16 U. S. COAST Al^D GEODETIC SURVEY. 



proximate value of (0.05)^ or 0.0025, and a fraction of the third 

 order having an approximate value of (0.05)^ or 0.000125, etc. These 

 formulas of the second order should, therefore, give the results cor- 

 rect to the third decimal place. 



In Table 2 are given the numerical values of a number of astro- 

 nomical relations which will appear in the following development of 

 the subject. A knowledge of these values will enable us to deter- 

 mine what terms may be safely neglected in the development. 



TIDAL COMPONENTS. 

 6. TIDE-PRODUCING FORGE. 



As the sun and moon are similar in their action in the production 

 of the tide, the force of either may be considered by itself, and the 

 resulting forms of expression may then be readily adapted to the 

 other. 



The tide-producing force of the moon is that portion of its gravita- 

 tional force which is effective in changing the water level on the 

 earth's surface. This effective force is the difference between the 

 attraction for the earth as a whole and the attraction for the differ- 

 ent particles which constitute the yielding part of the earth's sur- 

 face; or, if the entire earth were considered to be a plastic mass, the 

 tide-producing force at any point within the mass would be the force 

 that tended to change the position of a particle at that point relative 

 to a particle at the center of the earth. That part of the earth's 

 surface which is directly under the moon is nearer to that body than 

 is the center of the earth and is therefore more strongly attracted 

 since the force of gravity varies inversely as the square of the dis- 

 tance. For the same reason the center of the earth is more strongly 

 attracted by the moon than is that part of the earth's surface which 

 is turned away from the moon. 



The tide-producing force, being the difference between the attrac- 

 tion for particles situated relatively near together, is small compared 

 with the attraction itself. It may be interesting to note that, al- 

 though the sun's attraction on the earth is nearly 200 times as great 

 as that of the moon, its tide-producing force is less than one-half 

 that of the moon. If the forces acting upon each particle of the 

 earth were equal and parallel, no matter how great those forces 

 might be, there would be no tendency to change the relative posi- 

 tions of those particles, and consequently there would be no tide- 

 producing force. 



The tide-producing force may be graphically represented as in 

 Figure 7. 



Let = the center of the earth, 

 0= the center of the moon, 

 P = any point within or on the surface of the earth. 



Then OC will represent the direction of the attractive force of the 

 moon upon a particle at the center of the earth and PC will represent 

 the direction of the attractive force of the moon upon a particle 

 at P. 



Let the magnitude of the moon's attraction at P be represented by 

 the line PC. Now, since the attraction of gravitation varies in- 



