1. 关于on condition that和on the condition that

英语通常用 on condition that，很少用 on the condition that。如：

I will come on condition (that) she is invited too. 如果邀请她来, 那我就来。

You can go out on condition that you wear an overcoat. 你要穿上外衣才能出去。

You can take my car on condition that you drive carefully. 你可以用我的汽车，只要你小心点儿开。

I’ll lend it to you on condition that you bring it back tomorrow. 我把它借给你，条件是明天你还给我。

I’ll give you the day off on condition that you work on Satiny morning. 我让你休一天假，条件是周六上午你上班。

I will do it on condition that you help me. 只要你肯帮助我，我就愿做此事。

We’ll let you use the room on condition that you keep it clean and tidy. 我们可以让你使用这个房间，条件是你把它保持整洁。

但我在《朗文英语语法》上查到了on the condition that 的用法，见以加下粗部分：

下列连词也能引导条件句，但这些连词并非总是与 if 完全同义：as long as (只要)，assuming (that) (假定)，even if (即使)，if only (只要)，on (the) condition (that) (如若)，provided / providing (that) (只要)，so long as (只要)，unless (除非)；还有suppose (that) (假设)，supposing (that) (假定)，这最后两个连词通常后接疑问句。

2. 关于用于虚拟语气

on condition that 通常用于引出陈述语气，但有时也接should+动词原形这样的虚拟语气：

He can use the car on condition that he should return it tomorrow. 如果明天车能还回来，他就可以把车借去。

He can use the bicycle on condition that he should return it tomorrow. 如果他明天能还回来，就可以借用这辆自行车。