110 W. Harkness—Color Correction of Achromatic Telescopes. 
and consequently his subsequent reasoning is fallacious, for in 
that case 7 does not have to be infinite to cause equation (27) to 
vanish. (III) I may oe that the origin of the confusion is in 
making the ratio D ~ n equation (9) constant ; it may be, and 
of course should be cue BRC 
= (LY) Professor ‘Harkness has made another merge founded 
Mathematics. (V) His eo eer however (p. Sot direotly 
contravenes this principle, for he finds that the focal plane does 
not aie to the minimum focal distance, but to something 
greater. (VI) The source of error is the introducti ion of a varia- 
itself differently in observing the star and its spectrum. Had the 
writer used eye-pieces of successively higher power, thus lessening 
seaciacechs the power of accommodation of the system, with 
his pri u i 
serve to control the eye. 
“(VII) Finally, cae re conclusion (p. yh is strictly true, 
though we are not to conclude, as would seem from the text, that 
the detriment due to the secondary spectrum aupunie either solely 
upon the aperture or varies inversely as the focal length ; 
Let us examine this criticism in detail; referring- to its 
clauses, and to wes equations of my original paper, by their 
respective num 
Clause [ wnaally asserts that three quantities can be arranged 
in two classes otherwise than by putting one in one class an 
two in the other. To prove this we remark that equation (12) 
may be written 
0 = A,(d, + 2¢,y,2) + A,(b, + 2¢,7,”) + A,(b, + 2¢,7,") (36) 
For all glasses of which I have any knowledge, é is positive, 
and very much larger than c. The latter quantity is some- 
times negative ; but when this happens, it is exceedingly small. 
7 cannot be otherwise than positive. From these conditions it 
results that the quantities (b+ 2cyo’) are invariably positive, 
and therefore the sign of each term in (36) depends solely upon 
the sign of its A. But in order that (36) may be true, one of 
its terms must have a different sign from the other two; and 
just because the properties of a system of infinitely thin lenses 
