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Dec 12, 2022 at 18:33 comment added Francesco Polizzi Or, perhaps, one should consider the resolution of indeterminacy of the map $\bar{\phi}$.
Dec 12, 2022 at 18:27 comment added Francesco Polizzi My guess is that Burnside is implicitly considering the projectivized map $$\bar{\phi} \colon \mathbb{P}(V) \dashrightarrow \mathbb{P}^{n-1}.$$ Such a map is induced by a linear system on $\mathbb{P}(V)$, whose moving part gives the "variable systems" and whose fixed part (if it exists) gives the "fixed systems".
Dec 11, 2022 at 7:32 history asked benblumsmith CC BY-SA 4.0