GAUSS’S DIOPTRIC RESEARCHES. 495 
direction in the last passes through N. Let the inclinations of 
such a ray to the axis in the first and last media be «, y respec- 
tively. 
B2,42 = ky, + 1Bo, w. vn = hy, + lwe, 
oa ky, =me(L—J), yn = we. 
Hence if a small object at the distance w from M subtend an 
angle < at M, its image will be at the distafice v from N, and will 
subtend an angle y at N. 
. linear mag. image su _m—e_ f—q 
** linear mag. object” vu p—e f—w 
The image will be erect or inverted according as the numerator 
and denominator in the above expressions have the same or dif- 
ferent signs. 
8. When k= 0, the points M, N, E, F are indefinitely distant. 
In this case let the first and last surfaces meet the axis in A, B. 
Let AO = a, BO = 3; then, since k = 0, the expressions for 
Y254 12 B2s +2 become 
Yrst1 =IJIi +2Bo; Basro = 1B, gl =1, 
Bo _ Bost2  Yos4i 
eo p—@ wa gO 
whence a 
Z =gh+ g" ? 
B 
Bieta Stee 
When p—a= om, g—b=~%, therefore when the rays of a 
pencil in the first medium are parallel, the rays in the last medium 
are also parallel. The ratio of the angles the incident and emer- 
gent rays make with the axis is in all cases invariable. 
A telescope adapted to an object indefinitely distant and to a 
long-sighted eye, is an example of the preceding case. If the 
first and last media are the same, and if 6, y denote the angles 
which the incident and emergent rays make with the axis, 
Yai ee ata 
BE eo 
On this property depends the method of determining the 
power of a telescope proposed by Gauss in the second volume 
of the Astronomische Nachrichten, as well as the dynamometers 
of Ramsden and Plossl. 
