In my textbook, with $S=|\mathbf{S}|$, from the picture below, they derive
$\int_{x}^{x+h}\rho_lu_{tt}''d\lambda=S(x+h)\sin(\alpha(x+h))- S(x)\sin(\alpha(x))+\int_{x}^{x+h}Fd\lambda$, where $F$ is [N/m], $\rho_l$ is [kg/m] and $u$ the position in y-axis of a point of the string.
Why is it though, that they put a minus sign between the internal forces of the string in the expression above? I understand the concept (except this addressed confusion) and further steps of differentiating (dividing by $h$ and limit) in other derivation, although what if the tangential force $S(x)$ is not facing in negative direction, but in a positive one? - then the expression would be fault? My textbook says that it is general.