# Orthogonal Sentence Examples

- Such a determinant is of importance in the theory of
**orthogonal**substitution. - 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. - 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. - The inconvenience of
**orthogonal**illumination, which certainly gives better results, is avoided in the coaxial apparatus. - 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. - 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.**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. - 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. - 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+. - The problem was to find the
**orthogonal**trajectories of a series of curves represented by a single equation. - The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the
**orthogonal**projection of a circle, or by means of the calculus. - With the .
**orthogonal**arrangement for illuminating and observing the beam of light traverses an extremely fine slit through a well-corrected system, whose optic axis is perpendicular to the axis of the microscope; the system reduces the dimensions of the beam to about 2 to 4 in the focal plane of the objective. - 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. - By means of the
**orthogonal**relations (15).