HABMONIC ANAL,YSIS AND PREDICTION OF TIDES. 33 



For the third form we may obtain from (88) and (93) to the second 

 order of approximation the following : 



' cos (2Z + q;) = (1-5/2 ^2) cos (2(T + a) 



(IJ 



+ 7/2 e cos (2(7 + a + s - p) - 1/2 e cos (2a + oc-s + p) 



+ 17/2 e^ cos {2<T + a + 2s-2p) 



+ 105/16 me cos {2a + a + s -2h + p) - 15/lQ me cos {2(r + a-s + 2Ji~p) 



+ 23/8 m^ cos (2o- + a + 2s - 2h) + 1/8 m^ cos {2(T + a-2s + 2h) (96) 



Substituting (88) , (95) , and (96) in (84) , and letting a equal ± 2x, 

 (±X— 7r/2), or 0, as the case may be, we may obtain the following 

 equation for the equilibrium height of the lunar tide. For conven- 

 ience in reference each term is designated by the letter A with a 

 subscript. Following each term is a numeral giving its maximum 

 value in feet when / has its mean value. These numerical values 

 include the general coefficient and are calculated from the constants 

 in Table 2. They serve to indicate the relative importance of each 

 term. 



.-3/2 fix 



(J.)i [1/2 cos^ X cos* I /((I -5/2 e^) cos(2(r-2x) (0.7869) 



iA)2 +7/2 e cos (2(r-2x+s-p) (0.1524) 



(^)3 -1/2 e cos (2cr-2x-s+p) -. (0.0218) 



(^)4 +17/2 e2cos (2(r-2x+2s-2p) (0.0203) 



<A)5 +105/16 me cos (2cr-2x+s-2/i+p) (0.0214) 



(A)t -15/16 me cos (2(r-2x-s+2/i-p) (0.0031) 



U)7 +23/8 m2 cos (2(r-2x+2s -2/1) (0.0128) 



{A)s +1/8 m2 cos (2,T-2x-2s+2/i)} (0.0006) 



{A)g +1/2 cos2 X sin* | I {(1-5/2 e^) cos (2o-+2x) (0.0015) 



{A)io +7/2 e cos (2<r+2x+s-p) (0.0003) 



iA)n -1/2 ecos (2cr+2x-s+p) (0.00004) 



(A)i2 +17/2 e2 cos (2<r+2x+2s-2p) (0.00004) 



{A)u +105/16 me cos {2cr+2x+s-2h+p) (0.00004) 



(A)i4 -15/16 me cos i2<r+2x-s+2h-p) (0.00001) 



<^)is +23/8 m2 cos {2cT+2x+2s-2h) (0.00002) 



(A)i6 +1/8 m2 cos (2(r+2x-2s+2/i)} (0.000001) 



iA)n +1/4 cos2 Xsin2 /{ (1+3/2 e2) cos 2x - (0.0686) 



<A)i8 +3/2 e cos (2x+s-p) — (0.0056) 



(A)i9 +3/2 e cos (2x^s+p) . (0.0056) 



(.4)20 +9/4 e2 cos (2x+2s-2p) (0.0005) 



(^)2i +9/4 e2 cos (2x-2s+2p)_ (0.0005) 



(A)22 +45/16 me cos (2x+s-2/i+p).__..___.i_ (0.0008) 



iA)23 +45/16 me cos {2x-s+2h-p) (0.0008) 



{A)2i +3/2 m2 cos (2x+2s-2/i)„l (0.0006) 



{A)25 +3/2 m2 cos (2x-2s+2/i)}_ (0.0006) 



(^)26 +1/2 sin 2X sin 7 cos^ f /{(1-5/2 e^) cos (2(r - x - 7r/2) - (0.3266) 



iA)27 +7/2 e cos (2<r-x+s-p-7r/2 , (0.0632) 



iA)28 -1/2 e cos (2(7-x-s + p-7r/2) (0.0090) 



(^)2s +17/2 e^ cos (2<7-x+2s-2p-7r/2) .._ (0.0084) 



(A)3o +105/16 me cos (2<r-x+s-2/i+2J-.x/2)__.:. (0.0089) 



(A)3i -15/16 me cos (2(r-x-s+2/i-p-ir/2) (0.0013) 



(^)32 +23/8 m2 cos (2<7-x+2s-2/i-7r/2) (0.0053) 



{A)s3 +1/8 m^ cos (2<r-x-2s+2/i-7r/2)} (0.0002) 



(.4)34 +1/2 sin 2 X sin / sin^ ^ I {(1-5/2 e^) cos (2(r+x -W2) - (0.0141) 



(A)35 +7/2e cos (20-+X+S-P-W2) (0.0027) 



• (41)36 -1/2 e cos (2<7+x-s+p-W2) (0.0004) 



(.4)37 +17/2e2 cos (2a+x+2s-2p-7r/2)__ (0.0004) 



(A) 38 +105/16 me cos (2cr+x+s -2/i+p-7r/2) (0.0004) 



(^)39 -15/16 me cos (2(r+x-s+2;i-p -7r/2) (0.0001) 



(^)4o +23/8 m^ cos (2<r+x+2s -2/1 -7r/2) (0.0002) 



(A)ii +1/8 m^cos (2(r+x-2s+2/i-7r/2)} (0.00001) 



