but at the origin these quantities become infinite. For any closed surface not including the origin, the surface-integral is zero. If a closed surface includes the origin, its surface-integral is .
If, for any reason, we wish to treat the region round as if it were not periphractic, we must draw a line from to an infinite distance, and in taking surface-integrals we must remember to add whenever this line crosses from the negative to the positive side of the surface.
On Right-handed and Left-handed Relations in Space.
23.] In this treatise the motions of translation along any axis and of rotation about that axis, will be assumed to be of the same sign when their directions correspond to those of the translation and rotation of an ordinary or right-handed screw [1].
For instance, if the actual rotation of the earth from west to east is taken positive, the direction of the earth s axis from south to north will be taken positive, and if a man walks forward in the positive direction, the positive rotation is in the order, head, right-hand, feet, left-hand.
If we place ourselves on the positive side of a surface, the positive direction along its bounding curve will be opposite to the motion of the hands of a watch with its face towards us.
This is the right-handed system which is adopted in Thomson and Tait's Natural Philosophy, § 243. The opposite, or left-handed system, is adopted in Hamilton's and Tait's Quaternions. The operation of passing from the one system to the other is called, by Listing, Perversion.
The reflexion of an object in a mirror is a perverted image of the object.
When we use the Cartesian axes of , we shall draw them
- ↑ The combined action of the muscles of the arm when we turn the upper side of the right-hand outwards, and at the same time thrust the hand forwards, will impress the right-handed screw motion on the memory more firmly than any verbal definition. A common corkscrew may be used as a material symbol of the same relation.
Professor W. H. Miller has suggested to me that as the tendrils of the vine are right-handed screws and those of the hop left-handed, the two systems of relations in space might be called those of the vine and the hop respectively.
The system of the vine, which we adopt, is that of Linneus, and of screw-makers in all civilized countries except Japan. De Candolle was the first who called the hop-tendril right-handed, and in this he is followed by Listing, and by most writers on the rotatory polarization of light. Screws like the hop-tendril are made for the couplings of railway-carriages, and for the fittings of wheels on the left side of ordinary carriages, but they are always called left-handed screws by those who use them.
