362 Mr. E. J. Nanson on the Relations between 



the analogue of radiating bodies is to be sought at all in 

 ordinary mechanical or acoustical systems vibrating about 

 equilibrium. For the latter, even when gyratory terms are 

 admitted, give rise to equations involving the square of the 

 frequency ; and it is only in certain exceptional cases, e. g. 

 (31), that the frequency itself can be simply expressed. On 

 the other hand, the formula? and laws derived from observation 

 of the spectrum appear to introduce more naturally the first 

 power of the frequency. For example, this is the case with 

 Balmer's formula. Again, when the spectrum of a body 

 shows several doublets, the intervals between the components 

 correspond closely to a constant difference of frequency, 

 and could not be simply expressed in terms of squares of 

 frequency. Further, the remarkable law, discovered inde- 

 pendently by Bydberg and by Schuster, connecting the con- 

 vergence frequencies of different series belonging to the same 

 substance, points in the same direction. 



What particular conclusion follows from this consideration, 

 even if force be allowed to it, may be difficult to say. The 

 occurrence of the first power of the frequency seems suggestive 

 rather of kinematic relations* than of those of dynamics. 



XLVI . On the Relations between the Coaxial Minors of a Deter- 

 minant. By E. J. Nanson, M.A.] 



1. TT has been shown by Major MacMahonJ that the 

 JL coaxial minors of any determinant of order n are con- 

 nected by 2 n — n 2 + n — 2 relations, the determinant itself being 

 included under the term coaxial minor. In this Journal Dr. 

 Muir§ has given a simple proof of this theorem and, in the 

 case of an inversely symmetrical determinant, has obtained 

 one of the two relations which connect the coaxials of a deter- 

 minant of the fourth order. 



In the present communication it is proposed, first, to find 

 in several forms the second relation between the coaxials of 

 the special determinant considered by Dr. Muir ; and second, 

 to find the relations between the coaxials of the general 

 determinant of the fourth order. 



2. The special determinant to be considered may be written 



* E. ff. as in the phases of the moon. 

 t Communicated by the Author. 

 I Phil. Trans, clxxxv. (1894) p. 146. 

 § Phil. Mag. Dec. 1894, p. 537. 



