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Write $r=x-\bar{x}$, $A(t)={\int }_0^t\alpha (s)ds$, and use the usual reduced rigid-recoil representation $h=Z+r\theta -\alpha r^2$, with zero initial recoil and a smooth stroke returning $\alpha$ and $\dot{\alpha }$ to zero. For this representation,Define the fluid impulse and its first moment byThe force law from part (i) gives, by integration by parts,These are the endpoint momentum flux in elongated-body recoil. PutThe endpoint quantities are needed in addition to the integrals $M_j$ unless further end-shape assumptions are made. Lateral force and yaw torque balance areExpanding gives the requested simultaneous second-order equations, with the retained endpoint fluxes:Substituting $\ddot{Z}=\ddot{\gamma }-U\dot{\theta }$ and integrating once from the initially straight, unrecoiling state yieldsThese are first-order coupled equations for $\gamma ,\theta$ with known forcing.

The claimed mass-independent final angle requires a qualification. LetAfter the stroke, $\alpha =\dot{\alpha }=0$ and $A=A_T={\int }_0^T\alpha dt$. If the post-stroke dynamics are stable, the first-order system relaxes toLet $C=B_0(B_2-M_1)-B_1(B_1-M_0)$. Solving this two-by-two system givesThis is quadratic-bend turning with finite body mass. At the limiting state $\dot{\theta }=\dot{\gamma }=0$, so $\dot{Z}=-U{\theta }_{\infty }$. The forward velocity is $-U$ in the chosen $x$ direction, and its lateral component has precisely the body's slope: the fish glides parallel to itself. The angle is generally reached after a relaxation transient, not necessarily immediately at $t=T$.

The advertised result is obtained in the thin-body approximation that neglects $M_b$ relative to fluid added mass, or in the special geometry $B_1=M_0$:For example, take negligible nose added mass and nonzero tail added mass $m_T$, so $B_j=m_T(L-\bar{x}{)}^j$. In the zero-body-mass approximation, the post-stroke equations for $(\gamma ,\theta -2U{A}_T)$ are homogeneous. The inertia matrix is positive definite by Cauchy-Schwarz inequality, andThe two eigenvalues of ${\mathsf{M}}^{-1}\mathsf{K}$ therefore have positive real parts: their sum and product are positive, whether they are real or conjugate. For $U>0$ the transient decays, proving parallel gliding and the displayed turn in this approximation.

With finite $M_b$, the angle is not a consequence of the printed force law for arbitrary $m(x)$. A concrete stable counterexample uses $L=1$, $\bar{x}=1/2$, and $m(x)=m_{\ast }{x}^2$. Then $M_0=m_{\ast }/3$, $M_1=m_{\ast }/12$, $B_0=m_{\ast }$, $B_1=m_{\ast }/2$, $B_2=m_{\ast }/4$, and $C=m_{\ast }^2/12$. Hencewhich differs from $2U{A}_T$ for positive body mass. This example has positive determinant and positive matrix trace of the relaxation matrix. Another immediate check is that if both endpoint added masses vanish, integrating the force law conserves total lateral momentum. A finite-mass fish cannot then end with $w=0$ and a nonzero new transverse velocity. Thus endpoint and inertia assumptions cannot silently be omitted.

If $Z$ is literally the centre of mass position of a deforming body, use a mass-centered deformation rather than the raw quadratic. For body mass density ${\mu }_b(x)$, its mean is $-\alpha J_2/M_b$, with $J_2=\int {\mu }_b{r}^{2}dx$; in the slender approximation $J_2=I_b$. The reference translation above is then $Z_r=Z+\alpha J_2/M_b$. Likewise the active angular momentum contains $-\dot{\alpha }{J}_3$, where $J_3=\int {\mu }_b{r}^{3}dx$, which vanishes for symmetric mass distribution. These corrections modify derivative forcing during the stroke. Since the shape and its time derivative return to zero, they do not remove the finite-body-mass qualification of the final impulse relation.

The principal physical limitations are small slopes and small total turning angle, prescribed approximately constant forward speed, slowly varying slender geometry, and neglected viscosity, separation and detailed three-dimensional wake interactions. Fast large-amplitude C-starts need more than this linear theory. Neglecting body mass is most reasonable for a laterally thin body whose crossflow added mass greatly exceeds its displaced mass. Stable relaxation also requires an appropriate tail/end geometry; it is not guaranteed for every added-mass profile.