324 Prof. G. H. Darwin. [June 19, 



The sheets of heights (c) are pinned opposite to the IVs on the 

 Tables of Angles (b), and the heights are entered successively into the 

 columns corresponding to their T/s in a table like (e), which was used 

 for sorting according to values of V m - The sorting is carried as far 

 as the end of an exact multiple of a semi-lunation, in this case to the 

 end of 6 semi-lunations. No sub-division is necessary, but for the 

 purpose of verification it is useful to break the entries into groups of 

 about 40. This is conveniently done by a division after each third 

 ^-lunar-anomalistic period, so that i, ii, iii would be the first group ; 

 iv, v, vi the second ; vii, viii, ix the third ; and x, xi, xii, and all but 

 the end of xiii, the last. 



In this case the entries fall into all the four quadrants with about 

 equal frequency. 



We next sum the four times 18 columns, just as with M 2 in (g), 

 and form 1 st 3 rd and 2 nd 4 th , reversed, in the same way. 



Next we write the 18 cosine numbers, (1 st 3 rd ) (2 nd 4 th , 

 reversed) in vertical column, multiply by cosine factors, add, and 

 divide by the total number of entries, which is 342. Afterwards 

 write the sine-numbers (1 st 3 rd ) + (2 nd 4 th , reversed), multiply by 

 sine factors, add, and divide by 342. 



The results are : 



A * = TJ 2 ^ cos V, = +21-08. B, = -gi-gSli sin V, = +3'62. 



(1.) Sorting according to Values o 



The whole process is precisely parallel to the sorting according to 

 values of F OT in (e) ; the thirteen divisions are, however, given by the 

 quarter-lunar-periods I, II, .... XIII. The only difference lies in the 

 substitution of the factor * (for XIII equal to 0'00298) for <D. It is 

 unnecessary to give an example. 



The results are : 



Sh cos \V m = -10-50, Sfc sin \V m = +8'04, 



S*fc cos iV )tt = + 0-40, &*h sin |V = + 3-74. 



(m.) Sorting of \ V m . 



It is required to find what the sums in (1) would be if every H.W. 

 height had been unity, and every L,W. the same both in magnitude 

 and sign; in fact to find S cos ^F OT , S**" cos ^V m , &c. 



This is done by counting the entries in the preceding sorting in 

 (1) without regard to magnitude, taking the L.W. entries as actually 

 positive, instead of being (as they are) negative quantities with the 

 negative sign suppressed. 



Since in this case we have simply to count entries which are all 

 treated as positive, the table of sums of H. and L.W. may be written 

 together. The following example gives part of the work 



