Proposition 6

The map $t \mapsto \mathbf{v}(t)$ of J into $\mathrm{E}^*$, a primitive of a regulated function on J, is such that $\langle \mathbf{u}(t), \mathbf{v}(t) \rangle$ is constant on J for every solution $\mathbf{u}$ of the equation (4) of IV, p. 183 if and only if $\mathbf{v}$ is a solution of the homogeneous linear equation

where $-B(t)$ is the transpose of $A(t)$.

Equation (16) is called the adjoint of (4); clearly (4) is the adjoint of (16). The elements of the matrix $B(t)$ being regulated functions of $t$ on J, the results obtained above on linear equations apply to (16). In particular, the integral of (16) taking the value $\mathbf{x}_0^*$ at the point $t_0$ can be written as $H(t, t_0).\mathbf{x}_0^*$, where $H(t, t_0)$ is a bijective linear map of $\mathrm{E}^*$ onto itself, identical to the integral of the equation

which takes the value $I$ at the point $t_0$. As a result one has (with the notation of IV, p. 179)

for any $\mathbf{x}_0 \in \mathrm{E}$ and $\mathbf{x}_0^* \in \mathrm{E}^*$, which shows that

(the contragradient of $C(t, t_0)$). In particular, if one knows a fundamental system of integrals of the adjoint equation (16) the matrix $H(t, t_0)$ is determined, so is $C(t, t_0)$ also, and as a result, so are all the integrals of equation (4).