Part 8

CENTRAL VORTEX ASCENDING.

Greenwich time of passage 2d. 3h. 1m. Mean longitude of moon's node 78d 29' True " " 79 32 Mean inclination of lunar orbit 5 9 True " " 5 13 Obliquity of ecliptic 23 27 32" Mean inclination of vortex 2 45 0

Then in the spherical triangle PEV,

PE is equal 23d 27' 32" EV " 7 58 0 E " 100 28 0 P " 18 5 7 PV " 26 2 32

Calling P the polar angle and PV the obliquity of vortex.

To find the arc AR.

By combining the two proportions already given, we have by logarithms:

M.R.V. minor = 3256 Log. 3.512683 M.S.D. of moon = 940" " 2.973128 P.S.D. of earth = 3950 A. C. 6.403403 Radius 10.000000 T.S.D. of moon 885".5 A. C. 7.052811 Log. Cosine arc AR = 28d 57' 3" 9.942025 ---------

As the only variable quantity in the above formula is the "True" semi-diameter of the moon at the time, we may add the Constant logarithm 2.889214 to the arithmetical complement of the logarithm of the true semi-diameter, and we have in two lines the log. cosine of the arc AR.

We must now find the arc RK equal at a maximum to 2d 45'. The true longitude of the moon's node being 79d 32', and the moon's longitude, per Nautical Almanac, being 58d 30', the distance from the node is 21d 2', therefore, the correction is

-2d 45' x sin 21d 2' -arc RK = --------------------- = -59' 13" R

To find the correction for displacement.

True longitude of sun at date 100d 30' " of moon " 58 30 Moon's distance from quadrature 48 0

As the moon is less than 90d from the sun this correction is also negative, or

-90' x sin 48d Arc Kq = --------------- = -1d 6' 46". R

Arc AR = 28d 57' 3" RK = - 0d 39' 13" Kq = - 1d 6' 46" Sum = 26d 51' 4" = corrected arc AQ.

We have now the necessary elements in the Nautical Almanac, which we must reduce for the instant of the vortex passing the meridian in Greenwich time.

July 2d. Meridian passage, local time, at 9h. 5m. A.M. " in Greenwich time 2d. 3h. 1m. Right ascension same time 56d 42' 45" Declination north " 18 00 1 Obliquity of the vortex " 26 2 32 Polar angle " 18 5 7 Arc AQ " 26 51 4

PA = 17d 59' 59" } P = 128d 37' 38" PV = 26 2 32 } VA = 89 3 0 V = 47 59 44 VQ = 62 11 56 A = 20 3 42 PQ = 47 14 22 Q = 26 22 55 Latitude of Q on the sphere = 42d 45' 38"

CORRECTION FOR PROTUBERANCE.

We have hitherto considered the earth a perfect sphere with a diameter of 7,900 miles. It is convenient to regard it thus, and afterwards make the correction for protuberance. We will now indicate the process for obtaining this correction by the aid of the following diagram.

Let B bisect the chord ZZ'. Then, by geometry, the angle FQY is equal to the angle BTF, and the protuberance FY is equal the sine of that angle, making QF radius. This angle, made by the axis of the vortex and the surface of the sphere, is commonly between 30d and 40d, according as the moon is near her apogee or perigee; and the correction will be greatest when the angle is least, as at the apogee. At the equator, the whole protuberance of the earth is about 13 miles. Multiply this by the cosine of the angle and divide by the sine, and we shall get the value of the arc QY for the equator. For the smallest angle, when the correction is a maximum, this correction will be about 20' of latitude at the equator; for other latitudes it is diminished as the squares of the cosines of the latitude. Then add this amount to the latitude EQ, equal the latitude EY. This, however, is only correct when the axis of the vortex is in the same plane as the axis of the earth; it is, therefore, subject to a minus correction, which can be found by saying, as radius to cosine of obliquity so is the correction to a fourth--the difference of these corrections is the maximum minus correction, and needs reducing in the ratio of radius to the cosine of the angle of the moon's distance from the node; but as it can only amount to about 2' at a maximum under the most favorable circumstances, it is not necessary to notice it. The correction previously noticed is on the supposition that the earth is like a sphere having TF for radius; as it is a spheroid, we must correct again. From the evolute, draw the line SF, and parallel to it, draw TW; then EW is the latitude of the point F on the surface of the spheroid. This second correction is also a plus correction, subject to the same error as the first on account of the obliquity, its maximum value for an angle of 30d is about 6', and is greatest in latitude 45d; for other latitudes, it is equal {6' x sin(double the lat.)}/R.

The three principal corrections for protuberance may be estimated from the following table, calculated for every 15d of latitude for an angle of 30d, or when the correction is greatest.

Latitude. 1st Corr. 2d Corr. 3d Corr. 0 + 20' + 0 - 2 15 + 19 + 3 - 1.5 30 + 15 + 5 - 1.5 45 + 10 + 6 - 1. 60 + 5 + 5 - 1 70 + 1 + 3 - 0.5

We can now apply this correction to the latitude of the vortex just found:

Latitude on the sphere 42d 45' 38" n. Correction for protuberance + 14 22 ---------- Correct latitude 43 00 00

MILWAUKIE STORM, JULY 2.

As this example was calculated about ten days before the actual date, we have appended an extract from the Milwaukie papers, which is in the same longitude as Ottawa, in which place the calculation was made. It is needless to remark that the latitude of Milwaukie corresponds to the calculated latitude of the centre of the vortex. It is not intended, however, to convey the idea that the central line is always the most subject to the greatest violence--a storm may have several centres or nuclei of disturbance, which are frequently waning and reviving as the storm progresses. Generally speaking, however, the greatest action is developed along the line previously passed over by the axis of the vortex.