﻿Mr. J. J. Sylvester on the Quadrature of the Circle. 373 



The values of the constants //,, A, B may be determined by ob- 

 serving the quantities Q 1? Q 2 , Q 3 of fluid discharged through the 

 same channel with three different supplies, and therefore three 

 different depths of the fluid in the channel or values of b. 

 These constants being determined, the velocities r 2 and z q at 

 the bottom and sides are known from equation (1). That 

 equation shows the velocity to increase from the bottom to the 

 surface and from the sides to the centre, which corresponds with 

 observation. 



LIII. Note on a new Continued Fraction applicable to the Quadra- 

 ture of the Circle, By J. J. Sylvester*. 



I 



N a recent note inserted by the author in the Philosophical 



Magazine it was virtually shown, and indeed becomes almost 



u 

 self-evident as soon as stated, that the equation u x+1 = — H-w^—i 



possesses two particular integrals, a Xi (3 X) which are the products 

 of x terms of the respective progressions 



n 1 s r 5. ] .z 1 • 



n a i it i s i i 



I 1 } 1> x > 3' x > b> x > ' ' 'J' 



Now any continued fraction whose partial quotients are 

 -, - — =r, •••*"" w ^ De e q ua l to the ratio of some two particular 



tC K -J- X X 



values of u x in the above equation, i. e. of two linear functions of 

 a X) /3j. ; and in especial when k = 1 it will be found very easily that 



this fraction is — -• 



But, on supposing x infinite, — becomes equal to the well- 



IT 



known factorial expression for ^, viz. f-f.f.f.f... Hence 



IT 



we may deduce the following value for - under the form of a 

 continued fraction, viz. 



-=i+- i 



3 * -f- -r— ad infinitum, 

 Reverting to pure integers, the above equality maybe written 

 * Communicated bv the Author. 



