"In reply to your letter of . . . I would inform you . . . " is a common way of beginning official letters that do not need to be in the full-dress official style of "I am directed . . . " In this phrase would is not a mere auxiliary expressing the conditional mood; it retains the now archaic meaning of "I should like to". On another page I have deprecated the use of this expression on the ground that, since it is archaic, it cannot help being stiff.
Because would has this meaning, grammarians condemn such phrases as "I would like to", "I would be glad if", "I would be obliged if", and so on. Should, they say ought always to be used: to say would is tantamount to saying "I should like to like to", "I should like to be glad if", "I should like to be obliged if", and so on. Here, as in the use of will and shall with the first person (see Shall and Will), practice, under the lead of the Celts and Americans, is fast getting the better of the grammarians. But the grammarians should be respected by the British official, and the intrusion of would upon should ought not to be encouraged.
"It would appear" and "I should think" are less dogmatic, and therefore more polite, ways of saying "it appears" and "I think".