compound lenses and object-glasses. 237 
in contact, whose thicknesses are respectively t, t', t", &c. s 
and powers (neglecting their thicknesses) L, L', L", &c. 
their refractive densities being respectively p, p r , |x", &c. 
and the curvatures of their anterior surfaces, r, r', r", &c. 
then will the reciprocal distance of the image from the pos- 
terior surface of the last lens be given by the equation. 
/= L + L'-f L"+ &c. — D 
+ m j (jtA — 1 ) r — D yt 
+ w , {L+ {yJ—i) r'— D jV 
+ m"{L + L'+ (p"-i) r" — D|Y'-J- &c (y) 
continued to as many terms as there are lenses. In this equa- 
tion m, m , m ", &c. are the reciprocals of jx'', &c. 
General theory of the aberrations of spherical surfaces for rays 
incident in the plane of the axis. 
7. Let us next proceed to investigate the spherical aberra- 
tion of any system of surfaces. Suppose the ray, after passing 
through the n th surface, to be incident on the (» + i) tfc : its 
aberration here will arise from two causes ; 1 st. that after 
traversing the n preceding surfaces, instead of converging to, 
or diverging from the focus for central rays, its direction was 
really to or from a point in the axis, distant from that focus 
by the total aberration of those n surfaces; and, 2dly that, 
being incident at a distance from the vertex of the (w-f-i)th 
surface, a new aberration will be produced here, which (be- 
ing, as well as the other, of small amount) the principles of 
the differential calculus allow us to regard as independent of 
it, and which, being computed separately and added to it, 
gives the whole aberration of the system of w + i surfaces. 
