ON THE COASTS OF IRELAND. 25 



baha tide. The declinations were those for the times of transit at Greenwich one 

 day and three days previous to the transit next preceding the tide in question. For 

 the low waters of the first division, which follow the high waters of the first division 

 by ^th of a tidal day, the right ascensions and declinations ought to be taken for tran- 

 sit over the 6^ meridian ; this was done most conveniently by correcting the coeffi- 

 cients when combined in groups, by the following formulae. If the computed terms 

 containing w and x for the moon are W.ir+X.o? {W and X containing 1/ and z}, then 

 the corrected terms as altered for the change of right ascension, are 



Wcos ^+Xsin^)/^4-(Xcosj— Wsin^jj'. 



And if the computed terms containing 3/ and z are Y.?/-f-Z.z (Y and Z containing w 

 and x), then the corrected terms as altered for the change of declination, are 



^ ^iK28 ^shr28/^+ \^ sm2S ^^lm2d/^- 



No notice was taken of the changes of parallax; nor were the hour-angles referred to 

 the moon's place one or three days previous (as in strictness of theory they ought) ; 

 but as the observations extend over two whole lunations, it was supposed that the 

 effects of these omissions would nearly disappear. 



The factors of the unknown quantities were computed on the supposition that 

 a=0, and also on two other suppositions. It is easily seen, however, that the factors 

 for any assumed value of a can be readily formed from those which hold for a=0; 

 and this computation is made most conveniently for the groups. 



The numbers for high water were divided into groups related to the changes of 

 sign of the factors of w and x. These groups were then combined in the order 

 lst + 2nd — 3rd — 4th + 5th + 6th— &c. to form one equation, and in the order 

 1st— 2nd— 3rd-|-4th+5th— &c. to form another equation. The numbers for low 

 water were treated in the same way. In subsequent operations, the groups were 

 formed and combined in difi^erent orders. 



But, in whatever way the groups were formed, they were so combined as to form 

 four equations, each of which has the following form : 



A.w-\-B.x-{-C.wi/+D,wz-\-E.xi/-\-F.xz=G. 



To solve a system of four such equations is evidently no easy matter. Two me- 

 thods of solution were principally relied on. 



The first (and easiest) was, to make trial-substitutions to a great extent. The 

 numbers —2, —1, 0, +1. +2, were substituted for w, the same numbers were 

 substituted for x ; the same numbers were also substituted for^ and for z ; and every 

 possible combination of these numbers was used ; making 625 trial-substitutions in 

 each of the four equations. And when there seemed a probability of success, the 

 substitutions for one or two of the numbers were greatly extended. Calling the re- 



MDCCCXLV. B 



