The first circuit (A) is the one given in my textbooks and also the one used in labs to measure resistance using Wheatstone bridge. I know how this works and all the derivations associated with it.
But recently, I came across some other worksheet materials with circuit (B) in which the way the readings are taken differs. The galvanometer is connected across different resistances compared to A. Wouldn't all the equations change for this arrangement? It appears to me that it's connected in the wrong manner.