XX.] METHOD OF A'ARIATIONS. 451 



ill a period has probably been on the increase. "At first 

 the variability was more rapid, then it became gradually 

 slower ; and this decrease in the lent'th of time reached 

 its limit between the years 1840 and 1844. During that 

 time its period was nearly invariable ; at present it is again 

 decidedly on the decrease.^ The tracing out of such 

 complicated variations presents an unlimited field for in- 

 teresting investigation. The number of such variable stars 

 already known is considerable, and there is no reason 

 to suppose that any appreciable fraction of the whole 

 number has yet been detected. 



Principle of Forced Vibrations. 



Investigations of the connection of periodic causes and 

 effects rest upon a principle, which has been demonstrated 

 by Sir John Herschel for some special cases, and clearly 

 explained by him in several of his works.- Tlie principle 

 may be formally stated in the following manner : " It one 

 part of any system connected together either by material 

 ties, or by the mutual attractions of its members, be con- 

 tinually maintained by any cause, whether inherent in 

 the constitution of the system or external to it, in a state 

 of regular periodic motion, that motion will be propagated 

 throughout the whole system, and will give rise, in every 

 member of it, and- in every part of each member, to 

 l)eriodic movements executed in eqiial periods, with that 

 to which they owe their origin, tliough not necessarily 

 synchronous with tlieni in their maxima and minima." 

 The meaning of the proposition is that the effect of a 

 periodic cause will be periodic, and will recur at intervals 

 equal to those of the cause. Accord ingl}' when we limi 

 two phenomena which do proceed, time after time, through 

 changes of the same period, there is much prolmbility 

 tliat they are connected. In this manner, douljtless, Pliny 

 correctly inferred that "the cause of the tides lies in the 

 sun and the moon, the intervals between successive high 

 tides being equal to the intervals between the moon's 



' Ifiiiiil)i)l(lt'.s Cosmos (Bohn), vol. iii. p. 229. 



'^ Encyiloimdla Mdropolitdua, art. ,Suuud, § 323 ; Outlinfj^ of 

 Astniaoiiui, 4tli edit., § 650. pp. 410,487 — 88; Meteorology, Etiri/- 

 vl'Uxiidia JJrilunnica, Reprint, p. 197. 



G G 2 



