Under the general heading "Geometry" occur the subheadings "Foundations," with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; "Elementary Geometry," with the topics planimetry, stereometry, trigonometry, descriptive geometry; "Geometry of Conics and Quadrics," with the implied topics; "Algebraic Curves and Surfaces of Degree higher than the Second," with the implied topics; "Transformations and General Methods for Algebraic Configurations," with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; "Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry," with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; "Differential Geometry: applications of Differential Equations to Geometry," with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces.
Such a determinant is of importance in the theory of orthogonal substitution.
We can eliminate the quantities S l, E2, ï¿½ï¿½ï¿½ In and obtain n relations AbXi = (2B 11 - Ab)'ï¿½k1 +2B21x2+2B31x3+ï¿½ï¿½ï¿½, AbX2 = 2B12x1+ (2B22 - Ab) x2 +2B32x3+..., and from these another equivalent set Abx1 = (2B11 - X1 +2B12X2+2B13X3+ï¿½ï¿½ï¿½, Abx2 = 2B21X1+(2B22 - Ab)X2+2B23X3+ï¿½ï¿½ï¿½, and now writing 2Bii - Ab 2Bik - aii, Ob = aik, Ob we have a transformation which is orthogonal, because EX 2 = Ex2 and the elements aii, a ik are functions of the 2n(n- I) independent quantities b.
We may therefore form an orthogonal transformation in association with every skew determinant which has its leading diagonal elements unity, for the Zn(n-I) quantities b are clearly arbitrary.
Similarly, for the order 3, we take 1 v Ab= -v 1 A =1 +x2 + 1, 2 + ï¿½ - A 1 and the adjoint is 1+A v +Aï¿½ -ï¿½ +Av -v +Aï¿½ 1+11 2 A +/-tv pt+AvA +ï¿½v 1 +1,2 leading to the orthogonal substitution Abx1 = (1 +A 2 - / 22 - v 2) X l +2(v+Aï¿½)X2 +2(/1 +Av)X3 1bx2 = 2(Aï¿½ - v)Xl+(1 +ï¿½2 - A2 - v2)X2 / +2(Fiv+A)X3 Abx3 = 2(Av +ï¿½)X1 +2(/lv-A)X2+(1+v2-A2- (12)X3.
Orthogonal System.-In particular, if we consider the transformation from one pair of rectangular axes to another pair of rectangular axes we obtain an orthogonal system which we will now briefly inquire into.
This is called the direct orthogonal substitution, because the sense of rotation from the axis of X i to the axis of X, is the same as that from that of x i to that of x 2.
If the senses of rotation be opposite we have the skew orthogonal substitution x1 =cos0Xi+sinOX2r x 2 = sin °Xicos OX2r of modulus -1.
It may denote a simultaneous orthogonal invariant of forms of orders n i, n2, n3,...; degree 0 of the covariant in the coefficients.
Previous to continuing the general discussion it is useful to have before us the orthogonal invariants and covariants of the binary linear and quadratic forms.
Ignoring temperature effect, and taking the density as a function of the pressure, surfaces of equal pressure are also of equal density, and the fluid is stratified by surfaces orthogonal to the lines of force; n ap, dy, P d z, or X, Y, Z (4) are the partial differential coefficients of some function P, =fdplp, of x, y, z; so that X, Y, Z must be the partial differential coefficients of a potential -V, such that the force in any direction is the downward gradient of V; and then dP dV (5) ax + Tr=0, or P+V =constant, in which P may be called the hydrostatic head and V the head of potential.
The curves 0 = constant and 4, = constant form an orthogonal system; and the interchange of 0 and 4, will give a new state of uniplanar motion, in which the velocity at every point is turned through a right angle without alteration of magnitude.
Thus for m =2, the spheres are orthogonal, and it can be verified that a13 a2 3 aY3 i f /' = ZU (I - 13 - 7.2 3 + 3) ' (8) where a l, a2, a =a l a 2 /J (a 1 2 +a 2 2) is the radius of the spheres and their circle of intersection, and r 1, r 2, r the distances of a point from their centres.
The corresponding expression for two orthogonal cylinders will be With a 2 = co, these reduce to / y /, = Uy (I ra 2 p22 +-C24)..
The orthogonal projection of a section of this surface by a plane containing one of the perpendiculars and inclined to the axis is the quadratrix.
We may describe, through all the points in an electric field which have the same potential, surfaces called equipotential surfaces, and these will be everywhere perpendicular or orthogonal to the lines of electric force.
Now if the atoms are regarded as points or spherical bodies oscillating about positions of equilibrium, the value of n+3 is precisely six, for we can express the energy of the atom in the form (9 2 v a 2 v a2v E = z(mu 2 +mv 2 +mw 2 +x 2 ax2 + y2ay2-fz2az2), where V is the potential and x, y, z are the displacements of the atom referred to a certain set of orthogonal axes.
That there exists a point such that the tangents from this point to the four spheres are equal, and that with this point as centre, and the length of the tangent as radius, a sphere may be described which cuts, the four spheres at right angles; this "orthotomic" sphere corresponds to the orthogonal circle of a system of circles.
In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I', where the orthogonal circle to the system intersects the line of centres; these points are named the " limiti,ng points."
Regarded as a statement concerning the orthogonal projections -~ ~ -~ -~
In the first place, each of these figures may be conceived as an orthogonal projection of a closed plane-faced polyhedron.
Clerk Maxwell, who showed amongst other things that a reciprocal can always be drawn to any figure which is the orthogonal projection of a plane-faced polyhedron.
If we project both polyhedra orthogonally on a plane perpendicular to the axis of the paraboloid, we obtain two figures which are reciprocal, except that corresponding lines are orthogonal instead of parallel.
Work.The work done by a force acting on a particle, in any infinitely small displacement, is defined as the product of the force into the orthogonal projection of the displacement on the direction of the force; i.e.
A, /3 be the orthogonal projections of A, B on AB, we have AaB~=ABaf3=AB(I cos~,) =4AB.~2,
Tion follows at once from the fact that the sum of orthogonal ~--~ ->
If we imagine a point Q to describe a circle of radius a _________________ with the angular velocity ~, its A - 0 P orthogonal projection P on a fixed diameter AA will execute a vibration of this character.
We have seen that a true simple-harmonic vibration may be regarded as the orthogonal projection of uniform circular motion; it was pointed out by P. G.
In (1,3) satisfy the conjugate or orthogonal relations anaiai+aiiaiai+.
By means of the orthogonal relations (15).
The problem was to find the orthogonal trajectories of a series of curves represented by a single equation.