GAUSS’S DIOPTRIC RESEARCHES. 497 
the second lens, and .*. also in the plane S L M; / the focal length 
of the second lens. Let perpendiculars to the axis through L, M, 
N meet PSin R,S, @Tin T. Then 
whence, observing that 
SM_RL TN_RL_ SM 
Pw Pi An Pi f°’ 
RL 
SM=RL + PL ML. 
11. Let M, N be the focal centres of the nth of a series of lenses 
having a common axis, f),, 4 ; its focal length. 
TN SM 
SM= Yon+ Is QN = Bon +4 PM = Bons 
TN = SM, and 
also 
i 
LM = éo,,; 5 an = @2n4 1° 
J Zn 
Then 
Bon +9 = Bon + G9n+1 Y2n+1 
Yon+1 = Yon—1 + fon Bons 
whence, as in the former case, y, h, k, / retaining the same signi- 
fication, 
Yost1 =IYi + hBo 
Bos49 = ky, + 1 Bo. 
By a process exactly similar to that which has been employed 
in the case of a system of surfaces, if A be the first focal centre 
of the first lens, B the second focal centre of the last lens; M, N 
the first and second focal centres of the system of lenses ; P, Q the 
foci of incident and emergent rays; E the principal focus of rays 
coming in the contrary direction; F the principal focus of rays 
coming in the same direction; O a fixed point in the axis be- 
yond the last lens; a, b, m, n, p, g, e, f the distances of A, B, M, 
N, P, Q, E, F from O; u, v the distances of P, Q from M, N 
respectively ; then 
LE Ree ae m= 6+ 
1 1 
SATs 
