492 GAUSS’S DIOPTRIC RESEARCHES. 
spherical surfaces having their centres in the same straight line, 
on its index of refraction, #2, 49 the index of the following 
medium BT :— 
ton 
cial a, AR=Yn-1 BS =yoniiy 
Bo 
tan SPB = —, tanSQB Ponte *9,41 the radius 
M2n P2n +2 
of B S, (M2, Se ae an) 
Then 
Ton + Hl e at Mi 
Ban+2 = Bon + 02n41 Y2n41 
Yon+i = Yan—1 + ton B2n- 
Therefore, substituting 1, 2, 3,.... s for n successively, 
By = Po tai % 
Y3 =I, + ty Bo 
By = Bo + @3 Ys 
Yos41 = Yos—1 + bos Bos 
Bos4 2 = Bas + @954.1 Yas tie 
Whence it is easily seen that 
Bo = 21 1 + Bos 
Y3 = (1 + @ fe) + be Bos 
By = (21 to @3 + 8) + @3) ¥1 + (1 + fo @) Bo» 
Yos41= 91 + ABo 
Bos49 = ky, + lB; 
where g, A are the numerator and denominator of the last but 
one; k, 7 the numerator and denominator of the last of the frac- 
tions converging to the continued fraction, of which the quotients 
are O15 bay Ogy oe evecsecy L255 62s+ 1° 
Hence, by a well-known property of converging fractions, 
gl—hk=1, 
I) = lyos41 —hBas4 95 Bo = 9 P2549 — KYos41- 
4. Let A, B be the points in which the axis intersects the first 
and last surfaces; P, Q the points in which the axis is inter- 
