The narrow neck of the density bottle is to ensure that the volume of liquid inside the relative density bottle can be determined accurately.
Why is this necessary? The accuracy of the value you obtain when you are calculating the density of your liquid depends on how precisely accurate the volume of liquid in the density bottle is determined.
To ensure that the volume of liquid in the bottle is exactly what is measured, the bottle has a narrow neck through which a ground glass stopper with a fine hole is inserted. This ensures that any excess liquid rises through this hole and out of the bottle when the bottle is filled with liquid. This way, you are sure to get exactly the same volume of liquid each time it is filled.
Human error in gauging the level of liquid in the container is also eliminated, when compared with taking the same measurement using a jar with a larger cross sectional area.
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