Two lenses can be placed so that they work as one as long as they are attached *coaxially*, that is, with matching major axes. In this case, they will be called **juxtaposed**, if they are touching, or **separated**if there is a distance **d** separating them.

These associations are important for understanding optical instruments.

When two lenses are associated, it is possible to obtain a **equivalent lens**. This will have the same characteristic of the association of the first two.

Remembering that if the equivalent lens has positive vergence it will be convergent and if it has negative vergence it will be divergent.

## Juxtaposed Lens Association

When two lenses are juxtaposed, the **vergence theorem** to set an equivalent lens.

As an example of juxtaposed association we have:

This theorem states that the lens vergence equivalent to the association is equal to the algebraic sum of the component lens vergences. That is:

It can also be written as:

## Separate lens association

When two lenses are associated separately, a generalization of the **vergence theorem** to set an equivalent lens.

An example of a separate association is:

The theorem generalization says that the lens vergence equivalent to such an association is equal to the algebraic sum of the component vergences minus the product of these vergences by the distance that separates the lens. Thus:

Which can also be written as: