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after some trials, I obtained in a very convenient form, as 
follows :— 
** D being the deviation of the ray (not from one lens 
into another, but always) either from air into a lens, or from 
a lens into. air, a is as (sin? D). Then, calling the aberration 
of the entire combination A, and a, a., a3, &c., the aberrations 
(calculated from the above formula) at the Ist, 2nd, 3rd, &c., 
surfaces of the compound, 4 is equal to the sum of a, a, 
a, &e. (respect being paid to their signs), these being plus 
when the ray is deflected towards the axis, and minus when 
the contrary. 
«¢ When the spherical aberration of the compound under 
examination is corrected, the sums of the + and — aberrations 
will be equal, and 4 will evidently be zero, while in any 
other case, the + and — quantities will exhibit a fraction 
showing their relative proportion, and will indicate to the 
practical operator the means of a closer approximation to 
perfect correction. 
‘* The entire process may be shortly described as follows:— 
«‘ A diagram of the combination to be examined being 
made with care, and of a size as large as circumstances permit, 
—one single ray must be traced (in the usual manner) through 
the diagram; the most suitable distance for this ray from 
the axis of the combination will depend upon the angular 
aperture. For combinations of small angular aperture I prefer 
3-4ths to 4-5ths of the semi-aperture for this distance. In 
addition to this (should the combination have any two sur- 
faces in contact), the direction of the ray from these into air 
must be projected. 
‘“*'The diagram is now fitted for obtaining the sine of D 
for each surface; these, being measured on a scale of equal 
parts, and tabulated, and afterwards squared, give aj, a2, ds, 
&e., which being separated, as directed, into +-and -, give, in 
their sums, two- quantities, representing respectively the 
positive and negative aberration of the whole combination. 
