296 Prof. G. H. Darwin. [June 19, 



i(W'-Z') = B 1 cos (' + ' + & M) 



(X'+Y') =-E'sin 

 i(W-Z) = E'sin 



Hence 



= # ain 



(41). 



In these equations B' is equal to f ' H' and Ep is equal to Hp. 



The terms involving R' are clearly small, and approximate values of 

 B' and ", as derived from the first quarter, -will be sufficient to com- 

 pute them. Afterwards we can compute B p or H p and t"p ; then if 

 UP denotes /< O + ^TT, the equilibrium argument of P at the first 

 epoch, K P = p + u p . 



The values of H^, K P thus deduced ought not to differ very largely 

 from those assumed in the two independent reductions. 



The same investigation serves for the evaluation of the P tide from 

 any two sets of observations, each consisting of thirteen quarter-lunar 

 periods, and with a small change in the analysis we need not suppose 

 each to consist of thirteen such periods. But the two epochs must be 

 such that sin $h is small and cos &h is large, or the formulae, although 

 analytically correct, will fail in their object. 



8. The Disturbance of K 1? 0, P due to M 2 and S 2 . 



It has been remarked in 7 that the diurnal tides are perturbed 

 by the semi-diurnal. The general method has been given in 3, by 

 which to calculate the effect on any one tide, whose increment of 

 argument since epoch is V p and speed is p, due to a tide whose incre- 

 ment is V q and speed q. 



Since in the present instance all the diurnal tides have been 

 consolidated into one of speed <y <r, we have to calculate the effect 

 of the tides whose speeds are 2(<y <0 and 2(7 57) on the tide whose 

 speed is 7 <r. It follows, therefore, that the factor gjp or k g of (3) 

 is in the first case equal to 2(7 <0/(7~ <*) or 2, and in the second 

 case is 2(y-^)/(7-0 or 2"070 ; or k m = 2, Jc s - 2'070. 



