Coated plate compared with globe 97 



163] COR. III. If the plate of glass is not flat, and its thickness is very 

 small in respect of the radius of curvature of its surface at and near C, every- 

 thing else being as in Cor. I, the quantity of redundant fluid in AB will still 



B x CW 

 be very nearly equal to 



For as CE is very small in respect of the radius of curvature, the two 

 coatings will be very nearly of the same size, and therefore G differs very little 

 from M , and m + g is to W very nearly as CE to CW*, and moreover m and g 

 are both very small in respect of M and G f. 



164] COR. IV. If we now suppose that the density of the redundant fluid 

 in AB is greater at its circumference than it is near the point C, and that its 

 density at and near C is less than the mean density, or the density which it 

 would everywhere be of if it was spread uniformly, in the ratio of 8 to one, 

 and that the deficient fluid in DF is spread nearly in the same manner as the 

 redundant in AB, the quantity of redundant fluid in AB will be greater than 

 before in a ratio approaching much nearer that of one to 8 than to that of 

 equality, and that whether the glass is flat or otherwise. 



For by Lemma [XVI, Cor. II], m and g will each be less than before in the 

 above-mentioned ratio. 



165] COR. V. Whether the plate of glass is flat or concave, or whatever 

 shape the coatings are of, or whatever shape the canals CG and EM are of, 

 or in whatever part they meet the coatings, provided the thickness of the plate 

 is very small in respect of the smallest diameter of the coatings, and is also 

 sufficiently small in respect of the radius of curvature of its surface in case it 

 is concave, the quantity of redundant fluid in AB will differ very little from 

 B x CW 

 2.CE ' 



For suppose that the canal GC meets the coating AB in the middle of its 

 shortest diameter, and that the point in which ME meets DF is opposite to L, 

 as in Prop. [XXII, Art. 74], the thickness of the glass will then be very small 

 in respect of the distance of the point C from the nearest part of the circum- 

 ference of AB, and moreover, by just the same reasoning as was used in the 

 Remarks to Prop. XXII, it may be shewn that 8 will in all probability differ 

 very little from one, and consequently by Cors. I and III the redundant fluid 

 in AB will be as above assigned. But by Prop. XXIV the quantity of fluid in 

 the coatings will be just the same in whatever part the canals meet them, or 

 whatever shape the canals are of. 



166] COR. VI. On the same supposition, if the body H is a globe whose 

 diameter equals Ww, id est the diameter of a circle whose area equals that of 



* Lemma XVI. Cor. 



t As the demonstration of the sixteenth Lemma and its corollary is rather 

 intricate, I chose to consider the case of the flat plate of glass separately in Cor. I, 

 and to demonstrate it by means of Lemma XV. 



c. p. i. 7 



