, which in German is called an 

 Eigenfrequenz; but here we may as well keep to the English term 

 natural frequency. 



To each Eigenwert there corresponds a solution of the differential 

 equation, an Eigenfunktion. In the present instance the Eigen- 

 funktion corresponding to the Eigenwert m = UtJL is: 



yk = sm -j-x I Ck cos—j—t + T> k sm -j— t 1 • (119) 



