The perfect participle is a compound verb form consisting of an auxiliary (in the -ing form) and a verb.
Using the perfect participle emphasises that the first action was completed before the second action started. In the above example, the action of studying was before the action of going to play football.
It is sometimes correct (and more natural) to use the present participle in such sentences:
So, instead of writing:
We would write:
To avoid confusion, it is often better to use the perfect participle rather than the present participle:
The above sentence might give the impression that the two actions were simultaneous (happening at the same time). Here, therefore, the perfect participle would be better:
The perfect participle is, however, necessary when there is an interval of time between the two actions:
It is also used when the first action covered a period of time: