A X I 



ellipfe, is tlic axfs AP, laft defined ; being tliiis csllcd in con- 

 tradiftinftion to tlie conjuj^ate or fc com! ary axis. 



Or, in the ellipfe and" hyperbola, it is the diameter that 

 pafTcs through the two foei, and the two principal vertices of 

 the figure. 



The tranCvcrfe axis in the ellipfe is tlie longed; and in 

 thehyperbolait cuts the curve in the points A and P^J-32-) 

 and is the (liditeit diameter. 



Axis, con'm^^ate, or fcronJ nxis, of the ellipfe and hyper- 

 bola, is the diameter pading through the centre and perpen- 

 dicular to the tranfvtrfe axis. Such is ths luie FF 

 Cf^. 31.) drawn through the centre of the ellipfe C, parallel 

 to the ordinate MN, and perpeudiculai> to the tranfverfe 

 axisAP; being terminated at each extreme by the curve. 

 And fuch, in the hyperbola, is the right line FE (Jg.-^z.) 

 drawn through the centre parallel to the ordinates MN, 

 MN, perpendicularly to the tranfverfe axis AP. In the 

 ellipfe and hyperbola, the conjugate axis is the (horteft of all 

 the conjugate diameters. The axis of a parabola is of an 

 indeterminate length; that is, is infinite. The axis of the 

 tlhpfe is determinate. The parabola has only one axis; the 

 ellipfe and hyperbola has'e two. 



Axis of a Curve Line, in general, denotes that diameter 

 which has its ordinates at right angles to it, when that is 

 pofiible. For, as in the conic feftions, any diameter bifcfts 

 all its parallel ordinates, making the two parts of them on 

 both fides of it equal, and the diameter which is perpendi'- 

 lar to fuch ordinates is an axis ; fo in curves of the fecond 

 order, if any two parallel lines meet with the curve in three 

 points, the right line which cuts thefe two parallels fo that the 

 fum of the twopartsononcfideofthe interfcfting line, between 

 h and the curve, is equal to the third part terminated by the 

 curve on the other fide, then the faid line will in like man- 

 ner cut all other parallels to the former two lines, fo that 

 with refpeft to every one of them, the fum of the two parts, 

 or ordinates, on one fide, will be equal to the third part, 

 or ordinate, on the other fide. Such interfering line is 

 then a diameter ; and that diameter, whofe parallel ordinates 

 are at right angles to it, when that is poffible, is an axis. 

 The cafe is the fame with regard to other curves of ilill 

 higher orders. Newton, Enumeratio Linearum Tertii Or- 

 dinis, i 2. art. i. 



Axis of a Magnet, or Magnelical Jxis, is a line pafTing 

 through the middle of a mairnet lengthwife ; in fuch man- 

 ner, as that however the magnet be divided, provided the 

 divifion be made acc^irding to a plane, in which fr.ch line is 

 found, the magiiit will he cut or feparalcd into two load- 

 llones ; and the extremes of fuch lines are called the poles 

 of the ftone. See Magnet. 



Axis, in Mechanics. The axis of a balance is the line 

 upon which it moves or turns. See Balance. 



Axis of Ofcillalion, is a right line parallel to the horizon, 

 pafiing through the centre, about which a pendulum vi- 

 brates ; and perpendicular to the plane in which it ofcillates. 

 See Oscillation, and Pendulum. 



Axis in Pcn:rochio, or meel and Axk, is one of the five 

 mechanical powers, or fimple machines, contrived chiefly 

 for the raifmg of -.veights to a conuderable height. It con- 

 fills of a circle, reprcfented AB [Plate I. Mechanics, fig. 5,) 

 concentric with the bafe of a cyliiider, and moveable toge- 

 ther with it, about its axis EF. This cylinder is called the 

 axis; and the circle, the perit-ochium ; and the radii, or 

 fpokes, which are fomctimes fitted immediately into the 

 c'Ender, without any circle, the fcytalce. Round the axis 

 winds a rope, or chain, by means of which the weights, 

 Ice. are to be raifed, upon turrflne the wheel. 



The axis in peritrochio taites place in the motion of every 



6 



A X I 



machine, where a tirc'.e may be conceived as defcrlbeJ about 

 a fixed axis, concentric to the plane of a cylinder, about 

 which it is placed ; as in crane-wheels, miU-wheels, capftans, 

 &c.; a gimblct and an augre to bore with may alfo be re« 

 fcrred to the wheel and axis. 



Axis in Peritrochio, properties of the. I. If the power 

 applied to the axis in peritrochio, in the direftion AL 

 ffg. 6.), being a tangent to the pcripheiy of the wheel, or 

 perpendicular to the Icytala or fpokc, be to a weight W, a$ 

 the radius of the axis CE is to the radius of the wheel CA» 

 or the length of the fpoke ; the power will ]i\l\ fullain the 

 weight, i. c. the weight and the power will be in equU 

 11 brio. 



Deni. The fame power is required to fnpport W, what« 

 ever be the point of the axis to which it is applied, becaufe 

 the diilaiice from the correfpondiiig centre of motion is the 

 fame, and the v.heel and axis may be reduced to a bent 

 lever ; and confequently there will be an equilibrium, when 

 P : W : : W's dillance from the centre of motion, or ra- 

 dius of the axis, : radius of the wheel. Or, fince the di- 

 rcftions of P and W are perpendicular to their refpeftiva 

 diftances from their centres of motion, they are wholly ef- 

 ficient ; and P's velocity is to W's velocity, as the peri- 

 phery of the wheel to ti-.e periphery of the axis ; and cors- 

 i'equcntly, when there is an equilibrium, P : W : : peri- 

 phery of the axis : periphery of the wheel : : radius of the 

 axis : radius of the wheel. 



If the thickncfs cf the rope, to which W is appended, 

 be not inconfidcrable, it ought not to be negkc^ed ; for 

 when one or more coils or fpires of the rope are folded 

 about the axis, the diftance of W's direttion from the cen- 

 tre of m.otion is increafed, and becomes equal to the fum cf 

 the femidlameters of the axis and ropes ; and there is an 

 equilibrium when P ; W : : the wliolc diftance of W's di- 

 direftion from the centre of motion : femidiameter of the 

 wheel. 



2. If a power applied in F, pull down the wheel ac- 

 cording to the line of diretiion FD, which is oblique to 

 the radius of the wheel, though parallel to the perpendi- 

 cular direction ; it will have the fame proportion to a power 

 which afts according to the perpenaicular direftion AL, 

 which the whole fine has to the fine of the angle of direc- 

 tion DFC. For, fince FD is perpendicular to AC, DC 

 will be the diftance of the power applied at F from the 

 centre of motion; confequently the power at F : W : ; EC 

 : CD ; and the power at A : W : : EC : CA ; confe- 

 quently the power at F : power at A : : CA : CD. But 

 if CA or FC be taken lor the whole fine or radius, CD 

 will be the fine of the angle DFC ; and the power at F will 

 be to the power at A : : the whole fine is to the fine of the 

 angle of direftion DFC, in cafe of an equilibrium between 

 the power and weight. 



Hence, fince the diftance of the power in A is the radius 

 CA, the angle of direftion DFC being given, the diftance 

 DC is eafily found. 



3. Powers applied to the wheel in feveral points, F and 

 K, according to the direftions FD and KI, parallel to the 

 perpendicular one AL, are to each other as the diftances 

 from the centre of motion CD and CI, reciprocally. For 

 the power at F : W : : EC : CD ; and the power at K 

 : W^ : : EC : IC ; confequently the power at F : power 

 at K : : IC : CD. 



Hence, as the diftance from the centre of motion in- 

 creafes, the power decreafes, and -vice verfa, the w-eight 

 being the fame. Hence alfo, fince the radius AC is the 

 greateft diftance, and correlponds to the power afting ac- 

 cording to the line of direftion ; the perpendicular power 



will 



