174 Mi-. D. D. Heath on the Dynamical Theory of 



The actual force at the time t is then represented by the pres- 

 sure of a wave whose height at co is 



(H + A cos it) cos 2a> == H cos 2co 



||cos2(, a -|) + cos2(a, + ^)}, 



or, putting - = h; 



. H cos 2o) 4- h i cos 2f g> — 1=)+ A cos L^L.^+t^)' 



The first term is the one we have dealt with, representing the 

 effect of the mean moon moving at the rate m. The other terms 

 represent two waves of the same form as the principal one, but 



whose ridges are at any time t at distances -t to the cast and 



west of it, i. e. which are propagated so as to lose or gain on it 



• z ° . 



at the rate ^- They may be conceived as diie to two imaginary 



" ■ ■ & ■ 

 moons smaller than the mean moon in the ratio of h to IT, and 



t . . ' 



moving w T ith rates m+n> anc ^ the actual tide at any time will 



be found by the superposition of the waves due to these three 

 imaginary bodies. 



If there were no friction, these three waves would each lie 

 with their ridges at right angles to their respective luminaries, 

 and the highest tides would be when these bodies all coincide, 

 or when the force is a maximum. At other times, as the height 

 of the tidal wave in a shallow sea is greater the smaller its rate 

 of progress (i. e. the nearer K is to k), the eastern luminary dis- 

 turbs the main tide more than the western, and the time of high 

 water is slightly retarded after the time of maximum, with a re- 

 verse effect after the minimum. 



But when we take account of friction, the ridges are at un- 

 equal distances east of their respective luminaries, and the time 

 of greatest tides, which is when the ridges coincide, is not the 

 time of greatest force. When i is small, or the change of force 

 slow, the interval may be thus found, taking account only of the 

 first powers or differentials in the changes :- — 

 i 



m and m ± ~ being the rates of progress of the mean and sub- 



sidiary moons, if K and K + AK, $ and S + ebe the correspond- 

 ing quantities in our formulas, we have approximately 



(m±0 2 = wi 9 ±m*=0(K± AK) ; or AK= — = tyf*- 



