CONTEMPORARY ADVANCES IN PHYSICS 675 



harmonics. To each value of n belongs a "spherical harmonic of 

 order «," which itself is a sum of (2w + 1) terms, each multiplied 

 by a constant which is at our disposal and can be adjusted to fit initial 

 conditions or to emphasize particular modes of vibration. These 

 terms are products of sine-functions of (p by peculiar functions, the 

 Legendrian functions Pn, «, of the variable 6; so that the Eigenjunktion 

 for a permitted value n{n -\- \) oi the parameter X has this for its most 

 general form: 



n 

 Yn{d, <p)=an, oPn, o(cOS ^) + Z dn, s COS {Sip)Pn, s(cOS 6) 



'" n (135) 



+ Z ^n, s sin {S(p)Pn, s(cos 6). 

 s=l 



Each term by itself describes a particular mode of vibration of the 

 fluid; the sum represents a superposition of divers modes of vibration. 

 If we isolate one of these modes by giving to n some particular value 

 Wi, and to 5 some particular value 5i, and causing all the constants a 

 and b in (135) to vanish except a„,, si and b„^^ s^; we then find that Y, 

 and consequently "if, and consequently the motion altogether, vanishes 

 at Si values of (p and at Wi — Si values of d. If we draw a sphere 

 centred at the origin, we find that its surface bears 5i nodal meridian- 

 circles, and Wi — 5i nodal latitude-circles, along which there is per- 

 petual rest. If we consider all the spheres at once — if, that is to say, 

 we consider the entire volume of the fluid medium — we see that when 

 the fluid is vibrating in the mode distinguished by the integers (I had 

 almost said "quantum-numbers"!) tii and Si, it is divided into com- 

 partments by 5i nodal planes intersecting along the axis 6 = 0°, and 

 Wi — 5i double-cones having that axis for their axis and the origin 

 for this apex. 



We have not yet considered the dependence of the wave-motion 

 on the radius r; but the close analogy between this and the corre- 

 sponding stage of the problem of the tensed membrane will make the 

 task easy. The differential equation (133) ior f{r) resembles Bessel's 

 equation (127), and has the somewhat similar solution 



/(/-) = J- /„^, (,;;,). (136) 



■\r 



used five or six times; the values of the constant 5 in equation (135) are the Eigeii- 

 werte of the latter of these two. I thought it desirable not to overload the exposition 

 by carrying through all stages of the process of solution, especially as the splitting 

 of Yn{d, 4>) into the two functions is of secondary importance in the atom-model to 

 which all this leads up; nevertheless the reader may find it advantageous to supply 

 the lack. 



