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Full text of "Scientific Papers - Vi"

1919]
ATMOSPHERIC  BEFEACTION
599
At this point it is convenient to re-introduce 6, where cc = /3 cot 9, and
dec = — /3 sin~2 Od6.
We have
d
£sin-20d0
The upper limit oo for x corresponds to 0=0; the lower limit a corresponds to what after the integration we may still denote by 6, where a = /3 cot 0. Thus
Also ' and
.  1* xzdx
*=         8^0^0=(10-8^20
s                                  5 w
.(6)
cos2 0 sin2 v2dx      1
(0 _ £ sin 40),
0 now referring, as above, to the point for which x = a.    Using (6) and (7) in (5), we have
(t * - si
sn
sn
= ™ (sin 2
sin 20 + £ sin 40), ...... (8)
in which, if we please, we may substitute r sin 0 for /3, so that r, 0 are the polar coordinates of the point of observation.
Thus
c2 sin 20- A sin 40     c4|0 - sin 20 + A sin 40
fv\   -—.   ____   r_    _._____________&__   ______.__... ^    __  ____    ^   ^          _                        ________ O         ________
«9             01 n2V?                 r4                   Qin4 (9
/                Cull    U                   I                       olll    I/
.......................................(9)
as we write it for brevity.    It now remains1, to discuss (9) as a function of r and 0, under the limitation, however, that r > c.
When 0 is small,
/(0) = 40,    F(6) =
so that (9) becomes
.(10)
vanishing when 0 = 0, as was to be expected, since this is a line of symmetry. Inasmuch as .r > c, (10) takes its sign from 0,
The table gives values of/ and F for certain angles of 0. In the fourth column are entered the values of (9) when r — c, that is on the surface of the cylinder. So far as 0 = 50° these are the highest admissible. For example at 50°
= -8173 {(1-2049)2 - (1-2049 - c'/r8)3}.
So long as r > c, the value of (9) increases as r diminishes, and the greatest admissible value occurs at the limit r — c.    This state of things continues so-velocity U, is well known*. The problem may conveniently be reduced to one of "steady motion" by supposing the cylinder to be at rest while the fluid flows past it—a change which can make no