1890.] Harmonic Analysis of 'lidal Observations. 295 



Then if u = h v TT, the equilibrium argument at epoch 



of K lf 



*' = ' + it'. 



We have also H, - ?H' = 0-3309 H', * f = *'. 



Returning to equations (34) and (35), we compute R' = f'H', 



and hence R' V S ('+i), and then L and M. 



Sill 



Having these, we compute R and from (35). 

 Then, if u = h v 2(s f) +^TT, the equilibrium argument at 

 epoch of 0, 



TT 



*o = + M O , and R = , 



IG 



where f is a certain function of the longitude of the moon's node, 

 tabulated in Baird's Manual. 



A Long Series of Observations. Suppose that there is a half year of 

 observation, or two periods of thirteen quarter-lunar periods, each of 

 which contains exactly the same number of tides. 



Then each of these periods is to be reduced independently with 

 the assumption that q = 0'3309 and K P = *'. If this assumption he- 

 found subsequently to be very incorrect, it might be necessary to 

 amend these reductions by adding KJ,K' to the value of 0, and by 

 multiplying p n by U. f -f- 0'3309 H', but such repetition will not 

 usually be necessary. 



From these reductions we get independent values of H' cos K', 

 H' sin ', H cos *T O , H sin K O from each quarter year, and the means 

 of these are to be adopted from which to compute H', K, H , *<,. 



It remains to evaluate Hp and K P . 



The factor f and the angle v vary so slowly that the change from 

 one quarter to the next may be neglected, although each quarter is 

 supposed to have been reduced with its proper values. 



Let h , h' be the values of the sun's mean longitude at the two 

 epochs ; then since the second epoch is nearly a quarter year later 

 than the first, h' will exceed h by about 90. 



Let h' = h + ^?r + Sh, so that 6 h is small. 



If ' + ', P + fip he the values of f', p at the second epoch, we 

 have ' + ' =h' + v' +%* + *', f =_7, +i/ + jr+*-', and therefore 



Again, &+*& = &'$*+ <^i P= ho a^ + fyj and therefore 



a?, =+&. 



Let i+8i, k + $k he the values of i and k corresponding to the 

 second epoch, and let W, X', Y', Z' be the values of those quantities 

 in the second quarter. Then, replacing (2/n + l)e by 87'5, since 

 that is its value when 2m + 1 is 13, we have from (33) 



