8 U. S, COAST AND GEODETIC SURVEY 
the epoch taken as*the basis of the computation would have resulted 
in differences of less than 0.02 of a degree in the tabular values. 
The table may therefore be used without material error for reductions 
pertaining to any modern time. ~ 
25. Looking again at figure 1, it will be noted that when the 
longitude of the moon’s node is zero the value of the inclination J will 
equal the sum of w and 2 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 J will be 
equal to angle w—angle 7. The angle J will be always positive and 
will vary from w—7 to w+7i. 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 ares being considered as positive when 
reckoned eastward from Y and Y’, respectively. For all positions of 
the node south of the Equator, as the longitude changes from 360 
to 180°, v and é will each be negative, since the intersection A will 
then lay to the westward of T and T’ 
DEGREE OF APPROXIMATION 
26. The problem of finding expressions for tidal forces and the 
equilibrium. height of the tide in terms of time and place does not 
admit of a strict solution, but approximate expressions can be ob- 
tained which may be carried to as high an order of precision as desired. 
In ordinary numerical computations exact results are seldom ob- 
tained, 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) percent of the quantity itself, while 
the use of two significant figures will reduce the maximum. error to 
less than 5 percent 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 
ae is popularly expressed by two significant figures as 93,000,000 
miles. 
27. 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 
of the increase in the labor without a corresponding increase in util- 
ity, it will be usually found advantageous to limit the degree of 
precision in accordance with the prevailing conditions. 
28. 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- 
