ABCTIC SEAS.— PAET III. FREDEEIKSDAL. 641 



The Ranges of Spring and Neap Tides contained in the following Tables, are laid 

 down on the diagram (p. 643), which is intended to show the Parallactic Inequality 

 deducible from the observations. This Inequality may be also found from the columns 

 of differences of Ranges given in Tables I. and II. 



It appears, both from the diagram and from the Tables, that the maximum amount 

 of this Inequality, plus and minus, is 1"4 foot. 



The expression for the Tidal Range is 



R=2S(^^ycosVcos2(s— ?,)+2M(^ycos>cos2(m-i„), . . . . (1) 



where 



S and M are the Solar and Lunar coefficients. 



P and p the Solar and Lunar parallax ; and P„, p„ the mean values of the same. 



ff, jU., the declinations of Sun and Moon. 



s, m, the hour-angles of Sun and Moon. 



^/, i^, the Solitidal and Lunitidal intervals. 



As the Sun's declination may be regarded as constant for a fortnight, and as the 

 Moon's declination only changes sign in that period, it is plain that the differences of 

 successive spring and neap ranges are due altogether to the parallactic inequality. 



The Solar and Lunar coefficients are found as follows. When the observations are 

 spread over an entire year, as in the present case, the means of all the spring and neap 

 ranges give the sum and difference of the Lunar and Solar Tides, cleared of parallax 

 and referred to the mean declinations of the Sun and Moon — that is, to their declinations 

 at 45° from the intersection of their orbits with the equinoctial. 



Let ff„ and |«/„ denote these declinations ; then we have 



sin <r„ = sin 45° x sin 23° 28'= sin (16° 2r)| 



sin|«,„=sin45°X8in20° = sin (14°) J ^' 



Using these values of ff„ and [jb„, we obtain from Tables I. and II. 



2Mcos« (14°)+2Scos'' (16° 2r)=9-40 feet, 



2M co8» (14°)-2S cos^ (16° 21')=4-51 feet, 

 or 



4Mcos^(14°) =13-91 feet, 1 



4S cos^(16°2r)= 4-89feet J (3) 



From which we obtain, finally, 



M=3-693 feet ( L) 



S =1-328 feet (IL) 



The Lunitidal and Solitidal Intervals are found from the means of the hour-angles 



given in Tables I. and II., and are 



h m 



Lunitidal interval at Springs . . . ' . 2 59 

 Lunitidal interval at Neaps 2 48 



Mean 2 53^ 



4t2 



