- Smith, at the request of a member of the commission by which the prize was proposed, undertook in 1882 to write out the demonstration of his general
**theorems**so far as was required to prove the results for the special case of five squares. - As a geometer he is classed by Eudemus, the greatest ancient authority, among those who "have enriched the science with original
**theorems**, and given it a really sound arrangement." - So far these
**theorems**on vortex motion are kinematical; but introducing the equations of motion of § 22, Du + dQ =o, Dv+dQ =o, Dw + dQ dt dx dt dy dt dz and taking dx, dy, dz in the direction of u, v, w, and dx: dy: dz=u: v: w, (udx + vdy + wdz) = Du dx +u 1+.. - These
**theorems**, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest. - If, however, we defined the strength of the source by the statement that the strength divided 1 The beginner is often puzzled by the constant appearance of the factor 47r in electrical
**theorems**. It arises from the manner in which the unit quantity of electricity is defined.

» more...