Choice B is correct. It's given that the graph of $y = g (x)$ is produced by translating the graph of $y = f (x)$ $3$ units down and $4$ units to the right in the xy-plane. Therefore, function $g$ can be defined by an equation in the form $g (x) = f (x - 4) - 3$ . Function $f$ is defined by the equation $f (x) = \frac{a - 19}{x} + 5$ , where $a$ is a constant. Substituting $x minus 4$ for $x$ in the equation $f (x) = \frac{a - 19}{x} + 5$ yields $f (x - 4) = \frac{a - 19}{x - 4} + 5$ . Substituting $\frac{a - 19}{x - 4} + 5$ for $f left parenthesis x minus 4 right parenthesis$ in the equation $g (x) = f (x - 4) - 3$ yields $g (x) = \frac{a - 19}{x - 4} + 5 - 3$ , or $g (x) = \frac{a - 19}{x - 4} + 2$ . Therefore, the equation that defines function $g$ is $g (x) = \frac{a - 19}{x - 4} + 2$ .

Choice A is incorrect. This equation defines a function whose graph is produced by translating the graph of $y = f (x)$ $3$ units down and $4$ units to the left, not $3$ units down and $4$ units to the right.

Choice C is incorrect. This equation defines a function whose graph is produced by translating the graph of $y = f (x)$ $4$ units to the left, not $3$ units down and $4$ units to the right.

Choice D is incorrect. This equation defines a function whose graph is produced by translating the graph of $y = f (x)$ $4$ units to the right, not $3$ units down and $4$ units to the right.