How does Clifford Cocks 'Non-Secret Encryption' work?
The two last equations don't directly give you the value of $C_i$, they are telling you the values of the remainder of Ci when divided by $P$ and $Q$. You then use the Chinese Remainder Theorem with this information to produce the value of $C_i$ (modulo $N$) that you are looking for.

See en.wikipedia.org/wiki/Chinese_remainder_theorem (there is an algorithm for solving it for the case of two equations)