xx.] METHOD OF VARIATIONS. 453 



leaves the cistern in a given time depends upon the 

 aggregate result of that velocity, and is only to be 

 ascertained by the mathematical process of integration. 

 When one gravitating body falls towards another, the 

 force of gravity varies according to the inverse square 

 of the distance; to obtain the -velocity produced we 

 must integrate or sum the effects of that law ; and to 

 obtain the space passed over by the body in a given 

 time, we must integrate again. 



In periodic variations the same distinction must be 

 drawn. The heating power of the sun's rays at any 

 place on the earth varies every day with the height 

 attained, and is greatest about noon; but the tempera- 

 ture of the air will not be greatest at the same time. 

 This temperature is an integrated effect of the sun's 

 heating power, and as long as the sun is able to give 

 more heat to the air than the air loses in other ways, 

 the temperature continues to rise, so that the maximum 

 is deferred until about 3 P.M. Similarly the hottest day of 

 the year falls, on an average, about one month later than 

 the summer solstice, and all the seasons lag about a month 

 behind the motions of the sun. In the case of the tides, 

 too, the effect of the moon's attractive power is never 

 greatest when the power is greatest ; the effect always 

 lags more or less behind the cause. Yet the intervals 

 between successive tides are equal, in the absence of dis- 

 turbance, to the intervals between the passages of the 

 moon across the meridian. Thus the principle of forced 

 vibrations holds true. 



In periodic phenomena, however, curious results some- 

 times follow from the integration of effects. If we strike 

 a pendulum, and then repeat the stroke time after time at 

 the same part of the vibration, all the strokes concur in 

 adding to the momentum, and we can thus increase the 

 extent and violence of the vibrations to any degree. We 

 can stop the pendulum again by strokes applied when it 

 is moving in the opposite direction, and the effects being 

 added together will soon bring it to rest. Now if we 

 alter the intervals of the strokes so that each two suc- 

 cessive strokes act in opposite manners they will neutralise 

 each other, and the energy expended will be turned into 

 heat or sound at the point of percussion. Similar effects 



