178 Mr. D. D. Heath on the Dynamical Theory of 



.,. .,, , . , ds d0 , 



transverse to the meridian will be respectively -7-, -77, ancl 



— — for -j- angular), and the true velocity eastward will be 



( n — — \ sin 0. The centripetal force set free by the west- 

 ward velocity is therefore (4(a)), 2n sin — directed inwards 



along a perpendicular to the axis; and this being resolved ver- 

 tically and horizontally, the latter component (with which alone 



we have to do) is 2n sin cos -r- northward. 



' dt 



Again, the velocity of the side of the canal at [0, co) is n sin 0, 

 and at the other extremity of ds it is n sin (0 + d0), or 



n sin + n cos 0d0. 



The pressure of the side of the canal (which is normal to the di- 

 rection of ds) carries, therefore, the particle which is moving 

 along it, in the time dt, through the space n cos d0 dt eastward 

 of the place it would have reached by its own unrestrained velo- 

 city. Now the southern component of this normal pressure is to 

 the eastern as sin 6 da is to d0 ; it would therefore be competent 

 in the time dt to push the particle southward through the space 

 n cos sin dcodt, exactly the space through which the centrifugal 



/ ft*\ 



force would ( by the formula s=z — ) have pushed it northwards. 



Therefore these forces counteract each other ; and the motion of 

 a free wave in any canal is determined by the depth as before. 



Next, as to the forced wave. The moon's horizontal disturb- 

 ing force in the direction of the path ds may be determined di- 

 rectly ; or it may be immediately deduced, from one of the fun- 

 damental propositions of physical astronomy (which is very easily 

 proved), that this force may be found by the differentiation with 



M 



reference to ds of a known function ~ P 2 , where, if 8 be the 



moon's N.P.D. and co' its longitude west, 



P 2 =(fcos 2 8-i)(f cos 2 0-i) + fsin2Ssin20cos(a>-a>') 

 + j sin 2 8 sin 2 cos 2(g) — co 1 ) . 



If we expand this, it takes the form 



A + B cos 6)' -f C sin co' + D cos 2co ( + E sin 2g/, 



in which the coefficients are given for each point in the canal, and 

 co' alone depends on time (treating 8 as constant). We may 

 then treat each of these terms separately, exactly as we have 

 done with the several forces in (7). Each of them may be re- 



