ON THE COASTS OF IRELAND. 49 



If we unite the adjacent numbers already formed by union, so as to have the sum 

 of four adjacent means together, and combine them thus, 



(Ist+2nd+3rd4-4th) + (5th+6th+7th + 8th)-(9th + 10th + llth + l2th) 

 -(13th + 14th + 15th + 16th), 



the sum is — A, + 7- A,. 



If we combine them in this manner, 



(lst+2nd+3rd4-4th)-(5th+6th+7th + 8th)~(9th+10th + llth+12th) 

 -f(I3th+14th + 15th-t-16th), 



the sum is — Bi — ^Bo. 



Then if we divide the complete tide into twelve equal parts, and take the mean 

 height in each, we shall have 

 Mean height in the first part = 



Ao+^Ai(^cos 0-cos ^) +-A2(^cos 0-cos -^) +3^A3(^cos 0- cos ^) 4-|^A4(^cos 0- 



+^Bif^sm g-sin0j+2^B,(^sm -g--sm0j4-3^B3(^sm -_sm0j4-4^B,(^sin --smOJ, 



and so on. And combining these in the following manner, 



(lst+5th4-9th) + (2nd+6th + 10th)-(3rd + 7th + llth) — (4th + 8th + 12th), 

 24 4 32 



the sum will be — Ao, or the sum X ^ = — Ao. 



And if we combine them in the following manner, 



(lst+5th+9th)-(2nd4-6th + 10th)- (3rd+7th + llth)4-(4th+8th + 12th), 



24 4 32 



the sum will be — Bo, or the sum x^= ~ B.. 



32 



By applying one-third of these to the expressions last found, we shall obtain — Aj 



32 



and — B,. - 



The mean of all the means, either in the division by sixteen or in that by twelve, 

 is Ao. 



The whole of these operations (after taking the means of the original observations) 

 are performed with great facility, and without the possibility of mistake, by means of 

 a printed skeleton form, of which a specimen will be given shortly. 



The next thing to be considered is, how we shall correct these numbers for the 



effect of diurnal tide, which is included in the observations, but from which our 



formula for semidiurnal tide is to be freed. Suppose that the tide begins with high 



water, and suppose a to be the effect of diurnal tide at that high water, b the effect 



of diurnal tide at the low water following, or that which occurs in the middle of the 



tide. Then the complete effect of diurnal tide is represented by 



phase , , . phase 

 a . cos ^-g ht' . sm — g— ; 



MDCCCXLV. ■ H 



