What is a “surface” and the “likelihood”?

On Neyman & Pearson, 1933, page 302,

Then the family of surfaces of constant likelihood, $$\lambda$$, appropriate for testing a simple hypothesis $$H_0$$ is defined by

$$p_0 = \lambda p(\Omega_{\text{max.}}).$$

It will be seen that members of this family are identical with the envelopes of the family

$$p_0 = kp_t,$$

Which bound the critical regions.

Probably because this paper was written nearly a century ago, and many terminologies have since been changed, I don't understand what is a "surface" (a critical region?), or the "likelihood $$\lambda$$" (a probability?). Even worst, I don't know which region is defined by these two formulas. Can someone explain this in modern terminologies?

By the way, just to be sure, here "envelopes of the family" means their union, right?

Reference:

Neyman and Pearson. On the problem of the most efficient tests of statistical hypotheses. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character (1933) vol. 231 pp. 289-337

• "Surface" and "likelihood" not only remain current, but appear to have the same meanings they had in 1933. "Envelope," too, is a geometrical concept whose meaning hasn't changed. It's not a union, though. The "illustrative example" that follows this passage clarifies the meanings. – whuber Jul 17 at 20:01
• @whuber Just want to say that I learned math/statistics in Chinese, and thus are not familiar with these English terminologies. Can you point me to a Glossary or something like that? – nalzok Jul 19 at 17:59
• @whuber I have reached P.307 where the authors try to illustrate the idea with an example. I can grasp it from the diagram, but the obscureness of terminologies is too serious for me to fully understand it. They talked about "the envelope for which $\lambda = k$", but what does $\lambda$ and $k$ even mean, a likelihood function or what? Also, speaking of the "surface", it is a synonym of "boundary"? – nalzok Jul 19 at 19:19