AND THE MEAN SPECIFIC GRAVITY OF THE EARTH. 
593 
The latitudes thus obtained bein^ affected by the errors of the assumed declina- 
tions of the stars, the amplitudes to be adopted as final are obtained in the following 
manner. Let (p^ be the values of the amplitudes SA, AN to be determined, and let 
the stars observed at S and A only, give these values — 
<pi=a, <?,=«', <p, = a" 
Let stars observed at A and N only, give the values — 
<p2=b, <p^=b', (p^=b" 
Let stars observed at S and N only, give the values — 
<P^-\-(p.2=c', (pj-j-(p^Z=.c" 
And let stars observed at S, A, and N give the values — 
(p, = a„ <p^ = a\, <p, — a".... 
‘Pi—blf ‘p2“^15 ^2~bi .... 
Let d, e, and the same letters accented, be taken to denote the number of times the 
stars of the first set are observed at S and A respectively. Let f, g and h, k repre- 
sent the same quantities for the stars of the second and third set; and let n, p, q, 
and the same letters accented, be taken to denote the numbers of times the stars of 
the fourth and last set are observed at S, A, N respectively. 
The values of cp^ <p^ adopted are those which render the quantity 
a minimum. Making the differential coefficients of this quantity with respect to 
and ^2 respectively =0, we obtain 
in which equations 
H^i “L — 0 
K(p, + M^2— N=0, 
H=2 
L=2 
M=2 
N=2 
K=S 
If !«/ be any number, the value of <pi-\-[/j(p 2 is 
, (-f;.K + M)L-F(-K + iaH)N 
^ i - t (^^2 — HM — ’ 
hence the error of (pi+(i<p 2 depends upon the manner in which the errors of the quan- 
tities aa^...b b^...c enter into this expression. 
4 I 2 
