Why doesn't the selector h3:nth-child(1):contains('a') work?

:contains() is not was going to be a CSS3 selector (thanks T.J. Crowder for the link), but it didn't make it, most likely because the way it works tends to lead to severe performance and over-selection issues. For example, if an element E matches :contains() for a given string argument, then all of its ancestors would also match; using it with a universal selector would lead to unexpected results with certain style properties, on top of being slow for the browser.

There is no other CSS selector that serves a purpose like :contains(). So you'll have to find some other way, either by modifying your HTML or even by using jQuery's :contains(), to achieve the effect you want:

Select an h3 element
if it is the first child of its parent
and its text contains the letter 'a'.

For jQuery and Selenium RC users: :contains() is implemented in the Sizzle selector engine used by jQuery, which is also used in Selenium RC (but not Selenium WebDriver). It works as described in this decade-old revision of the CSS3 spec, but again, due to how the spec describes it, you need to use it with care or it may lead to unexpected selections.

On a final note, h3:nth-child(1) can be replaced with h3:first-child, which as a CSS2 selector has better browser support.

If you're trying to use :contains(a) to find an anchor tag (rather than the letter A), you could use:

h3:nth-child(1) a

or

h3:first-child a