IANICS. 



180 



corresponding centre and perpendicular of tho body to 1 



roaistanoe distributed equally over 



f tho first body, prevents its moving down- 



.uioo at every point ia perpendicular to 



r, these reaiatanoes, taken together, are a system of 



: fureos, and have a parallel centre somewhere in that 



baso. Lot this centre bo o. Join now o p ; and, aa the same 



reasoning holds good of 

 the body to tho left, l.-t 

 o, P, bo tho corresponding 

 lino in it. Moreover, let 

 x, x,, bo tho points in 

 which the lines O p, o, P, 

 cut tho circumference, or 

 boundary of tho bases 

 x T z, x, T, z,. Tho body 

 to tho right is thus acted 

 Fig. 22. on by two forces ; tho re- 



sistance at o upwards 



supporting it, and the weight at o pulling downwards. But, 

 as tho point p falls, in this case, outside the base x T z, tin n- 

 is nothing to prevent tho body obeying it by turning over on 

 its edge at x. 



But, in the other case, whore P t is within tho base, tho weight 

 at QI tenda to make the body fall inwards, turning on its edge 

 at x r But then, there is tho resistance of tho tablo at o l , acting 

 upwards to prevent that motion ; and consequently tho body 

 remains at rest, or is in equilibrium. 



And this statement holds equally good when the piano on 

 which tho body rests is sloped or inclined to tho horizontal 

 piano ; aa is evident from Fig. 23, whore tho cylindrical body 

 on tho slope A B must upset if a p falls outside tho baso x Y z. 

 We may, therefore, conclude generally both as to horizontal 

 and inclined planes that a body will rest in equilibrium on a 

 plane, if tho vortical line, passing 

 through its centre of gravity, meets 

 the plane within the baso. If it 

 meets it outside tho base, the body 

 will overturn. 



Between these two, it should be 

 observed that there is an interme- 

 diate case, in which the perpendicu- 

 lar meets the piano neither within 

 nor without the baso, but on ita 

 circumference. When this happens, 

 the body is equally disposed to 

 Btand or upset ; but, in fact, it will 

 overturn ; for in such an unsteady Fig. 23. 



position the slightest touch or shake 

 would send it over. It is a case of unstable equilibrium. 



In interpreting and applying this principle to practice, you 

 must bo on your guard as to tho meaning of tho word " baso ; " 

 else you may imagine some day yon have discovered that a body 

 does not upset when the vertical from tho centre of gravity 

 falls outside the base. Suppose tho baso to be bent inwards 

 into a horse-shoo form, as in tho cone, a (Fig. 24), or into the 

 form of the semi-circular wall, b, in which latter case the centre 



of gravity is u-ith- 

 out the substance 

 of the body ; then 

 the point p is on 

 tho floor, outside 

 the spaces along 1 

 which the bodies 

 arc in contact with 

 it. Still, neither 

 body will upset; for 

 the advanced spurs 



Fig. 24. of the bases at T 



and z will act as 



props, and in order to upset they must turn over the line 

 T z joining them. This shows that the real baso includes 

 all tho open space within T z ; and you learn that, when- 

 ever the base of contact bends inwards, you must measure 

 the base of support from one projecting point to another 

 all round, making no account whatever of the inward bends. 

 A common table touches tho floor only at four points, and a 

 round table at three; but in both the baso of support is 



all tho space within the oblong or triangle got by joining these 



There is another class of oases to be noticed, those which are 

 round all ov.-r their surface like a ball, or egg, or sea-shore 

 pebble, and have no jiat bases to rest on that is, which oan be 

 supported at only one point of their surface ; or, where there are 

 hollows on them, along a line of points surrounding the hollow. 

 This latter case we , 



need not consider, for ct -^*- 



such bodies rest, like 

 those we hare already 

 examined, so far as tho 

 hollows are concerned 

 (as in d, Fig. 25), on 

 wide bases. 



Confining attention, 

 therefore, to cases in ^^^B^BI 

 which there are no Iff* 25. 



hollows, or the surface 



is convex all round, if yon place such a body, say an oral, 

 in the position represented at a (Fig. 25), the perpendicular, 

 a P, from its centre of gravity, o, on the plane will fall out- 

 side its base, or point of support, o, and it will roll over 

 until, after rocking for a few turns, it settles into the position 

 b, in which o is above o. Vow movo it further from this until 

 it reaches the position c, in which again o will be over the 

 point of support, o ; and again you will havo a pouible equi- 

 librium, that is, possible in imagination, for tho body it supported 

 from below. But actually to produce equilibrium in this case is 

 tho celebrated problem of Columbus, which that great navigator 

 solved after so summary a fashion. So unsteady is it, that the 

 body drops immediately into the position b. 



Of this unsteady, or unstable equilibrium, we shall have more 

 in the next lesson ; my object hero is to point out the fact that 

 in both positions, b and c, the lino o o ia perpendicular to the 

 surface of the body. It is evidently perpendicular to the plane 

 on which the oval rests ; but, since the latter' s surface touched, 

 or coincides at o, with that plane, o o must bo perpendicular 

 also to that surface. Hence wo learn that, whatever bo the 

 number of points at which a convex body can rest, steady or 

 unsteady, on a horizontal plane, for every one of these points 

 the lines connecting them with the centre of gravity must pierce 

 its surface at right angles ; or 



Tho number of positions of equilibrium of a convex body, 

 supported on a horizontal plane, is equal to that of the perpen- 

 diculars to its surface which can bo drawn from ita centre of 

 gravity. 



A few instances in illustration of the principles explained in 

 this lesson will now bo useful. When a man stands upright, the 

 baso by which he is supported is so much ground under him as 

 is covered by his feet, together with the space between them. 

 If he widens that space to left and right, ho makes himself more 

 steady as to being thrown sideways, but ia more easily cast on 

 his face. If he puts one foot before the other, he becomes 

 steadier at front and back, but less so to his sides. A two- 

 wheeled gig, or Hansom, to be properly balanced, should have 

 its centre of gravity over the lino joining the points at which 

 the wheels touch the ground. If it bo in advance of that line, 

 it will throw a weight on the horse's back ; if behind it, the gig 

 will upset backwards should the 

 belly-band break. 



A body may be made to roll up 

 an incline by loading it at one 

 side. Take a round boll of cork, 

 for instance, and put some lead 

 into a hollow scooped out near its 

 surface, closing the hole so as to 

 leave the ball perfectly round. 

 The centre of gravity will then 

 no longer be at the centre of tho 

 boll, but to one aide, let it bo at 

 a (in a, Fig. 26). Put tho ball now on the incline, with the 

 leaded side looking up tho slope ; tho perpendicular o P will 

 meet the incline above o, and tho ball will roll upwards until o 

 comes over the point of support. 



This experiment may be tried in another form without the 

 use of the lead, by simply scooping a hollow on one side, or as 

 in the following example: Get a round cylinder of cork a 



