352 Prof. Gr. H. Darwin. On an Apparatus for [Dec. 15, 



If the number of days n be large, A,, B^, will be small unless the 

 denominator of one of the two terms in (4) be very small. This last 

 case can only occur when p = q and when p is small. Hence, in the 

 analysis of a term of the form under consideration, we may neglect 

 all the harmonics except the 2 th one. Accordingly (2) and (5) are the 

 only formulae required. 



A case, however, which there will be occasion to use hereafter is 

 when n = 30, q = 2,. when (4) becomes 



tt + 8691 p) ~ 



(6). 



For the present w.e haye to apply (5) in the two cases 2 = 1, 

 P = 0-0410686 and q = 2, p = C 'Q821372 ;. now the ratios of cosec p 

 to cosec (15g i/3) in these two cases are 722 to 1 and 697 to 1. In 

 both cases the first teem of (5) is negligible compared with the 

 second. 



Nowwrite * 



and (5) becomes, with, sufficient exactness, 



If this be compared with (2),. we see that when q = this formula 

 also comprises (2). 



In the applications to be made ^3 is very small, so that jf is approxi- 

 mately a function of the form cosec 0. This function increases very 

 rapidly when 6 passes 90, but for considerable values less than 90 

 it only slightly exceeds unity ; for- example-, when 6 = 60, $ = 1'2, 

 but when = 180, <f = infin. 



It fojlows, therefore, that if the number n of days in the series is 

 such that 12w/3 is less than say 60, the magnitudes of A qt B q are but 

 little diminished by division by <Jf ; but if I2n,p is nearly 180, A q , B q 

 become vanishingly small. 



If the typical tide here considered be the principal lunar tide M 2 , 

 and if the number of days be as nearly as possible an exact multiple 

 of a semi-lunation, 12n is nearly 180, and the corresponding A 2 , B 2 

 become very small. No number of whole days can be an exact mul- 

 tiple of a semi -lunation, so that A 2 , B 2 corresponding to M 2 cannot be 

 made to vanish completely. For the present they may be treated as 

 negligible, and we return to this point in the next section. 



The above investigation shows that in the expression for the whole 



