Magnifying glasses are often used for viewing three-dimensional objects.
A brief reference must suffice to the theory of three dimensional frames.
Hence any three-dimensional system can be reduced to a single force R acting in a certain line, together with a couple G in a plane perpendicular to the line.
A geometrical proof of this theorem, which is not restricted to a two-dimensional system, is given later (If).
Again, a rigid three-dimensional frame can be rigidly fixed relatively to the earth by means of six links.
Again, for a three-dimensional system, in the notation of ~ 7, 8, ~(X5x+YIy+ZIz)
For example, in a one-dimensional system such as a string or a bar, we have one dependent variable, and two independent variables x and t.
~ Three-dimensional Kinematics of a Rigid Body.The position of a rigid body is determined when we know the positions of three points A, B, C of it which are not colljnear, for the position of any other point P is then determined by the three distances PA, PB, PC. The nine co-ordinates (Cartesian or other) of A, B, C are subject to the three relations which express the invariability of the distances BC, CA, AB, and are therefote equivalent to six independent quantities.
In a two-dimensional frame, each joint may be conceived as consisting of a small cylindrical pin fitting accurately and smoothly into holes drilled t(~trough the members which it connects.
Three-dimensional Statics.A system of, parallel forces can be combined two and two until they are replaced by a single resultant equal to their sum, acting in a certain line.
The analytical reduction of a three-dimensional system can now be conducted as follows.
For a two-dimensional system we have, in the notation of ~ 3, 4,
Accordingly, at first sight it seems reasonable to define geometry in some such way as "the science of dimensional quantity."
Three-dimensional kinematics of a rigid body.
In general, however, a three-dimensional system of forces cannot be replaced by a single resultant force.
For a dimensional ratio of 400, N =o 00028, and therefore H = Ho - o 00028 I.
A similar procedure applies to a three-dimensional system, Thus if, 0 being the origin, OH represent any force P of the - system, the planes drawn through H parallel to the co-ordinate planes will enclose with the latter B N a parallelepiped, and it is evident H that OH is the geometric sum of 0 ~ ~----~ ~ ~---~>
Du Bois (Magnetic Circuit, p. 33), the demagnetizing factor, and the ratio of the length of the ellipsoid 2c to its equatorial diameter 2a (=c/a), the dimensional ratio, denoted by the symbol nt.
Retained by a bar of ferromagnetic metal after it has been removed from the influence of an external field produces a demagnetizing force NI T, which is greater the smaller the dimensional ratio..
Units, but since the dimensional ratio of his bars was comparatively small, the actual magnetizing force H must have been materially below that value.
A2 Since e= I -72= V the above expression for N may be written 4 7r nt + 1/m 2 - I N = 11t1-I (2011-- I in - Jnt2 - I I/ 4 7r C = 111 2 - I 1/111 t - log (in + Im 21) - I, from which the value of N for a given dimensional ratio can be calculated.
The specimen upon which an experiment is to be made generally consists of a wire having a " dimensional ratio " of at least 300 or goo; its length should be rather less than that of the magnetizing coil, in order that the field Ho, to which it is subjected, may be approximately uniform from end to end.
He lays stress on the dimensional relations of the problem, pointing out that the phenomena which occur with large vanes in highly rarefied gas could also occur with proportionally smaller vanes in gas at higher pressure.
The hydroxyl group also resembles the methyl group in its morphotropic effects, producing, in many cases, no change in symmetry but a dimensional increase in one direction.
(35) Du Bois has shown that _when the dimensional ratio in (= length/ diameter) exceeds t00, Nm 2 =constant=45, and hence for long thin rods N = 45/ m2.