104 MR. AIRY ON THE LAWS OF THE TIDES 



Put M for the lunar effect when the square of the cosine of the moon's declination 

 is 0*9 and her parallax is 57' ; m for the quantity by which this is increased for every 

 increase of 1' in parallax ; s for the mean effect of the sun (the square of the cosine 

 of his declination being 0*9) for one-fourth of a lunation, positive in the large tides 

 and negative in the small tides, which bears to the absolute effect of the sun a rela- 

 tion explained in page 34. The effect of the variations of the moon's parallax and 

 declination upon the luni-solar tide, as is well known, is nearly the same as that on 

 the simple lunar tide ; and therefore it will be correct to refer the mean of the luni- 

 solar tides to the mean of the moon's parallaxes and square of cosine of declinations. 

 The variations depending on the moon's declinations are not strictly in the propor- 

 tion of the squares of the cosines of her declination, but in the present instance, 

 where the means of the squares of the cosines are very nearly equal, may be assumed 

 to be so without sensible error. Forming then an equation from each of the lines in 

 the Table above by these considerations ; reducing them to four equations by retain- 

 ing the second, taking the mean of the first and third, the mean of the fourth and 

 sixth, and the mean of the fifth and seventh, and combining these so as to form three 

 favourable equations, by adding all, by subtracting the sum of the third and fourth 

 from the sum of the others, and by subtracting the sum of the second and third from 

 the sum of the others, we obtain the following equations: — 



377-15= Mx4-024-l-mX0-33-5X0008 

 69-61 = -Mx0008-i-mXl-29-F*X 3-938 

 31-87 = — M X 0-036-l-m X 7-83 — 5X 0-026. 

 From these we obtain M = 93-4, m=4'56, *= 16*37. The moon's effect, therefore, 

 for the parallax 57'+'^' niay be represented by 93-4-|-4-56X?2. If the moon's hydro- 

 dynamical effect varied as the cube of her parallax (which is the law of variation of 

 her statical effect), the formula would be 93-4-1-4-92 X n. The result of the movement 

 of the water has therefore been, to reduce the elliptic variation of lunar effect by 



0-36 , , 3 

 ^T^part, or by^part. 



Now it is shown in the Encyclopaedia Metropolitana, Tides and Waves, Art. 448, 

 that if the tides were created by the effect of the moon on the water in a uniform 

 channel surrounding the earth, and if h were the earth's radius, k the depth of the 

 water, g the acceleration produced by gravity in the unit of time, w' the moon's appa- 

 rent angular motion round the earth (as estimated by a spectator who supposes that 

 the earth does not revolve on an axis), and h the moon's angular motion from her 



4 Tih'^h 

 perigee ; then the elliptic variation is changed by ^ * n^ '^—ak V^^'^- 1 hus we obtain 



4 rJb'^h _ _ 3 

 ^' n'^b--gk~~''4:l 



Makmg - = gg' we find ^^ = — = - nearly ; and « = 3 • -7— Observmg that lib 



