78 TABLES OP NEPTUNE. 



Fact. T gives the change in Afl'go) during a century : so that the change in any 

 180-day j)eriod within one or two centuries of the epoch may be found. by mul- 

 tiplying Fact. T by the fraction of a century after 1850.0 at which the 180-day 

 period commences, and applying it to A^i^o)- 



A[i|o) gives the second difference for any series of 180-day periods within one or 

 two centuries of 1850 : so that, knowing the first value of A{}Jo), we can find a series 

 of values by successive addition. 



The period of 180 days has been selected as a convenient one for computing a 

 heliocentric ephemeris. If any other period, represented by iVdays, be preferred, 

 the corresponding values of A^'^ and A<^' are found by multiplying 



N 

 A^Jso) by jg^, 



and , 



^ESo) by jg^. 



Table II. gives the change of each longitude and argument for the first day 

 of each month during a four-year cycle. The change in I is given for that cycle 

 which begins with 1900 and ends with 1904. Column V gives, in units of the 

 second decimal of seconds, the change in column I during one cycle. Hence, 

 multiplying I' by the whole number P of the preceding table, and adding the 

 units of the product to the hundredths of seconds of 1, we have the change of 

 mean longitude during the cycle numbered P in Table I. The correction is 

 positive for years before 1900, because the mean motion is diminishing. 



Q must be corrected in precisely the same way ; but here the correction is nega- 

 tive before 1900. 



Rigorously, both y and require correction similar to I. But it is not requisite 

 that either of these quantities should be accurate within a second, so long as their 

 sum is exactly equal to the precession diminished by 1° 1'. The four-year changes 

 of both y and d, which destroy each other, are, therefore, neglected ; but the change 

 in d due to the secular variation of the constant of precession (0".0227) is allowed 

 for by the correction PQ'. 



Table III. gives the reduction from the first to the subsequent days of any 

 month, or the motion of the epochs and arguments during a number of days one 

 less than those on the left of the table. 



Table IV. gives the corrections to be applied to the longitudes and arguments 

 for the epochs 1800 + t to reduce them to the epochs IGOO -f t, 1700 + if, and 

 1900 + t, respectively. They are expressed in the form 



a,+ TX Fact. T+ T'X Fact. T% 



in which 7" is the fraction of a centuiy. 



Table V. gives the expressions for the perturbations of the longitude produced 

 by Uranus. To each of the expressions P,i and P,.^ 14" has been added, and to 

 P,.2 and P,o 3" has been added. Hence, when these quantities, as given in the 



