1892.] liquid condition of the Earth's interior. 343 



At an angular distance of 45° from the moon the water is 

 at rest, and its depth is the mean depth of the canal, viz. h. This 

 makes z the depth at P ; 



.-. = -pgh + C. 



Hence the work on a unit mass of water in the column at 

 Pis 



p ^cos2f- g(z-h) 



= P ( y cos 2 ^ ~ 9y) 



= p (^ cos 2f + g —M cos 2f[ 



pa a 2 co 2 _ . 



= P ^-tt ^—9 T COS Z-vZr. 



H 2 a'co'-gh r 



Therefore the work on the whole column of unit width is 

 pa a 2 co 2 a ,, s 



pa a 2 co 2 a , fi P a h o , N 



= P ^ -^—5 ; COS 2^1r [h ■— 7j- V^ f cos 2 T 



^ 2 aW-gh r \ 2 aW-gh T j 



(pa , a 2 to 2 o , />a aw \ 2 A /1 .} 



= P1^' l -r^ r COs2i|r- [ r ~ -^— z -(l + COs4-f)k 



r [ 2 aw—gh T \2 a'to—ghj 2 X r 7 J 



To obtain the work done on a length J.P of the canal we must 

 multiply this by acZ-v/r and integrate, whence, putting for -yfr its 

 value ^ — &)£, and taking the integral from t = to £ = £, we get 



work on AP — p 



t-r- h -s— » , {sin 2(6- cot) — sin 2(6] 



4 aco—gh 1 - ^ r> 



(pa aco \ /ia 

 + lT«V^AJ 2 w * 



1 Aua aaj \ 2 ha , . . , , jX . . , ' 





The work on the whole canal will be given by putting (6 = 27: , 

 and will be, 



work on the whole canal = p \——t- h—»-T, j sin 2cot 



r { 4 aV - gh 



(pa aa> \ 2 ha 



+ {Y aW-gk) ~2 at 



1 (pa aco \ 2 ha . . ,) 



4 V 2 aV - gh 2 ) 



