2. It is a sub-group of the general projective group, i.e. of the group of which any transformation converts planes into planes, and straight lines into straight lines.
3. An infinitesimal transformation can always be found satisfying the condition that, at least throughout a certain enclosed region, any definite line and any definite point on the line are latent, i.e. correspond to themselves.
4. No infinitesimal transformation of the group exists, such that, at least in the region for which (3) holds, a straight line, a point on it, and a plane through it, shall all be latent.
The property enunciated by conditions (3) and (4), taken together, is named by Lie “Free mobility in the infinitesimal.” Lie proves the following theorems for a projective space:—
1. If the above four conditions are only satisfied by a group throughout part of projective space, this part either (α) must be the region enclosed by a real closed quadric, or (β) must be the whole of the projective space with the exception of a single plane. In case (α) the corresponding congruence group is the continuous group for which the enclosing quadric is latent; and in case (β) an imaginary conic (with a real equation) lying in the latent plane is also latent, and the congruence group is the continuous group for which the plane and conic are latent.
2. If the above four conditions are satisfied by a group throughout the whole of projective space, the congruence group is the continuous group for which some imaginary quadric (with a real equation) is latent.
By a proper choice of non-homogeneous co-ordinates the equation of any quadrics of the types considered, either in theorem 1 (α), or in theorem 2, can be written in the form 1 + c(x² + y² + z²) = 0, where c is negative for a real closed quadric, and positive for an imaginary quadric. Then the general infinitesimal transformation is defined by the three equations:
| dx/dt = u − ω3y + ω2z + cx (ux + vy + wz), | (A) |
| dy/dt = v − ω1z + ω3x + cy (ux + vy + wz), | |
| dz/dt = w − ω2x + ω1y + cz (ux + vy + wz). |
In the ease considered in theorem 1 (β), with the proper choice of co-ordinates the three equations defining the general infinitesimal transformation are:
| dx/dt = u − ω3y + ω2z, | (B) |
| dy/dt = v − ω1z + ω3x, | |
| dz/dt = w − ω2x + ω1y. |