given by more general smooth measures $u(x)dx$ and $v(y)dy$ on $X$ and $Y$. A direct calculation yields the following

\[P:bian\] Suppose conditions (a), (b), and (c) hold. If the measures vary by $dx'=u(x)dx$, $dy'=v(y)dy$, and $d\mu'=\lambda d\mu$, where $u, v$ are positive smooth functions on $X, Y$, respectively, $u$ is $G$-invariant, and $\lambda$ is a positive constant, then $J(y)$ varies by $$J'(y)=\frac{u(y)}{\lambda v(y)}J(y).$$

Now we suppose that there is a Riemannian structure on $X$ such that $dx$ and $dy$ are the induced Riemannian measures on $X$ and $Y$, respectively. We suppose the following orthogonality condition holds

0.3cm (*orthogonality condition*) $T_yY\perp T_yO_y$, $\forall y\in Y'$.

0.3cm [Then]{} we can compute the factor $J(y)$ in a simple way by the following theorem.

0.3cm Let ${\mathfrak{l}}$ be a linear subspace of the Lie algebra ${\mathfrak{g}}$ of $G$ such that ${\mathfrak{g}}={\mathfrak{k}}\oplus{\mathfrak{l}}$, where ${\mathfrak{k}}$ is the Lie algebra of $K$. Let $\pi:G\rightarrow G/K$ be the natural projection, then $(d\pi)_e|_{\mathfrak{l}}:{\mathfrak{l}}\rightarrow T_{[e]}(G/K)$ is an isomorphism. We endow a Riemannian structure on $G/K$ such that the associated Riemannian measure is $d\mu$, then it also induces an inner product on $T_{[e]}(G/K)$. For $y\in Y$, we define a linear map $\Psi_y:{\mathfrak{l}}\rightarrow T_yO_y$ by $$\label{E:Psi}
\Psi_y(\xi)=\frac{d}{dt}\Big |_{t=0}\sigma_{\exp t\xi}(y) , \quad
\forall 