08 THE PRINCIPLES OF SCIENCE. 



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

 obtain the space passed over by the body in any given 

 time, we must again integrate with regard to the variable 

 velocity. 



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 it does not follow that 

 the temperature of the air is greatest at the same time. 

 This temperature is an integrated effect of the sun's heat- 

 ing power, and as long as the sun is able to give more 

 heat to the air than the air loses in any other way, 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 sun's or moon's attractive power is 

 never greatest when the power is greatest ; the effect 

 always lags more or less behind the cause. Yet the in- 

 tervals between the successive tides are exactly equal, in 

 the absence of disturbance, to the intervals between the 

 passage of the sun or moon across the meridian. Thus 

 the principle of forced vibrations holds true of all such 

 cases. 



In periodic phenomena, however, very curious results 

 will sometimes follow from the integration of effects. If 

 we strike a pendulum, and then repeat the stroke time 

 after time when it is in the same part of the vibration, 

 every stroke concurs with every other one 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 successive effects being 

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



