## Coordinates Sentence Examples

- With this end in view he expounded to the Berlin academy in 1849 a mode of determining an elliptic orbit from three observations, and communicated to that body in 1851 a new method of calculating planetary perturbations by means of rectangular
**coordinates**(republished in W. **Coordinates**referred to this system are termed equatorial.- It flickered on for her, and she correlated their
**coordinates**with two touches. - The other two
**coordinates**, which define the direction of a body, admit of direct measurement on principles applied in the construction and use of astronomical instruments. - - Employ the elliptic
**coordinates**n,, and -=n+Vi, such that z=cch?, cchncos,y=cshnsin-; (1) then the curves for which n and are constant are confocal ellipses and hyperbolas, and -d(n,) =c 2 (ch 2 n - cost) = 2c 2 (ch2n-cos2) = r i r 2 = OD 2, (2) if OD is the semi-diameter conjugate to OP, and ri, r 2 the focal distances, rl,r2 = c (ch n cos 0; r 2 = x2 +y2 = c 2 (ch 2 n - sin20 = 1c 2 (ch 2 7 7 +cos 2?). - Kirchhoff's expressions for X, Y, Z, the
**coordinates**of the centre of the body, FX=y 1 cos xY--y 2 cos yY-{-y 3 cos zY, (18) FY = -y l cos xX -Hy2 cos yX+y 3 cos zX, (Ig) G=y 1 cos xZ+y 2 cos yZ+y 3 cos zZ, (20) (21) F(X+Yi) = Fy3-Gx3+i /) X 3epi. - Then the result proves that the values of the
**coordinates**and momenta remain distributed in this way throughout the whole motion of the systems. Thus, if there is any characteristic which is common to all the systems after the motion has been in progress for any interval of time, this same characteristic must equally have been common to all the systems initially. - Similarly if we have F more unknowns than we have equations to determine them, we must fix arbitrarily F
**coordinates**before we fix the state of the whole system. - In this case the left-hand radial line passes through the point at which the
**coordinates**meet, showing that the reservoir will just equalize the flow of the driest year. - To measure the difference between the longitudinal co-ordinates of two objects by means of a graduated circle the instruments must turn on an axis parallel to the principal axis of the system of
**coordinates**, and the plane of the graduated circle must be at right angles to that axis, and, therefore, parallel to the principal co-ordinate plane. - He gave the
**coordinates**to Dan and replaced the micro in his pocket. - Webs serve not only for pointing on stars to determine their
**coordinates**(in manner afterwards described), but also for estimating the diameters of the star-images in terms of these 4" intervals. - This, and the inclination of the orbit being given, we have all the geometrical data necessary to compute the
**coordinates**of the planet itself. - The
**coordinates**thus found will in the case of a body moving around the sun be heliocentric. The reduction to the earth's centre is a problem of pure geometry. - To form a conception of this problem it is to be noted that since the position of the body in space can be computed from the six elements of the orbit at any time we may ideally conceive the
**coordinates**of the body to be algebraically expressed as functions of the six elements and of the time. - For instance, those of a ternary form involve two classes which may be geometrically interpreted as point and line co-ordinates in a plane; those of a quaternary form involve three classes which may be geometrically interpreted as point, line and plane
**coordinates**in space. - He had not attempted to include in his calculations the orbital variations of the disturbing bodies; but Lagrange, by the happy artifice of transferring the origin of
**coordinates**from the centre of the sun to the centre of gravity of the sun and planets, obtained a simplification of the formulae, by which the same analysis was rendered equally applicable to each of the planets severally. - The method of "generalized
**coordinates**," as it is now called, by which he attained this result, is the most brilliant achievement of the analytical method. - If ABCD is a tetrahedron of reference, any point P in space is determined by an equation of the form (a+13+ - y+5) P = aA+sB +yC +SD: a, a, y, b are, in fact, equivalent to a set of homogeneous
**coordinates**of P. For constructions in a fixed plane three points of reference are sufficient. - The integrations may also be effected by means of polar
**coordinates**, taking first the integration with respect to 4) so as to obtain the result for an infinitely thin annular aperture. - Phil.): - Let x, y, z be the
**coordinates**of P in the orbit,, r t, those of the corresponding point T in the hodograph, then dx dy _ dz c= ' 71 - a' - at therefore Also, if s be the arc of the hodograph, ds = v = V V1 1) j dt + (dt2) dt Equation (1) shows that the tangent to the hodograph is parallel to the line of resultant acceleration, and (2) that the velocity in the hodograph is equal to the acceleration. - Similarly, with the function (19) (2n+ I) 3 ch (2n+ I) ITrb/a' (2) Changing to polar
**coordinates**, x =r cos 0, y = r sin 0, the equation (2) becomes, with cos 0 =µ, r'd + (I -µ 2)-d µ = 2 ?-r3 sin 0, (8) of which a solution, when = o, is = (Ar'+) _(Ari_1+) y2,, ? - Uniplanar motion alone is so far amenable to analysis; the velocity function 4 and stream function 1G are given as conjugate functions of the
**coordinates**x, y by w=f(z), where z= x +yi, w=4-Plg, and then dw dod,y az = dx + i ax - -u+vi; so that, with u = q cos B, v = q sin B, the function - Q dw u_vi=g22(u-}-vi) = Q(cos 8+i sin 8), gives f' as a vector representing the reciprocal of the velocity in direction and magnitude, in terms of some standard velocity Q. - These equations are proved by taking a line fixed in space, whose direction cosines are 1, then dt=mR-nQ,' d'-t = nP =lQ-mP. (5) If P denotes the resultant linear impulse or momentum in this direction P =lxl+mx2+nx3, ' dP dt xl+, d y t x2' x3 +1 dtl dt 2 +n dt3, =1 ('+m (dt2-x3P+x1R) ' +n ('-x1Q-{-x2P) ' '= IX +mY+nZ, / (7) for all values of 1, Next, taking a fixed origin and axes parallel to Ox, Oy, Oz through 0, and denoting by x, y, z the
**coordinates**of 0, and by G the component angular momentum about 1"2 in the direction (1, G =1(yi-x2z+x3y) m 2-+xlz) n(y(y 3x 1 x3x y + x 2 x) (8) Differentiating with respect to t, and afterwards moving the fixed. - Qr,, be the generalized
**coordinates**of any dynamical system, and let pi, P2, - An algebraical curve is a curve having an equation F (x, y) = o where F(x, y) is a rational and integral function of the
**coordinates**(x, y); and in what follows we attend throughout (unless the contrary is stated) only to such curves. - It is possible, and (not so much for any application thereof as in order to more fully establish the analogy between the two kinds of co-ordinates) important, to give independent quantitative definitions of the two kinds of co-ordinates; but we may also derive the notion of line-co-ordinates from that of point-
**coordinates**; viz. - It is a form of the theorem for the case D = r, that the
**coordinates**x, y, z of a point of the bicursal curve, or in particular the co-ordinates of a point of the cubic, can be expressed as proportional to rational and integral functions of the elliptic functions snu, cnu, dnu; in fact, taking the radical to be r -0 2 .r - k 2 0 2, and writing 8 =snu, the radical becomes = cnu, dnu; and we have expressions of the form in question. - If the origin of rectangular axes fixed in the lamina be shifted through a space whose projections on the original directions of the axes are X, u, and if the axes are simultaneously turned through an angle e, the
**coordinates**of a point of the lamina, relative to the original axes, are changed from x, y to X+x cos ey sin e, u+x sin e+y cos e, or X + x ye, u + Xe + y, ultimately.