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



.2 lld quadrant is changed to IV th , I st , II nd ; the 3 rd to III rd , IV th , 

 I st ; and the 4 th to II nd , III rd , IV th . Hence eighteen entries of 1 st 3 ld 

 are converted into three sixes, !* Ill 1 *, II nd IV to , -{I 8t -III rd } ; 

 atid eighteen entries of 2 nd 4 th are converted into three sixes, 



-{IP*- IV th }, i*-iii rf , II nd -IV th . 



Hence a new I st III rd of six entries is made up thus : 

 first six of former 1 st 3 rd 



-f second six of former 2 nd 4 1 ' 1 



third six of former 1 st 3 rd . 

 And a new II nd IV th of six entries is made up of 



first six of former 2 nd 4 th 



+ second six of former I st 3 rd 



4- third six of former 2 nd 4 th . 



These I st III rd and II nd IV lh may now be treated just like the 

 other ones. Thus, without calculating | V m , we have from the former 

 ]_8t ord an( j 2 nd 4 th the results of a fresh grouping according to 

 values of %V m . 



It is true that there is a considerable loss of accuracy, because all 

 angles within 15 are now treated as having the same sine and 

 cosine. 



10. Rules for the Partition of the Observations into Groups. 



It appears from the preceding investigations that it, is required to 

 divide up the observations into groups. This may be done, with all 

 necessary accuracy, and with great convenience, by dividing the tides 

 just as they would be divided if every H.W. followed L.W., and vice 

 versa, at the mean interval of 6 h '2103. 



Now a quarter-lunar-anomalistic period is 165 h '3272, a quarter- 

 lunar period is 163 h '9295, and semi-lunation is 354 h 3670. Hence, 

 dividing these numbers by 6 h '2103, we find that there are 26'62145 

 tides in a quarter-anomalistic period, 26'3964 in a quarter period, and 

 6 7 '06 12 in a semi-lunation. 



It may be remarked in passing that these results show that the 

 n + 1 of (10), 4, is 53-243, and the n + l of (25), 7, is 52793. 



It is, of course, impossible to have a fractional number of tides, 

 and, therefore, we make a small multiplication table of these numbers, 

 and take the nearest integer in each case. For example, in the case 

 of the semi- lunations, we have 



1. 57-0612 57. 4. 228'2448 228. 



2. 114-1224 114. 5. 285-3060 285. 

 3.171-1836 171. 6. 342-3672 342. 





