Here's a simple lens situation for you to look at MrBrody. Let's make a convex-plano lens out of glass with an index of 1.5. Make it 1cm diameter (not critical) and the convex first surface (on the left) has a radius (R) = 5 cm, and it is a very thin lens.
It iseasy to show that such a lens has a focal length given by:
f = R /(n-1) = 5 / (1.5 - 1) = 10 cm so parallel rays from - infinity focus 10 cm to the right of our lens. The Petzval curvature is 1/nf = 1/(1.5 x 10) =0.0667 .
So the Petzval surface has a radius of 15 cm, and is curved in to the left towards the lens. Well we won't worry about any other aberrations like astigmatism.
Now suppose we made a concave-plano (negative ) lens with the concave surface being -5cm and using our 1.5 index glass. Well this lens will have a focal length of -10 cm, and it will have a Petzval curvature of -0.0667, so a -15 cm Petzval radius. Now it too, is a thin lens.
So now I take my negative lens, and I place it at the focus of the first lens, 10 cm to the right of the first lens. Well we'll put it just short of the 10cm, so the marginal rays can pass right through the negative lens, just before they reach the focal point on the axis. Remember these are both very thin lenses.
Well the rays had almost focused before they got to the negative lens, and it will try to defocus them a little bit, but the middle of the lens is nearly flat so the negative lens doesn't do much, but it does slightly increase the focal length of the combination.
In the limit of really thin lenses, the second lens does almost nothing and the total focal length remains about 10 cm.
BUT !! A miracle has happened. The second lens has completely cancelled the Petzval curvature of the first lens, while leaving virtually all of its optical focusing power intact. This is often called a "field flattener" lens, and is a very common approach to fixing up field curvature problems.
Now you could also do the reverse. If instead of putting our -10 cm negative lens at the focal plane, what if I put a second +10 cm positive lens there ??
Once again, the lens placed right at the image plane will not change the optical power significantly; but I just doubled my Petzval sum, and halved the radius of curvature of the system Petzval surface. Now these simple lenses wouldn't be much good in a real situation, specially with chromatic and other aberrations, but you can see how the focal surface curvature can be manipulated.
I'll let you draw the pictures for yourself, and play with it to see what happens.
If you look at the CERN monster accelerator, you will find, that all around that huge ring, there are magnetic focusing lenses, interspersed with magnetic defocussing lenses, exactly the same as I just described. Now Accelerator designers call this "Strong focusing"; they probably have never ever heard of the Petzval sum, or the Petzval curvature, but that is exactly what they are doing, to keep those particle beams confined to their track, without curling up into a pretzel. The same optical scheme is used to pipe an image a long way, such as in an old time optical periscope.