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For binary forms a notation such as $$ (a_0,a_1,\ldots,a_n)\\!\\!\\!\\!(x,y)^n $$ means $$ a_0 x^n+ a_1 \left( \begin{array}{c} n \\\ 1\end{array} \right)x^{n-1}y+ a_2 \left( \begin{array}{c} n \\\ 2\end{array} \right)x^{n-2}y^2+\cdots+a_n y^n $$ If Cayley uses the notation with the pointy arrow on one of the parenthesis he means the same thing without the binomial coefficients. For $p$-ary forms, I believe there must be a choice of ordering of monomials hopefully specified in the paper under consideration.