240 
yiR. S. S. PIOUGH ox THE APPLICATIOX OF HAR^IOXIC 
and therefore 
= • 26 G 9 P 3 - -IGTSP, + •0485Pg - -OOSlPg + 'OOOOPio - •OOOIP 12 + . . . ’ 
In the same manner, when hg/iartt^ = 
= -TOZOP^ - •IGZlP.i + •0285Pe - •0()27P8 + •OOO 2 P 10 - • • • ; 
W-^3 
when hgliorrir — 
^ = -oGOZP. - •1388Pi + -OlSlPc - -OOOGPg + . . . ; 
bo/ 
and when /((/jAoj-a- — 
= • 72 O 8 P 2 - -OOZSP^ + •0048P6 - -OOOlPs + • ■ • 
bo/-*- 3 
The hmar-fortnig’htly dedinational tides have been evaluated by Professor Darwin'^ 
for depths which correspond with the first and third cases given above, the results 
being expressed in series proceeding according to ascending powers of the variable [x. 
If we replace the various powers of p, by their values in terms of the zonal 
harmonics,! we may deduce the followfing series from those given b}’- Professor 
Darwin ; when hgjifohi'^ — we find 
= • 2889 P 3 - T755P, -}- •0490Pg - •0079Ps + -00091^10 - . . . , 
bo/ -* 2 
while, when hgjiojhi- = y,-, 
^ = •59G9Po - •1385Pj,+ •012GPg - -OOOGPs + . . . 
The difierence between these expansions and those we have given above, is to be 
explained by the fact that we have included in our analysis the eftects due to the 
attraction of the Avater on itself. I have re-computed the lunar-fortnightly tides, 
starting with the assumption that pjcr = 0 , and obtained practically identical results 
by the two methods. 
We see then that the effect of the gravitational attraction of the water is to diminish 
the tides, as compared with the equilibrium tides, in the first case by about 8 per 
cent., and in the second by about 5 per cent. 
* ‘ Eucyc. Brit.,’ Art. “ Tides.” § 18. 
t Ferrkks : ‘ Spherical Hariuonics,’ p. 27. 
