| ∂F | δx1 + | ∂F | δx2 + ... + | ∂F | δxn. |
| ∂x1 | ∂x2 | ∂xn |
If the differential operator
| ∂ƒ1 | ∂ | + | ∂ƒ2 | ∂ | + ... + | ∂ƒn | ∂ | |||
| ∂ai | ∂x1 | ∂ai | ∂x2 | ∂ai | ∂xn |
be represented by Xi, (i = 1, 2, ..., r), then the increment of F is given by
(e1X1 + e2X2 + ... + erXr) Fδt.
When the equations (i.) defining the general operation of the group are given, the coefficients ∂ƒs/∂ai, which enter in these differential operators are functions of the variables which can be directly calculated.
The differential operator e1X1 + e2X2 + ... + erXr may then be regarded as defining the most general infinitesimal operation of the group. In fact, if it be for a moment represented by X, then (1 + δtX)F is the result of carrying out the infinitesimal operation on F; and by putting x1, x2, ..., xn in turn for F, the actual infinitesimal operation is reproduced. By a very convenient, though perhaps hardly justifiable, phraseology this differential operator is itself spoken of as the general infinitesimal operation of the group. The sense in which this phraseology is to be understood will be made clear by the foregoing explanations.
We suppose now that the constants e1, e2, ..., er have assigned values. Then the result of repeating the particular infinitesimal operation e1X1 + e2X2 + ... + erXr or X an infinite number of times is some finite operation of the group. The effect of this finite operation on F may be directly calculated. In fact, if δt is the infinitesimal already introduced, then
| dF | = X·F, | d2F | = X·X·F, ... |
| dt | dt2 |