———_—SS—SSSS ee 
SS 
GAUSS’S DIOPTRIC RESEARCHES. 493 
sected by the directions of a ray in the first and last media, or 
the conjugate foci of a small direct pencil refracted through all 
the surfaces; O a fixed point beyond the last medium; a, 4, p, q 
the distances of A, B, P, Q from O; then 
Gant 15 Vi Pos42  Yos4i 
Mo P—@ Pasyo g—o 
2 
7 Yeti agyy + mage 
Has 2) 92s 1 Ko M1 
gb <a ahr ears 
Whence, writing », v for wo, #25 4 9 respectively, 
Bema Oy me Le 
aa rar aaa 4 
pe y y pb 
Let p—a=u+p6, g—b=v+ 4, and we get 
ee Repke tks pen ! 
i +1 Palaat! z = 
where gp =g—ko,UJ=1+hi, W=h+gb—lo—kbo. 
Let A= 0 and g'=/, «. g—k G=14+k4, O=h+gb—1o—kb 9, 
whence/+ k§=1, g—kO=1, 
yo fe 
= as 
where u=p—(a+n E+), 0=9— (0+ 227), 
In the above equation u, v are the distances of the conjugate 
foci from two fixed points M, N, such that m, n being the di- 
1 ; i nab +9, 
M, N are called the first and second focal centres of the sy- 
stem of surfaces. 
5. Let the directions of the ray in the first and last media meet 
the axis in P, Q, and perpendiculars to the axis through M, N 
in $,T. Then 
stances of M, N from O, m=a+ p 
B B l=] 1—1 
SM =") PM= "2 (p—a—p =, = fo 
k 
Bo542 B2s+2 Fin ar 
TN= —_ QN= ie (9-3-9 +) 95.41 Breq oo 
