60 
(and which were constant for all the surfaces of a combina- 
tion), were usually three and sometimes six inches ; the arcs 
were struck and the sines measured at both sides of the centre 
(to avoid errors of eccentricity), and the measurement of the 
sines taken with a scale of fiftieths of an inch, using a magni- 
fier, and estimating to tenths of divisions, or say 1-500th of an 
inch. i 
«¢ By tracing—with these precautions, and with care—two 
rays, one passing through nearly the margin of the combina- 
tion, and the other at one-half that distance from the axis, an 
approximation to the state of the spherical aberration of the 
compound, adequate for such combinations as are used in 
the construction of the objective of the microscope, may be 
obtained. 
‘‘Some practice in delicate manipulation is, however, 
required to arrive at the precision indicated, and even for 
approximate investigations a more accurate method would 
be desirable. 
‘¢ When, however, combinations of small aperture, com- 
pared with their foci (for example, such proportions of these 
as are adapted to the object glass of the telescope), are under 
examination, the method of diagram fails entirely in the ac- 
curacy requisite to estimate the correction of the spherical 
aberration. 
‘Such difficulties in the way of practical research in- 
duced me to consider how far the objections, arising out of 
the intricacy of the one mode of investigation, and the inade- 
quacy of the other, might be obviated by adopting a mixed 
method of proceeding. In short, it appeared that the diagrams 
not only included those conditions arising out of the thickness 
of the lenses, but that they would serve to furnish with suffi- 
cient accuracy for all cases that point in each surface where 
the ray impinges, and also the angle of its incidence. Little 
more thus remained to be done than to find a simple expres- 
sion for the aberration of each surface for the ray, and this, 
