434 C. S. Hastings—Triple Objectives 
Of course in its practical application this process should be used 
to yield a first approximation only, since the thicknesses and 
distances of the lenses are neglected ; but having this there is 
no difficulty, other than the laborious character of the compu- 
tations involved, in determining by successive approximations 
the values of all the radii requisite to secure complete color 
correction and at the same time eliminate spherical aberration. 
As in the case of a double objective, after satisfying the condi- 
tions of given focal length, of color correction, and elimination 
of spherical aberration, we have one arbitrary condition to 
impose, so in a triple objective we have two arbitrary condi- 
tions to impose. In my opinion, were we using materials that 
required large curvature sums, it would be advantageous to 
utilize these two conditions in making two of the lenses respec- 
tively biconvex and biconcave, thus rendering the necessary 
thickness of the materials a minimum 
These results are directly opposed to those of a recent writer 
in this Journal.* B is conclusions arise from erroneous 
calculation. Not only does his interpretation of his equation 
(12) imply the manifest absurdity that in a system of infinitely 
thin lenses in contact its properties are determined by the 
order of the lenses, but the interpretation is impossible. True 
A, should have an opposite sign to A,+A,, but that asserts 
nothing as to likeness of the latter symbols in sign. Thus ” 
in equation (16) may be negative and consequently his 
subsequent reasoning is fallacious, for in that case n does no 
have to be infinite to cause equation (27) to vanish. I may 
add that the origin of the confusion is in making the ratio i 
in equation (9) constant ; it may be, and if course should be, 
indeterminate. 
inadequate experiment, which has so important a bearing on 
not correspon 
thing greater. The source of error is the introduction of & 
* Professor Harkness, in the September number, pp. 191-193. 
