Cartan explained the generality index like this:

“[...] the general solution of a given deterministic system in involution depends on r2 + 2p - n arbitrary functions of three variables in the sense that the 3-dimensional solution (x4 = 0) that determines the most general solution can be obtained by arbitrarily taking r + 2p - n 2 of the unknown functions to be functions of x ,x ,x 1 2 3.”

Here, n is the number of equations of the system for p unknown functions; r 2 denotes the number of existing linear relations among the 2p derivatives of the unknowns with regard to two arbitrary directions in 3-space.