  The J\'{a}nossy density for a determinantal point process is the probability
density that an interval $I$ contains exactly $p$ points except for those at
$k$ designated loci. The J\'{a}nossy density associated with an integrable
kernel $\mathbf{K}\doteq (\varphi(x)\psi(y)-\psi(x)\varphi(y))/(x-y)$ is shown
to be expressed as a Fredholm determinant
$\mathrm{Det}(\mathbb{I}-\tilde{\mathbf{K}}|_I)$ of a transformed kernel
$\tilde{\mathbf{K}}\doteq
(\tilde{\varphi}(x)\tilde{\psi}(y)-\tilde{\psi}(x)\tilde{\varphi}(y))/(x-y)$.
We observe that $\tilde{\mathbf{K}}$ satisfies Tracy and Widom's criteria if
$\mathbf{K}$ does, because of the structure that the map $(\varphi,
\psi)\mapsto (\tilde{\varphi}, \tilde{\psi})$ is a meromorphic
$\mathrm{SL}(2,\mathbb{R})$ gauge transformation between covariantly constant
sections. This observation enables application of the Tracy--Widom method to
J\'{a}nossy densities, expressed in terms of a solution to a system of
differential equations in the endpoints of the interval. Our approach does not
explicitly refer to isomonodromic systems associated with Painlev\'{e}
equations employed in the preceding works. As illustrative examples we compute
J\'{a}nossy densities with $k=1, p=0$ for Airy and Bessel kernels, related to
the joint distributions of the two largest eigenvalues of random Hermitian
matrices and of the two smallest singular values of random complex matrices.
