ON THE COASTS OF IRELAND. 51 



As, in the analysis of all the tides, this transformation was to be performed about 

 6000 times, it was highly important to devise an easy method of effecting it. For 

 this purpose the following mechanical arrangement was contrived. Upon a nearly 

 square piece of pasteboard were carefully traced two scales at right angles to each 

 other, with graduations of equal parts proceeding from the point of union. Upon 

 the edge of another narrow piece of pasteboard was traced a graduation whose parts 

 were to the parts of the former in the proportion of 3-2 to 3'1416. The commencing 

 point of this graduation was made the centre of a quadrant, of which one radius was 

 in the line of graduation produced. The divisions of the quadrant, proceeding from 

 the line of graduation, were marked from to 90°, and also from 360° to 270°. The 

 method of using it was ; to insert a needle at the centre of the quadrant, and to 

 plant its point upon one of the lines of the large pasteboard at the graduation corre- 



32 

 sponding to — A4 ; then to plant a second needle in the other line of the large paste- 



32 

 board at the graduation corresponding to -^ B4, and to turn the graduated edge of the 



long piece till it touched this second needle ; the reading of the graduated edge, with 



a shift of the decimal point, gave ^ y/ j (-;^ A4J -{-f'^B^j i or \/{Ai^+B4^) ; and 



the division of the quadrant cut by the straight line on the large pasteboard gave p. 



32 32 



When the signs of ^ A4 and — B4 are the same, the reading between and 90° is to 



be taken ; when different, that between 360° and 270° is to be taken ; and in either 



32 

 case, when '— A4 is negative, 180° is to be added or subtracted. 



Applying the same process to the four combinations of Aj and B^, Ag and Bg, A3 



and B3, A4 and B4, we have the height of the water at every instant expressed by 



the formula 



Aq+Ci sin (phase -l-^i)+C2sin(2 phase -\-(p2)-\-C.^s]n (3 phase +^3) 



+C4 sin (4 phase +^4), 



where the phase is an angle which is measured from the assumed commencement of 



. 1 • I . • 1 whole duration of tide _ 

 the tide, and may be converted mto time by multiplymg it by ^^^ It 



is evident however that the argument (phase +^1) commences at the time when the 

 water would be at its mean height before attaining its greatest height, if the oscillation 

 of surface were supposed to depend on that term only. The time of high water, on the 

 same supposition, would be given by making phase +^1=90°, or phase =90° — ^^; 

 converting the expression, when found, into time by the rule above. Or the time of 

 low water, on the same supposition, would be given by making phase +^1=270°, or 

 phase =270°- (pp It is convenient to choose, of these two expressions, that which 

 gives the smaller quantity. The quantity so found is to be added to the Greenwich 

 time of assumed commencement of tide, and it gives the Greenwich time of high 



H 2 



