W. Harkness—Oolor Correction of Achromatic Telescopes. 118 
the number of terms being unlimited. For the dispersion 
formula, we write 
= P(A) (40) 
The form of g(A) is regarded as unknown; but there will be no 
loss of generality if it is developed in a series arranged accord- 
ing to the powers of 24. We therefore have 
Maat ba"+ cA"+ el? + Ke. (41) 
in which a, 0, c, ete., are constants, and the number of terms 
may be taken as great as is desired. Also, let us put 
O=A(a,— 1) + A(a,—1) +A,(a—1) + he. 
D=Ab,—Ab,4 A+ &e. 
E= Av.e,+ A,c, + A,c, + &e. (42) 
F>=Ae, + A,e, + A,e, + &e. 
&. &. kk. & 
right hand member of each of them, being the same as the 
number of terms in the right hand member of (41). Then, by 
@ simple transformation (39) becomes 
f= C+ DA" 4 EA* +4 Fa + &e. (43) 
This is the equation of the focal curve; A being the abscissa, 
and f the ordinate. Its first derivative is 
=~ f'(mDIA™ + nEN 4 pF 4 fe.) (44) 
the number of these equations, and the number of terms in the 
which, as is well known, expresses for every point of the curve 
the tangent of the angle made by the tangent line with the 
axis of abscissas. The number of rays of different degrees of 
refrangibility, which can be brought to a common focus, will 
evidently be the same as the number of times the focal plane 
intersects the focal curve. But the focal plane is necessarily 
parallel to the axis of abscissas; and therefore the greatest 
possible number of intersections of the curve with the plane 
can only exceed by one, the number of tangents which can be 
drawn parallel to the axis of abscissas. To find these tangents, 
we equate (44) to zero, and obtain 
0=mD + nEA*™ + pFAP + &e. (45) 
As 1 can never be either zero, imaginary, or negative, we 
have to consider only the real positive roots of this equation; _ 
each of which corresponds to a tangent. To make the number 
OF roots as great as possible, the quantities D, E, F, etc., must 
be ‘Independent of each other; which will be the case when 
the right hand members of the equations (42) contain as many 
Sas there are powers of 4 in (41). Hence it is evident that 
the number of real positive roots in (45) will be one less than 
the number of powers of 4 in (41), and we conclude that— 
