Codition for achromatism

Combination of two or more lenses such that there will not be a chromatic aberration is known as achromatic lense and the condition is called as achromatism
Mathmatically, df = fr-fv = 0
Now let us take two lenses having focal length f1&f2 and are separated by a distance x. Then theequivalent focal length of combination will be
1/f=1/f1 + 1/f2 - x/(f1*f2)
Diffrentiating this equation we have
-df/f2 + -df1/f1^2 - df2/f2^2 - x(-df1/f1^2f2 - df2/f1f2^2)
But we have
-df1/f1=w1 and -df2/f2 = w2
where w1 and w2 are the dispersive power of 1st and 2nd ;ems
-df/f2 = w1/f1 + w2/f2 - x( w1/f1f2 + w2/f1/f2)
For achromatism df = 0
w1/f1 + w2/f2 = x(w1/f1f2 + w2/f1f2) = 0
When two lense are kept in contact then x = 0
w1/f1 + w2/f2 = 0
f1/f2 = -w1/w2
The negative sign here indicates combination of lense must be of one convex and another concave