1886.] Dynamical Theory of the. Tides of Long Period. 341 

 so that 



Bs=~-ZEN x Nto B 5 = -ZEN^JV* £ 7 = -2M#jft &e. 

 and 0=-p7+2^1. 



Then the height of tide Ij is equal to h cos (2w/£ + a), the equilibrium 

 tide .e is equal to jE(^—/u?) cos (2nft + cc), and we have 



&=«+#(£ -^ 2 ) 

 = + ^-^ + 1/^ + ^1-/^)^ + 1^-/^)^ + . - • 



±J*£k - (i +/2jy i >2+i^ 1 (i +M> 6 + . . 



Now when /3=40, we have ^=^~x4ima=~a= 7 260 feet; so that 

 73=40 gives an ocean of 1200 fathoms. 



With this value of /3, and with /= -0365012, which is the value for 

 the fortnightly tide, I find 



^=3-040692, iV 2 =l-20137, JV" 3 =-66744, iV 4 =-42819, iV 5 ='29819, 

 jV 6 = -21950, N 7 = -16814, iV 8 =-13287, ^ 9 ='107, N 10 ='l, &e. 

 These values give 



jb^- 15203, l+ZW^l-0041, JJV 1 (1+/2JV 2 ) =1-5228, 



£Mifl +/ 2 JV3)=r2187, i^ 2 F 3 (lt/2JV 4 )=-6O90, 

 . . . iV 4 (l+/2iY 5 ) = -2089 iN-L . . . ^(1+/2JV 6 ) = -0519, 

 1 • • • ^(1 = -0098, pT x . . . iV 7 (l+/2JV 8 )=-0014, 

 . . . JV 8 (1+/2JV 9 ) = -00017, Ac. 



So that 



^ = •1520-l-004V + l-5228 / a 4 -l-2187 y a 6 + -6099^ 8 --2089^ 10 



+ -0519 / t 12 --0098^ 4 + -0014 / a 16 --0002 / ai8 + ... . 



At the pole, where /<=1, the equilibrium tide is — §£7; at the 

 equator it is + 



Now at the pole h = -Ex -1037= —\E x -1556, 



and at the equator 7* = +jB7x*L520= JEx -4561. 



In a second case, namely, with an ocean four times as deep, so that 

 /3=10, I find 



