W. Harkness—Color Correction of Achromatic Telescopes. 111 
in contact are independent of the order of the lenses; the 
choice of this term is arbitrary. Taking advantage of this 
circumstance to follow the usual practice of opticians, I made 
the middle lens different from the other two, and wrote 
es a A, (4, rs 2¢,y,) — A,(4, +: 2c,y,') oe A,(6, - 2¢,y,') (37) 
But Clause I declares, ‘True A, should have an opposite sign 
to A; + As, but that asserts nothing as to likeness of the latter 
symbols in sign.”—A statement which is manifestly untrue, 
unless it can be shown that three quantities can be arrang 
in two classes otherwise than by putting one in one class an 
two in the other. 
Clause IT asserts that n, in equation (16), may be negative. 
This is absurd, because n = A,~ A,, and it has j een 
shown that the signs of A, and A, are always similar. 
Clause III declares that D ~ E should be indeterminate ; and 
that all my alleged errors arise from making it constant. 
Referring to equations (6), we see that 
- = wee + Ab, + A,d, (6) 
= Ave, + A,c, + A,ec 
so far arbitrary that any glasses, and any curves, may 
chosen; but when the objective is completed I certainly do 
hold that its curves, and the physical properties of the pieces 
of glass composing it, are constant. If I am right in this, it 
follows that both D and E, and also their ratio are constant; 
Clause III to the contrary notwithstanding, 
Clause IV admits the accuracy of my statement that an 
objective is pro erly corrected for any given purpose when its 
minimum focal distance corresponds to rays of the wave-length 
which is most efficient for that purpose; but says the statement 
Tequires proof, and is not self-evident. With the law of dis- 
persion assumed in equation (2), the focal curve can have but 
One tangent parallel to the axis of abscissas; and I did not 
Suppose it necessary to tell the readers of this Journal that the 
parts of such a curve nearest the tangent line are those adja- 
Cent to the point of tangency. That consideration proves m 
Proposition, and it is so elementary that I thought it. self- 
evident. If more than two lenses, and a dispersion formula 
Involving more than two powers of the wave-length, are 
assumed ; I venture to say that the condition for color corree- 
ton stated above cannot be proved. It may be true in special 
Cases; but in general, the focal curve will have such a form as 
‘0 give more than one minimum focal distance. 
