THE ORBIT OF URANUS. 197 



u will then bo the true argument of latitude, autl log /• the logarithm of the radius 

 vector with seven places of decimals. 



Under w write 0; enter Table XVIII with the argument u and take out the 

 reduction to the ecliptic. Add it to u and 0, and the sum of the three quantities 

 will be the heliocentric longitude of Uranus referred to the mean equinox and 

 ecliptic of the date. Applying nutation the longitude will be reduced to the 

 true equinox. 



Enter Table XIX with m as the argument, or, when u exceeds 180°, with 

 u — 180°, and take out the principal term of the latitude, which will be positive 

 when u is less than 180°, and negative when it is greater. 



Enter Tables XX, XXI, XXII, and XXIII with their respective arguments, 

 the argument for the last being the date, and add up the various quantities having 

 the same designation, noticing that in the first three tables all the quantities arc 

 positive, while in the last they are all negative except (Z*.c.O). Then form the 

 expression, 



(i.c.O) -\- {Ji.sA) sin g -\- [h.c.l) cos g -\- {h..s.2) sin 2j -f {h.c.2) cos Qg, 



and add it to the principal term of the latitude, with regard to tlie algebraic signs. 

 The sum will be the heliocentric latitude of Uranus above the ecliptic of the date. 



When an ephemcris of Uranus is to be computed for a series of years, some 

 moditications may be introduced, which will save the computer labor. In tlie first 

 place an equidistant series of dates being selected for computation, it will be suffi- 

 cient to compute g, u, 0, and the arguments for every sixth, eighth, or tenth date, 

 and to fill in the arguments for the intermediate dates by adding the nearly con- 

 stant differences corresponding to the adopted intervals. The agreement of the 

 numbers thus obtained for the last date with those found by the original computa- 

 tion Avill prove the whole process. This interval may be as great as 120 days 

 without detracting from the accuracy with which the places for the immediate 

 dates can be interpolated, and the differences for this interval may be deduced 

 from the numbers at the bottom of Table II. If these numbers are used without 

 change the values of (j and for the last date may not always come out riglit. 

 But these errors, if less than a second, will be of no importance if the one quan- 

 tity comes out as much too great as the other is too small, and they may be avoided 

 entirely by making a small change in the constant difference to be added. 



Tables XI to XVI, inclusive, need be entered only for every third or fourth date, 

 and the sums of the quantities can be then interpolated to every date, and added 

 up with the corresponding quantities from the other tables. 



Again, it will be found convenient to compute the sum of the small terms 

 (v.s.'S) sin Sg -\- (r.c.3) cos 3/ -J- (v.sA) sin 4:g -\- (r.c.4) cos -ig, as well as the corre- 

 sponding terms of the radius vector, and all the terms of the latitude, not for the 

 dates adopted, but for every fourth entire degree of g. Having a scries of values 

 computed in this way, the sum can be interpolated to the value of g corresponding 

 to the date. To facilitate the formation of the smaller products for entire degrees 

 of g, a table of products of numbers by the sine and cosine of every degree is 

 appended to these tables, by which the products in C|Ucstion can be formed at sight 



