REPORT ON HYDRAULICS. PART II. 439 



the pressure of the tributary waters agahist the water of the 

 recipient. 



But, besides the law of acceleration, there remained other 

 elements to take into account, one of which related to the mo- 

 tion arising from the junction of two or more rivers. 



In the Memoirs of the Academy of Sciences for the year J 738, 

 M. Pitothas used the same principle to determine the mean di- 

 rection which the waters of the two rivers will take when freely 

 united together, and this he does according to the resultant of 

 the collision of hard bodies, where the same quantity of motion 

 is invariably preserved ; and from this hypothesis he draws as 

 its consequent that the common velocity of the united rivers is 

 equal to the quantities of motion in the two separate rivers di- 

 vided by the sum of those quantities of water. Grandi has en- 

 deavoured to decide by the same principles of the composition 

 and resolution of forces, not only the direction, but the absolute 

 velocity of the waters which either unite or divide. For this 

 purpose he constructed a float which gave the resultant of the 

 two confluences, from which he concluded that the course of 

 the river would naturally take an intermediate direction ; but, if 

 the banks of the recipient remained firm, its stream would pre- 

 serve the same direction as before, increasing, however, its for- 

 mer velocity by a part, in proportion to the velocity of the tri- 

 butary stream, as the cosine of inclination of the river is to the 

 radius : whence it would follow, that if the thread of water in 

 the tributary stream should second by its direction the thread 

 of water in the recipient, in making with it, as is generallj'^ the 

 case, a very acute angle, the velocity in the common bed would 

 be equal to the sum of the velocities of the recipient and affluent 

 streams. If this principle were admitted, it would follow, that 

 the sections of the receiving stream could not be considerably 

 augmented by the junction of the tributary, for this reason, that 

 the quantity of water augmenting, the velocities would be com- 

 pounded of this augmentation, and the flow of the current be 

 more rapid than it was before. 



Guglielmini, in the seventh chapter of his work, in consider- 

 ing the celebrated phaenomenon of the Po (of Venice), which 

 receives the branch of the Ferrara and the Panaro without any 

 enlargement of its bed, has stated in general that a smaller river 

 might enter into a larger one without increasing either its 

 breadth or height; and he was of opinion that this might happen 

 without any lateral dispersion, because the whole of the in- 

 creased body continued in motion by following the direction of 

 the thread of the stream. On the hypothesis that all the sections 

 were efi'ective, and that the velocities before and after the con- 



