1886.] Equilibrium Theory of Tides for the Continents. 305 



— 1 |cos 2 Xcos'2ZcZs=cos 2 X 2 cos 2Z 2 , — !— cos 2 X sin 2ZcZs=cos 2 X 2 sin 2Z 2 , 



ttQJJ 4?rQJJ 



i— J [sin 2X cos Zc?s=sin 2\ 1 cos \ t -^—p\ sin 2X sin ZcZs=sin 2X 2 sin Z 1? 



tQJJ " 47rQJJ 



i ^jj(|sin 2 X-i)^=|sin%.-l (5) 



the integrals being taken over the oceanic area. 



These five integrals are called by Sir William Thomson %, ^, (£, 

 $9, but by introducing the five auxiliary latitudes and longitudes, 

 X 2 , Z 2 , X l5 Z 1? X we shall find for the conclusions an easily intelligible 

 physical interpretation. 



It may be well to observe that (5) necessarily give real values to 

 the auxiliaries. For consider the first integral as a sample : — 



Every element of ff cos 2 \ cos 21 ds is, whether positive or negative, 

 necessarily numerically less than the corresponding element of 47rQ, 

 and therefore, even if all the elements of the former integral were 

 taken with the same sign, (47rQ) _1 ^/cos 2 \ cos 21 ds would be numerically 

 less than unity, and a fortiori in the actual case it is numerically less 

 than unity. 



Now using (5) in obtaining the value of ff (cos 2 z—^)ds, and 

 substituting in (3), we have — 



!] 5 ' icos 2 a[cos 2 Xcos2(A-Z)-cos 2 X 2 cos2(7i-Z 2 )] 



a ' 2y(l -f<r/^)r 3 



-f sin 2£[sin X cos X cos (Ji — l)— sin X x cos Xj cos (Ji Zj) ] 

 + |-(J-sin 2 3)(sin 2 X -sm 2 X) (6) 



The first term of (6) gives the semi-diurnal tide, the second the 

 diurnal, and the third the tide of long period. 



The meaning of the result is clear. The latitude and longitude 

 X 2 , Z 2 is a certain definite spot on the earth's surface which has 

 reference to the semi-diurnal tide. Similarly X-,, Z x is another 

 definite spot which has reference to the diurnal tide ; and X is a 

 definite parallel of latitude which has reference to the tide of long 

 period. 



From inspection we see that at the point X 3 , Z 2 the semi-diurnal tide 

 is evanescent, and that at the point X 2 , 7 2 + 90° there is doubled tide, 

 as compared with the uncorrected equilibrium theory. At the place 

 X 1? l Y the diurnal tide is evanescent, and at —\, l x there is doubled 

 diurnal tide. 



In the latitude X the long period tide is evauescent, and in 

 latitude (sometimes imaginary) arc sin — sin 2 X } there is doubled 

 long period tide. 



