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The American journal of science and arts online

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temperature for instance, has been found for a given place by
the method of mean months, and it is required to interpolate
from it the mean temperature of a particular day, care snould
be taken to assign the proper value to the abscissa. Take for
example the 15tn day of March. The time elapsed from the
beginning of the year to the middle of that day is 8lH-28"2422
+14i= 78-7422 days, and the corresponding arc is found by the
proportion

366-2422 days : 73-7422 day8=360* : 72** 41'.

The value 72^ 41' being given to the abscissa in the equation of
the curve, the resulting value of the ordinate will be the mean
daily temperature at the given place on the 15th of March.
The abscissa must be reckoned up to the middle of the day^ be^

▲m. Joub. Sci. - SscoifD Sbribs, Vol. XLIII, Xo. 129.— Mat, 1867.
41



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S18 E. L. DeForett on Reducing Meteorological ObservoHone.

cause the small arc of the curve which belongs to any one day
may be regarded as approximately a straight line, so that the
mean of all its ordinates is equal to its middle ordinate, which
therefore represents the mean temperature of the day. The
subjoined table will be found to lacilitate computations. It
giyes the abscissa for the middle point of each day in the year,
correct to the nearest minute, one minute of arc here corres-
ponding to about twenty-four minutes of time.

Ares representing the mean interval of time from the beginning of the
year to the middle of each dag.



1


Jan.
8^


Feb.

sf's


M'ch.


April.


M.y.


Jana.


July.


Au,.


Sept


Oct


Nor.


Dee.


&46


11^9 'l


l&ii


17% '8


2^il


2&{5 2&l9


800^


8^^


2


129


32 2


59 52


90 26


120


150 88


180 71210 40


241141270 48


30121


880 55


8


228


38 1


60 51


9125


120 59


15132


181 6


21140


24218127147


302 20


83165


4


8 27


34


6151


92 24


12158


152 81


182 5


212 89


24312,272 46


308 20


832 54


5


426


34 59


62 50


93 28


122 57


153 80


188 6


218 88


2441l!278 45


80419


38858


6


525


35 59


63 49


94 22


123 56


154 30


184 4


214 87


24510 274 44


:i0518


884 52


7


624


36 58


64 48


95 21


124 55;155 29


185 8


215 86


246 9=27544 80617


885 51


8


724


37 57


65 47


96 20


125 55


166 28


186 2


216 85


247 9;276 43 30716


88650


9


828


88 56


66 46


9719


126 54


157 27


187 1


217 84


248 81277 42 30815


837 49


-10


922


39 55


67 46


9819;i37 53 158 26


188 218 34


249 7 278 41 309 14


388 49


11


1021


40 54


68 44


9918 128 52 159 25


188 59 219 83


250 6 279 40131018


839 48


12


1120


4158


69 44


10017 129 51


160 24


189 59


220 32


251 5 1280 39 131118


34047


18


1219


42 53


70 48


10116 i:% 50


16124


190 58


22181


252 41281 38 312 12


84146


U


1818


43 52


7142


102 15 181 49


162 23


19157


222 80


258 8!282 38;31811


842 45


15


1418


44 51


72 41


10814


132 48


168 22


192 56


223 29


254 8 283 87131410


34844


16


1517


45 50


73 40


10413


138 48


164 21


198 55


224 28


256 2


284 86 315 9


344 48


17


1616


46 49


74 89


10518


184 47


165 20


194 54


225 28


256 1


286 85 816 8


345 42


18


1715


47 48


76 38


10612


185 46


16619


195 58


226 27


257


286 84 317 7


84642


19


1814


48 47


76 38


107 11


136 46


167 18


196 58 227 26


257 59


287 83 318 7


847 41


20


1913


49 47


77 87


10810


137 44


16817


197 52 228 25


25858


288 32*819 6


34840


21


2012


50 46


78 86


109 9


13848


16917


196 511229 24


259 57


28932|320 5


849 89


22


2111


5145


79 85


110 8


139 42


17016


199 50 230 23


260 57


290 3ll821 4


850 88


28


2211


52 44


80 34


111 7


140 42


17115


200 49 23122


26156


29130i322 8


85187


24


28 10158 43


8133


112 7


14141


17214


201 48 232 22


262 55


29229!828 2


85286


25


24 9i5442


82 32


118 6


142 40


17813


202 47,283 21


268 54


29328;324 1


858 86


26


25 8 65 41


83 82


114 5


143 39


17412


203 46 234 20


264 53 294 27


:J25 1


354 85


27


26 7,56 40


84 31


115 4


144 38


17511


204 46 23519


265 521295 26


326


855 84


28


27 6^57 40


85 30


116 3


145 37


17611


205 45 23618


266 61


296 26


826 59


356 38


29


28 55816


86 29


117 2


146 36


17710


206 44 237 17


267 51


297 26 327 58


857 82


30


29 5




87 28


118 1


147 36


178 9


207 48 28816


268 50


298 24 328 57


858 81


81


30 4




88 27




148 35




208 42 28915




299 23


859 80



When the mean daily temperatures at a given place have
been tabulated for every day in the year, it seems to have been
the usual practice hitherto to omit giving the temperature for
the 29th of February. It cannot be said that the meteorologi-
cal phenomena which occur on a single intercalary day are less
important, or leas worthy of being observed and recorded and
combined with others to form the monthly mean, than are those
of any other single day. The interval of time between the
mean position of the beginning of the calendar year and the
mean position of the middle of the intercalary day is assignable
with great precision, and when an equation of temperatures has



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E. L. DeForett on Reducing MeU&rological Obiervations. S19

been found by the use of mean months, the mean temperature
of the 29th day of February can be interpolated with as much
accuracy as that of any other day whatever. The time elapsed
from the beginning of the year to the middle of the intercalary
day is 81+284-i of 0-2422=691211 daysj and the correspond-
ing abscissa is found to be 68° 16'.

The monthly means of temperature at New Haven, as given
in the Transactions of the Connecticut Academy of Arts and
Sciences, from 86 years' observations, are

26-53 46*84 71*66 51-10 '

28-11 67-28 70-32 4032

86*09 66*96 62*50 80*42

When reduced to mean months they become

26*5811 47-3076 71*7807 50*9438

28*2410 67*7031 70*2452 401992

36*5268 67*2872 62*3404 80*3442

From these data I have obtained the equation of mean daily
temperatures throughout the year, in the way already stated in
this Journal, xli, 878, except that instead of finally reducing it
from the usual form,

y=^a-\^^ 8in(ir+Ei)-f-a2 8in(2a:+E2)-|-a3 8in(3a;+E3)+ Ac,

into a form where the signs before the terms are sometimes plus
and sometimes minus, I have reduced it to

y=a-|-aj 8in(«— f i)-|-a2 6in2(«-e2)+a3 8in3(a:— ej)^- Ac,

in accordance with the formula

8in(««+E»)=8inn^ a? - (360*-E;) y.

This prevents confusion of signs, and at the same time preserves
the significance of the arc «„ making it measure the time elapsed
from the beginning of the year to the first ascending node oi the
term in which it occurs.

The New Haven equation of temperatures then is

y=49*112+22*902 8in(«^110* 39' 22")+*289 8in2(«-20* 56')
+-443Bin3(a;-57** 42')+-022 8in4(«— 76** 22')
4-'402 8in6(« - 3*' 53')4-*093 sin 6a?.

An equation of this kind, to be perfect, ought to express ac-
curately all the &ct8 implied in the observed series of monthly
means, so that the mean for any one of the calendar months
may be derived from it with precision, by integrating ydx be-
tween the proper limits for the beginning and end of the month,
and dividing by the arc which measures its length. Let ike
general form,

be treated in this way between the limits x' and of' correspond-



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S80 E. L. DeFarest on Reducing Meteorological OhertatioMt,

ing to the b^mung and end of a month whose mean is m, and
let OS make

then the monthly mean will be expressed thns :

«=tt+a, — - nB(/?-#,)+a, -^ 8in2(jJ-€2)+a, -^ «n3(/J^,)+ Ac

The valnes of — , -r — , Ac^ depend only on the length of the

^ month, and their logarithms are given in the subjoined table,
for months of all the di£ferent lengths.





Mootlial
31 days.


MooikoT

30 days.


Fabrauy.


Mmo

BMMtkiL


, sioa


9-994840


9-996189


9-996719


9^6027


, tin 2a


9-979214


9-980647


9-982777


9-979971


^^


9-962661


9-966722


9-980862


9-964396


•-^^


9914288


9-919946


9-929887


9-917602


^ 6.


9-882728


9-872016


9-887409


9-868006


, (inAa


9-796782


9-810048


9-888688


9-808880



The valaes of the arc P^ which measures the time from the
beginning of the year to the middle of a month, are for the cal-
endar months

16* 16' 39" 103* 43' 64" 193* 66' 6" 284*» 35' 68"
44 28 26 133 47 38 224 28 26 814 39 37

IB 40 10 163 61 22 264 32 9 344 43 21

and for mean months they are 15°, 45°, 75°, &c

Now in the expression for the monthly mean m, let the con-
stants «,«,,«,,«,,«,, &c., take those values which have been
found for them in the New Haven equation, and the folloMring
monthly means for the calendar months may be obtained.
26*630 46-870 71*668 61*093

28*111 67-278 70*313 40*324

36066 66*949 62*612 30*424

The errors of these computed values as compared with the
monthly means actually observed are

•0 +-030 +-008 - 007

+*001 —•002 — *007 4.-004

-*034 -'Oil +012 +00*

This example shows the degree of accuracy with which an
equation obtained by the method of mean months may be ex-
pected to represent any observed series of means for calendar



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E. L. DeForesi on Reducing Meteorological Ohservationt. 821

months. The reason why the computed values do not agree
exactly with the observed ones is, that the curve of the fovn

by means of which my system of twelve equations was obtained,
is only an approximation to the true curve for any month, ana
is not the same as the computed curve whose equation contains
twelve constants. The two curves approach each other closely,
and intersect at several points, but they do not coincide. They
both include the same monthly mean for the mean month, but
not for the calendar month. It is probable, however, that no
other method of reduction of equal simplicity will give an equa-
tion which expresses the means for the calendar months so accu-
rately as this.

It should be noticed that when monthly means of rain-fall are
to be corrected for the inequality of the months by my system
of equations, the correction must be applied not to the mean
total amount of rain for any month, but to the mean daily
amount for that month. Take, for instance, the results of 24
years' observation at Albany, fipom 1826 to 1849 inclusive, given
by F. B. Hough in the '* New York Meteorologv." The mean
total amounts of rain and melted snow and hail K>r the calendar
months, in inches of depth, are



2-91


2-88


4-09


8-76


2-62


4-04


8'44


3-80


3-02


4-50


3-47


2-98



Dividing each of these by the number of days in the month, we
have the following values of the mean daily rain-feJl, for calen-
dar months, in decimals of an inch :

•0939 -0960 •1319 '1213

•0928 -ISOS •lllO -1100

•0974 1600 ^1157 -0961

Now applying the correction, we obtain the mean daily rain-fell

for mean months^

•0939 -0968 -1312 -1213

•0929 -1314 -1108 -1098

•0976 •ISOO •1168 '0960

and the equation of the curve is found to be

y='1123+'0202 8in(a?-106'' 43')+-0106 8in2(«-107° 10')
+•0112 sin 3(«- 14^ 11')+'0031 8in4(ir-61* 7')
- - 0024 8iD6(a;-66*' 40')Hh*0016 8in6x.

If we assign to x the value appropriate for any given day in
the year, the resulting value of y will be the average depth of
rain-fsJl at Albany for that day, expressed in decimals of an
inch.

After an equation has been obtained, there ought to be some



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322 Rese4ttrche$ on Solar Physics.

check to show whether it is free from enors of computatioo.
Thiavnaj be seeared by deriying fit>iii it the mean for any one
of the mean months. When the constants in the Albany equa-
tion are transferred to the general expression for the monthly
mean, and « and P take the values appropriate for the thim
mean month for instance, the result is m='0975; the agreement
of this with the daily mean for the third mean month as previ-
ously found, is evidence that the equation of the curve has been
computed correctly.

Jammrj Xlfh, 1S67.



Abt. XXXVL — Besearches on Solar Physics; by Wabben Dk
La Rub, Esq., Pres. R A.S., Balfoub Stbwabt, Esq., Super
intendent of^ the Kew Observatory, and Bbnjamin Lobwt,
Esq., Observer and Computer to the Kew Observatory.

Second Seriet {in continuation of First Series),* Areorfneasurement of
the Sun-spots observed by Carrington during the seven yfors from
1854-1860 inclusive^ and deduction iherefrom.

84. In our first paper (Art. IS) we stated that Mr. Carrin^n
had very kindlv placed at our disposal all his original drawings
of sun-spots. Our first step was to arrive at some estimate of the
accuracy of these sketches, and we requested Dr. von Bose, who
assisted Mr. Carrington in the greater part of his observations,
to give us a short outline of the methcd employed in obtaining
them.

From his account, it would appear that the sun's disk was
thrown upon a screen, and that each group as represented on the
screen was separately drawn on a sheet of paper. The groups on
piBiper were tnen each separately compared with those on the
screen and modified where faulty ; and this process was contin-
ued until the paper sketches agreed as nearly as possible with
the groups on the screen. It would thus appear that very great
care was taken with these sketches. [Engravings of several of
Carrington's sketches alongside of those of corresponding groups
as taken hj the Kew Heliograph are given in tne original me-
moir, showing that Carrington has obtained by the method above
described a very great accuracy of delineation.]

35. The trustworthiness of Carrington's sun-pictures being
thus established, it seemed to us that the labor of measuring
for each group the amount of spotted area would be well be-
stowed, inasmuch as the method nitherto employed, namely, the
mere statement of the number of sun-spots occurring at any pe-

* From A memoir printed for private ctreqlation ; tablet and plates, and manj
IMngraphs omitted. For First Series, see p. 1*79.



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Researcheg on Solar Phyrics. 823

riod, can only be supposed to afford very approximate means of
estimating toe extent of solar activity at that period; while,
again, if we wish to study the behavior with respect to size of
each group as it passes over the visible disk, this can only
be done accurately oy the laborious but sure method of meas-
urement.

36. Method adopted in measuring Carrington^s groups. — In order
to accomplish this task, the following method was adopted : — ^In
the first place, in order to obtain the apparent area of any group,
a piece of plate glass had a number of lines etched upon it, bv
means of which it was cut up into sauares, the side of each
square being ^V^b of an inch. In order to facilitate reading,
each fifth line was painted red.

This piece of glass was then applied (the engraved face toward
the drawing) to the group whose apparent area it was desired to
measure, and the number of squares and fractional parts of a
square occupied by the umbra, the penumbra, and the whole
spot was separately reckoned and noted down. If it was found
that the pumber of squares reckoned for the whole spot was
equal to the sum of those reckoned for the umbra and penum-
bra together, it was concluded that the measurement was correct.

This method of checking the accuracy of the measurement
had the further advantage of giving separately the areas of the
umbra and penumbra, thus affording determinations which may
be made use of in advancing our knowledge of the subject,
although not used by us in our present research.

87. But it is evident that after the apparent area of a group
has thus been correctly estimated, this apparent area will not
indicate the real sisse of the eroup, unless allowance is made for
the foreshortening occasioned by its angular distance from the
visual center of the disk.

JThe practical methods by which this allowance for foreshort-
emng was made are given in detail. The final results of the
measurements form an extensive table and give the material for
a graphical representation of the observed spotted area for each
clear day from the beginning of 1854 to the end of I860.]

40. Distribution of Spotted Area over Disk, — Our next inquiry
has reference to the relative distribution of spotted area over
different parts of the solar disk. We use the word disk in con-
tradistinction to surface, because it is evident that, on account of
the sun's rotation, the center of his visible disk on one day does
not represent the same portion of the solar surface as on another
day; indeed from this cause it is well known that sun-spots
travel over the visible disk from left to right. It is therefore
one inquiry to study from day to day the relative distribution
of spotted area over different parts of the sun's actual surface,
and another to study the same from day to day over different



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824 Researches on Solar Physics.

parts of his apparent disk. We have not hitherto attempted the
former inqaiiy (although the subject is not lost sight of, bat may
come within the range of our future researches), but have con-
fined ourselves entirely to the latter, and now proceed to des-
cribe the method of observation adopted.

41. Suppose the visible disk of tne sun to be cut up into se&
tions by great circles passing through these poles. These great
circles may be regarded as lines of longitude, only, in the present
instance, they are not supposed to move round with the sun's
sur&ce, but rather to be connected with the earth in such a man-
ner that the plane which passes through the earth is always
reckoned the zero or meridian.

Now it is well known that the pole of the sun differs very
little from that of the ecliptic, and therefore, in an approximate
investigation like the present, we may suppose the two to coin-
cide; these longitudes will thus denote ecliptical longitudes, and
the longitude in which the earth is placed being called zero, we
may with propriety reckon those to the left negative, and those
to the right positive. A sun-spot as it moves across thp disk on
account of rotation will thus appear at a longitude —90^, and
vanish at a longitude +90^

The same course will be pursued bv the inferior planets Mer-
cury and Venus, which move faster tnan the Earth ; while, on
the other hand, the superior planets, which move slower than the
Earth, may be supposed to pursue an opposite course, passing
across the circles of longitude from right to left

42. It will thus be apparent that, if the behavior of sun-spots
is at ail influenced by the positions of the planets, the fact is
likely to be discovered by this means. Thus if all the prominent

Elanets be in the same longitude as the Earth, if there be a bond
etween sun-spots and planets, we should be entitled to expect
in such a case some change in appearance or size when the sjpots
for that period pass the central line ; if, on the other hand, tnese
planets be together at 20° to the right of the Earth, we might
expect some change at 20° to the right, and so on. In fine, one
of our objects in the present research is to ascertain the compar-
ative size, at the different ecliptical longitudes in the visible cusk,
of the whole spotted area for any period, the mass of observations
being broken up for this purpose into periods embracing perhaps
three or four months, so as to comprehend in each a sufficient
number of groups.

48. For ULis purpose the following plan was adopted. A sub-
sidiary table was formed in which the whole visible disk was
portioned out into thirteen parts, each part denoting a day's
progress of a spot and embracing every 14° of longitude firom
—90° to +90°. Each of these parts had in this table two col-
umns allotted to it, in one of which the exact longitude of the



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Researches on Solar Physics. 325

spot (with reference to the earth or central point) was noted,
while in the other the area at this longitude of the whole spot,
including umbra and penumbra, was given. ITiis longitude of a
spot was determined in the following manner. In each of Car-
' nngton's large pictures the position of the sun's axis is given.
A circular sheet of transparent tracing-calico, of the size of Oar-
rington^s sun, had drawn on it lines of longitude for every 10®
from —90° to 4-90°. This sheet being applied in a proper man-
ner to each of Carrington's pictures, the longitude of a spot was
thus at once read off to the nearest degree.

The subsidiary table having been thus formed, it was then
carefully examined, and all those groups were rejected for which
(either on account of their exceedingly small size and conse-
quently doubtful area, or from paucity of observations) a reason-
ablv good line representing their behavior in passing over the
disK could not be obtained. Each non-rejected group was then
dealt with in the following manner. A curve was drawn, in
which the abscissae represented the longitudes of the visible disk
from —90° to 4-90°, while the ordinates denoted the correspond-
ing area of the group in millionths of the whole hemispherical
surface at each of these longitudes. This curve was formed
simply by connecting together by means of straight lines the
summits of the consecutive ordinates denoting observed areas.
From these curves a table was then formed denoting the proba-
ble area, of each non-rejected group from longitude —62° to lon-
gitude -f-64°, it being thought inadvisable to go nearer the sun's
border on either side. Finally, the groups of this table were
arranged into consecutive series, each series embracing two or
three months, it being supposed that during the course of any
one series the planetary configurations retained to a considerable
extent the same character. In the following table the results of
this subdivision are exhibited.

[The last two columns have been added from a plate in which
the positions of Venus and Jupiter are exhibited m connection
with a graphical representation of the series of numbers in the
table.]

45. Now, in (he first place, it is evident that during the time
embraced in a series the amount of spotted area which crosses
one ecliptical longitude is different from that which crosses
another, — that is to say, the average size of a spot varies with
the ecliptical longitude. This will be seen from a very cursory
glance ; thus in series ix, x, the average size of a spot attains
a maximum at about the longitude of the earth, while in series
XI this maximum is much to the right. Since most of these
series embrace a considerable number of spots, this behavior
may, we think, be considered to be an observational fact.

iM, JouB. Sci.— SxcoiTD S1ERIZ8, VoL. XLIII, Ko. 139.— Mat, 1867.
49



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326



Researches on Solar Physics.



2}Me exkibUing the area of the non-refected groupt for the different ediptieal Umgi-
tttde9 of the vuible disk {lotiffUude of Forth =0°).







Average lize of a groap in milllontht of the hemi-














Jnpim.


Naof
Mriet.


Averaee date
of Uriel.




Lmwitode


1 1 .' > I 1






.-62 - 48*,-34''


-20-, -6*,+8-+22'

1 ' '


+36»+60'+64''

1




I.


1864, Mar. 6; 81 90 113


156 1229 27i;311


330 368 366


+ 10


4120


II.


May 20 105 100 | 96


80


70 77i 78


72


67; 39


-f- 60


+ 50


III.


Not. 18|226 261,299


299


298j289 243


239


264 246


4-170


-110


IV.


1865, Aug. 26


28 1 24


21


22


d2| 43 64


73


84,161


— 20


- 10


y.il866,Aug.24


73 i 78


76


79


8:^, 67 84


60


^1! 83


-150


+ <o


VI. 1867. Mar. 16' 88 | 94 '117


126


167:178 206


211


187 168; - 80


-160


VII.


Aug. 16 111 1102:107


106


I20!l28il08


120


180 185


+ 70


+ 80


VIII.


Nov. 7 94;i04il24


182


121


110:108


97


881 81


+140


- 10


IX;i868,Feb. 1:361378 894


380


392


40 1 '406


374


326,261


-f-160


- 80


X.' Apr. 261141 .192 240


296


348


367'865


361


304:211



Online LibraryJohn AlmonThe American journal of science and arts → online text (page 38 of 102)