During this motion the element *QQ'* will generate an area in the form of a parallelogram whose sides are parallel and equal to *QQ'* and *PP'.* If we construct a pyramid on this parallelogram as base with its vertex at P, the solid angle of this pyramid will be the increment *d*ω which we are in search of.

To determine the value of this solid angle, let θ and θ' be the angles which *ds* and *d*σ make with PQ respect ively, and let φ be the angle between the planes of these two angles, then the area of the projection of the parallelogram *ds* . *d*σ on a plane perpendicular to *PQ* or *r* will be

and since this is equal to *r*^{2}*d*ω, we find

(2) |

Hence | (3) |

420.] We may express the angles θ, θ', and φ in terms of *r*, and its differential coefficients with respect to *s* and σ, for

(4) |

We thus find the following value for Π^{2},

(5) |

A third expression for Π in terms of rectangular coordinates may be deduced from the consideration that the volume of the pyramid whose solid angle is dω and whose axis is *r* is

But the volume of this pyramid may also be expressed in terms of the projections of *r*, *ds*, and *d*σ on the axis of *x*, *y* and *z*, as a determinant formed by these nine projections, of which we must take the third part. We thus find as the value of Π,

(6) |