400 NEWTON S PRINCIPIA. 



mean and actual distances, A the coefficient of 3 cos 2 6 1 

 in the preceding formula when the moon is at her mean 

 distance. Accented letters apply to the sun in the same 

 way as unaccented do to the moon. 



In this expression we see three kinds of terms. First, 

 those of very long period, and which are not dependent on 

 the hour of the day. These arise from the first line in the 

 above series, and cause differences in the mean elevation 

 of the water, depending on the declination and distances 

 of the disturbing bodies. As the expression contains the 

 factor 3 sin 2 A 1 there will be no such terms or inequali 

 ties for any place whose latitude is sin&quot; 1 *J\. 



Secondly. The terms in the second line going through 

 their values in about a day, they form therefore a diurnal 

 tide. This has no existence for any place on the equator 

 or at the pole, and is greatest in latitude 45. There will 

 be variations in the magnitude of this tide, depending on 

 the changes of declination and distance of the heavenly 

 bodies. 



Thirdly. The terms in the third line go through their 

 values twice in about a day ; they form together, therefore, 

 a semi-diurnal tide. This has no existence at the poles, 

 and is greater the nearer the place is to the equator. 



The value of this theory may be best stated nearly in 

 the words of the Astronomer Royal : &quot; The most con 

 spicuous tide on the coasts of Europe at least is the semi 

 diurnal. The acceleration or retard of this tide on the 

 moon s transit does not at one port in a hundred agree in 

 any measure with the result of this theory.&quot; &quot; The abso 

 lute elevation of the tide is great at one port and small at 

 another, without any relation to the quantity calculated 

 from theory. The proportion of the elevations, however, 

 at the same port in different stages of the lunation agree 

 pretty well with the theory (though not equally at all 

 ports), yet the critical phenomena (spring and neap tides) 

 occur later than the theory gives them.&quot; &quot; The peculiar 

 phenomena of river tides are not touched on by this theory.&quot; 

 &quot; The diurnal tide ought in these latitudes to be equal, 



