please judge my sentence

[hanky]

Here M(X) and M'(Y) are the number of particles containing
in the systems X and Y, respectively.

If this isn't a good sentence, could you please kindly suggest another?
Thanks.

 

[2006]

Here M(X) and M'(Y) are the number of particles contained in [STRIKE]the[/STRIKE] systems X and Y respectively. (no comma after Y)


Thanks.

2006

 

[hanky]

2006

Could you please explain why "the" is not used before "systems"?

Thanks.

 

[2006]

Could you please explain why "the" is not used before "systems"?

Thanks.

It's just not needed, so it's just in the way.

As another example, suppose you have 5 cars, numbered 1 through 5, in a race. You could say 'I think car 3 has no chance of winning.' You wouldn't say 'I think the car 3 has no chance of winning.'