1. [n]  a bevel gear that permits rotation of two shafts at different speeds; used on the rear axle of automobiles to allow wheels to rotate at different speeds on curves
2. [n]  a quality that differentiates between similar things
3. [n]  the result of mathematical differentiation; the instantaneous change of one quantity relative to another; df(x)/dx
4. [adj]  relating to or showing a difference; "differential treatment"
5. [adj]  (mathematics) involving or containing one or more derivatives; "differential equation"

1. \Dif`fer*en"tial\, a. [Cf. F. diff['e]rentiel.]
   1. Relating to or indicating a difference; creating a
      difference; discriminating; special; as, differential
      characteristics; differential duties; a differential rate.
            For whom he produced differential favors. --Motley.
   2. (Math.) Of or pertaining to a differential, or to
      differentials.
   3. (Mech.) Relating to differences of motion or leverage;
      producing effects by such differences; said of mechanism.
   {Differential calculus}. (Math.) See under {Calculus}.
   {Differential coefficient}, the limit of the ratio of the
      increment of a function of a variable to the increment of
      the variable itself, when these increments are made
      indefinitely small.
   {Differential coupling}, a form of slip coupling used in
      light machinery to regulate at pleasure the velocity of
      the connected shaft.
   {Differential duties} (Polit. Econ.), duties which are not
      imposed equally upon the same products imported from
      different countries.
   {Differential galvanometer} (Elec.), a galvanometer having
      two coils or circuits, usually equal, through which
      currents passing in opposite directions are measured by
      the difference of their effect upon the needle.
   {Differential gearing}, a train of toothed wheels, usually an
      epicyclic train, so arranged as to constitute a
      {differential motion}.
   {Differential motion}, a mechanism in which a simple
      differential combination produces such a change of motion
      or force as would, with ordinary compound arrangements,
      require a considerable train of parts. It is used for
      overcoming great resistance or producing very slow or very
      rapid motion.
   {Differential pulley}. (Mach.)
      (a) A portable hoisting apparatus, the same in principle
          as the differential windlass.
      (b) A hoisting pulley to which power is applied through a
          differential gearing.
   {Differential screw}, a compound screw by which a motion is
      produced equal to the difference of the motions of the
      component screws.
   {Differential thermometer}, a thermometer usually with a
      U-shaped tube terminating in two air bulbs, and containing
      a colored liquid, used for indicating the difference
      between the temperatures to which the two bulbs are
      exposed, by the change of position of the colored fluid,
      in consequence of the different expansions of the air in
      the bulbs. A graduated scale is attached to one leg of the
      tube.
   {Differential windlass}, or {Chinese windlass}, a windlass
      whose barrel has two parts of different diameters. The
      hoisting rope winds upon one part as it unwinds from the
      other, and a pulley sustaining the weight to be lifted
      hangs in the bight of the rope. It is an ancient example
      of a differential motion.
   

2. \Dif`fer*en"tial\, n.
   1. (Math.) An increment, usually an indefinitely small one,
      which is given to a variable quantity.
   Note: According to the more modern writers upon the
         differential and integral calculus, if two or more
         quantities are dependent on each other, and subject to
         increments of value, their differentials need not be
         small, but are any quantities whose ratios to each
         other are the limits to which the ratios of the
         increments approximate, as these increments are reduced
         nearer and nearer to zero.
   2. A small difference in rates which competing railroad
      lines, in establishing a common tariff, allow one of their
      number to make, in order to get a fair share of the
      business. The lower rate is called a differential rate.
      Differentials are also sometimes granted to cities.
   3. (Elec.)
      (a) One of two coils of conducting wire so related to one
          another or to a magnet or armature common to both,
          that one coil produces polar action contrary to that
          of the other.
      (b) A form of conductor used for dividing and distributing
          the current to a series of electric lamps so as to
          maintain equal action in all. --Knight.
   {Partial differential} (Math.), the differential of a
      function of two or more variables, when only one of the
      variables receives an increment.
   {Total differential} (Math.), the differential of a function
      of two or more variables, when each of the variables
      receives an increment. The total differential of the
      function is the sum of all the {partial differentials}.
   

differential line.