368 Mr. V. N. Nene. On a Method of Tracing 



( 82°*5\ 



we can show that when nb = 2\ 2tt-\- -_ — l we shall have anothei 



, , 7 / . something between 82' 5 and 90\ 

 maximum, and when nb—2\ 67r+. ^ J we 



shall have another minimum, and so on alternately ; for the general 

 equation from which these values are to be determined is tan x=J$7r + x, 

 where 1ST is zero or some integral number. 



We have taken the first factor only in dealing with the above sub- 

 ject, bat it will be seen by inspection that the same results would 

 have followed if the second factor had been taken. 



These changes of factors have been worked out for a particular case, 

 and curved in fig. 1 (Plate 4) to give the reader a general idea of 

 them. 



12. Now we shall repeat the equation (1) and proceed as follows : — 

 Y=u-\-u l sin ^U 1 + -^-J + ^2 sm ^U 2 -r--^-J-f u z sm^U 3 + — _J + ,&c, 



Let c be a small period or interval say of an hour, a day, a week, 

 or a month, &c, as the case may be, at which observations are avail- 

 able, then writing b Y for b 2 for b s for -— , &c, in the above 



Ko K 3 



equation (and appropriately modifying the constants TJ lt U 2 , &c, if 

 x and c are reckoned from different epochs), the successive observa- 

 tions will be equal to the sum of the terms of the following expres- 

 sions — 



1st obs. =u-\-u Y sin U^u^ sin TJ 2 + u s sin U 3 4- &c. 



2nd obs. = u + u Y sin (Uj + b{) + u 2 sin (U 2 + b 2 ) -f ^3 sin (U 3 -f b s ) + , &c. 



3rd obs. ~u+u l sin (U 1 + 26 1 ) +w 3 sin (U 2 + 25 2 ) +^ 3 sin (U 3 + 27> 3 ) + , 



&c. 



4th obs. =% + %sin (t^-hS^) -f u 2 sin (U 2 + 3& 2 ) +w 3 sin (U 3 + 3& 3 ) -)-, 



&c. 



(n + l) th =u + u x sin (U^ w&i) + w 2 sin (U 2 -f%& 2 ) sin (U 3 + %& 3 ) + , 



&c. 



In order to save room we shall henceforth use the factor — , — , — , 



2i % % 



m\ . mb 2 . m5 3 

 sin -g- sin -g- sin -g- 



&c, in place of , , , &c, respectively. 



fr, .69 • 



m sm — i m sm — i m sin —2 



2 2 2 



13. Suppose we have a table ruled with horizontal and vertical lines ; 

 in the first horizontal line let us enter the numerical values of the 



