stellar rnoTOGiumr. 



201 



When 8 = 0, the co-ordinates are rectansmlai 



o 



X 



Hence 



cos j) = — sin y cos t y and y = — sin 2 } y sin 2 t 



t- — - cin 2 i - 6i " 2 r ~ x * - sin 2 9 / - 4 ** (8in2 ^ ~ "5 

 1 — sin 2 v ' sm * — «;,,-■ v > sm « ' — .,,.4 7 



and 



snr 7 ' sm- y > -m< y 



^ 2 = i ^7 2 ' *~° ( sin2 r - *") = 2 sec * J y * 2 ( si » 2 y 



o 



This is the equation of a curve in the general form of a lemniscata. If y is suffi- 

 ciently small, it may be reduced to y 2 = 2 x 2 (-/ — x 2 ), whence X* — y 2 x 2 = — .] tf, 

 ^= } (y 2 ± \/y 4 _ 2 y 2 )- Accordingly, for any real value of x,y cannot exceed 



1 2 



-ft y , and # cannot exceed y; if y is infinitesimal, y must bo infinitesimal with re- 



ject to x, so 



b 



.- 



In all cases, when y is so small that we may substitute y for sin y and 1 for cos y, 

 the difference between \ tt and S+/>, which never numerically exceeds y, is a 

 quantity of the same order; so also, accordingly, is x. Let c = \ -n — 8; 



sin (c — p) = cos 8 cos p — sin 8 sin jo = cos (8 -f- />) = ./ 



As # is small, we may also write x = c — p, p = c — x, and cos (c — •/>) = cos x = 1 



Since 



cos /? — cos ^ = 2 sin £ (<? — p) sin £ (c -f p), 



cos jt? — cos <? = 2 sin £ 2 sin (c — \ x) = x (sin c — \x cos c). 



Also, from (1), cos p — cos c = cos p — sin 8 = — y cos 8 cos/; hence 

 %== — -r-^ where the term of the denominator containing x may be 



*. ill C ■ -R- £C COS C 



omitted, so that x = — y cos /. From (4), y = y sin 8 sin /. Hence cos 2 1 = -5, 



,2 ~2 -. .2 



sin 2 i = y and -* + / 2t = 1. The curve is therefore an ellipse, becoming 



y sin J o 7 7 7 sm * *-*•«» 



a circle of the radius y when 8 = |- it, and a straight line, as already shown, when 



8 = 0, since in that case y = 0. 



The general results of the inquiry, accordingly, are that, without restriction as 

 to the amount of the error of adjustment, the curve described by the image of a 

 star situated on the equator is a species of lemniscata; with an increase in the 

 declination of the star, the lower branch of the curve becomes smaller and narrower, 

 and disappears in a cusp when the declination is equal to half the error of adjust- 

 ment. For greater declinations, the curve is ovoid, and at the pole it is circular. 



If the error of adj 



sufficiently small, the curve is a circle at the pole 



o 



ht line at the equator, and an ellipse in intermediate declinations, as has 



been stated on page 195 



