22 Mr. W. Sutherland on a 



straight line. AC = BD = 27rca>/(© + X2); CB = DA = 2<7rcft)/ 

 (&)•— XI). With a? as abscissa measured from A the curve in 

 dots is composed as far as C of y == sin irx/AC, and from C 

 to B of ?/= sin7r^/CB, while the curve of dashes consists 

 from A to D of y = sin 7r^/DA, and from D to B of 

 t/= sin7r#/BD. The full curve is got by summing the 

 ordinates of these two. It represents a periodic, but not 

 simple harmonic, function of wave-length 



AB = 27rcco { 1/ (« + O) + I/O - O) } = 4<7rc6> 2 /(ft> 2 - O 2 ) 



derived from the original wave-length 4,ttc. "With w=l/\ 

 Ry dberg's extension of Balmer's formula is n=n — b/ (m + /x.) 2 , 

 reducing to Balmer's when //, = 0. If we compare this with 

 the formula just derived for wave-length we get, if V is 

 velocity of light and v that of disturbance in atom, 



n = Y/47rcv and {b/n^l(in + f^ — D./w. . (9) 



Let us suppose that this disturbance travelling with angular 

 velocities co and —ft) is a deformation caused in the stuff of 

 the atom, that is in the medium formed by the constitutive 

 pairs of electrons, and that the circle of radius c is the 

 equator of the spherical atom whose poles are the ends of its 

 electric axis, or a small circle parallel to the equator, giving 

 0)>O. This deformation is caused by a displacement of the 

 special pair of electrons which brings its negative electron 

 nearer to the most deformed part of the spherical surface of 

 radius R. This negative electron is the cause of the dis- 

 turbance and also determines the node, so that its angular 

 velocity is O. Since v is the velocity of propagation of the 

 disturbance through the stuff of the atom, then v = cm. Next 

 consider the small circle parallel to the equator whose radius 

 is c/m, where we shall first suppose m to be an integer. Then 

 a disturbance travelling round this circle with velocity v will 

 have angular velocities rruo and —"moo. In this way by con- 

 sidering the special pair of electrons as cause of elastic dis- 

 turbance in the atom, and as controlling the position of nodes 

 in the circles of radii c/m, we can give a simple kinematical 

 account of Balmer's formula. To pass to Rydberg's form we 

 would need to find a kinematical reason for angular velocities 

 which are proportional to m + fx,, where //, is a fraction cha- 

 racteristic of a spectral series. In my previous paper I have 

 shown that m may take the form m + l/n where n is an 

 integer, and even m+p/q where p and q are integers. But 

 for the present we shall confine our attention to the essentials 

 as they appear in the formula of Balmer. We have a funda- 



