( 355 ) 



q q' cos (nt -{- c) cod (71't -\- c') =z ^ q q' cos [{n — ti') t -\- {c — c')] -{- 



+ i y 7' co« [ (n + 7i') i + (c + c')]. 



Of the two vibrations, I shall only consider the one whose fre- 

 quency is n—n'. 



§ 13. It is easily seen, and may be verified by working out some 

 example, that we can obtain a secondary vibration of the first order, 

 i. e. one which really emits light, by combining a vibration of the 

 second order with one of the first order, these primary motions being 

 executed either by the same sphere, or by two concentric shells. 



Let us now imagine the three vibrations corresponding to the 

 functions Yx, Yy and Yz, and the five vibrations determined by Y^y, 

 yxy = ^{Yyy — Yxx), Yxj) ^i/3) ^2-- Let the factor p that has been intro- 

 duced in § 4 be of the form (15) for one of the former vibrations, and 

 of the form (16) for one of the latter. By considering the symmetry 

 of the system, it may be shown that a secondary vibration in the 

 direction of one of the axes of coordinates can only be produced by 

 the combination of these two, if, among the three indices of the two 

 harmonics, the one that corresponds to the axis considered, appear 

 an uneven number of times. Thus the mutual action of a Vj-y- and 

 a Fj -vibration will call forth only a vibration in the direction of OF. 



Another question is to determine the amplitudes of the derived 

 vibrations taking place along OX, OT and OZ. In every special case 

 the amplitude must be proportional to qq' ; we may therefore denote 

 it by multiplying qq' by a certain amplitude factor. 



These factors are not independent of one another ; they may all 

 be expressed in terms of one of them. To understand this, it must 

 be kept in mind that, if the first of the two combined primaries 

 a and b be decomposed into some components, say into aj, aj, etc., 

 the secondary vibration [a, b| may be considered as the resultant 

 of |ai, b}, {aj, b}, etc. If we denote the amplitude factors by 

 [I'.r.v, Yx]-K, etc., the last index indicating the direction of the 

 secondary vibration, we shall have 



and 



The last formula is a consequence of the relation 

 Yxjc + Tyy + y,, = . 



