146 Lord Kayleigh on the Maintenance of 



vigorous vibration whose 'period is the double of that of the 

 point of attachment "*. Other examples of acoustical interest 

 are mentioned in the paper. 



My attention was recalled to the subject by Mr. Glaisher's 

 Address to the Astronomical Society f, in which he gives an 

 interesting account of the treatment of mathematically similar 

 questions in the Lunar Theory by Mr. Hill J and by Prof. 

 Adams§. The analysis of Mr. Hill is in many respects in- 

 comparably more complete than that which I had attempted ; 

 but his devotion to the Lunar Theory leads the author to pass 

 by many points of great interest which arise when his results 

 are applied to other physical questions. 



By a suitable choice of the unit of time, the equation of 

 motion of the vibrating body may be put into the form 



^ +2&J + (©0 + 2©! cos 2t)w=0- . . . (1) 



where k is a positive quantity, which may usually be treated as 

 small, representing the dissipative forces. (® + 2© x cos 2pt) 

 represents the coefficient of restitution, Avhich is here regarded 

 as subject to a small imposed periodic variation of period it. 

 Thus © is positive, and © x is to be treated as relatively 

 small. 



The equation to which Mr. Hill's researches relate is in 

 one respect less general than (1), and in another more general. 

 It omits the dissipative term proportional to k ; but, on the 

 other hand, as the Lunar ' Theory demands, it includes terms 

 proportional to cos At, cos 6£, &c. Thus 



^ + (® o +2© 1 cos2* + 20 2 cos4* + ...)w = Os . (2) 



^+©^ = 0, (3) 



where 



© = 2 W ©„^, (4) 



n being any integer, and i representing N /( — 1). In the 

 present investigation ©_„=©„. 



* " See Tyndall's l Sound,' 3rd. ed. en. iii. § 7, where will also be found 

 a general explanation of the mode of action." 



t Monthly Notices, Feb. 1887. 



% " On the Part of the Motion of the Lunar Perigee which is a Func- 

 tion of the Mean Motions of the Sun and Moon," Acta Mathematica, 8 :1, 

 1886. Mr. Hill's work was first published in 1877. 



§ " On the Motion of the Moon's Node, in the case when the orbits 

 of the Sun and Moon are supposed to have no Eccentricities, and when 

 their Mutual Inclination is supposed to be indefinitely small." Monthly 

 Notices, Not. 1877. 



