582 



SCIENCE 



[N. S. Vol. LII. No. 1355 



venieut combination of the conditions. The 

 set of equations (1) can be written in the 

 form 



X 



-7- Qidt = riih, 

 dqi 



rrdT. 

 Jo ^3= 



! = %/i, 



(2) 



■)h. 



Adding, we obtain 



By Euler's theorem for homogeneous func- 

 tions, the integrand of the left hand member 

 is equal to twice the kinetic energy. Con- 

 sequently this integral is equal to the action 

 of the system for the type of motion under 

 consideration. Denoting the sum of the in- 

 tegers n.j, TCj, etc., by n, we have 



A 



■■ r2Tdt = ' 



n = (0), 1, 2, 



(3) 



This integral is invariant of the choice of 

 coordinates and can be evaluated easily if 

 the orbit and potential energy function are 

 known. Equation (3) is not equivalent to the 

 quantum conditions (1), but it is a deduction 

 from them for the type of problem under con- 

 sideration, which is suiBcient to fts the pos- 

 sible energy values of the atom or molecule. 

 In the normal state the atom will have the 

 least energy possible and the quantum number 

 n should therefore be small, though the value 

 zero must be ruled out if there is to be any 

 dynamic equilibrium at all. In the case of 

 the helium atom or the hydrogen molecule, it 

 is to be expected that n will be either one 

 or two. 



I have carried tlirough the numerical evalu- 

 ation of the action integral for the helium 

 atom model and regret to say that the calcula- 

 tion shows that if the atom is given an energy 

 corresponding to its ionization potential, the 

 quantum condition (3) is not satisfied. 



In making the calculation I have used an 

 approximate expression for the path of the 

 electron. This is permissible, since, by the 

 principle of least action, the variation in the 

 integral produced by a small variation in the 



path, holding the total energy constant, van- 

 ishes to small quantities of the first order. 

 The determination of the approximate path 

 was based on the data furnished by Dr. Lang- 

 muir. He says that the path of each electron 

 is very nearly an arc of an eccentric circle 

 subtending an angle of 155° 56' at the nucleus. 

 The radius vector from the nucleus to the 

 midpoint of the orbit is 0.2534 X lO-s cm. for 

 an ionization potential of 25.59 volts, and the 

 radius vector at the end of the orbits is 1.138 

 times as great. By expanding the expression 

 for the radius vector into a power series in 9 

 (the angle between the momentary radius vec- 

 tor and the radius vector to the midpoint), 

 and discarding higher power terms, it is easy 

 to show that an equation of the form 



r = !-o(l + ke^) (4) 



can be used to define an approximate orbit. 

 Here r„ is 0.253 X 10"^ cm. and h is easily 

 calculated from the known values of r and 6 

 at the end of the path. 



The expression for the potential energy of 

 the system is 



(5) 



* = -4£! + 



2r cos e ' 



where e is the charge on the electron. The 

 total energy W is easily calculated from the 

 above equation by inserting the values of 

 r and 9 for the end of the path. The kinetic 

 energy of the two electrons is 



T = TF - * = mi>2. (6) 



By means of equations (5) and (6) the ex- 

 pression for the action is easily transformed 

 into the form 



A = 4 / " 2mpds 

 Jo 





(7) 



where s is the rectified length of the path from 

 its midpoint to the point (r, 9), and s,n is the 

 maximum value of s. The graphically deter- 

 mined value if the integral which forms the 

 right hand member of (7) is 1.57 h. This re- 

 sult is in conflict with the quantum condition 

 (3) and shows that if the quantum conditions 

 (1) are correct, the Langmuir model of the 



