300 PHILOSOPHICAL TRANSACTIONS. [aNNO 1739. 



time, as the weight of mr^A, is to that resistance. Hence this resistance is 

 = mr^ a(1 — 39^). 



In like manner, by putting « and e for the radius of the hole, and the altitude 

 of another vessel, and 2pnis^E for the water issuing in the same time, in which 

 a body falls in vacuo through the height e, the resistance in this new vessel 

 will be = ms'^E(l — 3/)^). 



But, by the cor. to prob. 8, these two resistances are to each other, as qrx^ 

 to psE*. Making this proportion then gives this equation, jbrE(l — 3^^) = qsx 

 (1 — 3/>^) ; hen ce p = 



\/I^^^7-i^, or, = x/i+(irJ£„)._-V„ by putting 

 rE := nsA. And hence 2pms'^E is the measure ot the water issuing from the 

 second vessel, in the time that a body falls freely through the altitude e. q.e.i. 



Corol. 1 . If the diameters of the holes be in the ratio of the altitudes of the 

 water, the ratio of the measures will be the same, as if the water issued with- 

 out any resistance. For, if r : « :: a : e, or rE = sa, and n = 1, then h p=.q ; 

 and hence 2qmr^\ : 2pms^E :: 2mr'^A : ttw^e, which is the ratio of the measures 

 when void of all resistance. 



Corol. 1. If E be considered as nothing in respect of the altitude a, then will 

 n be as nothing also, and hence/) =■/•!-. Therefore the smaller the altitude e 

 is taken, the nearer p approaches to V -r. 



Corol. 3. If s be infinitely great in respect of r, then \s p = V -^. There- 

 fore the greater s is taken, the nearer p approaches to v^^. 

 Prob. 10. — The Water Jlmving into the Air; to determine the Ratio between 

 the Diameter of the Hole and of the Contracted Vein. 



This ratio cannot be determined without experiments. By prob. 7, p» = 



£Zl! X {v -\- 6q\ — l*^ 3qvv -j- gq^v^—lv^) ; hence p is determined, when q 



and V are known. 



But no experiments are known, by which q and v may be measured. Poleni's 

 experiments show the measure of the effluent water, whence q is known : 

 but they do not show the greatest distance to which the water is carried, when 

 issuing horizontally from the hole ; nor the distance to which the middle part 

 of the vein reaches, that issues with the velocity v. And Mariotte's experi- 

 ments measure the greatest perpendicular height, to which water jets, when 

 its motion is turned upwards, whence f' is known ; but they do not show the 

 measure of the effluent water. Therefore, for want of proper experiments, 

 the ratio can only be determined approximately, as follows. 



By schol. 2, prob. 7j it was made probable, that the ratio is constant be- 

 tween these two radii, or at least that it is very little varied. From Mariotte's 

 experiments it appears, that the difference between the altitude to which the 



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