HYDROSTATICS. 



square inch of the surface of the third layer, in 

 either vessel, has to bear the weight of two cubic 

 inches of water, and the particles at that depth 

 push upwards and laterally with corresponding 

 intensity. The pressure thus increases with^the 

 tlepth equally in the narrow vessel and in the 

 wide. 



If we suppose the vessels to be each a foot 

 deep, it is evident that on every square inch of 

 the bottom of the cylinder there will rest a pile of 

 twelve cubic inches (weighing about half a pound 

 troy), and therefore the pressure on the whole 

 bottom will be the weight of ten such columns. 

 But from the shape of the other vessel, the lower 

 part of which is enlarged, it is not so evident 

 what the pressure on the bottom is. One thing is 

 clear, that, on the square inch in the middle of 

 the bottom immediately under the tube, there rests 

 a column of twelve cubic inches. Now, there 

 must be the same pressure among the particles of 

 the water lying around the bottom of this column 

 as among its own particles, otherwise there would 

 be a flow towards the part where the pressure 

 was less. Throughout the whole of the enlarged 

 space D, there is the same pressure as there is 

 at the same level immediately under the tube. 

 Therefore, if the enlarged part of the vessel is of 

 the same width as the cylinder, having an area 

 of ten square inches, the pressure on the whole 

 bottom is equal to the weight of ten times twelve 

 cubic inches of water, exactly as in the cylinder 

 at B. 



It thus appears that the pressure on the hori- 

 zontal bottom of a -vessel is as the area of the 

 bottom and the perpendicular height of the liquid, 

 and that without regard to the shape of the vessel. 



Take the case of two vessels of equal capacity, 

 and shaped as in figs. 6 and 7. In fig. 6, the 



Fig. 6. 



Fig. 7. 



bottom is pressed with the weight of the column 

 of water ACDB. In fig. 7, the portion CD, equal 

 to the whole base of the other, sustains an equal 

 column ; but from the law of liquids the same 

 pressure is extended over the whole bottom. Now, 

 if the diameter EF is three times the diameter 

 CD, the larger bottom is nine times the area of 

 the smaller. Therefore, the pressure on the 

 bottom of fig. 7 is nine times the pressure on that 

 of fig. 6, although both vessels contain the same 

 quantity of water. If the bottoms of the vessels 

 were separate from the sides, and kept in their 

 places by springs or weights, it could be shewn 

 experimentally that such is the case. 



A strange consequence would seem to follow 

 from this. If the two vessels in fig. 5 were placed 

 -on the opposite scales of a beam, the bottoms of 

 both being equally pressed, it would seem as if 

 they must balance each other. Similarly, we 

 might expect fig. 7 to weigh nine times as much 



as fig. 6. But that such cannot be the case appears 

 from the following considerations : In the upright 

 cylinder in fig/ 5, the lateral pressure being 

 directly horizontal, neither presses the sides down 

 upon the bottom nor lifts them up ; but with the 

 irregular vessel it is different. Where the small 

 tube C enters the wide part D say two inches 

 from the bottom the pressure of the column is 

 equal to ten cubic inches of water on the square 

 inch ; and as this pressure acts upwards as well 

 as downwards, the horizontal cover of the wide 

 part is pressed upwards with a force of ten cubic 

 inches on the square inch. Its area, the tube 

 being one inch, is nine inches, so that the whole 

 upward pressure is 9 X 10 cubic inches. The cover 

 being attached to the sides, and the sides to the bot- 

 tom, a part of the pressure on the bottom is thus 

 counteracted, and only the difference the differ- 

 ence between 10 X 12 and 9 X 10 or thirty cubic 

 inches of water remains to weigh down the scale. 

 The confined water is in this case like a person in 

 a covered box who should place his shoulders 

 against the top and press against the bottom with 

 his feet ; he would thus add nothing to the weight 

 of the box beyond his own dead weight With 

 regard to figs. 6 and 7, the pressure on the sides 

 EC and ED of the former being at right angles to 

 those sides, is partly downwards, and thus adds 

 to the direct pressure on the bottom ; while in fig. 

 7 the pressure on the sides is partly upwards, so 

 as to relieve a part of the bottom pressure. The 

 result is, that the pressure or weight on the scale 

 is the same in both. 



Pressure of Liquids on the Sides of Vessels. 

 We begin with the case of an upright side as the 

 simplest. Let AB represent a vessel with upright 

 rectangular sides filled with water, and suppose 

 the perpendicular depth, AC, of the water to be 

 six inches. A square inch of the bottom at C</ is 

 evidently pressed with the weight of a column of 

 six cubic inches of water. Now, as water presses 

 equally in all directions, the square inch from C to 

 5 of the side adjacent, will also be pressed; but 

 evidently not to 

 the same ex- 

 tent, owing to 

 the gradual di- 

 minution of the 

 weight of the 

 water, as we ad- 

 vance upwards. 

 The particle of 

 water in the f_ 

 corner at C 

 presses equally 

 downwards and c 

 outwards, and Fig. 8. 



as its pressure 



downwards arises from the weight of the column 

 of particles above it from C to A, we may repre- 

 sent its outward pressure by the line Ce, made 

 equal to CA. At 5, the downward pressure, and 

 therefore also the outward pressure, is one-sixth 

 less, and will therefore be represented by 5/, made 

 equal to 5A. If el is joined, the figure Gr/5 

 represents the sum of the outward pressures of all 

 the particles from C to 5, and also the mode of 

 their gradual diminution. Had the pressure 

 continued constant, the sum would have been 

 expressed by the parallelogram C/, which is equal 

 to Artf, the weight on the bottom. From this it 



