THE TIDES. 



disturbing force in magnitude and direction. The proof 

 is as follows : 



Let it be borne in mind that ME is about sixty times 

 the radius of the earth. Hence, if we consider Ml 



the error cannot exceed = th part. For the 



error is greatest when x coincides with D (fig. 2). But 

 in that case ML* = ME* + r* = (60r) 2 +r* = 3601 r* 



,-. MD = rv/3601 = r ( 60 + -!-) nearly. Therefore 



Again, Mn, Mm, Ml, ME, being arithmetical proportionals 

 with a difference varying from to ~th, may be regarded 



as geometrical proportionals ; the greatest possible error 

 being about the same as before. Hence, Mn : Mloi MX : : 

 Ml* or MX* : ME*, i.e. as the moon's force at E : force at x ; 

 therefore, if MX represent the moon's force at x, Mn will 

 represent the force at E in magnitude and direction, and 

 the difference or disturbing force will be represented in 

 magnitude and direction by xti. In order to have a fixed 

 scale we must represent the force at the centre by ME. 

 On this scale xn is in the nearer hemisphere too small, 

 and in the more remote too large, in the proportion of 



o -I 



Mn to ME. This error is at most ^ths = -r^th. This will 



bO 20 



be considered by-and-by, but for the present it may be 

 overlooked. 



The vertical component of xn is xh. The tangential 



