Final answer:
Dupin challenges traditional methods and criticizes the police's lack of imagination, causing him to investigate the windows further.
Step-by-step explanation:
Dupin uses the process of elimination in sentences 60-64 to cause him to investigate the windows further by challenging the traditional methods used by the police. He criticizes their lack of imagination and limited perspectives, leading them to overlook alternative hiding places. Dupin, on the other hand, considers the possibility that the hidden letter may be located in a more obvious and accessible place, such as in plain sight on a window sill.