STELLAR PnOTOGRAPnT, 201 



Wlien 8 = 0, the co-ordinates are rectangular ; 



X = cos j) := — sin y cos t, and ^ = — sin^ } y sin 2 t. 

 Hence 



COS- t = ^-^^ ; Sin- t ^= ^, ; siu" 2 r = ^ . / ; 



sin- y ' sm- y ' sin* j- ' 



and 



/ = gi„4 ., ^ (sin- y — .^-) = 2 sec* -} y x- (snr y — a-). 



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

 ciently small, it may be reduced to ^- = 2 3^ (y* — a^), whence x* — y^ x^ = — i y*, 

 and ar =: i (y- ± \/y* _ 2 //^)- Accordingly, for any real value of x, y cannot exceed 

 -j= -f, and X cannot exceed y ; if y is infinitesimal, y must be infinitesimal with re- 

 spect to X, so that the lemniscata becomes a straight line. 



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 + p, which never numerically exceeds y, is a 

 quantity of the same order ; so also, accordingly, is x. Let c = | - — S ; 



sin (c — p) = cos S cos jo — sin S sin^ = cos (8 -\- p) = x. 



As X is small, we may also write x = e — p, p ^^^ c — x, and cos (c — p>) ^ cos x =^\ 



Since 



cos^; — cos c = 2 sin I (e — jo) sin | (c +^^), 



cos p — cos c = 2 sin ^ X sin (c — ^ x) ^ x (sin c — h x cos c). 

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



*' SlTl C COS i 



a- = — . ' ^ — ; , where the term of the denominator containingf x may be 



sin c — i! X cos c ' o ./ 



omitted, so that .r = — y cos /. From (4), ^ := y sin 8 sin t. Hence cos- ^ = 71, 



2 2 2 



sin- i = a y -> x , and ^, + ., ^ ., „ = 1. The curve is therefore an eUipse, becoming 



y^ Sin- d ' y- ' ;- sin- 5 r ' o 



a circle of the radius y when 8 = 4 v, 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; witli 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 adjustment is sufficiently small, the curve is a circle at the pole, 

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

 been stated on page 195. 



