# FG Sentence Examples

- If ai, bx, cx be different forms we obtain, after development of the squared determinant and conversion to the real form (employing single and double dashes to distinguish the real coefficients of bx and cz), a(b'c"+b"c'-2 f'f") +b(c'a"+c"a'-2g'g") +c(a' +a"b'-2h'h")+2f(g'h"+g"h'-a' + 2g (h ' f"+h"f'-b'g"-b"g')+2h(f'g"+f"g'-c'h"-c"h'); a simultaneous invariant of the three forms, and now suppressing the dashes we obtain 6 (abc+2fgh -af t - bg 2 -ch2), the expression in brackets being the S well-known invariant of az, the vanishing of which expresses the condition that the form may break up into two linear factors, or, geometrically, that the conic may represent two right lines.
- Clebsch to take the form T= 2p(x12 +x22)+2p'x32 + q (xiyi +x2y2) +q'x3y3 +2r(y12+y22)+2r'y32 so that a fourth integral is given by dy 3 /dt = o, y = constant; dx3 (4 y) (q + y) _ (y y) dt - xl 'x2 xl Y Y x l 2 - 1, y2 () = (x12 +x22) (y12 + y22) = (X 1 2 + X 2) +y22)-(
**FG-**x3y3)2 = (x 1 y32-G2)-(Gx3-Fy3) 2, in which 2 = F 2 -x3 2, x l y l +x2y2 =**FG-**x3y3, Y(y1 2 +y2 2) = T -p(x12 +x22) -p'x32 -2q(xiyi 'x2y2)- 2 q ' x = (p -p') x 2 + 2 (- q ') x 3 y 3+ m 1, (6) m1 = T 2 i y 3 2 (7) so that dt3) 2 =X3, (8) where X3 is a quartic function of x3, and thus t is given by an elliptic (8) (6) (I) integral of the first kind; and by inversion x 3 is in elliptic function of the time t. - Now (x1 - x21) (y 1 +y21) = xl l +x2y2 + - (' r 1 2 - x2y1) =
**FG-**x3y3+iV X3, yi+3 7 21_FG-x3y3+2V X3 xl+x21 X12 +X22 (x 1 +x 2 i) = - i{(q' - q)x3+r'y3]+irx3(y1+y21), =**FG**- x3y3 +ZJ X3 dt2log(x1+x22) - - (q g) x 3- r y3+rx3 F2x32 (12) d dl2 log V x1 ± x2 2 (q'-q)x3-(r'-r) y3FrFF2-x 2 3 ' (13) requiring the elliptic integral of the third kind; thence the expression of x1-f -x21 and yl-}-y21. - Introducing Euler's angles 0, c15, x1= F sin 0 sin 0, x 2 =F sin 0 cos 0, xl+x 2 i =iF sin 0e_, x 3 = F cos 0; sin o t=P sin 4+Q cos 0, dT F sin 2 0d l - dy l + dy 2x = (qx1+ryi)xl +(qx2+ry2)x2 = q (x1 2 +x2 2) +r (xiyi +x2y2) = qF 2 sin 2 0-Fr (
**FG**- x 3 y 3), (16) _Ft (**FG**_x 323 Frdx3 (17) F x3 X3 elliptic integrals of the third kind. - (22) Y (F2 x2) Suppose x 3 -F is a repeated factor of X3, then y 3 = G, and X 3 = (x 3 -F)2 [P' _ P(X3+F)2+2' _ G(X +F) -G 2 ], (23) nd putting x3-F=y, (y) 2= 7'3'2- [41' r 1' F 2 -{-4 g r qFG - G2 +2 (2P'r 19F+9 r q G) y+ r y (24) o that the stability of this axial movement is secured if A = 4 P' r ?'F 2 + 4 Y q
**FG**- G 2 (25) s negative, and then the axis makes r J l (-A)/7r nutations per second. - It is terminated by a well-developed structure (
**fg**) corresponding with the apical sense-organ of ordinary Trochospheres, and an excretory organ (nph) of the type familiar in these larvae occurs on the ventral side of the stomach. **Fg**, Apical sense-organ.- Till W 1 has advanced a distance a only one load is on the girder, and the curve A"F gives bending moments due to W 1 only; as W1 advances to a distance a+b, two loads are on the girder, and the curve
**FG**gives moments due to W 1 and W2. - The pulley or sheave
**FG**has a weight W - A cord has one end made fast a to and wrapped round the barrel AE; it passes from A under the sheave
**FG**, and has the other end wrapped round and made fast to the barrel BD. - Transmitted by wrapping connectors to
**FG**, and combined by that sheave so as to act on the fol w lower W, whose motion is the same with that of the centre of**FG**. - Hence the instantaneous axis of the sheave
**FG**is in the diameter**FG**, at the distance**FG**ACBC - And, consequently, the velocity of its centre is
**FG**ACBC ai(ACBC) - Gh and
**Fg**, Gastral filaments (phacellae). - At the time of the last great subsidence, in glacial times, an arm of the sea extended across Sweden, submerging a great part of the littoral up to the Gulf of Bothnia, and including the Plies period the this of themnorthe and Baltic were s
**fg**ficiently salt for oysters to flourish.