Simpson's two formulae also apply if u is of the form px 3 - }- 5x 2 + rx -}- s.
If (f,4) 1 be not a perfect square, and rx, s x be its linear factors, it is possible to express f and 4, in the canonical forms Xi(rx)2+X2(sx)2, 111(rx)2+1.2 (sx) 2 respectively.
If 4) = rx.sx, the Y2 =1 normal form of a:, can be shown to be given by (rs) 4 .a x 4 = (ar) 4s: 6 (ar) 2 (as) 2rxsy -I- (as) 4rx; 4) is any one of the conjugate quadratic factors of t, so that, in determining rx, sx from J z+k 1 f =o, k 1 is any root of the resolvent.
If the liquid is stirred up by the rotation R of a cylindrical body, d4lds = normal velocity reversed dy = - Rx- Ry ds (5) ds 4' + 2 R (x2 + y2) = Y, (6) a constant over the boundary; and 4,' is the current-function of the relative motion past the cylinder, but now V 2 4,'+2R =o, (7) throughout the liquid.
An angular velocity R, which gives components - Ry, Ix of velocity to a body, can be resolved into two shearing velocities, -R parallel to Ox, and R parallel to Oy; and then ik is resolved into 4'1+1'2, such that 4/ 1 -R-Rx 2 and 1//2+IRy2 is constant over the boundary.
Example 3.-Analysing in this way the rotation of a rectangle filled with liquid into the two components of shear, the stream function 1//1 is to be made to satisfy the conditions v 2 /1 =0, 111+IRx 2 = IRa 2, or /11 =o when x= = a, +b1+IRx 2 = I Ra2, y ' 1 = IR(a 2 -x 2), when y = b Expanded in a Fourier series, 2 232 2 cos(2n+ I)Z?rx/a a -x 7r3 a Lim (2n+I) 3 ' (1) so that '?"
Let s be the perpendicular from 0, the join of C and T on the direction of S; t the perpendicular from A, the join of C and S on the direction of T; and c the perpendicular from B, the join of S and T on the direction of C. Taking moments about 0, Rx - W 1 (x+a) - W 2 (x+2a) =Ss; taking moments about A, R3a-W 1 2a-W 2 a =Tt; and taking moments about B, Rea-W I a = Cc.
For this purpose it was fixed that there should be an annual provision of Rx.i,50o,000, to be spent on: (1) relief, (2) protective works, (3) reduction of debt.
The value of the imports from Kabul to India in 1892-1893 was estimated at 221,000 Rx(or tens of rupees).
In 1899 it was little over 217,000 Rx, the period of lowest intermediate depression being in 1897.
We have also the geometrical relations x = (a/c) (qz ry), 5 = (a/c) (rx p1), = (a/c) (pyqx).
In 1894-1895 it had sunk to 274,000 Rx, and in 1899 it figured at 294,600 Rx.
In 1898-1899 the imports from Kandahar to India were valued at 330,000 Rx, and the exports from India to Kandahar at about 264,000 Rx.