Hello, I'm reading a textbook named "Introduction to Rocket Science and Engineering" by Travis S. Taylor.

I'm now in a general cosepts about orbits. I have a question, what the book says in here?

I can't understand the bold formatted sentence. What the author try to mean? If you can, can you describe this sentence by using images, datas or diagrams, please?

This is my interpretation...
M is the mass of the body whose sphere of influence we are determining.
m is the mass of a smaller body orbiting the larger body, and which may serve to limit the larger's sphere of influence. I.e., when a third body is sufficiently far from M (>r_{SOI}) its orbit may be significantly affected by m.
a_{p} is the semimajor axis of m's orbit about M.

I see it differently. Since the sphere of influence decreases as the mass of the larger body increases I am lead to believe that it refers to the distance from the smaller mass. Thus the larger the big mass gets the closer you have to get to the smaller mass for it to have a larger affect than the big mass.

Well this description is good but, I can't image what is going on, sadly. Can you describe this with some paper & pen drawings or an paint image, please? My English and Google Translator's Turkish is not enough to describe three things' relations :P

Thanks for your reply, I got what you tried to mean, thanks!

It specifically mentions the sphere of influence of the earth being constrained by the sun and the moon, but you are right that the form of equation only makes sense if the sphere of influence in question is that of the smaller body.
Thanks for the correction.