TRANSACTIONS OF SECTION A. 



569 



On the rigid of AF we take 



dy' 

 dv 



~ J _ = v= — arhin ^6 + 2c'„€~^"^'sin pny 



■■ u = ar- cos ; 



' + 2c'„e-J"'^'< 



COS ptii/ 



(1) 



Where c'„ is an arbilrary constant and 2 indicates summation with regard to n 

 for all integral values from 1 to infinity. 

 On the left of AF we take 



dJA 

 dy 





e"™' sin my) — 2CniP"^' sin |j?iy 



(2) 



J 



Where a,„ c,,, and A are arbitrary constants and S indicates summation with regard 

 to m for -Ajimte number of values of m, the largest of which is supposed to be small 

 compared with p. 



Since the velocities must be continuous on each side of AF, we must have 

 along AF. 



S(rt,„co3 my) = « - A + 2(e,/ - c„)cos jmy \ ,„, 



-S(amSin ??iy) + ^a«/ = 2(c,/ + c„)sin^?iy J • w 



These must hold from y = o to y = Trjp. But if we expand the left-hand sides by 

 Fourier's Theorem we get c„ and e/ as functions of n. Since the left-hand side of 

 the second equation of (3) must vanish when y = nlj), this furnishes one relation 

 among the constants. We can then show that c„ and c,/ are small quantities at 

 most of the order 1/^'. 



It is easy to form from (2) the equation to the outer stream-line BEF. If the 

 vessel be of finite breadth at infinity, A will be a small quantity of the order Ijp. 



Looking now at (2), we see that if OE be the surface of the liquid, ti and v must 

 when x' = —1 be small quantities at most of the order Ijp. A and the 2 term 

 already satisfy that condition. In the S term m has several values. Suppose the 

 particular m in (2) to be the smallest of these values, and suppose m = logp, then 

 when .v' = — 1 the S term also satisfies the surface condition, and the more accurately 

 the larger^ is, since log pjp diminishes asp increases. 



If FJ is a jet we must have, since AF is small, at every point of the jet, 

 nearly 



M^ + y- = 2gr. 



But we see at once from (I) that this condition is nearly fulfilled, the error being 

 of the order Ijp"^. 



As a particular case, if we give to m in (2) the two values 8 and 9, p will be 

 about 2980 and the maximum error (which will be at F) will be about 

 + •0000143. 



