HAEMOlSriC ANALYSIS AISTD PEEDICTION OF TIDES. 15 



sum of CO and i and will be at its maximum. In this position the 

 northern portion of the moon's orbit will be north of the ecliptic. 

 When the longitude of the moon's node is 180°, the moon's orbit 

 will be between the Equator and ecliptic, and the angle / will be 

 equal to angle w — angle i. The angle / will be always positive and 

 will vary from co — i to co+i. When the longitude of the moon's node 

 equals zero or 180°, the values of v and ^ will each be zero. For all 

 positions of the moon's node north of the Equator as its longitude 

 changes from 180 to 0°, v and ^ will have positive values, as indi- 

 cated in the figure, these arcs being considered as positive when 

 reckoned eastward from T and T', respectively. For all positions of 

 the node south of the Equator, as the longitude changes from 360 

 to 180°, V and ^ will each be negative, since the intersection A will 

 then lay to the westward of T and t'. 



Tables 1 and 2 contain a collection of astronomical constants and 

 formulas to which reference will frequently be made in this work. 



5. DEGREE OF APPROXIMATION. 



The problem of finding an expression for the equilibrium height 

 of the tide in terms of time and place does not admit of a strict so- 

 lution, but an approximate expression may be obtained which may 

 be carried to as high an order of precision as may be desired. In or- 

 dinary numerical computations exact results are seldom obtained, 

 the degree of precision depending upon the number of decimal places 

 used in the computations, which, in turn, will be determined largely 

 by the magnitude of the quantity sought. In general, the degree of 

 approximation to the value of any quantity expressed numerically 

 will be determined by the number of significant figures used. With 

 a quantity represented by a single significant figure, the error may 

 be as great as 33 1 per cent of the quantity itself, while the use of 

 two significant figures will reduce the maximum error to less than 

 5 per cent of the true value of the quantity. The large possible error 

 in the first case renders it of little value, but in the latter case the 

 approximation is sufficiently close to be useful when only rough 

 results are necessary. The distance of the sun from the earth is 

 popularly expressed by two significant figures as 93,000,000 miles. 



With three or four significant figures fairly satisfactory approxi- 

 mations may be represented, and with a greater number very precise 

 results may be expressed. For theoretical purposes the highest at- 

 tainable precision is desirable, but for practical purposes, because 

 ^f the increase in the labor without a corresponding increase in util- 

 ity, it will be usually found advantageous to limit the degree of pre- 

 cision in accordance with the prevailing conditions. 



Frequently a quantity that is to be used as a factor in an expres- 

 sion may be expanded into a series of terms. If the approximate 

 value of such a series is near unity, terms which would affect the 

 third decimal place, if expressed numerically, should usually be re- 

 tained. The retention of the smaller terms will depend to some ex- 

 tent upon the labor involved, since their rejection would not seri- 

 ously affect the final results. 



The formulas for the moon's true longitude and distance in Table 

 1 are said to be given to the second order of approximation, a frac- 

 tion of the first order being considered as one having an approximate 

 value of 1/20 or 0.05, a fraction of the second order having an ap- 



