In Riemann's "Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe", Riemann mentions taking a factor as "constant" in "partial integration", which seems to me to be referring to the factor $u$ in the standard integration by parts formula $\int u\,dv = uv-\int v\, du$:
...durch zweimalige partielle Integration, indem man zuerst $\lambda(x)$, dann $\lambda'(x)$ als constant betrachtet... (pp. 23-24 here, a TeX transcription; or pp. 30-31 here, reprinted from Riemann, Abh. Gesell. der Wiss. zu Göttingen 13, 87-132 (1867)
Does anyone know where the terminology for "partial integration" is discussed? Why is a function, which is clearly not a constant function, called "constant"? Is there a reference where this is discussed?
I have a theory that it comes from a partial-derivative form of the formula for integration by parts, namely, $$\int {\partial\over\partial v}(uv)\;dv = uv-\int {\partial\over\partial u}(uv)\; du \,,$$ since in ${\partial\over\partial v}(uv)$, the factor $u$ is treated as constant. But I made that formula up and have not found it anywhere. Indeed, googling through the 19th century, the standard approach to integration by parts of today seems dominant in Cauchy and in English calculus textbooks.
Update
Replacing $uv$ by $f(u,v)$ in the above equation seemed an obvious, if unfruitful, move. Zermelo made a similar remark to Brendel, who applied it to various integrals in "Ueber partielle Integration", Math. Ann. 55, 1901 (248-256), half a century after Riemann wrote his paper. While Brendel starts with the standard integration by parts formula above, he switches the roles of $u$ and $v$ without comment and treats $v$ "als constant"; he then replaces $u$ by $x$. Be that as it may, it seems to suggest the view was what I surmised above, and presumably that was Riemann's view, too. It leaves open the question of when and where such a view of integration by parts or partial integration, in which one factor is treated as constant, arose and was common.
(By the way, Brendel remarks that Hilbert suggested that the standard formula should be called "integration of a product." (249))
