NEWTON S PRINCIPIA. 405 



ference is simply this. Laplace s theory aims at deter 

 mining the tides as formed in extensive sheets of water, and 

 therefore a solution of the Hydrodynamic Equations is 

 taken that is adapted to such a case. The wave theory 

 aims at discussing the tides as formed in long canals, and 

 therefore a solution of the equations is taken as suited to 

 such a case. It is to the labours of Professor Airy that 

 much of the progress that has been made in this subject is 

 due. I may refer to his work in the Encyc. Met. for the 

 demonstration of most of the theorems alluded to in the 

 following sketch. 



Let us begin by considering Newton s case of the mo 

 tion of water in an equatorial channel acted on by the sun, 

 supposed to revolve in the equator. That part of the dis 

 turbing force which acts perpendicular to the water, will 

 produce but little effect compared with the tangential force, 

 the weight of a column of fluid equal to the depth acted on 

 by the central force being infinitely less than that of so long 

 a column as a quarter the circumference of the globe acted 

 on by the tangential force. We shall therefore neglect the 

 central force. On applying Laplace s equations it can be 

 proved that the motion in a canal will be the same as if the 

 earth were reduced to rest, and an equal but opposite 

 angular velocity impressed on the disturbing body. By 

 Lunar Theory, the tangential force is known to be 



where b is the radius of the earth, ^ the sun s mass, D its 

 distance from the earth, p the relative angular velocity of 

 the sun and earth, and &amp;lt;p any angle determining the position 

 of the particle. Then, by the proposition demonstrated in 

 the chapter on waves, the altitude will be 



3 p kb* 



-- COS ^- 



showing that if the depth k of the sea be less than , 



D D 3 



