494 GAUSS’S DIOPTRIC RESEARCHES. 
—1 1—/ 
= gy, + bBo — (ka — 18.) F— = y, — By. 
e TN= SM. 
This result, combined with the equation expressing the rela- 
tion between uw, v, shows that, when pz, y are unequal, the path 
of the ray in the last medium is exactly the same as if the di- 
stance between the focal centres M, N had been annihilated, 
and the first and last media separated by a spherical surface 
— passing through the point in which 
M, N are supposed to coincide. When gp, v are equal, the path 
of the ray in the last medium is the same as if a thin lens, ha- 
having for its radius 
ving - for its focal length, had been placed, in vacuo, at the 
point in which M, N are supposed to coincide. 
6. Let E be the place of P when Q is indefinitely distant, F the 
place of Q when P is indefinitely distant ; e, / the distances of 
ks [fe 
E, F from O, w=p—m, v=q—n 5-5 =h 
when g= ©, p=e, whenp=0,qg=/f,.. m—e=—,f—n=—, 
er ae. 3. of — i. ee 
-e=a—p7,f=bt+r~, jan pe 
Whence (p—e) (f—g) = (m—e) (f—2). 
Or PE.FQ = ME.FN, PE, FQ being measured in dif- 
ferent directions from E, F. 
x S 
Nese s = 
Let PS, the path ofa ray in the first medium, and a parallel to 
it through E, meet a perpendicular to the axis through M in S, H. 
Draw ST, H G parallel to the axis meeting perpendiculars to 
the axis through N, F, in T, G. Then GT will be the path of 
the ray in the last medium ; for 
OPN: Pal’ EM 
¥FG_ SH’ SH’ MB’ 
7. When u = 0, v = 0, therefore M, N are conjugate foci, or 
when the path of a ray in the first medium passes through M, its 
“, PE.FQ=ME.EFN. 
