On a generalization given by Laplace of Lagrange’s theorem

The theorem which we are about to explain may be enunciated as follows : “Let F(x 1, x 2, ..., x n) denote any given function of the n variables x 1, x 2, ..., x n , which by means of the n equations (1) x m = t m + α m φ m ( x 1 , x 2 , ... , x n ) , [ m = 1 , 2 , 3 , ... , n ] $${x_m} = {t_m} + {\alpha _m}{\varphi _m}\left( {{x_1},{x_2},...,{x_n}} \right),\quad \left[ {m = 1,2,3,...,n} \right]$$ depend on the 2 n variables t 1, t 2, ..., t n , α 1, α 2, ..., α n, consider these as the independent variables and assume that the functions F, φ 1 φ 2, ..., φ n , in their explicit form, contain no other variables than the dependent ones x 1, ..., x n . Then a 1, a 2, ... , a n being positive integers, we have d a 1 + a 2 + ⋅ ⋅ ⋅ + a n F d α 1 a 1 d α 2 a 2 ... d α n a n = d a 1 + a 2 + ⋅ ⋅ ⋅ + a n − n d t 1 a 1 − 1 d t 2 a 2 − 1 ... d t n a n − 1 [ d n F d α 1 d α 2 ... d α n ] $$\frac{{{d^{{a_1} + {a_2} + \cdot \cdot \cdot + {a_n}}}F}} {{d\alpha _1^{{a_1}}d\alpha _2^{{a_2}}...d\alpha _n^{{a_n}}}} = \frac{{{d^{{a_1} + {a_2} + \cdot \cdot \cdot + {a_n} - n}}}} {{dt_1^{{a_1} - 1}dt_2^{{a_2} - 1}...dt_n^{{a_n} - 1}}}\left[ {\frac{{{d^n}F}} {{d{\alpha _1}d{\alpha _2}...d{\alpha _n}}}} \right]$$ where the brackets may indicate that, after having transformed the included derivative into a rational and integral expression comprising only derivations with regard to t 1, t 2, ..., t n , the quantities φ 1, φ 2 , ..., φ n are to be replaced by φ 1 a 1 , φ 2 a 2 , ... , φ n a n $$\varphi _1^{{a_1}},\varphi _2^{{a_2}},...,\varphi _n^{{a_n}}$$ .”