## Analytically Sentence Examples

**Analytically**the hyperbola is given by ax2+2hxy+by2+2gx+ 2fy+c=o wherein ab>h 2 .- Viewed
**analytically**in its developed nature, magic is a wonder-working recognized as such, the core of the mystery consisting in the supposed transformation of suggested idea into accomplished fact by means of that suggestion itself. **Analytically**expressed ff+ co x I 2 d dn=ff dxdy= A (9) We have seen that Io (the intensity at the focal point) was equal to A 2 /X 2 f2.- By applying the method of the differential calculus, we obtain cos i= { (µ 2 - 1)/(n24-2n)} as the required value; it may be readily shown either geometrically or
**analytically**that this is a minimum. **Analytically**thus (Thomson and Tait, Nat.- For commercial purposes iron is universally employed and works well; but it is not available
**analytically**, because a superficial oxidation of the empty part of the vessel (by the water and air) cannot be prevented. - The ellipsoid is the only shape for which a and (3 have so far been determined
**analytically**, as shown already in § 44, so we must restrict our calculation to an egg-shaped bullet, bounded by a prolate ellipsoid of revolution, in which, with b =c, Ao= fo (a2 + X)V [4(a2+X)(b +X)2]-J0 2(a2 +X)3/2(b2+X), (13) Ao+2Bo = I, (t4) _ B 0 t - A 0 I a?I-A0' Q I - Bo I-{- A o I-? - Failing
**analytically**to probe its nature, historically we seek relief to our perplexities by tracing its origin.. - The area BCDdb under the path represents the external work done by the substance in expanding from B to D, which is
**analytically**represented by the integral of pdv taken along the given path. - Swanton - have studied many of these languages
**analytically**and comparatively. - The rectangle, for instance, has so far been regarded as a plane figure bounded by one pair of parallel straight lines and another pair at right angles to them, so that the conception of " rectangularity " has had reference to boundary rather than to content;
**analytically**, the rectangle must be regarded as the figure generated by an ordinate of constant length moving parallel to itself with one extremity on a straight line perpendicular to it. - But when the rate of change of aethereal strain - that is, of (f,g,h) specified as Maxwell's electric displacement in free aether - is added to it, an
**analytically**convenient vector (u,v,w) is obtained which possesses the characteristic property of being circuital like the flow of an incompressible fluid, and has therefore been made fundamental in the theory by Maxwell under the name of the total electric current. - These circuital relations, when expressed
**analytically**, are then for a dielectric medium of types = (dt + x) (f',g',h')+dt(f,g,h), dR dQ = da dy dz dt' ' I See H. - The object of the above sketch has been to embrace in constructive outline the ground usually covered
**analytically**and on a far larger scale by Introductions to the New Testament, and by Histories of the New Testament Canon. - It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and,
**analytically**considered, it is not so different in principle from the logarithmic method. **Analytically**, the Cartesian equation to a coaxal system can be written in the form x 2 + y 2 + tax k 2 = o, where a varies from member to member, while k is a constant.- 22, 84 a 7-8) he distinguishes two modes of investigation,
**analytically**(avaXurtK&r) and logically (Xo'yu&,·). - The wave at a depth x is represented
**analytically**by the equation 0 - 0 0 = Ae mx sin (21rnt - mx). - Adopting a hypothetical law of the dispersion of differently coloured rays of light, he proved
**analytically**the possibility of constructing an achromatic object-glass composed of lenses of glass and water. - Deduction is analysis when it is regressive from consequence to real ground, as when we start from the proposition that the angles of a triangle are equal to two right angles and deduce
**analytically**that therefore (i) they are equal to equal angles made by a straight line standing on another straight line, and (2) such equal angles are two right angles. - Newton did indeed first show synthetically what kind of motions by mechanical laws have their ground in a centripetal force varying inversely as the square of the distance (all P is M); but his next step was, not to deduce synthetically the planetary motions, but to make a new start from the planetary motions as facts established by Kepler's laws and as examples of the kind of motions in question (all S is P); and then, by combining these two premises, one mechanical and the other astronomical, he
**analytically**deduced that these facts of planetary motion have their ground in a centripetal force varying inversely as the squares of the distances of the planets from the sun (all S is M). - Really, we first experience that particular causes have particular effects; then induce that causes similar to those have effects similar to these; finally, deduce that when a particular cause of the kind occurs it has a particular effect of the kind by synthetic deduction, and that when a particular effect of the kind occurs it has a particular cause of the kind by analytic deduction with a convertible premise, as when Newton from planetary motions, like terrestrial motions,
**analytically**deduced a centripetal force to the sun like centripetal forces to the earth. - Treated
**analytically**, but we shall only require the formula for infisiitely small displacements. **Analytically**we have d d~**Analytically**we have from (5),**Analytically**we have tm(~+~2+~.1)l=1/2~(m).**Analytically**, it is convenient to put Q~ equal to eirt multiplied by a complex coefficient; owing to the linearity of the equations the factor e~ni will run through them all, and need not always be exhibited.- This connexion is only supplied by theories which treat aberrations generally and
**analytically**by means of indefinite series. - This may be readily accomplished geometrically or
**analytically**, and it will be found that the envelope is a cardioid, i.e. - But it can be shown,
**analytically**or geometrically, that if the given curve has a node, the first polar passes through this node, which therefore counts as two intersections, and that if the curve has a cusp, the first polar passes through the cusp, touching the curve there, and hence the cusp counts as three intersections. - Thirdly, for the double tangents; the points of contact of these are obtained as the intersections of the curve by a curve II = o, which has not as yet been geometrically defined, but which is found
**analytically**to be of the order (m-2) (m 2 -9); the number of intersections is thus = m(rn - 2) (m 2 - 9); but if the given curve has a node then there is a diminution =4(m2 - m-6), and if it has a cusp then there is a diminution =6(m2 - m-6), where, however, it is to be noticed that the factor (m2 - m-6) is in the case of a curve having only a node or only a cusp the number of the tangents which can be drawn from the node or cusp to the curve, and is used as denoting the number of these tangents, and ceases to be the correct expression if the number of nodes and cusps is greater than unity. - The expression for the number of inflections 3m(rn - 2) for a curve of the order m was obtained
**analytically**by Plucker, but the theory was first given in a complete form by Hesse in the two papers " Uber die Elimination, u.s.w.," and " Uber die Wendepuncte der Curven dritter Ordnung " (Crelle, t. - This last quantity is
**analytically**defined by the equation r 2 = x 2 - + y2. **Analytically**the elements are determined from these data by solving the four equations just given, regarding a, b, c and d as unknown quantities, and x, y, x', y' and t as given quantities.