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OF THE

NATIONAL ACADEMY OF SCIENCES

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NATIOlSrAL ACADEMY OF SCIENCES.
Volume XIV.

FIRST INIKMOIR.

REPORT ON RESEARCHES ON THE CHEMICAL AND MINERAL-

OGICAL COMPOSITION OF METEORITES, WITH ESPECIAL

REFERENCE TO THEIR MINOR CONSTITUENTS.

BY

GEORGE PERKINS MERRILL,

HEAD CURATOR OP GEOLOGT, UNITED STATES NATIONAL MUSEUM.

OOITTENTS.

Page.

I. Introduction and scope of investigation? 7

II. Elements doulitfully reported or of unusual occurrence 8

III. Detailed chemical and mineralogical determinations 9

1. Canon Diablo 9

2. Casas Grandes 10

3. Mount Joy 11

4. Perry ville 11

5. Mount Vernon 12

6. Krasnojarsk 12

7. Bishop\alle 12

8. Collescipoli *. 13

9. CulUson 14

10. Elm Creek 15

11. Fisher 16

12. Holbrook 17

13. Indarch 17

14. Juvinas 19

.15. McKinney 19

16. Monroe 20

17. Ness County 21

18. Selma 21

19. Stannem 22

20. Ballinoo, Glorieta and Misshof 22

IV. Discussion of results 23

Gold and the platinoid elements 23

Phosphorus 23

Silicon . - 24

Sulphur 25

Oldhamite 25

Tin 26

Other elements reported 26

V. R^sum^ 26

VI. Table of analyses and disctasaion 27

5

I (

A REPORT ON RESEARCHES ON THE CHEMICAL AND MINERALOGICAL COMPOSI-
TION OF METEORITES, WITH ESPECIAL REFERENCE TO THEIR

MINOR CONSTITUENTS.

By GEORGE PERKINS MERRILL,

Head Curator of Geology , United States National Museum.

1. INTROD0CTION AND SCOPE OF INVESTIGATION.

In June, 1909, in view of the current speculations regarding earth history, the writer
published a paper on the composition of stony meteorites compared with that of terrestrial
igneous rocks." In the preparation of this paper he was impressed with the comparatively
small number of satisfactory chemical analyses available, but 99 being found which were con-
sidered sufficiently complete and accurate for his purpose. A second fact was the apparent
similarity in, and simplicity of, meteoric composition, there being shown scarcely any of those
elements which recent rock analyses have found to be common constituents, though in small
quantities, of terrestrial rocks. These facts, coupled with the occasional reported occurrences
of such elements as platinum, gold, lead, zinc, etc., and the high degree of perfection reached
by modern analytical chemistry, suggested to him the advisability of undertaking a systematic
investigation of the chemical nature of both stone and iron meteorites, with particular reference
to the occurrence of such elements as had been reported as doubtful or found only in traces.
On mentioning the matter to Prof. Morley, he was encouraged to make application for
financial assistance from the J. Lawrence Smith fimd of the National Academy of Sciences.
This was promptly granted, and a preliminary report of progress was published in 1913.*
An application for further assistance being granted, the work has been continued down to
approximately the present date, the analytical work, as before, being placed in the hands of
Dr. J. E. Whitfield, of Booth, Garrett & Blau-, in Philadelphia.

As is well known, and was stated in the preliminary report, the nongaseous elements
characteristic of meteorites, the presence of which has been established by quantitative methods
beyond controversy, are silicon, aluminum, iron, chromium, manganese, nickel, cobalt, mag-
nesium, calcium, sodium, potassium, sulphur, phosphorus, and carbon. In addition there
have been reported, usually under such conditions as to need authentication or at least
corroboration, antimony, arsenic, copper, gold, lead, palladium, platinum, tin, titanium,
tungsten, uranium, vanadium, and zinc. The preliminary investigation above referred to was
made mainly for the purpose of fixing the presence or absence of these last named in amounts
sufficient for determination by a skillful analyst, though the possible occurrence of other
elemental constituents of terrestrial rocks was not ignored, and during the final researches
on feldspathic types great care was taken in searching for barium, strontium, and zirconium.
Wherever possible samples of the same meteorites in which an element had been doubtfully
reported were analyzed. In other cases meteorites were taken which had not before been
subjected to analysis, or the analyses of which were unsatisfactory for one reason or another.
Whenever possible, too, an amount of material was taken sufficient to warrant a representative
selection; as a rule 50 grams and upwards were thus utilized. In a few instances, particularly

o Amer. Journ. Sci., vol. 2;, June, 1909, pp. 469-174.

» On the Minor Constituents of Meteorites, Amer. Journ. Sci., vol. 35, May, 1913, pp. 509-525.

8 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [%'ol.xiv.

in the case of the older falls, the absurdly high price demanded by holders of the material
necessitated a lower limit, which, however, was rarely less than 10 grams. In all cases the
meteorite was made the subject of careful microscopic study, and the purpose ever held in
view of not merely ascertaining the presence or absence of any constituent but of relegating
the same, as well, to its proper source.

n. ELEMENTS DOUBTFULLY REPORTED OR OF UNUSUAL OCCURRENCE.

Before discussing the results it will be well to repeat what was given in the preliminary
paper regarding the previously reported occurrences of the unusual elements.

Arsenic. — The first determination of arsenic in meteorites of which I have record is that
of Karl Rural er who, in 1S40, reported <* getting distinct arsenical reactions from the olivine-
like mineral occurring in both the Atacama, Bolivia, and the KJrasnojarsk, Siberia, pallasites.
It is difficult to detect possible sources of error in Rumler's method as given. The fact, however,
that no one has since been able to corroborate his work would suggest some possible impurity
in his reagents. SiUiman and Hunt also reported traces of arsenic (and copper) in the iron
of Cambria, N. Y.* The only other reported occurrence of the element known to me is that
of Fischer and Duflos in the Bramiau iron.'' This determination can to-day scarcely be
considered satisfactory. The solution remaining after the precipitation of the copper was
evaporated, the dry residue mixed with soda and heated before the blowpipe; result, a garUc
odor. In stating the analysis, copper, manganese, arsenic, lime, magnesium, sihcon, carbon,
chromium, and sulphur are all thrown together as amounting to 2.072 per cent.

Antimony. — Traces of this metal were reported by Trottarelli iii the stone of CoUescipoli.
I have not seen the original paper, but an abstract by Max Bauer <* gives, among other con-
stituents, lead, antimony, tin, and lithia, as occiirring in traces, palladium to the amount of
0.7745 per cent, and soda (Na,0) to the unheard of amount of 10.386 per cent (!). I have there-
fore a natural feeling of skepticism regarding the results as a whole. (See new analyses, p. 14.)

Copper. — Copper in amounts from traces up to weighable quantities has been reported by
such authorities as Rammelsberg, Rose, J. L. Smith, and many others, and should be removed
from the doubtful list.

Gold. — Gold, so far as I am aware, was first plausibly suggested as a meteoric constituent
by A. Liversidge,* who thought to find it in an iron from Boogaldi, New South Wales. Not-
withstanding the fact that the work of Prof. Liversidge seems to have been performed with
proper care, there exists a lingering doubt in the minds of many as to the actual occurrence of
this element as an original constituent of the hon. It is to be noted, however, that more recent
investigations by J. C. H. Mingaye are confirmatory.^

Lead. — Ti'ottarelli, whose analysis is above referred to, reported traces of lead in the Col-
lescipoli stone. R. P. Greg also reported e native lead lining the cavities in an iron from the
Tarapaca desert of Chile. J. L. Smith, however, concluded from his own examination * that
the metal was altogether foreign to the stone when it fell.

Lithia. — Lithia was reported by Story Maskelyne ' to the amount of 0.016 per cent in the
enstatite and m traces in the augitic constituent of the Busti stone. J. L. Smith likewise re-
ported^ traces of lithia in the stones of Waconda, Kans., and BishopviUe, S. C. Others report
it determined by spectroscopic methods.

Platinum, palladium, and indium. — Platinum, palladium, and iridium come in for occa-
sional reference as meteoric constituents, but almost invariably in amounts too small to weigh,

o Pogg. Ann. Phys. Chem., vol. 49, 1840, p. 591.

6 Amer. Journ. Sci., vol. 2, 1846, p. 376.

c Pogg. Ann. Phys. Chem., vol. 72, 1847, p. 479.

d Neues Jahrb. tiir Min., etc., 1891, vol. 2, p. 238.

« Joum. Proc. Eoy. Soo. of. N. S. Wales, vol. 36, 1902.

/ Records Geol. Sun-. N. S. Wales, vol. 7, 1904, p. 306.

s London, Edinburgh & Dublin Philos. Mag., vol. 10, 1855, p. 12. .\Iso Amer. Journ. Sci., vol. 23, 1S57, p. 118.

» .\mer. Joum. Sci., vol. 49, 1870, p. 305.

• Philos. Trans. Eoy. Soc., vol. 160, 1870, pp. 206-7.

i Amer. Joum. Sci., vol. 13, 1877, p. 212, and vol. 38, 18<i9, p. 226.

No.L) RESEARCHES ON METEORITES— MERRILL. 9

and often in analyses made under such conditions as to give rise to a feeling of doubt as to their
correctness. Trottarclli's reported finding of palladium has already received attention. J. M.
Davison" obtained from 608.6 grams of the Coahuila iron 0.014 gram of platinum; from 464
grams of the Toluca iron a few crystals of potassium platinic chloride were obtained which
showed a reddish color and probably contained iridium. Tassin '' doubtfully reported the acid
soluble portion of the Persimmon Creek iron as containing traces of platinum too small to weigh.
Mallet's work on the Canon Diablo iron is confirmatory, however, of its occasional- occurrence
(see p. 10).

Tin. — Tin to the amount of 0.17 per cent SnO^ was reported <^ by Stromeyer and Wahn-
stedt as long ago as 1825 as occurring in the olivine of a pallasite. Unfortunately some doubt
exists as to whether this was the pallasite of Kxasnojarsk or Steinbach. Rammelsberg in 1884 <^
reported findmg 0.08 per cent Sn in the metallic portion of the Klein-Wenden aerohte, and he
also tabulates ^ 0.57 per cent Sn in the analysis of the Nashville ( ?) Tenn., iron. Jackson f
thought to have found 0.063 per cent Sn in an iron from Dakota, while C. A. Joy reported 9 0.44
per cent SnOj in the mineral portion of the Atacama pallasite, and Mallet reported '^ 0.002 per
cent to 0.003 per cent Sn in the iron from Staunton, Virginia. Still more recently traces of the
metal have been reported in the Barraba and Cowra, New South Wales, irons, by J. C. H. Min-
gaye.* Coming from such a source the statement might well be considered as conclusive (see
further p. IS). Numerous other occurrences of like small amounts are mentioned in the litera-
ture, the copper and tin being frequently undifferentiated. Although not so stated, the infer-
ence may be drawn, with the possible exception of that found in the olivine above noted, that
tin, if present at all, occurs mainh^ if not wholly in the metallic portion.

Titanium. — Rammelsberg J foimd 0.16 per cent titanic oxide (Ti02) in the insoluble residue
from the Juvinas stone. This is the first reported occurrence. Davison reported '' traces of
titanium in the stone from Estacado, Tex. Everhart ' found 0.09 per cent in that of Pickens
County, Ga.; and Stokes'" found 0.08 per cent in the stone of Allegan, Mich. These occur-
rences are all dwarfed by Tschermak's " determination of 2.39 per cent in the stone of Angra
dos Reis. The amounts are, however, mostly very small, and knowing the difficulties in the
way of determmation, it seems unquestionable that it is a common and \videspread constituent.
Vanadium. — Apjohn reported " finding immistakable evidences of vanadiimi in the
Limerick stone, but in amoimt too small for quantitative determination. The occasional
presence of the element seems now confirmed beyond doubt.

Zinc. — E. PfeifiFer, in the report p of his analysis of the ParnaUee stone, included traces
of copper, tin and zinc. J. L. Smith ' found in the schreibersite from the Tazewell iron a trace
of zinc. I find no other recorded occurrence of tliis element.

ra. DETAILED CHEMICAL AND MINERALOGICAL DETERMINATIONS.

In the following pages are given in considerable detail the results of my investigations,
the chemical analyses upon which most reliance is placed being mainly those of Dr. J. E.
Whitfield, as before stated, though in the final tabular statement I have brought together the
work of such other analysts as seems sufficiently detailed or given in such forms as to merit
attention. The results given in the first paper are here repeated. The oi'der of arrangement
is alphabetical, imder the three heads Irons, Stony-irons, and Stones.

(1) Iron. — Canon Diablo, Ariz. A coarse octahedral iron with nmnerous interlamina-
tions of schreibersite and -inclusions of graphite and troihte, the latter sometimes an inch or

o Amer. Joum. Sci., vol. 7, 1899, p. 4. J Pogg. ^Vnn. Phj-s. Chem., vol. 73, 1848, p. 585.

6 Proc. U. S. Nat. Mus., vol. 27, 1904, p. 959. * Amcr. Journ. Sci., vol. 22, 1906, p. 59.

c See Rose, Beschreibung u. Eintheilung der Meteoritcn, etc, 1864, p. 77. ' .Science, vol. 30, 1909, p. 772.

d Pogg. Asm. Phys. Chem., vol. 62, 1844, p. 449. ™ Troc. Wash. .\cad. Sci., vol. 2, July, 1900, p. 48.

'Die Chemischer Natm Meteoriten, 1870, p. 146. " Tsch. Min. Pot. Mittheil. , vol. 2S, 1909, p. 110.

/ .\mcr. Joum. Sci., vol. 36, 1863, p. 260. o Joum. Chem. Soo. London, vol. 27, 1874, p. 104.

B Amer. Joum. Sci., vol. 37, 1864, p. 245. p Silz. k. .\kad. Wiss. Wien, vol. 47, 1863, p. 401.

h .\mer. Joum. Sci., vol. 2, 1871, p. 13. t -\mer. Joum. Sci., vol. 19, 1855, p. 155.
* Records Geol. Surv. N. S. Wales, vol. 7, pt. 4, 1904, p. 305.

10

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

(Vol. xrv.

more in diameter. In the selection of samples for analyses these were avoided as far as possible.
Carbon is present in form of microscopic diamonds and also as graphite. It has been the
subject of nimierous analyses, yielding variable restdts owing to its coarse crystallization.
Whitfield's analysis is given in column I below. In column II is given the average of three
analyses by Moissan, Booth, Gari-ett & Blair, and Tassin.

Constituents .

Silicon

Sulphur

Phosphorus

Manganese

Copper

Nickel

Cobalt

Combined carbon .
Graph it ic carbon. .

Iron oxides

Iron protochloride
Iron

Total

Per cent.

Trace.

0.009

.261

None.

.015

7.335

.510

.105

.028

2.520

.097

S9. 167

100.047

Per cent.

0.032

.007

.159

None.

Trace.

5.828

.044

.465

In process of analysis schreibersite to the amount of 1.832 per cent separated out. This
afforded the following composition:

• Per cent.

Iron 55. 04

Nickel 29.58

Phosphorus 15. 38

Platinum was looked for in two portions of fifty grams each, but none was found. Neither is its
presence recorded in previous analyses by this same firm, by Derby," Tassin or Moissan.* J. W.
Mallet, working on a residue from the solution of 25 pounds of the iron in dilute hydrochloric
acid obtained residts representing 3.63 grams of platinum and 14.95 grams of iridiimi per metric
ton of the original iron, "with probably a trace of rhodium." He suggests, and this is in accord-
ance with our own results, that the platinoid metals are not uniformly distributed in the iron."
This may account for the failure to find it on the part of others. The quantity of material
worked upon is imdoubtedly an important factor, however.

(2) Iron. — Casas Grandes, Mexico. Medium octahedrite. Previously analyzed and de-
scribed by Tassin."* Tlie results of Dr. Whitfield's analyses are given in Column I below. In
Coliunn II are given Mr. Tassin's results as previously obtained.

Constituents.

I

II

Silicon

Per cent.

0.010

90.470

7.742

.604

.012

.166

.029

.145

.032

.794

Per cent.

95.13

4.38

.27

Trace.

.24

None.

\ Traces.

Nickel

Cobalt

Sulphur

Total

100.004

100.02

Whitfield's analyses were made on 50-gram samples, free from evident inclusions of troilite.
The following elements were especially looked for but not found; antimony, arsenic, tin, lead,
palladium, platinum, titanium, tungsten, vanadium, uranium, chromium, manganese, molyb-
denimi, and zinc.

a Amer. Joum. Sci., vol. 49, 1895, pp. 101-110.

b See contributions to the Study of the Canyon Diablo Meteorite, by G. P. Merrill and W. Tassin, Smithson
50, pt. 2, 1907, p. 209.

c Proc. Acad. Nat. Sci. Phila., Dec., 1905, p. 913. Footnote on p. 862.
(t Proc. U. S. Nat. Mus., vol. 25, 1902, pp. 69-74.

I Misc. Coll., Quar. Issue, vol.

N0.1.J RESEARCHES ON METEORITES— MERRILL. 11

(3) Iron. — ^Mount Joy,Pa. Coarse octahedrite; brecciated. Previous analysis by Eakins"
yielded :

Per cent.

Iron 93.80

Nickel 4.81

Cobalt 51

Copper 005

Phosphorus 190

Sulphur 01

99. 325
Further tests for the minor constituents by Dr. Whitfield on a 50-gram sample yielded :

Per cent.

Chromium 0. 006

Manganese 075

Copper 008

Chlorine 255

Platinum '. Trace.

No vanadium, molybdenum, tungsten, gold, silver, lead, or tin, nor in fact any other ele-
ment in amounts large enough to be determined by wet analysis, were found.

(4) Iron. — PerryvUle, Mo. Described by Merrill * as belonging to Brezina's group of finest
octahedrites (Off) . As the entire iron was in possession of the National Museum, and it had not
before been described, all the necessary material was sacrificed for a very detailed analysis, with
the results tabulated below:

Per cent.

Iron 89.015

Nickel 9. 660

Cobalt 545

Copper 025

Manganese None.

Phosphorus 365

Sulphur 002

Silicon 003

Carbon 015

Iridium

Palladium

Platinum

Ruthenium

Traces.

99.63

The amount of the rarer elements found in different samples of this iron was quite variable,
but always small. From one portion of 25 grams was obtained 0.004 gram of platinum and
from another portion of 100 grams weight but 0.0002 gram. The precipitates of ammonium
platinic chloride were in all cases faintly orange, indicating the presence of palladium but in
amounts too small for determination. In a 100-gram sample of the iron were found 0.014
gram of ruthenium and 0.028 gram of iridium, while another portion of equal weight yielded but
0.0009 gram of ruthenium and 0.0011 gram of iridium.

So far as I am aware this is the first recorded occurrence of ruthenium in a meteoric iron.
The probable presence of iridium in the Toluca and Coahuila irons was recognized by Davison,
as already noted, but is not elsewhere recorded. No chromium, vanadium, molybdenum,
or titanium were found.

The mineral schreibersite, constituting 2.61 per cent of the iron, was isolated and analyzed
with the following results:

Per cent.

Phosphorus 14. 00

Iron 51.10

Nickel 34. 13

Cobalt 30

99.53

" Amer. Joum. Scl., vol. 44, 1892, p. 416. 6 Proc. U. S. Nat. Mus., vol. 43, No. 1M3, 1913.

12 MEMOIRS NATIONAL ACADEMY OF SCIENCES. tvoL.xiv.

The high percentage of nickel shown by this analysis is comparable with that of the schrei-
bersite from the Magura iron. Even greater amounts have been reported as yielded by the
irons of Seelasgen, Germany (36.17 per cent), and Cranbourne, AustraUa (38.24 per cent).

(5) Stony-iron (Pallasite). — Mount Vernon, Ky. A coarse paUasito, consisting of large
blebs of olivine in a mesh of metal. Described by Tassin."^ No complete (bulk) analyses
made owing to the coarse nature of the stone. Tlio nickel aUoy yielded Tassin as follows:

Per cent.

Iron 82.520

Nickel 14.044

Cobalt 949

Copper 104

Sulphur 288

SiUca 808

Aluminum 410

Carbon 465

Phosphorus 390

Chlorine Trace.

99. 978

He also gave analyses of the included tsenite, schreibersite, troihte, chromite, and olivine
separately, but found no constituents of miusual occm'rence.

A 50-gram sample, badly oxidized, submitted to Dr. Whitfield for tests for minor elements,
yielded :

Per cent.

Chromium 0. 300

Copper 016

Nickel 2.960

Cobalt ■ 090

Manganese 151

Vanadium Trace.

No trace of molybdenum, tmigstcn, antimony, tin, lead, zinc, gold, silver, or platinum
was found.

(6) Stony-iron {Pallasite). — Krasnojarsk, Siberia. A well-known historic meteorite, stated
by previous workers to contain arsenic and tin. The material being too coarse for bulk or mass
analysis, the metal and olivine were examined independently. The metallic portion yielded :

Per cent.

Fe 89.90

Ni 9.52

Co 60

P 085

100. 105
The sihcato portion (ohvine) yielded:

Per cent.

SiO. 37. 22

FeO 15.21

AlA 46

MgO 47.07

99.96

with no traces of arsenic, chromium, manganese, sulphur, or tin, in either portion.

(7) Meteoric stone, CMadnite. — BishopviUe, S. C. This unique stone fell on March 25,
1843, and was first described by Shepard in 1846.'' It has since been the subject of numer-
ous writings (see Wiilfing, pp. 29-31), and in several instances subjected to partial anal-
yses. The widely variant results obtained are due as much to imperfect sampling as to
incoiTect determinations. The Museum collection possessing several grams of fragmental
material, it was deemed advisable to sacrifice them to the searching methods of modern
analytical chemistry. The stone was described by Shepard as consisting in large part of a

, — — ■ — #

a Proc. U. S. Nat. Mus., vol. 28, 1905, p. 213. b Amer. Journ. Sci., vol. 2, 1846, p, 379.

No. 1.1 RESEARCHES ON METEORITES— MERRILL. 13

light gray material, rcgardod by liim as a pcrsilicate of magnesia, and to which he gave the
name chladnite in, honor of the ciiemist Cliladni. Researches in 1864 by J. Lawrence Smith
showed the mineral to be identical with enstatite. The stone was described in 1851 by W.
Sartorious von Waltershaiiscn, who thought to show that the siliceous portion was made up
of 95.011 per cent chladnite and 4.985 per cent labradorite. Rammelsberg in 1863 stated as
a result of his examinations that it contained no feldspar. Later microscopic investigations
by Wadsworth " and Tschermak '' show the stone to be a crystalline granular mass of enstatite.
plagioclase feldspar, and an iron sulphide identified as pyrrhotite. Wadsworth includes also
augite and metallic iron. The probable presence of oldhamite, confirmed by the present
researches, was suggested but not proven by Maskelyne." Below are given the results of Dr.
Wliitfield's analyses, made with especial reference to the presence or absence of the mineral
oldlaamite, and the elements barium, strontium, and zirconium. The commonly quoted analy-
ses of J. Lawrence Smith, it should be noted, were not of the stone in mass, but of the selected
wliite pyro.xenic constituent, tlie chladnite of Shepard.

Per cent.

Silica (SiOj) 57. 034

Alumina (AlA) 1-706

Ferric oxide (FeA) 1-406

Manganous oxide (MnO) . 189

Lime (CaO) 2.016

Magnesia (MgO) 33. 606

Cobalt oxide (CoO) Trace.

Nickel oxide (NiO) 538

Soda(Na,0) 1.027

Potash (K2O) 089

Ignition (HoO) 1.995

Iron(Fe) 181

Nickel (Ni) ■ .' 039

Sulphur (S) 297

100. 023
Minus O for S .147

99. 876

An amount of calciimi equivalent to 0.67 per cent calcium sulphide was hberated by
boiling the finely pulverized stone for two hours in distilled water. Inspection of the stone
in mass shows, in addition, occasional granules of an iron sulphide (troilite or pyrrhotite) which
were not included in the portion analyzed. No traces of barium, strontium, or zirconium could
be detected. This is worthy of note in view of the feldspathic nature of the stone. The amount
of material utihzed in the analyses was not as large as could have been desired.

(8) Meteoric stone, Chondrite (Cc). — Collescipoli, Italy. This stone, which fell on February
3, 1890, is of a gray color, somewhat friable, of a common chondritio type (Cc), composed
essentially of ohvine and enstatite with the usual sprinklmg of metal and metallic sulphide. It
presents no imusual features macro- or microscopically, and attention was given it here only on
account of the extraordinary array of the rarer elements reported in TrottareUi's analyses.'' It is
unfortunate that although this fall is supposed to have comprised some 4 or 5 kilograms, but
1,802 grams are known to-day and hence the prices quoted by dealers ($0.70 to .S1.27 per gram)
are so high as to place it out of reach for exhaustive investigation. Fortunately a few grams
were found which, on account of their minutely fragmental condition, could be purchased at
prices justifying sacrifice. Below are given the results obtained by Wliitfield:

Percent.

Metallic portion 18. 60

Silicate portion '. 81. 40

100. 00

o Lithologicsl Studies, p. 200, 1(IS4. c Proc. Royal Soc. Edinburgh, vol. 18, 1870, p. 146.

b Sitz. k. Akad. Wiss. Wien, vol. 88, pt. 1, 1883, p. 363. <l Gaietta Chiinica Italiana, vol. 20, 1890, pp. 611-615.

14

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

The metallic portion yielded :

Percent.

Iron(Fe) 91.60

Nickel (Ni) 8. 00

Cobalt (Co) , 49

100. 09
The silicate portion yielded :

Percent.

SiUca (SiOj) 42. 50

Alumina (AI2O3) 7. 90

Iron protoxide (FeO) 19. 50

Lime(CaO) 2.20

Magnesia (MgO) 26. 00

Potash (ICjO) 32

Soda (Na20) 1 . 80

100. 22

Careful search was made for other elements, and in particidar, for the reasons noted, for
arsenic, antimony, copper, and platinum. None were found.

In order to compare these results with those of TrottareUi, the analyses were recalculated
and the results given in colunon I below; in colimin II are given those of Trottarelli. I will
not attempt to account for the discrepancy.

Constituents.

I

n

Silica (SiOs)

Per cent.

34.59

6.43

15.87

21.17

1.79

.26

1.46

17.04

1.49

.09

None.

None.

None.

None.

None,

Percent.
31.0.57
.9304

.0186"
. 1169
Traces,
10.386
40. 983
1..544
Traces.
.7745
1.0060
. 5616
7.679
2.100

Alumina (AI1O3)

Ferrous oxide (FeO)

Lime ( CaO )

FoUsh C K"0 )

Soda(Na26)

Iron(Fe)

Nickel(Ni)

Cobalt (Co)

Palladium (Pd>

Sulphur (S) . ...

Total

100.19

1 97. 157

,. <■ With traces of lead, antimony, tin, lithia, sulphuric anhydride, and chlorine.

(9) Meteoric stone, Chondrite (Cc). — Cullison, Kans. Described by Merrill.'^ Thin sections
showed it to be of the normal chondritic type containing olivine, enstatite, monoclinic pyroxene,
plagioclase feldspar, with the usual sprinkling of metal and metallic sulphides. The separation
of the component parts by an electromagnet and treatment with iodine resulted as foUows:

Troilite

Metal

Silicate

Schreibersite.

The metallic portion yielded:

Per cent.

6. 07

19. 40

74. 50

10

100,07

Per cent,

SiUcon 0,129

Sulphur Trace.

Phosphorus 071

9. 207

507

040

160

088

Nickel.

Cobalt

Copper

Chromium .
Carbon

■> Proc. IT. S. Nat. Mus., vol. 44, 1913, pp. 325-330.

No.1.) RESEARCHES ON METEORITES— MERRILL. 15

Per cent.

Manganese ! 080

Iron 89. 700

99. 982
with no traces of tungsten, vanadium, or molybdenum.

The silicate portion yielded :

Percent.

SiKcaCSiO,) 47.36

Alumina (ALOa) 5. 67

Ferric oxide (FcjOj) .10

Ferrous oxide (FeO) 11. 25

Lime (CaO ) 84

Magnesia (MgO) 31.72

Manganous oxide (MnO) .36

Soda (Na^O) 2. 42

Potash (K2O) 23

Titanic oxide (TiOo) None.

99. 95
Combining the metallic and nonmetaUic portions and recalculating with the usual assump-
tion that the mineral called troihte is the monosulphide FeS, and that the schreibersite conforms
to the formula FcjNiP, the following figures are obtained representative of the composition of
the stone as a whole :

Per cent.

Silica (SiOo) 35. 30

Alumina ( AUOj) 4.. 24

Ferric oxide (Fe203) 75

Ferrous oxide (FeO) 8. 38

Lime (CaO) 62

Magnesia (MgO) 23. 631

Manganous oxide (MnO) 268

Soda (NajO) 1. 804

Potash (KoO) 171

Sulphur (S) 2.184

Phosphorus (P) 0138

Nickel (Ni) 1. 80

Cobalt (Co) 098

Copper (Cu) 008

Chromium (Cr) 029

Carbon (C) 017

Manganese (Mn) 015

Irou (Fe) 21. 270

100. 5988

None of the rarer elements, other than those noted, were found.

(10) Meteoric stone, Chondrite. — Elm Creek, Kans. This stone was plowed up in a field
some time in May, 1906. Nothing is known of its fall. It was considerably oxidized on the
outside, indicating that it had lain some time in the soQ. It is described by Howard" as of a
dark gray, nearly black color, thickly studded on a polished surface with metaUic points and in-
distinct chondrules, which break, in part, with the groimdmass. The silicate portion, as shown
by the microscope, consists essentially of olivine and enstatite with a polysynthetically twinned
monochnic pyroxene. It had never before been subjected to chemical analysis, and was there-
fore open to critical investigation. Dr. Whitfield found:

Per cent.

Silicates 93. 18

Metal 6.82

100. 00
0 Amer. Journ. Sci., vol. 28, 1907, p. 380.

16 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv.

The silicate portion yielded:

Per cent.

Silica (SiOj) 36. 76

Alumina (AI2O3) 3. 10

Ferric oxide (FejOs) 13.23

Ferrous oxide (FeO) 14. 22

Chromic oxide {Ct^O^) 35

Lime(CaO) 1.62

Magnesia (MgO) 2.5.66

Water (HjO) 5.10

100. 04

No barium, strontium, zirconium, or other rare ek^ments coukl be detected.
The metallic portion yielded:

Per cent.

Iron(Fe) 87.13

Nickel (Ni) 11. 30

Cobalt (Co) : 1. 42

Manganese (Mn) 15

Copper (Cu) None.

100.00

The amount of metal available (1.35 grams) was not sufficient for an exhaustive examina-
tion for the rarer elements.

(11) Meteoric stone, Chondrite {Ci) . — Fisher, Polk County, Minn. This, the first and only me-
teoric stone thus far reported from Minnesota, is supposed to be the representative of a fall which
took place on the 9th of April, 1894. It was made the subject of an investigation by Prof. N. H.
Winchell," which was, however, not completed. The matter was subsequently taken up by the
present writer, and a detailed account of it pubhshed in the Proceedings of the U. S. National
Museum.* The stone is described as consisting of a confused aggregate of irregular crystaUine
granules of ohvine and pyroxene interspersed -with numerous imperfectly outlined chondrules
of the same mineral, throughout wliich are occasional interstitial areas of a colorless, pellucid,
isotropic material referred to maskelynite. The pyroxenes are wholly of the enstatite type and
devoid of twinned structure. The analyses yielded results as below:

Par cent.

Metallic constituents 11. 44

Silicate constituents 88. 56

100. 00
The silicate portion yielded :

Pep cent.

Silica (SiOo) 43. 70

Alumina (AloOj) 4.96

Ferrous oxide (FeO) 18.27

Manganous oxide (MnO) .38

Nickel oxide (NiO) 23

Lime(CaO) 2.19

Magnesia (MgO) - 29.38

Chromite (FeOCrA) ^^

99. 91
The metallic portion, freed from the last trace of siliceous matter, yielded:

T'er cent.

Iron(Fe) 85.00

Nickel (Ni) 14.15

Cobalt (Co) 74

Copper (Cu) Trace.

99.89

a American Geologist, vol. 14, 1S94, p. 389; vol. 17, 1898, p. 173; vol. 20, 1897, p. 316.
f> Vol. 4S, 1915, pp. 503-506.

No. 1.1 RESEARCHES ON METEORITES— MERRILL. 17

A recalculation of these results gives the bulk or mass composition of the stone as follows:

Per cent.

Silica (SiOj) 38. 70

Alumina (Al.Oj) 4.39

Ferrous ojdde (FeO) IG. 406

Manganous oxide (MnO) 336

Nickel oxide (NiO) 204

Lime(CaO) 1.939

Magnesia (MgO) 26.018

Chromic oxide (CrjOa) 482

Metallic iron (Fe) 9. 724

Metallic nickel (Ni) 1. 608

Metallic cobalt (Co) 084

99. 891

No traces could be discovered of barium, strontium, zirconium, or potassium.

(12) Meteoric stone, Ohondrite. Holbrook, Ariz. Described bj' Merrill." A gray ohon-
dritic stone, very fresh, having fallen on July 19, 1912. Contains veiy little metallic iron, but

is correspondingly rich in sulphide. Analyses yielded:

Per cent.

Schreibersite 0. 11

Troilite 7. 56

Metal 4. 85

Silicates 87. 48

100.00
The metallic portion yielded :

Per cent.

Nickel 8.68

Cobalt 64

Copper .29

Iron 90. 50

100.11
The silicate portion vi elded:

Per cent.

Silica (SiOj) 41. 93

Alumina ( ALOj) 4. 30

Ferrous oxide (FeO) 21. 85

Lime (CaO) 2.40

Soda(Na20) Trace.

M^nesia (MgO ) 29. 11

Manganous oxide (Mn.0) .25

Nickel oxide (NiO) 08

99. 92
Specific gravity at 22.6° C, 3.48.

None of the rarer elements under consideration were found, even in traces. The sulphide

occurs in such forms as to be readily separated mechanically, and yielded on analysis:

Per cent.

Iron 63. 62

Sulphur 36. 50

Nickel, cobalt, and copper None.

100.12

This shows the mineral to be troilite, though its specific gravity (4.61) is low. It is, however,
wholly imattracted by the magnet, and apparently there is no question as to its true nature.
Its occurrence in this fomi is interesting in a stone so low in the metallic constituent.

(13) Meteoric stone, Carbonaceous Chondrite (Cc). — Indarch, Elizabethpol, Russia. This
interesting stone fell, according to Meunier,'' on the 9th of April, 1891. Although made the sub-
ject of numerous brief papers, it seems never to have been previously analyzed, and was, there-

a Smithsonian Muse. CoU. vol. 60, 1912, No. 2149. 6 Comptes Rendus, .Vcad. Scl., Paris, vol. 125, 1897, p. 894.

21403°— 16 2

18 MEMOIRS NATIONAL ACADEMY OF SCIENCES. (vol.xiv.

fore, taken up in connection with the present work and a detailed description of it has been
given in the Proceedings of the National Museum." The stone is there described as of a dark
greenish gray color, firm and compact, admitting of a polish and thickly studded with small,
dark, almost black chondrules and nodular masses of metal and troUite. A microscopic exam-
ination shows it to consist of a dense black irresolvable ground, throughout which are scattered
iron and iron sulphide, together with abundant sharp splinters of pyroxene and numerous more
or less fragmentary chondrules of the same mineral. No olivine, feldspar, or other sihcate min-
eral was determined. The presence of carbonic acid, as shown by the analysis, suggested the
mineral breunnerite, but this could not be determined absolutely owing to the obscuring effect
of the abundant graphite. The sulphide of calcium, oldhamite, was detected both by chemical
means and by the microscope. The analyses yielded results as follows:
Metallic portion separated by mercuric chloride solution:

X 6r cent.

Iron(Fe) 90.44

Nickel (Ni) 8.26

Cobalt (Co) 18

Phosphorus (P) 08

Manganese (Mn) 1-04

100.00

Silicate portion, free as possible from the metal, sulphides, and graphite, yielded:

Per cent.

Silica (SiOj) 47. 970

Alumina (AlA) 2.647

Ferrous oxide (FeO) 19.283

Phosphoric acid (PA) 699

Manganous oxide (MnO) 175

Nickel oxide (NiO) 739

Cobalt oxide (CoO) 067

Lime(CaO) 1.559

Magnesia (MgO) 22.736

Carbonic acid (CO,) 363

Soda(Na20) Trace.

Potash (K2O) None.

Water (H2O) 3.762

100.00
A recalculation of these analyses gives the following, showing the composition of the stone
as a whole:

Per cent.
Silica (SiOj) 35.699

Alumina (AlA) 1- 9C9

Ferrous oxide (FeO) 2.5.790

Manganoiis oxide (MnO) 130

Nickel oxide (NiO) 549

Cobalt oxide (CoO) 049

Lime(CaO) 1.160

Magnesia (MgO) 16. 920

Carbonic acid (CO2) 271

Phosphoric acid (PA) - 520

Water (HA 2.799

Iron(Fe) 10.400

Nickel (Ni) 949

Cobalt (Co) 020

Phosphorus (P) 092

Manganese (Mn) 119

Carbon (graphite) (C) 310

Sulphur (S) 5.100

102. 846
Minus O for S 2. 54

„ , . . . . , , , , , 100. 306

No barium, strontmm, or zirconium could be detected.

o Proc. U. S. Nat. Mus., vol. 49, 1915, p. 109.

No. 1.)

RESEARCHES ON METEORITES— MERRILL.

19

The mineral composition so fax as determined by analysis and microscopic examination is:

Per cent.

Silicate (enstatite) 74.42

. Metal 11.50

Troilite 13. 296

Oldhamite 596

Graphite 31

100. 122
Specific gravity, 3.42.

(14) Meteoric stone, Eukrite (Eu). — Jnvinas, France. This stone has been widely circulated
and is represented iii aU the collections of importance throughout the world. As a result it
has been made the subject of numerous memoirs and briefer notices, Wiilfing recording forty-
seven titles in his catalogue. I find, however, no recorded analysis later than that quoted
below, which dates back to 184S. In view of this, and the additional fact that it is a feld-
spathic stone, it seemed worth the while to give it consideration here with especial reference
to the possible occurrence of barium, strontium and zirconium. Wadsworth, in his review of
the mineralogical- determiuations made by Rammelsberg, Tschermak, and others, states that
the stone consists of anorthite and augite with small amounts of pyrrhotite and nickel-iron.
Rammelsberg noted chromite and ilmenite, whQe Fouque and Levy detected also enstatite.

In Column I below are given the results obtained by Whitfield and in II those of Rammels-
berg.

Constituents.

I

II

Silica (SiO.)

Titanic oxide (TiOs)

Per cent.
47.99

.57
13.50

. 22
Traces.

.11
Trace.
18.63
10.60
7.20
None.
None.
None.
None.

.55
Trace.
None.

.054

.02

Per cent.

49.23

.10

12.65

l.?l

.16

Ferric o.xirle (FcjOs)

Iron (Fe)

Nickel oxide (NiO)

Cobalt oxide (CoO)

Ferrous oxide (FeO)

Lime(CaO)

20.33
10.23
6.44

Magnesia ( MgO)

Barium oxide (BaO)

Strontium oxide (SrO)

Zirconium oxide (ZrO)

Potash (K"0)

.12
.63
.24
.28
.09

Soda(Na26) ..

Chromic oxide (CrjOs)

Phosphoric acid (PjOs)

Sulphur (S)

Sulphuric anhydride (SOa).
Total

99.444

101.61

The amount of metal was so small as to make it practically impossible to determine the
proportional amounts of nickel, cobalt, and iron. The SO3 and a part of the FcjOj were doubt-
less derived from the iron sulphide through oxidation, and have so been considered in the final
tabulation. Tests were made for oldliamite by boUing the powdered material in water, but
no calcium could be detected. The absence of chromium in Whitfield's analysis is doubtless
due to the sporadic occurrence of the mineral chromite and the small size of the sample sub-
mitted, but 9 grams being available.

(15) Meteoric stone, BlacJc Chondrite (Cs). — McKirmey, CoUin County, Tex. Referred
to by Brezina" and relegat<?d to his Cs type, characterized by colorless chondrules firmly
imbedded in a dark gray to black ground. The mineral composition and structure are given,
but no analyses. Dr. Whitfield found the stone to consist of:

Per cent.

Troilite 6.26

Schreibersite .58

Metal 5. 70

Chromite .11

Silicate minerals 87. 35

100.00

o Ann. k. k. Holmus., Band 10. Heft 3 u. 4, 1895 (96), p. 252.

20 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv.

The silicate portion yielded:

Per cent.
Silica (SiOj) 43.30

Alumina (AI2O3) 15- 18

Ferrous oxide ( FeO) 8. 45

Liine(CaO) 1.88

Magnesia (MgO) 30.48

Manganous oxide (MnO) .2.5

Nickel oxide (NiO) 51

100. 05
The metallic portion yielded:

Per cent.

Iron (by difference) 85. 84

Cobalt 92

Copper .08

Nickel 13.16

100.00

Barium, strontium and zirconium, in addition to the other rarer elements, were looked
for but no traces discovered.

(16) Meteoric stone, Gray CJiondrite (Cg). — Monroe, Cabarrus County, N. C. This stone
has been briefly described by several writers and subjected to at least one previous chemical
analysis. The first examination was made by C. U. Shepard " who relied for his mineralogical
determinations upon the results of chemical analyses, thin sections at that date not bemg avail-
able. Wadsworth mentioned it briefly in his Lithological Studies, but added little excepting to
justly remark that "judging from its general character Shepard's analysis is incorrect and it is
hoped a new one may be made." Wiilfing in his catalogue places the stone in Brezina's group,
Cga, that is, with stones consisting of a tuff-like mass with variously colored chondrules firmly
embedded in the ground. It remains to be stated that the chondrules are in part of olivine and
in part of pyroxene, both varieties occurring in porphyritic or in barred or radiating forms. Two
varieties of pyroxene are recognizable, the one obscurely twinned and monoclinic — the kUno-
enstatite variety — and the other occurring frequently in large peUucid forms and orthorhombic
in crystallization. Metallic iron and iron sulphides with their oxidation products complete the
list of recognizable minerals. The wide discrepancies between the analyses of Shepard and
Whitfield can be accounted for on the supposition that the material utilized by the first named
was not representative and his methods imperfect.

Shepard's analysis:

Per cent.

SiUca (SiOj) 56. 168

Ferrous oxide (FeO) 18. 108

Magnesia (MgO) 10. 406

Alumina (AI2O3) 1.797

Nickel-iron with traces of chromium 6. 326

Magnetic pyrite 3.807

Lime, soda, potash, and loss 3. 394

100. 006
Whitfield's results as follows:

Per cent.

SiUca(Si02) 36. 7l'

Alumina (AI2O3) : 3.59

Chromic oxide (C-jOs) Trace-
Ferrous oxide (FeO ) 14. 80

Manganous oxide (MnO) 23

Nickel and cobalt oxides (NiOCoO) 46

lime(CaO) 2.27

Magnesia (MgO) 24. 54,

Iron (Fe) 12. 58

Nickel(Ni) 87^ Metal 13.54

Cobalt (Co) 09

0 Amer. Journ. Sci., vol. 10, 1850, p. 127.

Per cent.
Silicates.. 82.60

No.i) RESEARCHES ON METEORITES— MERRILL. 21

Iron (Fe) 2. 391 Percent.

Sulphur (S) 1.4lJ TroiUte... 3.80

9!). 94 99. 94

No trace of barium, strontium, lithium, soda, potash, zirconium, or copper, could be
discovered.

(17) Meteoric stone, CTiondrite (Ci).— Ness Comity, Kans. Described by H. L. Ward." No
analysis given. The stone was somewhat decomposed through weathering, but yielded approx-
imately 15 per cent of nickeliferous iron which showed:

Per cent.

Copper 0. 30

Nickel 7. 00

Cobalt 20

Iron 92.04

99.54

A bulk analysis in which all the combined and oxidized iron was determined as ferric oxide,
yielded:

Per cent.

Smca(Si02) 38.340

Ferric o.xide (FejOj) 8. 551

Alumina (AI2O3) 8. 259

Chromic oxide (Ct^O^) 587

Lime (CaO) 1. 180

Magnesia (MgO) 24. 040

Metal (FeNi) 15. 000

Loss on ignition 3. 500

99. 457

Recalculating the first analysis in order to include the components of the metallic portions,
and thus obtain the composition of the stone as a whole, we have:

Per cent.

Silica (SiOj) 38. 340

Ferric oxide (FejOj) 8. 551

Alumina (AI3O3) 8.259

Chromic oxide (CrjOs) 587

Lime (CaO) 1. 180

Magnesia (MgO) 24. 040

Loss on ignition 3. 500

Iron (Fe) 13. 860

Nickel (Ni) 1.050

Cobalt (Co) 030

Copper (Cu) 050

99.447

None of the rarer elements were found. The high ignition is due largely to the hydrous
sesquioxide of iron formed through weathering.

(18) Meteoric stone, Chondrite (Cc). — Selma. Ala. Described by Merrill* but no analyses
given. Examination of thin sections showed the presence of the usual sulphide particles to-
gether with olivine, enstatite, and a monoclinic pyroxene.

A bulk analysis as given in my preliminary report was so unsatisfactory, particularly
with reference to the iron and alkali content, that additional material was selected and new
analyses made with results as given below. No metallic iron could be detected, though whetber
or not this was due to oxidation, as seems probable from the high content of ferric oxide and
water, could not be determined.

a Amer. Journ. Sci , vol. 7, 1899, p. 233. t Proc. V . S. Nat. Mus., vol. 32, 1907, p. 59.

22

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

I Vol. XIV.

Per cent.
Silica (SiOj) 31. 06

Alumina (Al^) - 4.30

Phosphoric acid (P2O5) -25

Chromic oxide {Cifi^) 41

Ferric oxide (Fe^) 1^- 15

Ferrous oxide (FeO) 13.07

Manganous oxide (MnO) .26

Nickel oxide (NiO) 1. 45

Cobalt oxide (CoO) 15

Lime(CaO) 2.13

Magnesia (MgO) 21. 21

SodaCNa^O) 3.96

Potash (KjO) 07

Vanadium oxide (V2O3) Trace.

Water (H2O) 3.07

Troiiite{}f);^V;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;:;:::::::;::::::::::::::;:;::;;;::;; :f2

100. 05

No traces of other constituents than those mentioned.

(19) Meteoric stone, Eukrite. — Stannem, Moravia. This stone, which fell on the 22d of
May, 1808, has become, on account of its wide distribution in public and private, collections
throughout the world, one of the best known of meteorites. It naturally follows that it has been
repeatedly the subject of investigation and publication. Wiilfing's catalogue gives 74 independ-
ent publications between the date of fall and 1894, two of which included chemical analyses.
Of these only that of Rammelsberg, given in Column II below, needs consideration. The
latest analysis, by Whitfield, is given in column I. This was made with especial reference
to the possible presence of barium, strontium, or zirconium, it being a feldspathic stone.

Constituents.

I

II

Silica (SiOs)

Per cent.

47.94

.41

11.19

1.20

Trace.

.25

Trace.

18.97

10.36

7.14

Per cent.
48.30

Titanic oxide (TiOa)

12.65

Ferric oxide (FejOa)

NicVel oxide (NiO)

Ferrous oxide (FeO )

T.imo fCaO^

19.32

11.27

6.87

.81

Barium oxide ( BaO )

Strontium oxide (SrO)

Zirconium oxide (ZrO)

Potash. (K3O)

None.

None.

None.
.13
.75
.35
.55
.31
.14
.30

.23
.62
.54

Soda (Na^O )

Chromic oxide (CraOa)

Iron ^Fe'^

Trace.

Phosphoric acid ( PaOs)

Total

99.99

100. 61

(20) The three meteorites mentioned below were subject to partial analyses only.

Ballinoo, Australia. — Iron. Finest octahedrite. This iron, described by Sjostrom," so
closely resembles in its physical and chemical properties that of Perryville, Mo., that it was
thought advisable to test it for the rarer or minor constituents not reported in its published
analysis. A 30-gram fragment submitted to Dr. Whitfield showed neither platinum nor
iridium, but on the other hand did show unmistakable traces of palladium and ruthenium.

Glorieta Mountain, N. Mex. — Iron. Medium octahedrite. This u-on, described by G. F.
Kunz, with analyses by L. G. Eakins '' was stated to caiTy 0.03 per cent zinc. Dr. Whitfield,
however, working on Another portion, failed to find a trace of the metal.

o Sitz. k. Preus. Al^ad. Wlss. Berlin, 1908, p. 21.

b Ann. N. Y. Acad. Sci., vol. 3, 1885, p. 33).

N0.1.) RESEARCHES ON METEORITES— MERRILL. 23

MissTiof, Couiiand, Russia. — Stone, Cc. Johanson's analysis shows 0.18 per cent Cu
and 0.156 per cent SnO,." Two fragments, one of 7 and one of 28 grams, were submitted to
Dr. Whitfield, who reports 0.01 per cent Cu in the first and 0.008 per cent in the second, but
no traces of tin in either.

IV. DISCUSSION OF RESULTS.

Gold and the platinoid elements. — It will be noted that our work has failed to substantiate
the reported occurrence of gold in meteorites, either iron or the stony varieties, while the occur-
rence of platinum, palladium, iridium, and ruthenium in the irons appears proven beyond a
doubt. That these last are not more frequently reported is probably due to the careful, detailed
work involved in their determination, and perhaps also to their very irregular distribution,
noted on page 5. The close relationship existing between the gold and platinum metals
would render their association in the meteorites not surprising were it not that the mineral asso-
ciations of the two are in terrestrial rocks so unlike, gold being rarely if ever reported from rocks
as basic as the peridotites. I do not find, however, that the terrestrial peridotites have as yet
been subjected to the carefid scrutiny necessary to decide this absolutely.

While, however. Dr. Wliitfield's analyses fail to bring to light a trace of gold, it should be
noted that in addition to Liversidge's determinations, J. C. H. Mingaye,'' in a very thorough
analysis of the Momit Dyrring, N. S. Wales, pallasite found traces of gold, together with platmum,
iridium, and palladium. The same authority also reported platinum and iridium and traces of
tin in the Barraba iron.

Phosphorus. — The phosphorus shown by analyses to occur in meteorites is usually relegated
to the schreibersite of the metallic portion or to apatite of the silicate portion. So far as the
writer at present recalls, the presence, in a meteorite, of the mineral apatite has been satisfac-
torily demonstrated by optical means in but a single instance, though Shepard claimed to have
found in the Richmond, Va., stone particles of such size that he was able to isolate them for
qualitative tests."^ These results, so far as I am aware, have never been corroborated. Ber-
werth '' identified the mineral in granular, short, prismatic, and skeleton forms in the silicate
secretions of the Kodaikanal iron.« Numerous other supposed instances of its occurrence have
been reported, based mainly upon the presence of a soluble phosphate in the silicate or non-
metallic portions. That the phosphatic mineral is not schreibersite is conclusively shown by
its solubility, and the presence of lime in the same soluble portions is naturally suggestive of its
combination as apatite, the form in which it exists in corresponding igneous rocks. In my
own work I have repeatedly obtained reactions for phosphorus by digesting for a short time the
pulverized stony material in an acid solution far too weak to attack the schreibersite. This is
the case with the Alfianello, Bluff, Dhurmsala, EstherviUe, Farmington, Felix, Indarch, Queng-
gouk, and several other stones which have been examined.

Investigations made to settle the question of its occurrence in some other form of combina-
tion than that of apatite have yielded unexpected results which may be briefly mentioned here,
though elaborated elsewhere.^ It may be recalled that in the writer's description of the mete-
orite from Rich Moimtain, N. C,9 he mentioned the occurrence of a doubtful mineral occur-
ring in plates of irregiJar outline, faintly gray or almost completely colorless, showing very
faint, short, sharp cleavage lines with weak polarization colors, and which were optically biaxial.
This he referred to the monticeUite-like mineral described by Tschermak,^ though confessing to
a feeling of doubt as to its true nature. Since Tschermak's writing the mineral has been ob-

o .\rb. des Natur. Vereins zu Riga, Neue Folge, Siebentes Heft, 1891, p. 51.

6 Records Oeol. Surv., N. S. Wales, vol. ", pt. 4, 1904, p. 303.

c .\mer. Journ. Sci., vol. 45, 1M3, p. 102.

d Tsch. Min. Pet. Mittheil., vol. 25, 1906, p. 188.

t Tschermak in his paper on the .Vngra dos Reis meteorite describes a singly refracting, optically negative, colorless mineral concerning which
he remarks, " Man kann wohl als sicher annehmen, dass diese Komchen dem Apatite angehoren. " This occurrence was overlooked in my paper
on the monticellite-like mineral in meteorites elsewhere referred to.

/ See "On the Monticellite-Uke Mineral in Meteorites," Proc. Nat. Acad. Sci., vol. 1, 1913.

» Proc. V. S. Nat. Mus., vol. 32, 1907, p. 243.

» Sits. k. Akad. Wiss. Wien, vol. 8S, 1883, p. 353.

24 ItlEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol.xiv.

served by numerous other investigators, the more recent notices being those of Lacroix in the
stone of St. Christophe la Chartreuse," and Borgstrom in the stone of St. Michel, Finland.* In
connection with the present investigations it was determined if possible to settle the identity
of this mineral. With this in view, slides of both the Rich Mountain and Alfianello stones were
submitted to Dr. F. E. Wright, of the Geophysical Laboratory, who reported the refractive
indices of the mineral in question as a = L62.3±0.002, 7 = L627±0.002, birefringence weak,
less than 0.005, and interference colors not exceeding gray white of the first order. The addi-
tional data stiU left the exact mineral species undetermined, though the refractive indices sug-
gested that if a known mineral it is allied to the phosphate francolite. With this in mind, sev-
eral slides, embracing those of i\lfianeUo and Rich Mountain, were uncovered, and the mmeral
submitted to microchemical tests, which proved conclusively its phosphatic nature. The
objections to considering the mineral francolite are, that so far as known among terrestrial
rocks this mineral is of secondary origin, and a product of aqueous deposition, thus suggesting
conditions which are not supposed to prevail among meteorites.

Farrhigton,'' it will be recalled, thought to have found native phosphorus in the meteorite
of Saline County, Kans. Notwithstanding the care exercised in his determinations, one can
but feel that the observations need corroboration before acceptance. The stone had lain
some three years in the ground when found and the examinations were not made for a year
or so later. Under these conditions, when the nature of phosphorus is considered, it seems
well-nigh impossible that material so susceptible to change could have remained imaltered.

Silica or silicon is not infrequently reported in analyses of meteoric iron in amounts rarely
exceeding 0.2 of 1 per cent. The condition under which the element exists is in some cases
at least problematical. H\mt and SUliman, in describing the iron of Lockport (Cambria),
N. Y., ** refer to a reddish brown residue obtained by them as being "either silica with a trace
of carbon or silicon," which last, they add, "Prof. Shepard has already shown to exist in the
Oswego iron." Prof. Shepard, however, in his paper simply tabulates his results as "Silicon
0. 20 per cent" and does not commit himself as to the condition under which the element may
exist. Prof. Mallet, in his analysis of the Staunton, Va., iron, gives 0.067 per cent, 0.061
and 0.056 per cent Si02, but adds by way of explanation, "some of it (i. e., the Si) seems to
have in reality existed as a silicide of iron." « Cohen, in his Meteoritenkunde (p. 55), refers
the Ca, Mg, Al, K, and N very properly to the silicate minerals, and adds, "Das gleiche gilt
wohl auch in der Regel fiir Silicium; doch fiihrte Winkler in metallischen Theil von Rittersgriin
gefundene Kieselsaure auf Silicium zuriick, welches mit Eisen verbunden war, und nahm das
Vorhandensein eines Siliciumeisen von der Formel FcjSi an, dessen Menge er fiir das Nickelcison
zu 0.329 Procent berechnete. " Tassin in 1907 annoimced verbally in an informal communi-
cation before the National Academy of Sciences "the discovery of elemental silicon" in the
meteoric iron of Casas Grandes, Mexico, incidentally claiming it as "the first announcement
of the occurrence of this element in nature. " With reference to these reports it may be stated
that an examination of the insoluble residues from all of these irons reveals the presence of
minute particles of quartz, sometimes shreds of glass and simdry silicates.^ It seems most
probable, therefore, that the small percentage of this constituent foimd had existed either as
free quartz (SiOj) or as a silicide of iron. Until the element shall be actually isolated it is
unsafe to claim its existence in other form than that of combination with other elements.

a Bull. Soc. sei. nat. Quest, 2d ser., vol. 6, 1906, p. 81.

6 Bull. Com. Geol. Finlande, No. 34, 1912

<; Amer. Joum. Sci., vol. 15, 1903, p. 71.

d Amer. Joiirn. Sei., vol. 2, 1846, p. 374.

e Ame'. Joum. Sci., vol. 2, 1871, p. 14.

/ See Cohen & Weinschenk on the Toluca, Mexico, meteoric iron, Meteoreisen-Studien, Ann. k. k. Hof.-Mus., 1S91, p. 140. It should be added,
however, that personally I regard the preterrestrial origin of these particles as open to serious doubt. In residues from a quantity of shavings
from the Casas Grandes iron and from a 10-gram piece showing a portion of the original surface, though carefully cleansed, I found easily recogniz-
able, clear, glassy quartz, both in form of crystals and angular fragments, shreds of colorless glass and also undetermined silicate minerals. Two
other determinations on pieces cut from a depth of 2 centimeters below the surface yielded no such results, the residues being clean graphitic
particles and schreibersito flakes. A few minute, colorless, isotropic parti; le.s, too small to manipulate, were crushed under the microscope between
glass slides and were found to scratch and bite into the glass with all the energy of the diamond.

Noll RESEARCHES ON METEORITES— MERRILL. 25

Sulphur. — The siilphur in meteorites is unquestionably combined in large part with iron,
though the form of combination, whether as troilite or p3Trhotite or some intermediate com-
pound, is recognized as still open to discussion. The work of Dr. E. T. AUen, of the Geo-
physical Laboratory, has shown that in the presence of an excessive amount of iron appar-
ently only the mono-sulphide is possible of formation. That it sometunes occurs in this form
in stony meteorites poor in iron has been shown by the writer « and also by Ramsay and
Borgstrom.'' The subject is thoroughly reviewed up to date of publication by Cohen in his
Meteoritcnkunde (vol. 1, 1894, pp. 182-209), and it is evident that fm-ther work is needed
before the question can be considered fully decided. A portion of the sulphur, and one that
heretofore has been almost wholly ignored if not overlooked, is that combined with calcium
in the form of —

OUhamite. — The presence of this sulphide was first noted in 1870 by Maskelyne in the
meteorite of Busti. "^ Its probable occurrence was also suggested in that of BishopviUe. Since
that time the mineral has been determined both chemically and microscopically by Borgstrcim
in the meteorite of Hvittis, and by Lacroix and the present writer in that of Indarch. Its prob-
able presence as indicated by a soluble calcium-bearing mineral (oldhamite or its alteration
product gypsum) has been also shown by the present writer in the stones of Morristown <* and
Cullison, and in that of Allegan by Tassin. « These results rendering it probable that, as
suggested by Maskeljme, the mineral was more commonly distributed than the pubhshed
description would lead one to suppose, quantities of a gram or so from each of the stones listed
below were finely pulverized and boUed for an hour in distilled water, the solutions being then
tested for calcium by the ordinary ammonium oxalate method. The residues from this boiling
were in some cases boiled also in very dilute hydrochloric acid and the solutions tested for
phosphoric acid, with a view of deciding if the phosphorus shown in the bulk analyses was
from the schreibersite, which would be insoluble under these conditions, or existed in the form
of apatite or other soluble mineral. The results of the calcium tests were as follows:

Alfianello Positive reaction.

Allegan / Positive reaction.

Beaver Creek Negative reaction.

Bishop\'ille Positive reaction.

Cullison Positive reaction.

Dhurmsala Positive reaction.

Dores dos Campos Positive reaction.

Estherville Positive reaction.

Farmington Faint positive reaction.

Fayette Positive reaction.

Felix Positive reaction.

Forest City Positive reaction.

Hessle Faint positive reaction.

Holbrook Positive reaction.

Homestead Positive reaction.

Knyahinya Negative reaction.

L' Aigle Positive reaction.

Mocs Positive reaction.

Monroe Negative reaction.

New Concord Doubtful reaction.

Pamallee Faint positive reaction.

Pultusk Positive reaction.

Quenggouk Positive reaction.

Stannern Negative reaction.

Tennasilm Positive reaction.

a \ recent meteorite fall near Holbrook, Ariz., Smithsonian Misc. Coll., vol. 60, No. 9, 1912, p. 4.

<> Bull. Com. Oeol. Finlande, No. 12, 1912.

c Phil. Trans. Roy. Soc. London, vol. 160, 1870, pp. 189-214.

d Amer. Journ. Sci., vol. 4, 1S96, p. 149.

« Proc. U. S. Nat. Mus., vol. 34, 1908. p. 433.

/ Quantiutive tests on this stone by 'Whitfleld show but O.OM per cent of this constituent, as against upwards of 6 per cent as found by Tassin
(Proc. U. S. N. M., vol. 34, 190S, p. 433). Although not so stated, it seems probable that the latter ascd an acid solution and decomposed in part
the pbosptaatic mineral and iron sulphide. This would account for the present writer's Inability to detect the mineral in the thin section.

26 MEMOIRS NATIONAL ACADEMY OF SCIENCES. (Vol.xiv.

There being a question, which is suggested by Maskelyne's description of the Busti stone,
as to the sporadic occurrence of the calcium sulphide, three individuals from the Pultusk fall
were selected and tested, two of which yielded distinct traces of calcium in the water solution,
while the third showed not the slightest trace. The results then are apparently to the effect
that oldhamite, or its alteration product, is a fairly common constituent of meteorites, but
that it is by no means uniformly distributed throughout the mass of the stone. The cause of
its being overlooked is doubtless due in part to the small size of the granules, to their breaking
away in the process of cutting the section, or to the obscure form of its alteration products.
The most careful examination by the writer has failed to reveal it in distinct crystalline form
in any of the cases listed above.

Tin. — The occurrence of this metal has for a long time been regarded as open to question
by the writer, notwithstanding the apparent care and skill under which the various analyses
had been made. The skepticism was based in part upon the conditions under which the metal
occurs m terrestrial rocks, where, as is well known, it is limited almost wholly to acidic types ;
in but two exceptions has it been found to occur in rocks of intermediate (andesitic) type.
Genetically then it is fair to assume there is some connection. Among the common mineral
associations of terrestrial tin, in the form in which it usually occui's (cassiterite) are, further,
several very characteristic species such as fluorite, tourmaline, wolframite, topaz, etc., which
are utterly unknown in meteorites. It is of course possible that the metal, if present, is in the
form of the sulphide (stannite) or as an alloy with iron, but none of the recorded analyses of
meteoric sulphides show a trace of the element, nor do analyses of terrestrial irons, as those
of Ovifak, Greenland, or the various terrestrial nickel-irons as josephinite, awaruite, etc."

OtJier elements reported. — Concerning the occurrence and distribution of some of the other
less abundant elements, there is still a lingering doubt. The reported occurrence of titanium,
nickel, cobalt, and chromium in the silicate portions, freed from metal, may reasonably be con-
strued as indicating their combination in silicate compounds, particularly the pyroxenes, as
in terrestrial rocks. Dr. Whitfield in his analyses has aimed at deciding this by exercising
particular caution in separating the metallic from the nonmetallic portions. The analyses of
the latter, it will be noted, still show small amounts of nickel and cobalt. It may be recalled
in this connection that Tschermak ** reported 2. .39 per cent TiOj in the meteorite of Angra dos
Reis, all of which he relegated to the augitic constituent.

V. RfiSUMfi.

To sum up in brief the results of this investigation : So far as the minor elements are con-
cerned, we have not merely failed to confirm but in most instances have thrown grave doubts
on previous determinations of antimony, arsenic, gold, lead, tin, tungsten, uranium, and zinc.
The occasional presence of platinum is apparently confirmed beyond question, and in two
instances of vanadium.*^ Palladium, ruthenium, and iridium have also been found in traces.
It is very probable that further investigations on the iron meteorites would yield confirmatory
results. The presence of platinum was to be expected from the analogy with the terrestrial
sources of this metal. Vanadium and titanium were also not unexpected in view of their wide-
spread occurrence in terrestrial peridotites, as shown by HiUebrand's investigations.''

The apparent universal absence of barium and strontium may perhaps be accounted
for by the paucity of the meteorites examined in feldspathic minerals. It is unfortunate that
the National Museum collections are very poor in feldspathic types, and the prices per gram
asked by dealers, and even other museums and collectors, are practically prohibitive.'

<■ These analyses are brought together in convenient form and discussed on pp. 313-15, 2nd. ed. of Clarke's Data of Geochemistry.
6 Tsch. Min. Pot. Mittheil, vol. 28, 1909, pp. 110-114.

c Traces of vanadium are also reported by II. C. White (Records Geol. Surs-. N. S. Wales, vol. 7, 1904, p. 312) in the meteorite of Mount Browne.
d Bull. 167 U. S. Geol. Surv., pp. 49-55.

t Ward's Values of Meteorites quotes prices of the feldspar-bearing Eukrites and Howardites varying from $1 to t4 per gram. The Juvinas and
Stannem among the Eukrites alone drop to prices from 50 cents to S2 a gram.

Tahie of telttUd analyttt of stony meUorites.

I.ocallly,

Dal*.

AiuttyM.

at

SlOi.

TlOr

A1.0„

OiO,.

Fe.

FeO.

NI.

NIO.

Co.

CoO.

CkO.

MgO.

MnO.

NftiO.

K.O.

S.

P.

P,Oi.

H,0
-110».

+ 1I0«.

C.

Mist-,

Totals.

Remarks.

TolBl
known
wfielit n(

IKW

11. S.Blflkw

s»

Xl.&'i

n.i*

2-5i
1.7*
S.73
2.411
2.17

i.Ton
2.ns

2.17
2.27
fi.43
-1.34
1.74
2.B9
3.(10
1.70
3.34
4.24
3.08
3.30
l.fiS
3.70
l.liS
1.07
1,80
1.77

13,50
3,10

TlKW.

3.76
3.32
3,41
3.02
3,08
3.7D

13,20
1,80
.36

Traon.
a. 47
3.30

2 37
.3n
.T«
.00

■ lU

.04
3.31
I.Kt

i.on
3,S7
4,3U
3. IS
3.11
11.10
2.U

3.n

3 3t

O.W
.42
Tntf«.
..8
.31

.GO
.40

■■■;«■■

.33
.320
N..I.
.58
.SO
.48
.090

.a

.ST
.GO

Trace.
Trtiw.

.63

:::

.44

°;»

.07
.55
1.07
l.OG

'I'm™.
M

.10

.60
1.25
.43
.41
.41
.G3
.40
.35
.93
.63
.44

.an

1.81
1.14

Tree*.

0.15
,08

1.73
.80
31,01
1.70
1.70
/[.5201

1.82
2,20
1,38
1.70
.02
16.17
1.51
3,00
1.75
5.45
1,01
1.115
3.13

2. [18
2.10
1.41
1.16
1,23
1.31

10 60
1.80
2.20

3. S3
2.06
3,40

Tnicea,
1.40
1.40
1.06
1.03

21,00
23.43
10.05
24,71
23 73
.33 500
25.75
25.04
23.76
21.17
23.63
13.01
33.01
23.71
20,16
10.74
26.02
33,01
28.62
25,06
23,46

51'

2S34
21,59
7.20
23.06
32.63
33.64
18,63
25.72
25.07
32,74
23,79
36.00
37.30
19.33
15.95
36.45
24.51

2.1.35
31.04
1.^.31
35. 80

20.43
25.00

[ia.33

34.00
2.1.34
23.03
13.81
31.21
36.21
23.16
7.11
23.36
37.93
24.87
2.^10

0,18

.13

Truco.
.23
.189
.13
.45

{■':¥'■

.09
{.013 )
N.d.
.16
.08
.34

.33

}■■■'

.105
Trac*.

.07

.78
.20

^.

.31
.07
1.37

.23

Trow.

TtBM.

Trace.
.43

(■!■

.20
.12
.25

TraM.
.15

0.60
1.00
.20
.01
.79
1,02T
1.24

0.23
.21
.19
.04
.097
.080
.33

1,84
3.i4

.4.1
2.47
I.S4

.297
1.98
1.30
2.07

0.16

0.27

.13
.25
.25

0.00

0.19

rcii 0.01, l.io,
\ trace,
Cl,0, 0.103....

1 100.01
00,98
100.02
100.37

Oianu.
31.100

54.866

AIltwui.Mlrh

"m!

[4.321
S.13

[3.311

IS. .-a

.IHl

([3 40]

16.38

3.4T

rr3.«t)

[1071
34.43

&

a

(.24]
10.40

1..'.<1
} 13.80
33.82
t.22I
11.04
1,5.87
/ [-57]

Ainanello, Ilnly

iQld

}""

1M3
IMM

im

IMO
I KM
1911
191 1
IHM
IMa
IttfW
IVOO
IHM
IWO
1001
IMW
1013
IDOl
1801
I8NU
IHIH
1831
1809
|8jyi
IMa

ISW
IK.',*
1N20
1801
IN80
1870
IMS

t8S3
lOUl

mo
iiua

(IrtnlwjRWi.lT«ch«
\ mak.

O.T.Pflor

W. F. UUlvbniiKl. .

J.K.WhIinsW

3.M

......

13,01
H9.M
S7,13

5:.ai4

tl.OC
37.70
35.H
H,M
IV 30

tH.on
34. as

36. M
30,95

33, c;

38,000
SS.fl»
48.00
3n.9l
SO.AS
11.63
3S.T0

ao.o7

33.11
47.00
32.06
34.98
33,03
41.13

43. OS
30,45
3S.U
30.07
37.00
40.13
M.OO
43.74

44. IS

».). ;i

M.Hl
41. n

43. n

40. U
10.33

S0.&3

sr.oH

MiM
40. lU
Jl.DO
10.10
35.71
tT.M
W-Ol
44.75
30.30
41.13

2.30
.10
.07

1.500-3.000

AnBra "Iw n«l«. I*"""

,75
1.50
.030
.72
.05

( ,.«

1.49

. I.SO

l.«

.M
.30

1.10

} ■"
3.16
.42
1.06
.05
.83
.43
\

0.033
.538
.07

.OS
.00

.04
.09
.11
.09
.098

Traee.
.10

.17

Oil 0.00(1

Ilmvor CrMk, Itrlliih Co-

14,000

100.177
100,04
90,79
100,00
100.10
100.53
100.15
08.770
100.27
100.00
99,79
09.761
00,99
99.35
09.32
09.81
100.28
100. 071
09.205
08.66
09.42
100.77
90.79
100,05
100,22
99.09
100.305
99.95

on.no

100,004
100.74

. 99,728
90.51
90.46
89.04

ion.(w

100.00
100,00
90.02

9(..57
100.10
101.33
100.30
100.53
100.31
100.00
\ 100. 10
09.37
00.83
08.70
99.010
100,10
90.31

<t,ooo

niKhopvilto. 8. C

.11

J. E. Whlinelil

B. I,, rmifielil

3.M
3.117

.25
.39

.80
1.46

^l.S04
.41

None.
3.07
.73
.02
Trftoe.
.81
.96
1.57
Trace.

t.26
.13

.11
.30
.171
.11
None.
.33
.11
.14
.00
.056
.10

.33

.17

0.01

do

O.T. Prior

3*7

2.184
.00

.0138

.017

rii 0.008

.3ft

..-. { ill'

■■« '15:2'

14,08 1 15.53
(3.W1 1 ^ „

.n;

10,000

.32
1.01
.204
.14

.07
.65

1.77
.11

.00
.08
.06
.03
.084
.13
.01

Tmc*.
.03
.07
.02
.05
.01

Trace.

.040
Trace.

4.75

Cu0.00

Cu trace

do

Less 0 for 8, 0.68

J. U. Dovldon

3,00

Notdoi.

1.37
1.82
1.73
Trace.
2.25
1.61

.18
3.75
3.30
3.10
3.58
1,8S

.002
2.16
3.09
2.00
2.36
1.85
2.124
1.00

3.38
1.70
2.29
3 01

1.3S
1.41

■J, 02
.11

3.03
.15

l.OO
2.52
5.26
2.22
1.05
3.00
1.93
.10
1.01
3.27
.31
.025
.01
1.94
1,30

.16

290,000

P. FIremAn

J. B. WhltfleM

3.7S
3.37

.10

.36

ruo.oi

9.724

&l

3.37
10.30

p;

1 61
3.81

TriK*.

33-25

US

23.00
10.37

(li:g'

10.05
3.00
14,40
f[3.08|
lG.07
4.24

CS!

7.93

11.50

{&

10.33
0.33

4.1',
30.44

n,Ji

7. XII

UiiS'

i:i.77

(T)
10, 7U
31.10

t.HD
11,07

('?:SS'

( [Ml
1I1.2,S

,a'

13.43

\ 10,11

.34

\ 14.35

\ 21.30

37.07

( ,tSJ'

11,70

[ in. 61

0.04
17.43

) ,.,,.
( ,S;?y

33.85
13.18
7.40
[-751
3»i,S3
\ 11,85

( ASS'

15.37

1"-

12.51
34.72

i;i,.i5

HI. ,13
13.00
n.5o
13.44

( lis'
a

( .klS'
( kS'

1G,10
10,39

17, ill

le.04

( ,[S

,'.;:;'

123,037

Kon-lornonvlllo, X. C

w tJi"

.013
Trace.

.08
.093

6,000

Cu+Sn0.02...
CuO.Ol

21,895

J. E.WhItOcli]

a.*n

14, OM

J. i:. WhitOfiii

3.42

.52

2.80

.31

27,000

TT.

Cu trace

33,830

II.B.WMhliiKtun..

Tmw,
.S7

.65
,35
1.41

.17
1.39

] -^

.26
1.12

.28
.00
Trace.
.60
.38
1.33

.20

Trace.

.03

.43

.=7
.00

TiS

39,543

^ 1.55

( Nl+Co.

L 3.33

3.37

1.3U

.71

\ 1.707

.67

l.ltl

.92

} ■»'

1.35

1 1,38

.0.1

..S7

1.78
l.:il

} .05
.20

,41

1.07

1.84
1.10
.OS
1.3li

i.i;

Cu trace

Kon.10

8.67

)

.31
.03
.16
.07
.14
.06
.10
.05
.20

Kloln-Woml«ri,ITuMm....

KrUlioiiliMK, flovnrlft

I.tniini, <}«rmaiiy

Cu0.O5.Sn0,0«
SnO,0.I8

3,2S0

3.m

3.M

— Llniinor

.12

»,<xa

LoiiK I-liitirt, Kan»

I.iiiiiIiiKanl, BwoiImi

ir.W.NJchol.

A.NordoMkJoM...

3,01

.77
.05
.44

.01
.02
.05

.06

1.52
.."iO

.02

CuO,01

Cii 0.004

11,000

100,000
3,136

U. It, vonFinillni).

.41
3,W

)i.2«

.44
Niiiie,

.20

1,13

.21

Tmco.

Sum.

Mfuhor, RiiMlA

fCl 0.007, 8nO,
V 0.156, CuO.lH.
LiO, trace.....

iNiu+

1.87

Tm».
.011

i™

.20

.04

Tnu».

3.78
1.74
3.27

2.21
.03
3.<i:i
1.40

2.41
1.51

IK'

'l.64
2.03
1.87
3.31
2,13
1.75
1.01
10.30
1.374
3.33
1.48
3.13

.41

.03

.10

345,000
16,000
8,300

J. E.WIilinri.!

H.P.WWt*.

Muimt nrowno,NDsv8()ulli

Tmcu.

.03

.11

[ObyilJff.3.78.
l^iiondVjOi
[ truce.

NoDftjSr.Zr. Cl,Si.,s.,,l,f

35,350

^,11
3.&S

350,000
1,39)

,^, '/K. il. voii lIuiMD.

2.73

.45

NowoUrol.IluMla

l«yi
IKTrt
1<M!
IHKI
1010
180H
IBdO

ma

U.Jvrorul(<lTiuidI-.

UtMlilnolT.
J. I..8mmi

3:20

2,0M

.67
.61

(.«

1.32
.30
.85
2.20
3.00
.73
.03

.18
.23
.13
.08
.06
.73
.07
.23
.15

.74
1.15

'i'

.11

.21

-U
.04
.10

.04
.17

.07

.15
TiBoe.

■:

.05
.30

.00

.11

.073

.0(C
/ PiO

.30

CI 0.27

CiiO.Ol

i3,rea

7,000

HI. MMnl, rinlonil

L. 11. Ilonxtrilm.,.

.02

.3(1

1.21
.01

B. I,. l-oiifloM

8.«l

8,015
110,000
18,600

.35

3.07

I 'F««0fl8.1G.'

Contains tnu-es of Ya,. but no Oilier minor const linen I.i ihwi (hiw menltoned.

HlioUiiinio.riwailii

UH. llOTKilrDm.

.78
1.61

1.07

.33

1.34

.3

im

a. l.lnil*tT(km

3.7*

.11

1

ao.04

filniiiipni, Amlria

.14

.So

NoDa,8r,orZr. Nooldlumlte

53,000

38,50)
2,600

Unknown,

[,:«

.52

.01

1,13
1,30
1.10

.13
.15
.14

.41

.84

Culroce

Cu 0.025.

ISM l/*^' '^' ^^^ Uniim-

im

'•

.02

!

.16

3140:f-lfl. (Tofij.*i»p»37.)

Noll RESEARCHES ON METEORITES— MERRILL. 27

It may be added that there must invariably be more or less variation in the proportional
amounts of the different elements as found by analysis, owing to the difliculties in sampling,
without sacrificing what is felt to be too large a quantity of material, a rock in which metallic
iron is so prominent a constituent. This might account for the failure on the part of some to
detect platinum and allied elements, or such minerals as oldhamite and the problematic phos-
phate. An analysis of the stony forms, accompanied, as it always should be, by an examina-
tion in thin section, leaves, however, little excuse for lapses of this nature. The reported occur-
rences in the older analyses of elements not found in later investigations may very naturally be
ascribed, in part, to impure chemicals.

VI. TABLE OF ANALYSES AND DISCUSSION.

In the following table the writer has brought together the more satisfactory complete analyses
of stony meteorites made during the progress of this investigation, as well as such made
by others as seem up to the modern standard. In this work of compilation, analyses have been
ruled out in most cases in which the totals fell 1 per cent and more short of 100 per cent, and
also those that footed up approximately to 100 per cent, but in which certain elements which
were obviously present were not mentioned. It is, of course, possible that in all cases full
justice in the selections has not been done to the other workers, but the error, if there is one, is
of omission rather than commission.

The purpose of tabulation in this form is to render the analj-ses comparable with those of
terrestrial rocks. In the work of preparation it has been necessary to recalculate in part sev-
eral of them. Those constituents the statement of which required most frequent attention
have been the ferrous sulphide which has been tabulated as Fe and S; the ferric oxide, which is
mainly secondary, and has been recalculated as ferrous, and the chromite which has been recal-
culated as chromic oxide (CrjOj) and ferrous oxide (FeO) , a proceeding which is recognized as not
absolutely correct, since the meteoric chromites almost invariably contain several per cent of
alumina and appreciable quantities of magnesia. The phosphorus has been allowed to stand as
given, either as P or P2O5, though the probabilities are that it belongs in aU cases to the sihcate
portion, and to be consistent should be tabulated as P2O5. The recalculation and tabulation, of
the iron in the sulphide as Fe is also open to question, and hence in such cases the percentage
amounts are inclosed in brackets. In all these cases the sulphide is assumed to be in the form
of FeS. Where an element has been recalculated as an oxide or the reverse, allowance for the
gain or loss in the total percentages has been made in the customary manner.

It is to be noted that one of the most common and widely disseminated of the minor mete-
oric constituents — chlorine — has been ignored in this investigation, as in that of the majority
of workers on stony meteorites. This is due in part to the ready oxidation of the mineral law-
rencite, in which it mainly occurs, and in part to its seemingly trifling amount. Its presence
in other form of combination than with iron — and perhaps nickel — has yet to be satisfactorily
shown." As wiU be seen by reference to the table, I have found but five recorded determina-
tions from which to calculate averages.

» Sw Cohen's Meteorit«Qkunde, pp. So aad 227.

28

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

(Vol. XIV.

The above analyses, it Avill be noted, cover, to a fair extent, the entire range in composition
of the stony meteorites. Although the number is small, it will nevertheless be not without
interest to average them, as was done in my paper of 1909," and by Farrington at a later date. *
In making this average I have adhered to the plan first adopted of considering only the actual
determinations of any particular constituent. Further, it has seemed advisable to rule out
a few cases in which the percentage amounts of any constituent were so high as to be considered
anomalous, as that of 2.39 per cent TiO, in the stone of Angra dos Reis, or 6.33 per cent Cr^Os
in that of Long Island.

This elimination is of course open to serious objection, and the question may well be raised
as to the desirability of omitting entirely from the calculations the analyses in which such
anomalies occur. The only answer that can be given is that in so doing the total number
would be so reduced as to make any average of little interest. But it must be borne in mind
that the value of any average that can be made to-day lies not in its showing the actual average
composition, but rather in showing what has been done and inferentiaUy what remahis to be
done.

The figures given m colmnn I are therefore averages of 53 analyses with the 15 exceptions
noted. In column II is shown for comparison the average composition of terrestrial igneous
rocks, and m III that of the lithosphere, as given by Clarke."^ In columns IV and V are repeated
the meteoric averages given in my previous paper and that of Dr. Fari'ington.

Average composition of stony meteorites compared with terrestrial rocks.

Constituents.

I

II

in

TV

V

SiOa

Per cent.

38.68

.18"

None.

None.

2.88

Per cent.
59.93

.74

Per cent.

59.85

.73

Per cent.
38.732

Per cent.

39.12

.02

.02

Ti02

.03

14.97

2.58

.05

.02

.63

14.87

2.63

.05

.02

.MjOa

2.733
} .835

2.62
f .38
\ .41

Cr-Oa

.472

Trace.
11.98

1.15»

.07<

14.58

.483

.06«

2.42

None.
22.67
.29'

None.
.87"
.21'

Trace.
.75i«
.269

1.80"
.01412
.1513
.081*
?
?

VtOt

Fe

11.536

} 1.312

16.435

11.46
1 1.15
1 .05

16.13
.21

Ni

FeO

3.42
.03

3.35
.03

NiO

CaO

4.78
.11

3.85
.10
.04

3.40

2.99
.01

1.94
.26
.11

4.81
.10

3.77
.09
.04

3.29

3.02
.01

2.05
.25
.10

1.758

2.31

MgO

22.884
.556

'»2 42
"!l8

MnO

SrO

NajrO

.943

.328

.81
.20

KjO

Li"0

HsOCIgn.)

PoOs

.20

.12

1.98

sT

1.839

Cu

c

.03
.06
.10
.70
.02

.06

CI

.06
.10

.48

F

COj

Ni,Mn

\

.02

Cu Sn

/

Totals

100. 0-14

100.00

100.00

99.891

99.87

1 Average

2 Avera?o

3 Average
* Averapo
t Averatje
6 Avorage
' Average

of 46 determinations,
of 4'3 deierminations.
of 50 determinations,
of 41 determinations,
of 19 determinations,
of fi determinations,
of 33 determinations.

8 Average of 49 determinations.

9 Average of 44 determinations.
'(■Average of 15 determinations.
11 Average of 51 determinations.
'2 Average of 16 determinations.
'3 Average of 8 determinations.
'< Average of 5 determinations.

a Amer. Journ. Sci., vol. 27, p. 469.

c Bull. 491, U. S. Geol. Surv., 1911, p. 32.

b Publ. 151 Field Mus. Nat. Hist.,
Geol. Series, vol. 3, No. 9, 1911.

Noll RESEARCHES ON METEORITES— MERRILL. 29

It is, I think, commonly recognized by all who have givon the matter thought, that greater
uniformity in both method and statement of meteorite analyses is desirable. That the prevail-
ing practice of separation into soluble and insoluble portions is also desirable is, I believe, with-
out question, but the results of these processes can be accepted as little more than suggestive
and not final, unless accompanied by a careful and detailed microscopic examination.

The solution obtained by digestion in distilled water will show with a fair degree of safety,
the presence, or absence, of calcium sulphide and ferrous chloride as well as ferrous decomposi-
tion products, and the metallic and silicate portions be left practically untouched. The moment
an acid is added, however, care needs be exercised in the interpretation of results since one
has no means of telling how complete may have been the solution of the "soluble" constituents
or the extent to which the "insoluble" may have been attacked. To illustrate: It was found,
when working on the olivine of the Admire paUasite, that the material when not reduced to
an impalpable powder was not completely dissolved even through repeated digestions over
the steam bath in standard hydi'ochloric acid and sodium carbonate solutions, but left, without
previous evaporation to dryness, a skeleton residue of white and completely isotropic silica,
amounting to some 1.3 per cent of the original material. As showing the approximate propor-
tional amounts of tiie olivines and pyroxenes the method is, however, unquestionably useful,
particularly when accompanied by the microscopic examination. The late work of Borgstrom, "
Prior, *" and Cohen "^ furnishes an excellent example of what may be accomplished, and certainly
the carefully detailed analyses of MingayC* leave little to be desired. The practice, %vithin
the prescribed limits, of stating what elements were looked for and not found, as well as what
were foond, is worthy of commendation and should be followed. Such extremely careful and
detailed work as has in times past been done by Fletcher ^ is quite beyond the possibility of
general practice, and with the now almost universal use of the microscope is perhaps not essential.

<■ Der Meteorit von St. Michel, Bull. Com. Geologique de Finland, No. 34, 1912.

b The Meteoric Stones of Baroti, etc., Min. Mag., Dec, 1913.

c On the Meteoric Stone which Fell at the Mission ol' St. Marks, etc. Ann. South .\frican Museum, vol. 5, 1903.

rf Notes on, and analyses of, the Mount Dyrring, Barraba, and Cowra Meteorites. F.ecords Geol. Surv. N. S. Wales, vol. 7, 1904.

« Min. Mag., vol. 10, No. 43, 1894, and vol. 13, No. 59, 1901.

o

MEMOIRS

OF THE

NATIONAL ACADEMY OF SCIENCES

Volume XIV

SECOND ]vikm:oir

WASHINGTON

GOVERNMENT PRINTING OFFICE

1919

NATIONAL ACADEMiY OF SCIENCES.

Volume XIV.
SECOND m:km:oir.

COMPLETE CLASSIFICATION OF THE TRIAD
SYSTEMS ON FIFTEEN ELEMENTS.

BY

H. S. WHITE, F. N. COLE, and LOUISE D. CUMinNGS.

54061°— 19 1

TABLE OF CONTENTS.

Page.

Part 1. Triad syBtems on 15 elements whose group is of order higher than unity 5_

By H. S. White.

Part 2. Trains for triad systems on 15 elements whose group is of order higher than unity 27

By Louise D. Cummings.

Part 3. Groupless triad systems on 15 elements 69

By H. S. White and Louise D. Cummings.
Part 4. Structure as defined by interlacLngs, heads, and semiheads; a complete census of triad systems in 15

elements - 73

By F. N. Cole.

Pajit 5. Sequences and indices for all groupless triad systems on 15 elements 81

By Louise D. Cummings.

3

PART 1.

TRIAD SYSTEMS ON 15 ELEMENTS WHOSE GROUP IS OF ORDER HIGHER

THAN UNITY.

By H. S. White.

§ 1. INTRODUCTION.

The problem which first led to the study of triad systems (Tripel-systeme) was proposed
in the first place for 15 elements, Kirlonan's "Fifteen schoolgirls problem." ' In various
journals, during the past 60 j^ears, are described many different ways of consti-ucting triad
systems on 15 things. There was not known, prior to 1912, any short and decisive method
for comparing systems apparently different; accordingly duplicates were produced, and up to
1912 only 10 noncongruent systems had been found. A triad system, hke any other airange-
ment of elements, may have its appearance changed while its structure is unaltered by a per-
mutation among the elements — a substitution. When one system can be derived from another
by a substitution, the two are called congruent or equivalent; otherwise they are incongruent or
nonequivalent. If there are substitutions which transform a system into itself (usually per-
muting its triads among themselves), aU such substitutions together are called the group of
that triad system.

The number of elements in a triad system must be of the form 6n+l or Gn + 3, where n
is an integer.^ Such numbers are 3, 7, 9, 13, 15, 19, etc. For any number of elements under
15 the exact number of nonequivalent triad systems has been known for some time, namely,
one S3^stem for each of the numbers 3, 7, 9, while for 13 there are two systems. These two
systems on 13 elements have different groups, of orders 6 and 39, respectively. One might
anticipate that for numbers above 13 the same thing might happen, so that the group would
serve as a distinctive mark or characteristic for its sj^stem. Miss Cummings has shown, hovr-
ever, that for 15 the case is different; that sometunes the same group belongs to two or more
Incongruent S3^stems.'

.(Vnother test has been employed by E. H. Moore to prove the nonequivalence of two
systems.^ K among the 35 triads in 15 elements there occur 7 triads on any 7 elements, exclu-
sively, the larger system, which Moore denotes by A15, is said to contain the smaller, a A,.
Then if one Au does contain a A7, while another does not contain any A7, the two are obviously
incongruent. But this is of course not a conclusive test for equivalence, since Miss Cummings
has found 23 incongruent Aij's, each of which does contain a A,.

Probably it has been the lack of convenient and rehable tests for equivalence or non-
equivalence that has deterred investigators from the task of finding how many essentially
distinct triad sj^stems are possible in 15 elements. But now we have available two distinct
methods of comparison, both of which have given rehable results, positive as well as negative,
in all cases where they have been tried, although then- value as positive tests of equivalence
still lacks a priori demonstration. One of these, the method of trains, uses a given triad system

' T. p. ICirkman: On a problem in combinations. Cambridge and Dublin Mathematical Journal, vol. 2 (1847), p. 191. See also Note on an
unanswered prize question, ibidem, vol. 5 (1850), p. 255.

• Nctto: Substautiotunlheorie, p. 220, 5 192.

• L. D. Cummings: Note on the omupafot triple systems. Bulletin of the American Mathematical Society, vol. 19 (1913), p. 355.

• E. Hastings Mooro; Conurning triple at/stems. Mathematische Annalen, vol. 43 (IS93), p. 271.

5

6 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

as a transformer of aU possible combinations of the elements in threes, and so assembles these
combinations into self -evolvent aggregates or trains, analogous to the "path-curves" of a
point transformation.* The entire set of such trains, abstractly considered, belonging to any
triad system on 15 elements is characteristic for that system and for aU systems equivalent to
it. A second method, that of sequences and indices, is described and exemplified by Miss Cum-
mings in her dissertation.^ These two methods have given, so far, accordant results, and both
lead to the easy discovery of the substitutions which transform two equivalent systems into
each other.

With these two direct and simple methods available for testing triad systems it is reason-
able to attempt the complete enumeration of those on 15 elements. In this part, however,
I shall undertake an easier task, that of constructing all triad systems on 15 elements whose
groups are of order higher than unity ; aU which have gi-oups containing at least one operation
different from the identity. In this research the group, and indeed the particular substitution,
will furnish the starting point.

§2. SUBSTITUTIONS ADMISSIBLE IN THE GROUP OF A A15.

Not every type of substitution can occur in the group of a triad system. Consider in par-
ticular a system A15 whose elements are denoted by letters, and S a substitution of its group.
Represent S, as is uniquely possible, by a product of mutually exclusive cycles:

8^{A, A, .... Aa) {B,B, .... Bf,) {C,C, .... C,)

It is required by the definition of a triad system that every pair of elements shaU occiu' in some
triad, and that no pair occur twice. The pair AJi^ must occur in some triad, as A^BiC^. As a
special case, the third element 0^ may be in one of the cycles {A) or (J5). We note that in S the
order c of cycle {C) must be a factor of the L. C. M. of a and 6; otherwise the pair AJi^, would
occm* in two or more triads. Let a = mP, b = ma, where a and /3 are relatively prime; then

must

map = yc (7 an integer) .
For similar reasons

a is a factor of the L. C. M. {h, c), ml3 divides mac;

his & factor of the L. C. M. (a, c), ma divides mfic.
Hence, as a is prime to /3, c is a multiple of the product a/3, c = na^, or

7 c = 7/ia^ = 7na/3,

therefore m = yiJ.. Now we find, more exactly, that

ffl( = 7^1/3) divides the L. C. M. {yixa, tia^)

which is na . L. C. M. (7, P).

Therefore

7 /3 divides a. L. C. M. (7, /3)
and

7 o divides /3. L. C. M. (7, a)

Hence 7 is prime to both a and |3. We have accordingly for the three orders of cycles {A),

iB), and (C),

a = M/37, o = nya, c = naff;

and we have proved this theorem:

If in the group of a A15 there occurs a substitution containing two mutually exclusive
cycles of orders a = pi37, 6 = M7a respectively, then that substitution contains also a
cycle (possibly coincident with one of the first two) of order c = txaP; a being prime
to /3, and 7 some common factor of a and i but prime to a and /3.

• H. S. White: Triple systems as transformations, and their paths among triads. Transactions of the American Mathematical Society, vol.
14 (1913), pp. 6-13.

> L. D. Cimunings: On a method of comparison for triple systems. Transactions of the American Mathematical Society, vol. 15 (1914), pp.
311-327.

No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 7

Particular cases under this theorem are:

a= 1, giving cycles of orders /iff, ixy, nffy;
a = /3 = l, giving cycles of orders m7, mt, m;
o = ;8 = Y = l, giving cycles of orders fi, ix, n;
H = l, giving cycles of orders ^y, ya, a/3.
Wlien two of the numbers are equal, they may refer to the same cycle, under conditions which it
is not necessary to examine here.

For our present purpose, the important deduction from this theorem is the simplest case,
namely, that if a=l and &=1, then must c = l. In other words, if there are two letters (or
elements) A and B invariant under the substitution S of the group, the triad containing these
two must contain a third, C, also unaltered by the substitution S. An immediate extension
of this corollary gives the following rule:

AU the elements not altered by a substitution in the group of a triad system (cycles

of period unity) must constitute a complete triad system contained in the principal

system, a subordinate system. Hence, for a A15, the number of cycles of period

imity in any operation in its group can only be 0, 1, 3, 7, or 15.

The numbers 9, 13 are excluded, since a subordinate system can not contain half as many

elements as the principal systems.

If the substitution S has cycles of different periods both higher than unity, a<h, then a
power S°- of that operator wiU have invariant a additional elements. If S^ still contains cycles
of different periods, another power can be foiuid to dimmish the number of different periods;
and ultimately some power of S wiU be fomid with 0, 1, 3, or 7 cycles of period 1, and having
the rest of its cycles of equal prime period.

Every triad s}'stcm on 15 elements, whose group is not the identity, is invariant imder
at least one substitution of one (at least) of the following seven types :

1. (5) (5) (5)

2. (3) (3) (3) (3) (3)

3. (1) (7) (7)

4. (1) (2) (2) (2) (2) (2) (2) (2)

5. (1) (1) (1) (3) (3) (3) (3)

6. (1) (1) (1) (2) (2) (2) (2) (2) (2)

7. (1) (1) (1) (1) (1) (1) (1) (2) (2) (2) (2),

where the digit in any parenthesis indicates the period of a cycle.

These seven types of substitution offer a natural means of classifying triad systems on
15 elements, provided they admit groups of substitutions. Under each type I shall construct
aU possible invariant triad systems, omitting obviously equivalent repetitions. In one case,
tj'pe 4, no system can exist; aU the others have actual systems. It wiU still happen that certain
systems occur in two or more classes, their groups containing substitutions of two or more of
these seven types. Further reduction is uJidertaken by Miss Cummings (Part 2), who furnishes
the proof of nonequivalence of the net residue, 44 systems. Of these 44, 24 were known
previously, more than haK of them discovered by Miss Cummings. It will be noted that the
20 new Ais's contain no A?; they are not of the " odd-and-even" structure; they may be called
"headless, " while any A? contained in an earher known A15 is termed its head. One headless
system onlj'-, discovered by Heffter, has been known heretofore. The discussion of groupless
systems, and their enumeration, is deferred to Parts 3 and 4.

§3. CLASS I, INCLUDING THE KIRKMAN AND HEFFTER SYSTEMS.

For each substitution the work of constructing systems of triads must be special; no gen-
eral conclusions are to be developed. Three requirements guide us: (1) Every pair of elements
shall occur; (2) no pair shall occur twice; and (3) the triads shall be grouped in sets conjugate
under the particular substitution (operator).

Denote by (S, the operator of type I, and take for its three cycles of five, respectively,
English and Greek letters and Arabic numerals.

S,={a bcde) (a /3 7 5 «) (12 3 4 5).

8 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol. xiv.

Triads of a system may contain elements from one cycle only, or from two, or from three.
Further subdivision gives us 10 classes. Indicate the numbers of triads in these several classes
as follows :

5Wi of type abc, Bu^ of type a/37, Bu^ of type 123,

5v, of ty])e aba, 5^2 of type afil, Bv^ of type 12a,

5w, of type ahl, 5^2 of type a0a, Bw^ of type 12a,

5t of type aal.

As all possible pairs must occiu", we have the diophantine equations to satisfy:

15Ui+ SVi +5Wi = 10 (i =

= 1,2,

3),

10Vi + 10Wi+i + 5< =25 (i =

= 1,2,

or 3=0).

1st.

2d.

3d.

t = l, w, =«2 =Ug =0,

Vi =v^ =^3 =2,

1,

0,

Wi=w2=w3 = r),

1,

2.

The solutions are

Of these the third solution may be dropped, since it is related to the above arrangement of
elements in the operator Si exactly as the first solution is to the operator S^, (or S{~^). The
first and second solution yield two kinds of systems, as follows:

First Und.—t = l, Wi = 0, ^1=2, 1171 = 0 (i = l, 2, 3).

The triads enumerated by 5^ = 5 shall be these: aal, 6/?2, cy3, <?54, ee5. With the pair ab

must occur 5; for if the triad were ahy, then must follow in cyclic order hc5, , ea/3. There

remains, therefore, no letter of the second cycle to join with the pair ac, since we have abeady
the pairs aa, cy, and ay, a/3, c5, ce. The same reasoning excludes the combination ahe. This
leaves only the possibility ab5. Under 5, there follow from ab8 the pairs ce, ay, and there is
left for the pair ac only the letter /3, hence ac/3. Similar reasons apply to triads from the second
and third cycles and from third and first cycles. The entire system is accordingly determined
uniquely, and is given by the following seven triads and the conjugates derived from them by
the operator Si :

System II: aal; abS, ac0; a/34, a72; 12d, lob.
Second land. — 1 = \, Ui = 0, 'i'i=w'i = l (i = l, 2, 3).

After the five triads, aal and its conjugates under Si, the apparent possibilities are a&7
and abh.

Assume fii'st a triad a67. Its conjugates contain the pairs ac, ad, and /3a, /3f, so that there

remains a possibility of the triad affc. But also there are found the pairs yc, yb; whence the

pair a7 can join with no letter from the first cycle, but must be completed by either 2, 4, or 5.

We examine these successively.

(1) Triad a72.

Conjugate triads will involve the pairs U, 1/3, 2a, 27, 3/3, 35; and already we had la, 2/3, 37.
Therefore the pair 13 can not be completed from the Greek cycle, nor can its four conjugate
pairs. But ^3 = 1, and for the pair 12 we have available the letter 5 only. Necessarily one
triad is 125. We must complete

13 by either b, d, or e,
ac by either 2, 4, or 5.

The hypothesis of 136 would result in excluding ac from 2, 5, and 4. The hypothesis 13e leaves
open only the alternative ac2, having ebl as a conjugate, which is inconsistent with 13e. Dis-
missing, therefore, 136 and 13e, we have 13d, requiring ad. The remainder of the system
follows uniquely from these.

No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 9

System 12 : a.al, aby, ac4, a^c, a72, 125, 13(Z, and their conjugates under the operator S^.

(2) Triad a74.

By similar scrutiny it is found that triad a74 excludes the pair 1 2 from completion by the
Greek cycle, but allows the requisite triad 135 and its conjugates. To be tested are now 12c,
12(/, and \2e. Of these, 12c leads to ac5 and so to cc2 inconsistent with 12c; and the thirtl
alternative 12e woidd exclude the pair oc from the Arabic cj^cle altogether, and is hence inad-
missible. There remains I2d, which allows further ac2. Complete the system thus, uniquely,

aa\, aby, ac2, 12d, 135, afic, ayi.

This system difTors from 12, above, only in the interchange of the Greek cycle and the Enghsh.
Omit it therefore as a duphcate.

(3) Triad ay 5.

By considerations like the above we find that ay5 and aal with their conjugates will exclude
12 from completion in the Greek cycle, and wiU necessitate the occurrence of 13e, which in
turn requires 075. Next, 12 must be joined with either c, d, or e. The fii-st and last of these
lead to inconsistent triads, and we have remaining 12fZ, and therefore ac2. The sole admissible
system here is therefore that given by these seven:

aal, aby, ac2, ay5, a/3c, 13e, 12c?.

But this triad system is related to the operator S^,

S{'={a yeP5)(lS524)(acehd),

in exactly the same way as system 12, above, is related to the operator S^,

S^=(a 6 c (Z c) (a /3 7 5 e) (1 2 3 4 5),

We may therefore omit it as a duplicate.

This exhausts the possibihties under the fii-st assumption, i. e., that the second triad was
aby. Test now the other alternative: Assume abS. By trials similar to the foregoing, we con-
struct the following five sj-stems, exhausting the possibihties:

(a) aal, abS, ayb, ac4, o|83, I3d, I27.

(b) fflal, abS, ayb, ac5, a/33, 13c, I27.

(c) aal, abd, ayb, ac4, a/35, 13d, 12€.

(d) aal, abS, ayb, ac5, afi5, 13c, 12e.
System 13: aal, abS, ayb, ac2, ajS-i, 13/3, 12d.

Four of these systems are equivalent to the two already found. Notice first that two of
them, (d) and (c), reduce to (a) and (b), respectively, by the reversal of order in each cycle, i.e.,
by using for operator S^* in place of 5",. Then (a) is seen to become system 12 by exchange of
English and Arabic cycles. To show that (b) is congi'uent to S3'stem 12, replace operator (Sj
by fS'i' in a changed order of cycles, thus

S, =(a & c d e) (a j3 7 5 e) (1 2 3 4 5)
S','=(a 5/3 6 7) (14 2 5 3) (adbec)

The same substitution wdl change (b) into 12.

The net result of this section is therefore the construction of three systems, II, 12, and 13.
The fu-st and third of these are the well-kno^vn systems of Heffter and Kirkman, respectively, the
second hitherto unknown.

10 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

§4. CLASS n. EIGHT SYSTEMS INVARIANT UNDER A SUBSTITUTION OF THE TYPE (3)\

A first separation into two divisions is found when we distinguish tliree kinds of triads.
Denote the elements of the five cycles by the letters a, b, c, d, e, attaching to each in their order
the subscripts 1, 2, 3:

/S=(<i, ttj a,} (6, 62 63) (c, C2 C3) {di d^ d^) (e, Cj eg).

Denote any three different letters from among these five by Tc, I, m, leaving the subscripts
undetermined. Then obviously every triad that can occur is of one of the three types repre-
sented by

Iclvk (Denote their number by u) ,
IcTcl (Denote their number by 3u),
1dm (Denote their number by 3w).

In a given triad system, every possible pair of elements occurs once. There are in the entire 105,

15 pairs of type Tck,

90 pairs of tjrpe Tel, makmg in combination 35 triads.
Compare these with the numbers of each found in triads of each of the three types above. We
find the necessary relations :

3u + 3-y = 15,

6i; + 9w = 90,

« + 3i; + 3iy = 35

We can have, therefore, the two kinds of S3'stems, divisions 1 and 2 :

xt

V

10

o

5

3
0

8
10

Division 2

Division 1. — As u is 2, assume triads a^a^a^ and &1&2&3. Now subdivide further, and let
Ic represent either an a or a h, and Z, m represent any two of the letters c, d, c, subscripts being
disregarded. Since 311 = 9, let us distinguish —

Of tyi)e ahm, 9=3.3 triads.

Of type IcU, 3x triads,

Of type Hm, Sy triads.

Of type Imm, Sz triads,

Of type cde, Bt triads.

These numbers have to satisfy the following conditions:
Triads, 3a; + 3(/ + 32 + 3< + 2 + 9=35;
Pairs H, 18 + 6x + 6y = 54,
Pairs ZZ, 32; + 32 = 9,

Pairs Im, 3?/ + Qz + 9t = 27.

These conditions admit three solutions, which wUl be taken up in order.

Family a
Family 6
Family c.

X

y

z

t

3

3

0

2

2

4

1

1

1

5

2

0

NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 11

Family la.

The only possible schedule, prior to tlie assignment of subscripts, is this:
aaa, hhh; ace, add, ace;
abc, abd, abe;
bed, bee, bde;
cde, cde.
For the hypothesis ace, add, and bee, for example, would require a, to unite with two e's in
the one triad left for it after the three where it occurs with b's; whereas ee is found already
with b in bee, whence the impossibility.

With entire generality we fix, in the above schedule, the subscripts in eight typical triads:
a, a., a^; &, h^ b^; Oi c, C3, a^ d^ d^, a, Cj €3]
(?! &i Ci, ttj 62 di> «i ^3 ^v

Six trials now sufBce, to reduce the possible ways of supplying subscripts for the remaming
triads to the following two, supplementary to the above:

System II, Ij: c, di b^, c, fZ, e^, c^ d^ e^;

6, C3 fj, &i di Ci.

System II, 1,: c, d^ 6,, c'l <?, (!„ c^ d^ e^;

h <^2 «1) ^1 ^^2 «3-

Family lb.

Prior to assigning subscripts, the only possible schedule is this:
aaa, bbb; ace, add; cee; cde;
abe, abe, abc;
bed, bed; ode, bde.

(The other apparent possibihty, aec, bdd, cee, leads to the impossibility of five times three
pairs ce.)

Subscripts may be fLxed arbitrarily, in order, in the following triads (subject, of course,
to the cyclic permutation of 1, 2, 3):

a^a^a^, bfij)^; afi^Ci, afi^e^^, afi^Ci,
aiC^d^; OiC^Cy, Qid^d^.

The remaining five sets arc foimd by trial to admit two arrangements only. With the above
we may unite either of these:

System II, 1,: biC^d^, System II, 1,: b^e^d^,

^i<:A> b^c,d„

Cid^Cs, c^djCi,

Family Ic.

Tlio association of letters in triads, disregarding subscripts, may foUow two schedules.
The first possible schedule is this:

aaa, bbb;

ace, cdd, cee;

abc, abd, abe;

ode, ode, bde, bed, bee.

In the first six that follow aaa and bbb we may dispose of subscripts by fixing a,5if,, then fljCCs,
c.rfjt/j, 0,^2^3. So far letters d and e are exchangeable, also subscripts 2 and 3. This observation
reduces to five the number of essentially different ways of affixing subscripts to the next two
triads, abd and aic, viz:

(1) a^dfi^, (2) a,dfi„ (3) a,dj)„ (4) a.d.b^, (5) a,d,\,
0,6,63; o-ye^h; aiCz&a; O1C362; ciie^by

12 MEMOIRS NATIONAL ACADEMY OF SCIENCES. pol. xiv.

Of these five only the first and the fifth can be completed to full systems. They give each a
unique system.

System II, I5: afi^a^, bfij)^; System II, l^: afi^a^, hfi^b^;

'^I^2'"3> ''l''2'^3) ^1^2^3' ^1^2^31 ^fli^Z) ^l^i^it

(^i^'i&i, diCfls, diC^a^; dyC^a^, d^e^l^, d^e^a^;

CidJ>3, c^ej)^. CjdJ>2, Cicfis.

The second of the two possible schedules diverges from the first in its fifth triad and is the
following, with seven triads void of subscripts:

«ia2«3. bfi^h^ ^iCjCj, c,<Z2(?3, diC^e^;
6jC,(Zi; adb, adb, ode;
ceb, cch, cea; abe.

On trial only two ways are found for affixing subscripts to the triads still left blank, and these
differ only by the interchange of 2 and 3. Hence we have finally in this family Ic only the
one additional system:

System II, 1,: Oia^a^, bfij)^, a^CiCs, c^d^d^, die.j^, b^c^di
a^dfi^, a^djji, aid^e^;

^1^1^3J ^l^2^2J ^1^3^!^

Division 2. — In the second principal division of this class there are five triads formed from
single cycles of thi"ee letters, and these are necessarily

a,a2ffl3. ^i&2&3) C1C2C3, d^d^d^, c^e^c^.

The ten sets of three triads which are to complete the system contain, as we saw, each three
(UfTcrent letters; and no combination of three letters can be repeated, as is shown readily by
writing the diophantme equations of condition. There are only 10 combinations possible, so
that each wiU occur once, i. e., in one set of thi"ee triads. If we attend first to the triads con-
taining a, remembering that in each of the five cycles the subscripts ma}" be permuted cychcally,
it is found that there are but two distinct kinds of sets. To characterize them, we may best
use Cole's term, interlacing; either there is an interlacing in the triads containing sj^mbols a
or there is none. In the first case the four symbols concerned may be given index or subscript
1, b^c^di^e^, and the four triads may be:

afiiCi, a^d^e^, flj^i'^i «2fifi-

In the second case we find, Iiy permutations of letters and of the five cycles independently,
that we may denote four triads by

afiiC^, aidifi, ajb^d^, a^CjC^.

Each of these is completed, in two equivalent ways, to a full sj^stem.

System II, 2,: Oi?>,c,, a^d^e^, a^bid^, a2C,e,, afi^ei, a^c^d^,

biCJs, byC^e^, hid^e^, c^d^e^.
System II, Zj: afi^c^, ald^e^, aji^di, a^sC^, a^c^e..^, a^c^di,
he J 3, 6,C3fi, h^d.es, c^d^e^.
This last system is found to have no interfacings whatever, and so is evidently the exceptional
system in Cole's enumeration, the headless cyclical system of Heffter.

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 13

The preceding system, however. No. II, 2,, is found hy quite obvious indications to be
equivalent to System II, 1 , and the one is transformed into the other by the folk)wing substitution:
II, li: Giflja, 61^362 C1C2C3 dicZjtij 61^263.
II, 2i: bib-fi^ (tid^a^ <^i'h''i <'i<'2<^3 di^'i^i-
Further, the two systems II, I3, II, I4 are equivalent by the interchange of letters
(ffj «,) (6162) (Cs C2) W3 ^:) (^1^3)- In conclusion, therefore, this class II contains not more
than seven triad systems that are essentially distintit, Nos. 1,, I2, I3, I5, Is, I7, 23.

§5. TRIAD SYSTEMS INVARIANT UNDER AN OPERATION OF TYPE (1)(7)(7).

There are three distinct systems, and no more, wliich admit a substitution of period 7.
The proof is almost intuitional, no long analysis being needed. Indicate the 1+7 + 7 letters,
and the substitution, thus:

S={A) [a h c d efg) (12 3 4 5 6 7).

By diophantine equations of condition it is found that there must be seven triads — a system
A7, constituted within the one cycle, e. g., the cycle of letters {abed efg); seven triads like
Aal, and 21 triads composed of one letter and two digits.

Take S to be such a power of the cyclic operation that the included A? consists of the triad
ahd and its six conjugates. Of the 21 now remaining to be determined, thi-ee must contain the
letter a. Indicate on a circle in their order of sequence the seven digits at equal intervals. Since
a and 1 are already together hi the triad Aal, we have now to connect in pairs, by tlirce chords,
all six digits 2, 3, . . ., 7. No two of these chords can be of equal length, since then the rotation
effected by the substitution S would produce a repetition of a pair, contrary to the definition of
a triad system. Trial shows at once that chords 23 and 25 would lead necessarily to equality
of at least two chords, hence these are excluded; while 2i, 26, and 27 lead to one solution each,
as represented in figure 1.

Fig. 1.

Accordingly the three possible systems are given by the following, with the triads conjugate to
them imdcr the operation S.

System III, 1: Aal, abd, 024, a37, a56.

System ni, 2: ylal, abd, a26, a34, a57.

System III, 3: Aal, abd, a27, a36, a45.
The fu'st of these is evidently the one ordinarily constructed from the A^ by the method
for passing from n to 2n+l; viz. by substitution of coiTesponding elements from the second
cycle in triads of the first, there are formed from abd, for example, three others: a24, Ibi, I2d.
The 21 of this structure, the original 7, and the 7 like Aal make up the complete set. It is,
prima facie, the Kirkman system (No. III^ of Miss Ciuumings's dissertation).

§6. NO TRIAD SYSTEM CAN EXIST THAT ADMITS A SUBSTITUTION OF TYPE (1)(2)'.

If a triad system can be invariant under an operation of the type (1) (2)', denote the element
in the unit cycle by A, and in each duad cycle denote one element by a letter, the other by a
digit:

S=U) (a 1) (6 2) (c 3) (d 4) (e 5) (/6) (g 7).

14 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol. xiv.

Every pair al from one duad must occur in some one triad and can be associated with no third
element save A; hence seven triads are like Aal. The remaining 28 triads may be an-anged in
two pairs of classes, each pair equal in number by the symuietry of the operator S in letters
and digits, thus:

X triads like abc, x triads like 123;
y triads like abl, y triads like al2.

Since the nature of the system calls for 21 pairs of letters, as db, and 21 pairs of digits, while
there must be 42 mixed pairs such as a2, we have the equations of condition:

3a; + i/ = 21,47/ = 42,

insoluble in integers. Hence no triad system of this type can exist.

§ 7. SUBSTITUTIONS OF TYPE (1)^(3)* AND THEIR INVARIANT TRIAD SYSTEMS.

Denote by A, B, C three elements not affected by a certain substitution, and by (Oj a^ O3),
{h^ 6, 63), (Ci C2 C3), {di d^ d^, four cycles of period 3 in that substitution.

S^{A) {B) (C) {a, a, a,) {h, I, \) (c, c, c,) (rZ, d, d,).

Any triad system invariant imder S must contain the triad ABC; and equations of condition
show that of triads and sets like a^a^a,, b^h^c, there are respectively either 1, 3 or 4, 0. It
will be shown that this gives in letters, irrespective of their subscripts, five possible schedules.
There are 18 triads like Aah, falhng uito tliree sets of twice three. By the subscripts of these
latter each of the first five schedules is made a source of five subclasses.

The equations of condition show further that there must be either two or four sets of thi-ee
triads like abc, formed from three different cycles in S. Hence there are at least three different
pairs of letters, as ab, that can not occur more than twice (i. e., 2X3 times) in conjunction
with letters A, B, or 0. The complementary pairs are excluded thereby also ; e. g., a set of triads
Aah would imply another set Acd, since all possible pairs of elements occur in a system A15.
Where four triads of period 1 occur, like afi^a^, and therefore four sets like abc, no pair ab
can occur with two letters from the thi-ee A, B, C. We arrive by such considerations at the
first five main divisions.

Case 1. — Four triads like afi^a^, fom- sets like abc. Hence the schedule:
afi^a^, b^b^bs, c^c^c^, d^d^d^
abc, abd, acd, bed;
Aab, Acd; Bac, Bbd; Cad, Cbc.

Cases 2 and 3. — One triad of period 1, d^d^d^; two sets drawn from three different cycles.
Consider the triad sets containing pairs aa, bb, or cc. With these may occur the letter d in
3 2 1, or 0 sets. If in none, then we must have (if lettera are chosen suitably) aab, bbc, cca.
But tliis leaves us to construct triads in which d^, for example, shall be miited with all nine
letters a, b, c. Three of these are of course m sets Aad, Bbd, Ccd, implymg sets Abc, Bac, Cab.
In the remaining two sets of three, d must be united twice with each letter a, b, and c, a plain
impossibihty. Similar absurdity results from assmning two sets like daa, dbb. Ilyisotheses of
one such set, or of three such, are admissible, as follows:

Case 2: Assume two sets, a<id and bba. Trial of ccb leads to absurdity, whence we must
have cca. In full, therefore, this schedule is:

ABC; d.d^d^; Abc, Aad, Bac, Bbd, Cab, Ccd; aad, bba, cca; led, bed.

Case 3: Assimie thi-ee sets with d, aad, bbd, and ccd. The full schediile will be:
ABC; d^d^d^ Abc, Aad, Bac, Bbd, Cad, Cbc; aad, bbd, ccd; abc, abc.

Cases 4 and 5.— With ABC and d.dj^ as in the preceding case, take dupUcate pahs of
letters with two of the isolated elements ABC; e. g.,

Aab, Acd, Bab, Bed; Cac, Cbd.

There are yet to be constructed 15 ( = 5X3) triads. Of these five sets three have to contam a
doubled letter, as aa, while the other two consist of distmct letters. Listing the pahs that

No. 2) TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 15

must occur, we see that ad and be must occur in triads with a double letter, so that either ac
or hd, not both, will also occur in such a triad. There are accordinj^ly two possible schedules:

Case 4: Triads ABC, ddd; Aab, Acd, Bab, Bed; Cac, Cbd; aad, bbc, cca; abd, bed.

It is noticeable that in this arrangement no two of the letters abed are interchangeable;
the same is true in the next, the final case.

Case 5: Triads ABC, ddd; Aab, Acd, Bab, Bed; Cac, Cbd; aad, bbd, cob; abc, acd.

After the above distinction of five schedules there is for each case a subdivision into species
by means of the pairs of subscripts attached to letters in the triads with A, B, and C. The
numbers of such species, a priori, are for case 1, fom*; for case 2, four; for case 3, three; and
for cases 4, 5, four each. Some of these are realized by two completed systems, some by one,
or by none; so that, in all, 26 systems apparently distinct are found invariant under an operation
or substitution (1)^(3)^

Species in case 1. — In case 1 the sets of letters a, b, c, d are indistinguishable, and are dis-
tributed to the exchangeable letters A, B, C in all the complementary pairs. As a means of
abbreviating tabulation, write these triads in the order

Aab Acd
Bac Bbd
Cad Cbc,

and instead of rewriting these letters with subscripts, write the subscripts only. On each letter
independently the subscripts may be changed if we do not change the cychc order 123; and this
order may be reversed to 132 for aU four sets of letters simultaneously. The first three pairs of
subscripts may be fixed arbitrarily as 1, 1, thus: ajji, c^d^, a^Ci. We need only write the three
remaining pairs in their relative positions, without letters:

(a)
11

11, m
11 11

11, (7)
12 12

11, (S)
12 11

12
13

These four cases exhaust the possibilities; for we can reduce aU others to these four by permis-
sible interchanges of letters and subscripts. There should be nine cases where indices 2, 3 do
not appear in the first row or the second and we see that (a) represents one, (/3) four, and (7)
four. If the pair 1 2 or 1 3 occurs in the second line, the reduction is less obvious, but not
intricate, as one example will show. Let these six pairs in same order be afii, c^d^; aiC^, b^d^;
a^dj, biCj (representmg also, of course, their 12 conjugate pairs). Write subscripts only, and
change those of d cychcally by writing 1 for 2, etc. This leaves

11, 13; 11, 11; 11, 13.

Next, c is written for d and vice versa, as the letters are of equal significance; then for 31 write
12, one of its conjugates. Now we have

11, 12; 11, 13; 11, 11,

since second and third lines have exchanged subscripts. But this is case (5) by permutation of
letters A, B, C. By such verification we confirm the completeness of tliis list of four species.

Species in case 2. — In case 2 tliere is no distinction between letters b and c, but a and d are
not interchangeable with them or with each other. First we fix triads Ad^a^, Bdfii, Cd^c^, so
that all indices are of determinate meaning except for a choice between the orders 123 and 132.
Since the combination bed is to occur in two sets of three, these can onl}' be biC/l^ and biCjd^;
accordingly the triads Abc must include AbiC^. The pairs with B and C admit some freedom
of choice still, all alternatives being reducible to these four following:

(a) CaJ), m CaJ), (7) Ca,6j (5) Cafi^
BaiCi BttiC^ BuiCj Ba^c^

Species in case 8. — Letters a, b, c are not yet distinguishable separately in case 3. Fix
first, as in case 2, the triads Adfii, Bdfi^, Cd^Ci. Next, two sets of triads are to contain abc,

16

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV

there are three possibilities, and these serve to fix uniquely also the subscripts of the remaining
sets of triads. There are then three schedules:

(o) &,Ciai (/3) iiCiflj (7) 61^203

tiCjOj i^Cia^ hiCiO^

AhiCs Ab^c^ -46,c,

BCiUs Bcitts Bc^a^

Cafis Cafii CaJ)j^

Species in case 4. — With the letters A, B, C, there occur in case 4 twice the pairs ah and cd.
With either one of these the subscripts can be taken as 11, 11 ; then with the other the two can
be, in their order, either 12, 12, or 12, 13. In the third set, ac and hd, we are still free to fix
arbitrarily one pair of subscripts, as 6,(Zi, leaving two alternatives for the other pair ac. Four
partial schedules result, as follows, for the pairs with A, B, and C.

(a)

m

(t)

(5)

A

afii

Cidt

11

11

11

11

11

11

B

afis

cA

12

12

12

13

12

13

C

a^Ci

hA

12

11

11

11

12

11

Species in case 5. — The same four schedules, for the same reasons, are valid in case 5 as in
case 4.

The foregoing 19 schedules can be completed to actual systems, as is readily seen by inspec-
tion, in the following 26 ways only:

System V, la: ABC, a^a^a^, l^ljb^, c.c^c^, dA(h: Aafi^, Ac,d^, Ba^c,, 5&,J„ Ca^d^, Cb,c,;

afi^Cs, afiAt «iC2<^3> '^iCa'^a-

A second system, of course, equivalent to this, differs only in the cyclic order 132

instead of 123.
System V, 1^ is impossible.
System V, I7: ABC, a^a^a^, hfi^h^, c^c^c^, dAd^i- Aafi^, Ac A, Ba^c^, BhA, CaA, Ch^c„.

afi^c.,, afiA, fliC3<^i. &iCi<^3-

An equivalent system is converted into this by the substitution {ac) (bd) (23); or

two others similar.

System V, 15: ABC, a.a^a^, bfijb^, c.c.fi^, dA^z- ^«i^, ■^<^A, -SaiC,, BbA, OaA, Cb.c^.

afix^, afiAi <''i<^Ai b^cA-

' System V, 2a: ABC, dA^z: AaA, -^^iCi, BbA, Ba,c„ CcA, CaJ),; a,aA, ^^0-3, ^1^,03/

b^c^ds, b^cA-

System V, 2/3: ABC, dA^^ AaA, ^^i^i, BbA,.Ba,c„ CcA, CaJ),; a,aA, ^A^z, c.c^a^;
b^c-fli, b^cA-

System V, 2t: ABC, dAds,- AaA, Ab,c„ BbA, Ba,c„ CcA, CaA; «iM3. b^a^, c^cm^;

bicA, biCA-

System V, 28: ABC, dA^^; AaA, Ab,c„ BhA, Ba^c^, CcA, Cafi^; Oia^d^, b^a^, c^c.a^;

b^c^d^, bicA-

System V, 3o: ABC, dA^^; AaA, Ab,c.„ BbA, Ba,c., CcA, ^a^- a^^^A, ^Ms, CiCj^s/

System V, 3;3: ABC, dAda,' AaA, Ab.c^, BbA, Ba^c^, CcA, Ca^; "iMs, ^fiA, c,cA;

System V, 37: ABC, dAdz; AaA, Ab,c„ BbA, -BffliC,, CcA, <^aA; OiMs, &1M3, Ci^A;

0,6263, ffli&jCj.

System V, 4a (1 and 2) : ABC, dA^^;
Aafii, AcA, (1) aiaA, &1V2) CiMa,' <iJ>3^2, b^cA;
Bafi^, BcA, or

Ca,Ci, Cb,d^, (2) a,aA, ^A^s, c^c^a.,; aJ)A, \cA-

These two are equivalent; (1) is converted into (2) by the substitution Ahh) (d^dA)
(23) (AB).

No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 17

System V, 4/3 (1 and 2): ABO, d/lj,;
Aafii, Acidi, (1) a^a^d^, hfi^c^, CiC^a.^; afi^d^, b^Cid^.
BaJ)2, Bcid.^, or

Ca^c-i, Cb^di, (2) aiO^di, hfi^c.., CiC^a^; afi^d^, b^Cidj.

System V, Ay (1 and 2): ABC, d,d,d„-
Aafii, Ac^di, (1) a^ad3, ^Vu c^c^a^; aj)^^, h^Cid^.
Bafi^i Bcid^, or

CoyCi, Chidi, (2) a^a^du bfixj, CiC^a^; aj)d^_, hiC^d^.

System V, 43 (1 and 2") : ABC, d^d.^!
Aafii, Ac^d^, (1) a^a^^, hfi^Ci, CiC^a^; aiMi, b^Cid^.
Bafi^, Bcidg, or

CttiCj, Ch^di, (2) aiO^dj, hfi^c,, c^c^a^; afi^d^, h^c^d^.

Equivalent systems, by the substitution {a^a.fl^) (CiC^c^) (2;?) {AB).

Systems V, 5a (1 and 2) : ABC, dyd^d^;
AaJ)^, Acidi, (1) ttjajdi, h^h^d^, c^cjbj^; afi^c^, a^c^d^,
BaJ)^, Bc^d^, or

(7aiC„ CT,di, (2) aia^d^, ifi^d,, c^c.J>:,; aj).,c.^, a^^Cid,.

These two are equivalent in the same way as V, 4a, 1 and 2.
Systems V, 5/3 (1 and 2) : ABC, d,dj,;
Aafii, Ac^d^, (1) UjU^di, 6463^3, Cicjb^; afi^c^, a^Cid,,
Bafi^, Bcjd^, or

Ca^c^, Cb^di, (2) a^a^l^, bfi^d^, c^cfi^; afi^c,, a^Cid^.

Systems V, 57 (1 and 2) : ABC, did2d3;
AaJ>i, Ac^d^, (1) a^a^d^, bj)^.^, c^c^bi; afi^c^, a^Cid^,
Bafi^, Bcids, or

CttiC,, Cbidi, (2) a^a^d^, bfi^d^, Cicj)^; afi^c^, a^c^d^.

Systems V, 55 (1 and 2) : ABC, d^d^d^;
Aafii, Acid^, (1) afi^d^, bfi^d^, c^c^b.^; afi^c^, a^Cid^,
Bafi^, Bc^d^, or

Ca^c^, Cb^di, (2) aitt^di, bfi.^^, c^c^b^; afi^Cz, a^Cjd^.

Two equivalent systems, as under V, 45.
Beside the eqxiivalences already noted, one less obvious is that of systems V, 3a and V, lo.
which Miss Cummings wiU establish in Part 2. That done, we shall have invariant under this
tyi^e of substitutions (1)' (3)*, 21 distinct systems.

§8. TRIAD SYSTEMS WHOSE GROUP CONTAINS A SUBSTITUTION OF THE TYPE (1)' (2)».

Operations containing longer single cycles belong to fewer distinct types of triad systems,
While the fifth kind of substitution, (I)^ (3)*, gives rise to 21, the sixth, now to be examined,
will yield apparently more than 30. Actually the reduced number is the same, 21, for some
systems admit two or more substitutions of the same type. Tliis large number of systems might
weary the attention, were it not that novel points of difference are developed, in themselves
interesting.

Denote the 15 elements and the operation thus:

S={A) (5) (CO (a. &.) (a, \) {a, b,) {a, b,) (a, b,) {a, 6.)

Two triads conjugate imder S we shall call dual to each other; there wiU be six triads self-dual,
those containing pairs aj>i, aj)^, etc. According to the principles in section 2, ABC must be
one triad and the six self-dual pairs a J}, must be united with A, B, or C to form triads. Denoting
the three capitals generically by K, we could specify eight possible types of triads, but for
present purposes conjugates combine and form four types. With (or without) the aid of dio-
phantine equations, we find three sets of numbers for these four classes, as follows:
540G1°— 19 2

18 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

pe of triad.

Number

of triads in

a system,

/utian

5,

3,

1,

KaA

1,

3,

5,

ffliajOu

1,

2,

3,

ajajSit

7,

6,

5.

The doubles of these numbers, plus the 1 and 6 above mentioned, give the total of 35 triads for
a system. The second kind will be taken as standard ; the other two wiU be found to be reducible
to this.

By definition, each of the elements A, B, Cmust appear with six pairs of small letters a, h.
Since those not self-dual, as AaJ)],^ or ^aifflk, must occiu* in pairs, Aaih•^^, Aa\,hi, the self-dual
triads containing A (or B, C) must also be an even number G, 4, 2, or 0. We therefore divide
systems of this section into thi'ee principal classes.

In class VI, 1 : Six self-dual triads contain A;

In class VI, 2: Four self-dual triads contain A, two have B;

In class VI, 3 : Two self-dual triads contain A, two have B, two have C.

We can fix, for each class, these six self -dual triads and stiU retain freedom to exchange symbols
a, h in each pair; also to exchange certain subscripts. Beside ABC, assign to each class these
fundamental triads:

Class VI, 1 : Aafii, Aajb.^, AaJ}^, Aaja^, Aa^hr,, AaJ)^.

Class VI, 2 : Aajb^, Aafij Bajb^, Ba^h^.

Class VI, 3 : Aa^bi, AaJ)^; BaJ)^, Bafit,- CaJ)^, Ca^hg.

In class VI, 1 we are free to arrange that pairs with the symbol B shall be either aiaj^ or iib^
and that the pairs of subscripts shall be 12, 34, 56. Quite similar is the choice permitted in

VI, 2.

Class VI, 1: Ba^a^, Bb^b^,; Ba/i^, BbJ)j Ba^a^, BbJ)^.
Class VI, 2: i?ffl,a2, Bb^b^; Ba^a^, BbJ)j Aa^a^, AbJ)^.

In both classes it is still optional to exchange the Oi, bi of any conjugate pair as Ba^a^,
Bb^b^. Hence it residts that triads in Ccan all be put into a standard form Cuib],. Thus in classes
VI, 1 and VI, 2 the numbers of triads can be brought to agree with the second column in our
tabulation; that is, there will be two triads of type aiajay,, and six of type aiOjb^.

For both these classes, therefore, we \\Tite down at once the possible sets of triads containing
G, leaving for separate discussion the class VI, 3.

Class VI, 1 — Triads in C.

No. VI, li: CaJ>2, Oa^b^, Oa^b^, CaJ)^, Ca^b^, Ca^b^.
No. VI, la". CaJ)^, Ca^bi, Oa^bi, Cajb^, Cafi^, Ca^b^.
No. VI, I3: Oafi^, CaJ)^, Cajb^, Ca^h, CaJ)^, CaJ)^.

Class VI, 2 — Triads in C

No. VI, 2,: CaJ)^, Cajb^, CaJ>^, Cajb^, CaJ)„, CaJ)^.
No. VI, 22: CaJ)^, CaJ)^, Cajb^, Cap^; Ca^b^, CaJ)^.
No. VI, 23: CaJ)^, Cajb^; Oa^b^, Cajy^, Bajj^, OaJ)^.
No. VI, 24: CaJ)^, OaJ)^, Cajb^, Cafi^, Cafi^, Cajb^.

Class VI, 3 has two self-conjugate triads in A, two in B, and two containing C. There are
yet to be formed for each, four triads or two pau"s of conjugates. That is, for each of these
three letters we must combine four subscripts into two pairs. Notice that the six pairs are to
contaui each subscript twice. These may be grouped into one or more c}' cles ; for example, if 12,
23, 31 are among them, they constitute a cycle of thi-ee. Possible are apparently

Three cycles of two pairs;

One cycle of two, one cycle of four;

Two cycles of tluee;

One cycle of six.

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CITMMINGS.

19

By trials it is quickly proved that the third alternative gives a schedule which can not be com-
pleted to a full system. There remain then oidy cycles with two, four, or six pairs. Each cycle
can be divided into halves so that each half contains the same subscripts as the other (by taking
alternate pairs in the cycle) . This gives us twice three pairs involving all six subscripts. One
set of paii-s corresponding can be chosen of letters aa or hb, the other of unlike letters, ah. All
these possibilities, together with the self -conjugate triads, are outlined in the following; exhaust-
ing the possible schedules for triads in A, B, and C, except for nonessential substitutions.

Class VI, 3 — Triads in A, B, and C, after ABC.

Condensed tables nf subscripts.

'

ab

aa, bb

ab, ba

No. VI, 3,:

A.

11,22
33,44
55,66

11,22
33,44
55,66

11,22
33,44
55,66

11,22
33,44
55,66

34
12,56

56

B

C

12,34
56

No. VI, 3„:

A '

34
12,56

B

C

13,24
56

No. VI, 83:

A

34
15,26

B

C

13,24

45
26
13

No. VI, 3,:

A

36
15
24

B

C

The cycles above described are seen in the second and third columns of these tallies (under
aa, hb and ab, ba). There are: One each of the first two species, two of the fourth.

Altogether in the tlu-ee classes VI, 1, 2, and 3, we have therefore 11 schedules or partial
systems. These 11 can all be completed to full systems, most of them in two or more ways.
All the systems contain the triad ABO; the 18 triads containing a single A, B, or C are given
above; and the various supplementary sets of 16 triads, of the types aaa, bbh, aab, and ahb are
now to be listed in full.

sxtpplementary sets, to complete the foregoing systems.

(Of two conjugate triads only one is given.)

VI, l,a. aaa: 135, 246.

aab: 145, 164, 326, 361, 523, 542.
VI, l,a'. aaa: 135, 246.

aab: 146, 163, 325, 362, 524, 541.

Equivalent to VI, l,a by the substitution (16) (34) (25).
VI, l,/3. aaa: 135, 146.

aab: 235, 246, 254, 263, 361, 451.
VI, l,/3'. aaa: 135, 146.

aab: 236, 245, 253, 264, 361, 451.

Equivalent to VI, 1,|8 by the substitution (36) (45).
VI, L. aaa: 145, 235.

aab: 135, 162, 245, 263, 364, 461.

A second supplementary set is equivalent to this by the substitution
(12) (34).
VI, 13a. aaa: 135, 246.

aab: 145, 162, 324, 361, 523, 546.

An equivalent set is derived by the substitution (12) (36) (45).

20 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

VI, I3/3. aaa: 136, 145.

aab: 234, 246, 253, 261, 351, 465.
VI, 2,a. aaa: 135, 246.

aah: 145, 1G4, 326, 361, 523, 542.

An equivalent set is derived by the substitution (13) (24).
VI, 2,|3. aaa: 135, 146.

aab: 235, 246, 254, 263, 361, 451.
VI, 2,7. aaa: 135, 146.

aah: 236, 245, 253, 264, 361, 451.
VI, 2,5. aaa: 135, 245.

aah: 613, 624, 632, 641, 145, 235.

An equivalent set is derived by the substitution (13) (24).
VI, 22a. aaa: 145, 235.

aah: 135, 162, 245, 263, 364, 461.

A second supplementary set is transformed into tlus by the substitution
(14) (23) {a,h,) {a,h,).
VI, 23a. aaa: 136, 145.

aah: 234, 245, 256, 263, 351, 461.

An equivalent set is reduced to this by the substitution (34) (56).
VI, 23/3. aaa: 135, 146.

aah: 234, 245, 256, 263, 361, 451.

Another supplementary set comes from this by the substitution (34) (56).
No. VI, 2,18 is reduced to VI, 23a by the substitution {a^bj (ajj^) {a^hj

VI, 237. aaa: 135, 245.

aah: 145, 163, 234, 265, 362, 461.
VI, 2,a. aaa: 135, 246.

aab: 145, 162, 324, 361, 523, 546.
VI, 2^/3. aaa: 135, 246.

aah: 146, 165, 321, 362, 524, 543.

This is seen to arise from VI, 2^0 by the substitution (14) (23) (56).
VI, 2,7. aaa: 145, 136.

aah: 234, 246, 261, 253, 351, 465.
VI, 2,5. aaa: 145, 235.

aah: 612, 624, 631, 645, 135, 243.
VI, 2,6. aaa: 236, 245.

aah: 146, 165, 153, 132, 354, 462.
VI, 3ja. aaa: 135, 246.

aah: 145, 164, 326, 361, 523, 542.
VI, 3,a'. aaa: 135, 246.

aah: 146, 163, 325, 362, 524, 541.

Equivalent to VI, 3,0 by the substitution (BC) (aiaj)^)^) iajaj)^)^)
ia^afij)^).
VI, 3,|3. aaa: 135, 146.

aah: 235, 246, 254, 263, 361, 451.

This comes from VI, S,a by the substitution (AB) (13) (24) (aM.
VI, 3,/3'. aaa: 135, 146.

aah: 236, 245, 253, 264, 361, 451.
VI, 3i7. aaa: 135, 245.

aah: 613, 624, 632, 641, 145, 235.

This also is equivalent to VI, 3,0, by the substitution (AC) (15) (26)

(a A).
VI, 3i7'- aan: 135, 245.

aab: 614, 623, 631, 642, 145, 235.

NO. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 21

VI, Sj. aaa: 145, 235.

aab: 135, 162, 245, 263, 364, 461 ;
or 135,164,245,261,362,463.

These two alternatives are equivalent by the substitution (12) (34).
We shall refer to the first.
VI, Sja. aaa: 146, 235.

aab: 125, 136, 241, 362, 453, 564.
VI, 330. aa^: 146, 235.

aab: 126, 132, 245, 364, 451, 563.

Equivalent to VI, Sga by the substitution (12) (34) (56).
VI, 33T. aaa: 124, 136.

aab: 251, 352, 453, 654, 236, 461.
VI, SjS. aaa: 412, 465.

aab: 316, 325, 354, 362, 164, 251.
VI, S^a. aaa: 146, 235.

aah: 124, 132, 436, 453, 625, 651.
VI, 34/3. aaa: 146, 235.

aab: 125, 134, 432, 456, 621, 653.

Equivalent to VI, 3,a by the substitution {BC) (12) (36) (45).
VI, 3^7. aaa: 126, 134.

aah: 234, 251, 461, 352, 456, 563.
VI, 346. aaa: 312, 345.

aab: 142, 253, 614, 625, 643, 651.

Equivalent to VI, 3^7 by the substitution {AB) (13) (24) (56).

In the above enumeration some systems can still be omitted as redundant.
No. VI, Sit' is reduced to VI, \^a by the substitution iflp^ (bfi^-
Five systems are reducible to VI, 2ia, viz:

VI, 2i7 by the substitution {BC) {a^a^) (b^ag) (a^b^) (b^b^) (ajSJ;

VI, 3, a by the substitution (ACB) {a^a^a^ (b.b^aj {a,_aJ)J)J)J)^):
and the tlu-ee whose equivalence to VI, 3ia has been noted akeady.
Further, No. VI, 3,j3' is reducible to VI, 2i5 by the substitution

{B(7) (a^a^) {a J) J),) (a^bjy.ajt^b^).

Some of those equivalences are obvious on comparison of the structure as here described,
but othere would not have been found without the aid of some definite system of procedure.
The method actually used was Miss Cummings's method of sequences and indices.

2\iter these deductions for equivalence, there remam 21 systems apparently distinct,
automorphic under a substitution of the type (1)'(2)^

In wTiting down these supplementary sets of triads, the first step was to T^nite the two
required triads aia^a^ in all ways that are different as regards the schedide of triads in A, B,
and C; that is, in all possible ways not transformable into one another without alteration of
the preceding 19 triads of the proposed system. After each way of writing these two triads
OiOjak, it is easy to decide from mspection whether the number of ways of filling out the six
triads aab is 0, 1, or 2. Where possible pairs of triads a^afl^^ have been omitted, it indicates
the impossibility of filling out a system.

§9. THE SUBSTITUTION OF THE TYPE (1)"(2)'': INVARIANT TRIAD SYSTEMS.

The only remaining (reduced) type of substitutions is that which leaves unchanged 7 of
the 15 elements and exchanges the others in pairs. Denote the former by numerals or digits
1, 2, 3, 4, 5, 6, 7; the latter by the pairs of letters Aa, Bb, Cc, Dd. The operation to be con-
sidered is S :

5=(l)(2)(3)(4)(5)(6)(7)(^a)(56)((7c)(Z?(i).

22 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

The first 7 elements, as we have seen, must constitute by themselves a triad system A?,
while each one occurs in 4 additional triads with pairs of letters from the last S. Of these 8
letters aU possible pairs occur, 4 self-conjugate and the rest in 12 conjugate pau-s of pairs;
28 in aU, so that every such pair is joined in a triad with one of the digits. Two pau-s that are
conjugate must form triads with the same digit as third element. Hence the 4 self-conjugate
pairs, like Aa, are either all completed to triads b}' the same numeral, as 1, or else by two
numerals, as 1 and 2, each joined with two pairs. Triads not self-conjugate, as SAB, Sat, occur
two by two.

Assembling in colunms of four the pairs associated with the several digits, we shall have
a seven-by-four array. We shall find that there are five types of such arrays, aside from per-
mutations of entire columns. To complete them to triad systems, it remains only to annex
a triad system A? constituted upon the seven digits. This can be done in, a variety of ways,
so that several systems will result from each seven-by-four array. Triads composed of one
digit and two letters shall be termed mixed. First we tabulate the mixed triads, writing down
the pairs of letters only.

Pairs from Mixed Triads, Class VII 1.

1 2

3

4

5

6

7

Aa AB

Ah

AC

Ac

AD

Ad

Bb ah

aB

ac

aO

ad

aD

Cc CD

Cd

BD

Bd

BC

Be

Dd cd

cD

bd

bD

be

bC

Pairs from Mixed Triads, Class VII

2.

6

7

First five like

the

above.

AD

ad
Be

bC

Ad
aD
BC

be

Here explanation is necessary. Any column could be selected as the second, whence the
thii'd would follow. Beside the exchange of conjugate letters in independent pairs, there are
still permissible the substitutions

{AB){ab), {OD){cd), {AC)(,BD){ac){bd), (AD) (BC) (ad) (be),

this last a result of the others. Compared with the second or the third, any later column may
be either cross-tied or not. For example, the fourth is cross-tied to the second by the triads
2AB, 2 CD in the one and 4 AC, ABB in the other; hence also by the remaining pairs in the
two columns. Notice also that when it is cross-tied to the second column it is necessarily cross-
tied to its cognate column, the third. As an example of the opposite kind, the columns 6 and 7
in class VII2 are not cross-tied to columns 2, 3, 4, or 5.

If all four self-conjugate pairs stand in a single column, there are but two nonequivalent

classes of seven-by-four arrays, those having the other six columns all cross-tied,

and those having four aU cross-tied and the two others not cross-tied with them.

All others having column 1 can be reduced to either VIIl or VII2.

In the other alternative, when self-conjugate pairs of elements appear in two colunms, two

in each, there are three classes of schedules. Let the columns containing self-conjugate pairs

be the first and second ; the other two pau-s in the first column are cross-tied to the self -conjugates

in the second, and -vice versa. Therefore also there wiU be another column — ^let it be taken for

the third — cross-tied to both the fu'st anil the second. Compare the four subsequent columns

with the third. Either 4, 2, or 0 are cross-tied with this third. If two are not, select them for

the sLxth and seventh columns. The resulting arrays are the following:

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 23

Pairs from Mixed Triads.

For

all three c

•lasses.

For class VIIS.

1

2

3

4

5

6

7

Aa

AB

Ab

AC

Ac

AD

Ad

Bh

ab

aB

ac

aC

ad

aD

CD

Oc

Cd

BD

Bd

BC

Be

cd

Dd

cD

hd

hD

be

bO

For class VII4.

4

5

6

7

AC

Ac

AD

Ad

ac

aC

ad

aD

BD

Bd

Be

BC

bd

bD

bC

be

For class VIIS.

4

5

6

7

AC

Ac

AD

Ad

ac

aO

ad

aD

Bd

BD

Be

BC

hD

bd

bO

be

Class VIIS is reducible to class VII2. The array of VII2 has the column 1 unique, cross-tied
to (interlaced with) all the others, a character not found in anj' other column. In the array of
class VIIS, the column S is unique in the same particular. From this clue, one finds without
difficulty a transformation of the latter array into the former. This transformation does not
preserve, however, the pairs of conjugate letters; in other words, it does alter the substitution
S with reference to which the systems are constructed ; but it changes S into another substitu-
tion S' of the same type (1)'(2)^ The transformer is this: (A) {B d b a D c C) (16 5 4 2 7 3).
Since the array of a system of class VIIS with respect to a substitution S, of type
(1)'(2)^, can be transformed into an array of class VII2 with respect to a different
substitution S' of the same type, all systems belonging in the one class belong also
in the other; and hence class VIIS does not require a separate investigation.
Upon these arrays we are now to superpose triad systems, Ay's, constructed in all non-
equivalent modes from the seven digits. First, for the class VII 1, there is an immediate
deduction available. The columns are triply cross-tied (interlaced), so that they indicate an
inherent triad-system or Aj. Compared with this inherent system, the A, to be imposed must
have 7, S, 1, or 0 triads in common. As no column and no inherent triad is unique in this
array, no further distinction is possible, and there are precisely four essentially different systems
in this class.

Supplementary Sets, Ay's, fob Class VII 1.

System VIII,: 123, 145, 167, 246, 257, 347, 356.
Sj'stem VIII,: 123, 145, 167; 247, 256, 346, 357.
System VIIl^: 123, 146, 157, 247, 256, 345, 367.
System VIIls: 124, 136, 157, 237,256,345,467.

In the array for class VII2, as has been pointed out, column 1 is unique, and the two columns
6, 7 are unlike 2, 3 and 4, 5 in relation to cross-tying or interlacing. AU three of these pau's, or
else only one of them, or none at all, may be united with numeral 1 in the superimposed Aj.
If only one, that one may be either 2 3 or 6 7. There are thus four cases, and each can be com-
pleted in two ways, giving apparently eight supplementary sets of triads or Ay's.

24 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv

Supplementary Sets, Ay's, for Class VII2.

System VII2, 1 ir.7-p46, 257, 347, 356.

(System VII22)J ' ' '1247, 256, 346, 357.

System ¥112, !,.,„ . „ ,4fi.!245, 267, 347, 365.

(System VIl2,)j ' ' '1247, 265, 345, 367.

System VII2,1 (236,456,247,357.

(System VII2,)J ' ' '1237, 457, 246, 356.

System \qi2, 1 ifi,.f237, 453, 674; 256.

(System VII28)J ' ' '1235, 456, 672; 347.

These are equivalent, two and two. For systems VII2, and VII22 the transformer is
obviously {67){AB){ah){CD){cd). The same transformer relates VII25 and VIU^. SUghtly
more intricate is the transformation of VII27 and VIl2g, viz by the substitution (24) (35) (67)
{ABDOiahdc); and that which exchanges VII23 and VII2„ namely (23) ii5){Q7 )iABab){DCdc).
We omit, therefore, the even-numbered systems in the above list, as indicated by parentheses.

In the array for class VII4, column 3 is imique, and the others are paired by being cross-
tied or interlaced. The three pairs, 12, 45, 67, are distinct or imlike; for the first are triply
interlaced with cohunn 3 by sets of foiu- letters, the second are interlaced with each other, and
each by itseK vnth column 3, while the last pair are interlaced with each other but not with
colunm 3. It is important to observe that we can exchange simultaneously the members of all
three pau's, by the substitution (ADad) (BCbc) (12) (45) (67). Tliis allows us to omit one of
every two that have one of these pairs of numerals in a triad with 3, as for example 345.

Supplementary Sets, Ay's, for Class VII4.

System VII 4,: 312, 345, 367; 146, 157, 256, 247.
System VII 4^: 312, 346, 357; 145, 167, 247, 256.
System VII 43: 316, 345, 327; 142, 157, 652, 647.
System VII 4,: 314, 325, 367; 126, 157, 427, 456.
System VII 4^: 314, 326, 357; 125, 167, 427, 465.
System VII 4,: 314, 326, 357; 127, 165, 425, 467.

This list is complete. For we need only consider the triads containing the element 3.
Either aU tliree contain pairs whose columns are interlaced in the array, (VIl4i), or only one, or
none. Tliat should give us (l+6-l-4 = )ll systems, after making allowance for the automor-
pliism mentioned just before the list. A fm-ther reduction is effected by observing that each of
the thi-ee operations like {AB){CD){ab){cd) exchanges each of two pairs of columns, as (45) (67),

leaving the other colunms of the array unaltered. Accordingly the five systems VII42

VIl4o represent ten, and VIl4i makes up eleven, the fuU coimt.

•The array for class VII5 has the unique column 3, the imique pair of cohmins 1, 2 containing
conjugate pairs, and the interchangeable pairs of columns 45, 67. We shall take account of five
substitutions among letters in the array, and their effect in permuting columns and their
respective digits.

r,: (^B) (06) produces (47) (56).

T,: {CD){cd) produces (46) (57).

7;: iAB){CD){ab){cd) produces (45) (67).

T,: {AC){BD){ac)ibd) produces (12)(67).

T,: (AD) (BC) (ad) (be) produces (12)(45).

Hence we distinguish only four cases, different as regards the pairs associated in triads with
the unique numeral 3. Either all the pairs 12, 45, 67, or the pair 12 only, or one of the others
exclusively, as 45, or none of them, must occur with 3. Each of these admits evidently two
modes of completion, but two of the resulting eight systems are redundant, as wiU be explained.

NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 25

Supplementary Sets, At's, for the Class VII5.

System VII5. | ,^7.1146, 157, 247, 256.

System VIIS^ f ' ^ ' 'll47, 156, 246, 257.

System VII53 Qr:7.1145, 167, 247, 265.

(System VIl5,)r^^' '^^^' "^^'il47, 165, 245, 267.

System VII55
System VII5,

316,3^5,327;

(142, 157, 647, 625.
147, 125, 642, 657.

System VII5,. 1 (125,167,427,465.

(System VII5,). J ' "^ ' "^ '1127, 165, 425, 467.

In this list two equivalences can be detected. A distinction has been pointed out, in the
array of class VII5, between the set of columns, 123, and the others, columns 45 67. Substitu-
tions can be seen which will transform any one of these latter columns into itseK, permute
the tlu-ee others in cycle, and permxite cyclically also the first three columns. Select the pair
ACiroTQ column 4. Under 5, 6, 7 note the pairs containing A and C; similarly under 132.

6

6

7

1

3

2

Ac

AD

Ad

Aa

Ab

AB

Ca

Ob

OB

CD

Od

Oc

The substitution (A)(0) (abB) (cDd) is found to convert column 4 into itself, hence it is equivalent
to the operation on numerals :

(4) (132) (567), ■

and this transforms the supplementary system ¥115^ into VII 5,. In the same way is found the
operation :

(B) (D) (dcO) (bAa)=(5) (123) (467),

which shows an equivalence between VII55 and VII59.

Four other relations, found by the sequence method of Miss Cummings, are readily verified.

System VIl2,=System VIIU, by the substitution (<dl)(Dd).

System VIl4,^System ¥1123, by the substitution (23)(4c)(5C)(6d)(7Z?).

System VIl5i^ESystem VII25, by the substitution

1234567 AaBb Oc Dd^

f 1234567 Aa Bb Oc Dd\
\l7QbOBc 25 43 aD Ad)

System VIl53^System VIII 4, by the substitution

1234567 Aa Bb Oc Dd^^

/1 234567 Aa Bb Oc Dd\
\213ABba OD cd 46 75/

Accordingly there remain as distinct systems in this section the following 16:

Class VIIl, systems „ 2, 4, oi

Class VII2, systems 3, 5, ,;

Class VII4, systems 1,

Class VII5, systems

No proof of nonequivalence has been given, save in a few special cases. It is beheved that
aU redundances have been ehminated from each separate section. To Miss Cummings wdl fall
the discovery of equivalences in different sections, and the proof of essential difference in the
residue. As a summary result, we know that the 71 triad systems analyzed and listed in the
foregoing include certainly all that admit any substitution other than the identity.

II 3! 4) 5> e>
2> 5) 61 7-

PART 2.

TRAINS FOR TRIAD SYSTEMS ON 15 ELEMENTS WHOSE GROUP IS OF

ORDER HIGHER THAN UNITY.

Bj' L. D. CUMMINGS.

To investigate the 71 systems obtained in Part 1, and to determine the group for each
system, Mr. White's method of comparison ' for triad systems is employed. For this method
the triple system is regarded as an operator and certain covariants of that operator are deduced.
Those covariants can bo represented graphically and are called the trains of the system.

The trains show that the 71 systems are reducible to 44 noncongruent systems; of these
24 are completely known systems already fully discussed in my dissertation,^ but the remaining
20 systems have not been described heretofore. The trains for the 44 noncongruent systems
are exhibited, and for each of the 20 new systems the group is determined. The substitutions
which transform the 51 systems into their equivalent systems are also given below.

A triple system on n elements consists of triads so selected that every pair of elements
(or dyad) occurs once and only once in the chosen triad. If there are 15 elements, every
element occurs with 7 pairs of others, and there are in the system in all 35 triads. This property
qualifies the triad system to be a transformer of dyads into single elements, and since each
dj'ad occurs once and no more this duality is unique for dyads. Thus, if the system contains
the three triads 124, 135, 236, then it will transform the triad 123 which contains the pairs
12, 13, 23 into the triad 456.

From 15 elements 455 triads can be formed. Any system contains 35 of these, leaving
420 that may be called extraneous triads. Apply the system to transform them all; we shall
see, as in the example worked out below, that the 35 triads in the system are transformed into
themselves, but the 420 extraneous triads go either into extraneous triads or possibly into
triads of the system. Some triads will transform into themselves, some wiU be produced more
than once, and others may not be produced at all by the transformation. All that arefomid to
be produced by the transformation are called derivative; all that are missing after the transfor-
mation, if any, are called primitive. •

TRAINS OF TRIADS.

Under a given triad system as an operator, let a triad /, be converted into the triad t^.
Repeat the operation and continue indefinitely, so that /j becomes t^; t^ becomes t^. Since only
455 triads exist, either a triad of the system t^ is reached or else a triad that has already appeared
is repeated, namely, tm+k^tm- In the former case the triad tj. repeats forever, while in the
latter case the train beginning at tm constitutes a recurring c}'cle. If the triads of the system
are designated as one-term cycles, then every triad that is primitive with respect to a given
triple system initiates a train terminating in a periodic cycle. Triads that do not recur in the
terminal cycle are classified as forming appendices, and a complete train consists of one recurrent
cycle and aU its appendices.

Some substitution may transform the triple system into itself. Such a substitution evi-
dently must also transform each train into itseK or into a precisely similar train and therefore

' H. S. White: Triple systems as transronnations and their paths among triads. Transactions of the American Mathematical Society, vol. 14
(1913). pp. 6-13.

' I.. D. Cummings: On a method of comparison for triple systems. Transactions of the American Mathematical Society, vol. IS (1914), pp.
311-327.

27

28

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

must leave unchanged the totality of trains connected with the system. The totality of com-
plete trains (cycles with their appendices) forms accordingly an arrangement of triads invariant
imder those substitutions on the 15 elements that transform the triple system into itself and
facihtates the determination of the group of the system.

Example: The tnple system VI 3i,y on 15 elements. — For convenience the system is trans-
formed by the substitution

^/ABC a J), a^h, afi^ afi^ aj>, a J) A
^=\b acl234defg7S65j'

and is exhibited in the following 15 by 7 array:

a

6

c

d

e

/

3

1

2

3

4

5

6

7

8

be

ac

aft

ae

ad

ag

of

0.7

lis

a6

aS

ai

03

al

a2

de

rf.l

d2

hf,

W>

ftS

hi

62

61

64

63

bd

be

6?

bl

fg

ed

el

c2

cl

c3

ci

ce

cd

cl

eg

c6

c5

08

cl

17

rs

n

n

fi

dl

d3

dl

eg

dt

dl

el

d8

di

d6

28

s^

0

«3

q2

ti

a

q6

ti

fS

el

n

n

f5

f3

36

12

56

47

38

25

16

35

37

15

18

«8

«1

/6

qb

45

34

78

68

57

67

58

48

46

27

26

13

24

23

14

The transforming process is simple and may be shown in its application to a triad 458 which is
extraneous to this system. Its pairs 45, 48, 58 transform, respectively, into a, I, g, giving the
transformed triad alg. The triad alg transforms into the triad of the system 76/ which repeats
indefinitely. These three triads form the type of train which is exhibited graphically in figure
3. This system applied as an operator on the 455 triads yields the following set of covariants
(trains) :

Trains for the System VISiT.

Six classes of trains terminating in triads of the system: (1) 11 trains, figure 1; (2) 4 trains,
figure 2; (3) 12 trams, figure 6; (4) 2 trams, figure 206; (5) 2 trains, figure 210; (6) 4 trains,
figure 211.

One class of trains terminating in a cycle of period 4: (7) 1 train, figure 182.

Two classes of trains terminating in cycles of period 6: (8) 5 trains, figure 183; (9) 1 train,

figure 213.

Twoclassesof trains terminating in cycles of period 12: (10) 1 train, figure 214; (ll)4trains,

figure 215.

Determination of the Group for the System VI347.

The trains for this system separate the 35 triads into 6 distinct classes and every operation
of the group that leaves the system invariant must transform any train into itself, or into
another tram of the same class. Since only those elements may be permuted which occiu- the
same number of times in a class, the enumeration of the appearances of each of the 15 elements
in the 6 classes of trains, as m the following table, shows the possible sets of transitive elements.
An examination of the triads of the system belonging to class (1) shows that the 15 elements do
not enter symmetrically as members of the triads of the class ; for example, in these 1 1 triads
the element c appears 7 times but no other element appears 7 times.

a

6

c

d

e

/

»

1

2

3

4

5

6

7

8

2

(1)

1

1

7

1

1

1

1

3

3

3

3

2

2

2

(2)

4

1

1

1

1

1

1

1

1

3

4

3

3

3

3

2

2

2

2

3

3

3

3

(4)

2

1

1

1

1

(5)

2

1

1

1

1

(6)

1

1

1

1

1

1

1

1

1

'

1

1

The possible systems of transitivity for the group are therefore a; b; c; d, e,f, (/; 1 , 2, 3, 4 ; 5, 6, 7, 8.

The sots of possible transitive elements subdivide the classes into sets of triads which are

not transformable into one another by operations of the group of the system; the subdivisions

No. 2.)

TRIAD SYSTEMS— WHITE, COLE, CUMMINGS.

29

aro shown by lines separating the triads in a class. The system VI347 contains 1 1 nonpermutable
subdivisions given in the following table;

(1)

(2)

(3)

W

(5)

(«)

abc

cda

135

CIS

bdS

d68

%l

al7

ait

612

<i47

c/3

418

csa

6/S

/76

a\b

ttfg

643

/25

ccl

248

be6

«57

efi

a28

rt8

cg4

327

bur

gsb

,a

036

a\6

In determining the group we examine first for substitutions that transform into itself one
of the trains, and secondly for those that transform this train into the remaining trains of its
class. Substitutions when determined must be tested on the 35 triads of the system.

The substitutions may be determined from any class of trains in the system, but most
easily from the class containing the trains with the greatest nimiber of triads since these exhibit
more repetition of the elements. In the present case (4), figure 206 is selected.

/26~.
abf — cg8-

-457n

alg-

■ c/7 368-

ide

dbe — c<Z6 285s

d48-^

ahd—ceb 176-

(i) Examine for substitutions to transform the train ode into itself.
two similar parts and the substitution

t={a) (b) (c) ide) ifg) (12) (34) (56) (78)

-afg

The train consists of

permutes these similar parts. No other substitution exists which converts this train ade into
itself. Therefore only a subgroup of order 2 transforms this train into itself.

(ii) Examine for substitutions to transform the train of the triad ade into the other train
of its class. The substitution

s=(a) (6) (c) {dfeg) (1423) (5867)

transforms the train of ade into the train of afg. Since t^^ the substitution t is omitted. The
substitution s applied to the 35 triads of the system transforms the system into itself. There-
fore the group for this system is a cycUc group of order 4 and is generated by s^{a) (b) (c)
(dfcg) (1423) (5867).

Similar detailed study determines the group for each of the following 43 systems:

Trains for the System V4al.

Ten classes of trains terminating in triads of the system: (1) Seven trains, figure 1; (2) 6
trains, figure 2; (3) 3 trains, figure 3; (4) 3 trams, figure 6; (5) 1 train, figure 66; (6) 3 trains,
figure 205; (7) 3 trains, figure 207; (8) 3 trains, figure 208; (9) 3 trains, figure 209; (10) 3 trains,
figure 212.

One class of trains terminating in a cycle of period 12: (11) Three trains, figure 216.

Group for the System V4al.

The sets of transitive elements are A; B; C; afl^a^; bfifi^; CtC^c^: d^d^d^. These with
the trains separate the system into 13 nonpermutable subdivisions. The group is generated
by s^A) {B) (C) (aiajfflj) (bjf^is) (C1CJC3) (did^^j), and is of order 3.

The trains w^hich belong to the system "V13,7 are exhibited in Plato I, those of the system
V4ol in Plate II; even a casual inspection of these two plates establishes conclusively the
noncongruence of the two systems.

The same type of train may occur in several systems, and in order to avoid repetition
in the diagrams the 204 distinct types which occur in the remaining 42 sj'stems are numbered
and listed in a definite order. The most convenient arrangement seemed to be the following,

30 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

namely: AH trains which contain a principal or maximnm succession consisting of 1, 2,

3^ ,1c,... triads are placed together and numbered consecutively; the subordinate

arrangement of these is easily observed. In general, the trains are exhibited in full, but in
order to save space a few of the longest trains which are <livisible into parts cychcaUy repeated
are shown m one part only. These, on account of their greater length, appear toward the
end of the series of figures.

The noncongruence of the remaining 42 systems is shown in the dissimilarity in the number
and m the type of trains enumerated mider each system. Systems with different numbers of
trains in their classes are noncongruent. The distinctiveness of trains numbered differently
is evident, and the fact that no possible confusion of the trains as pictured can arise constitutes
the chief merit of this method of proof of the nonconvergence of triad systems.

The trains for the 44 systems contain 216 different types, among which appear trains
with cycles of period 4, 6, 9, 10, 12, IS, 20, 24, .30, and 72. Trains with cycles of period 2 or
of period 3 are impossible, but while no cycle of period 5 has appeared among the trams of
these systems, there is no evident reason why such a cycle may not occur in the trains of sys-
tems not yet investigated. The groups for the 44 systems have been determined and the
generators for each of the 20 new groups are exhibited.

The groups for the 24 systems lA, IB, IC, IIA, IIB, IIC, IID, HE, IIF, HIA, IIIB, IIIC,
HID, IVA, IVB, VA, VB, VC, VD, VIA, VIB, VIC, VID, VII have already been determmed,'
and for the sake of brevity are omitted.

Mr. Cole has pointed out an error in the order of the group for the system IIIB. The
order of this group is 192, and not 96, as previously stated.

Trains fok the System IA.

Three classes of trains terminating in triads of the system: (1) Seven trains, figure 20
(2) 7 trains, figure 1; (3) 21 trains, fig-ure 2.

One class of trains terminating in cycles of period 18: (4) Seven trains, figure 194.

Trains for the System IB.

Six classes of trains terminating in triads of the system: (1) Three trains, figure 24; (2) 4
trains, figure 20; (3) 3 trains, figure 4; (4) 6 trains, figure 3; (5) 9 trains, figure 2; (6) 10 trains,
figure 1.

Three classes of trains terminating in cycles of period 18: (7) Three trains, figure 195;
(8) 3 trams, figure 196; (9) 1 train, figure 197.

Trains for the System IC.

Six classes of trams terminating m triads of the system: (1) Six trains, figure 24; (2) 1
train, figure 20; (3) 3 trains, figure 4; (4) 12 trains, figure 3; (5) 6 trains, figure 2; (6) 7 trains,
figure 1.

Three classes of trains terminating in cycles of period 18: (7) One train, figure 198; (8) 3
trains, figure 199; (9) 3 trains, figure 200.

Trains for the System IIA.

Five classes of trains terminating in triads of the system: (1) One tram, figure 5; (2) 12
trams, figure 21; (3) 6 trains, figure 15; (4) 4 trains, figure 49; (5) 12 trains, figure 35.

Trains for the System IIB.

Eleven classes of trains terminating in triads of the system: (1) Two trains, figure 21;
(2) 1 tram, figure 14; (3) 2 trains, figure 15; (4) 4 trains, figure 17; (5) 2 trams, figure 16; (6)
4 trams, figure 85; (7) 4 trains, figure 44; (8) 4 trains, figure 77; (9) 4 trains, figure 74; (10)
4 trains, figure 39; (11) 4 trains, figure 31. ^___

> Cummings, L. D. (Loc. cit.)

No. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 31

Trains for the System IIC.

Twenty classes of trains terminating in triads of the system: (1) One train, figure 26; (2) 1
train, figure 25; (3) 1 train, figure 21 ; (4) 2 trains, figin-e 55; (5) 2 trains, figure 56; (6) 2 trains,
figure 90; (7) 1 train, figure 14; (8) 2 trains, figure 50; (9) 2 trains, figure 88; (10) 2 trains,
figure 16; (11) 1 train, figure 15; (12) 2 trains, figure 86; (13) 2 trains, figure 82; (14) 2 trains,
figure 44; (15) 2 trains, figure 83; (16) 2 trains, figure 36; (17) 2 trains, figure 74; (18) 2 trains,
figure 40; (19) 2 trains, figure 67; (20) 2 trains, figure 65.

Trains for the System IID.

Twenty classes of trains terminating in triads of the system: (1) One train, figure 26; (2) 1
train, figure 25; (3) 1 train, figure 21; (4) 2 trains, figure 19; (5) 2 trains, figure 59; (6) 2 trains,
figure 89; (7) 1 train, figure 15; (8) 1 train, figure 14; (9) 2 trains, figure 50; (10) 2 trains, figure
16; <11) 2 trains, figure 43; (12) 2 trains, figure 87; (13) 2 trains, figure 84; (14) 2 trains, figure
45; (15) 2 trains, figure 34; (16) 2 trains, figure 36; (17) 2 trains, figure 69; (18) 2 trains, figure
76; (19) 2 trains, figure 72; (20) 2 trains, figure 40.

Trains for the System HE.

Nine classes of trains terminating in triads of the system: (1) 1 train, figure 5; (2) 4 trains,
figm-e 26; (3) 6 trains, figure 25; (4) 4 trains, figure 21; (5) 8 trains, figure 55; (6) 2 trains,
figure 14; (7) 4 trains, figure 50; (8) 2 trains, figure 15; (9) 4 trains, figm-e 36.

Trains for the System IIF.

Thirteen classes of trains terminating in triads of the system: (1) 2 trains, figure 21 ; (2) 2
trains, figure 15; (3) 2 trains, figure 51 ; (4) 1 train, figure 14; (5) 4 trains, figure 52; (6) 2 trains,
figure 16; (7) 4 trains, figure 45; (8) 2 trains, figure 70; (9) 2 trains, figure 80; (10) 4 trains,
figure 39; (11) 2 trains, figm-e 33; (12) 4 trains, figure 75; (13) 4 trains, figure 64.

Trains for the System IIIA.

One class of trains terminating in triads of the system: (1) 35 trains, figure 5.

Trains for the System IIIB.

Three classes of trains terminating in the triads of the system: (1) 7 trains, figiu-e 5; (2) 24
trains, figure 25; (3) 4 trains, figure 14.

Trains for the System IIIC.

Four classes of trains terminating in triads of the system: (1) 1 train, figiu-e 5; (2) 12
trains, figure 25; (3) 16 trains, figure 23; (4) 6 trains, figure 14.

Trains for the System HID.

Two classes of trains terminating in triads of the system: (1) 28 trains, figure 23; (2) 7
trains, figure 14.

Trains for the System IVA.

Five classes of trains terminating in triads of the system: (1) 1 train, figure 5; (2) 8 trains,
figure 25; (3) 16 trains, figure 19; (4) 6 trains, figure 15; (5) 4 trains, figure 14.

Trains for the System IVB.

Eleven classes of trains terminating in triads of the system: (1) 1 train, figure 25; (2) 4
trains, figure 23; (3) 4 trains, figure 56; (4) 4 trains, figure 59; (5) 3 trains, figure 14; (6) 3
trains, figure 15; (7) 2 trains, figure 50; (8) 4 trains, figure 52; (9) 4 trains, figure 17; (10)
2 trains, figure 36; (11)4 trains, figure 39.

32 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv

Trains fob the Ststem VA.

Three classes of trains terminating in triads of the system: (1) 1 train, figure 5; (2) 18
trains, figure 15; (3) 16 trains, figure 13.

Trains for the System VB.

Five classes of trains terminating in triads of the system: (1) 1 train, figure 14; (2) 6 trains,
figure 15; (3) 12 trains, figui'e 47; (4) 4 trains, figure 12; (5) 12 trains, figure 39.

Trains for the System VC.

Nine classes of trains terminating in triads of tlie system: (1) 2 trains, figure 25; (2) 4
trains, figure 57; (3) 4 trains, figure 15; (4) 1 train, figure 14; (5) 4 trains, figure 50; (6) 8
trains, figure 54; (7) 4 trains, figure 46; (S) 4 trains, figure 36; (9) 4 trains, figure 39.

Trains for the System VD.

Seven classes of trains terminating in triads of the system: (1) 3 trains, figure 25; (2) 4
trains, figure 60; (3) 1 train, fig. 14; (4) 6 trains, figm-e 50; (5) 12 trains, figure 53; (6) 3 trains,
figure 15; (7) 6 trains, figure 36.

Trains for the System VIA.

Six classes of trains terminating in triads of the system: (1) 4 trains, figure 20; (2) 3 trains,
figure 15; (3) 6 trains, figure 132; (4) 12 trains, figure 64; (5) 6 trains, figure 109; (6) 4 trains,
figure 1.

Trains for the System VIB.

Twenty-one classes of trains terminating in triads of the system: (1) 1 train, figure 26

(2) 2 trains, figure 22; (3) 2 trains, figure 20; (4) 1 train, figure 15; (5) 2 trains, figure 104

(6) 1 train, figure 16; (7) 2 trains, figure 102; (8) 1 train, figure 130; (9) 2 trains, figure 83

(10) 2 trains, figure 3; (11) 2 trains, figure 72; (12) 2 trains, figure 69; (13) 2 trains, figure 35

(14) 2 trains, figure 75; (15) 2 trains, figure 125; (16) 2 trains, figm-e 67; (17) 1 train, figure

113; (18) 1 train, figure 119; (19) 2 trains, figure 112; (20) 1 train, figure 109; (21) 2 trains,

figure 1.

Trains for the System VIC.

Thirteen classes of trains terminating in triads of the system: (1) 3 trains, figure 22; (2) 1
train, figure 20; (3) 3 trains, figm-e 16; (4) 3 trains, figure 3; (5) 3 trains, figure 73; (6) 3 trains,
figure 74; (7) 3 trains, figure 122; (8) 3 trains, figure 124; (9) 3 trains, figure 126; (10) 3 trains,
figure 31; (11) 3 trains, figure 65; (12) 3 trains, figure 117; (13) 1 train, figure 1.

Trains for the System VID.

Thirteen classes of trains terminating in triads of the system: (1) 3 trains, figure 26; (2) 3
trains, figure 22; (3) 3 trains, figure 89; (4) 3 trains, figure 55; (5) 1 train, figm-e 20 ; (6) 3 trains,
figure 103; (7) 3 trams, figure 101; (8) 3 trains, figure 3; (9) 3 trains, figure 36; (10) 3 trains,
figure 76; (11) 3 trains, figure 96; (12) 3 trains, figure 98; (13) 1 train, figure 1.

Trains for the System VII.

One class of trains terminating in triads of the system: (1) 35 trains, figure 1.

Two classes of trains terminating in cycles of period 6: (2) 5 trains, figure 183; (3) 5 trains

figure 186.

Trains for the System I2.

Seven classes of trains terminating in triads of the system: (1) 5 trains, figure 32; (2)
5 trains, figure 159; (3) 5 trains, figure 94; (4) 5 trains, figure 163; (5) 5 trains, figure 146:
(6) 5 trains, figure 2; (7) 5 trains, figure 1.

NO. 2] TRIAD SYSTEMS— WHITE, COLE, CLTMMINGS. 33

Group for the system 12. — The sets of transitive elements are ah c d e; a fi y 5 e; 1234 5;
these with the trains separate the system into 7 nonpernmtahle subdivisions. The group is
generated by

s = (a 6 c <Z f ) (1 2 3 4 5) (a /? 7 3 e)
and is of order 5.

Tkains fok the System III 2.

Three classes of trains terminating in triads of the system (1) 15 trains, figure 1; (2) 2
trins, figure 48; (I!) 18 trains, figure 145.

One class of trains terminating in a cycle of period 4: (4) 9 trains, figure 182.
(rroupfor the system III 2. — The sc^ts of transitive elements are a, a^ a^ hi h^ h^; c, Cj ^3 ^1 <^2 d^
e, fj fj; these with the trains separate the system into 9 nonpermutablo subdivisions. The
group is generated by

s= (a,) («2) (^1 ^2 ^3) (Ci di 61) (Cj cZj e,) {c^ d^ e^) (d^),
t=(ai a^) {a^) (6, h^) (63) (c, e^) (c^ e^) (c, e,) W, d,) (d^),
10= {Ui 6, a^ 62) (03 63) (c, gj C2 di) (C3) (^2 63 Ci (Z3),
d is of order 36.

Trains for the System III3.

Four classes of trains terminating in triads of the system: (1) 1 train, figure 179; (2) 3
trains, figui-e 2; (3) 3 trains, figure 27; (4) 28 trains, figure 1.

Three classes of trains terminating respectively in cycles of periods 5, 6, and 6: (5) 1 train,
figure 203; (6) 1 train, figure 188; (7) 1 train, figure 185.

Group for the system III3. — The sets of transitive elements are Oj a^ a^; i, h^ b^; c, c^ Cg,
(?, (?2 d^; f , ^2 fit' these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

S=(«l 0,2 ds) (^I ^2 ''3) (Cl C2 ^3) (^1 <^2 ^^3) (^t ^2 ^3),

and is of order 3.

Trains for the System Ilia.

Nine classes of trains terminating in triads of the system: (1) 1 train, figure 106; (2) 3
train, figure 12; (3) 6 trains, figure 148; (4) 3 trains, figure 122; (5) 3 trains, figure 6; (6) 3
trains, figure 62; (7) 3 trains, figure 2; (8) 3 trains, figure 30; (9) 9 trains, figure 1.

Two classes of trains terminating in cycles of periods 6 and 4, respectively: (10) 2 trains,
figure 189; (11) 3 trains, figure 182.

(rroupfor the system IIl^. — The sets of transitive elements are a^ a^ a^; 6, h^ h^; Cj ^2 ''3/
di d^ d^ «! ^2 ^3/ these with the trains separate the system into 9 nonpermutable subdivisions.
The group is generated by

s=:{a^) (hi) (Ci) (aj 03) (b^ 63) (c, Cg) (d^ e,) (d^ e^) {d^ e^),
/=(fl, O2 «3) (^1 ^2 ^3) (<^i C2 ^3) (di dj dj) (e, e^ e^),
and is of order 6.

Trains for the System III,.

Eight classes of trains terminating in triads of the system: (1) 1 train, figure 171; (2) 3
trains, figure 178; (3) 3 trains, figure 93; (4) 3 trains, figure 162; (5) 6 trains, figure 2; (7) 3
trains, figure 6; (8) 13 trains, figure 1.

Two classes of trains terminating in cycles of period 6: (9) 1 train, figure 184; (10) 1 train
figure 187.

Group for the system IIlj. — The sets of transitive elements are Ui a^ a^; 6, h^ h^; c, fj c,;
d, (Zj ds; fi e, e,; these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

s^a^ Oj flj) (&, hj 63) (c, Cj C3) (di dj d,) (f i e, e,) ,
and is of order 3.

54061°— 19 3

34 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol.xiv.

Trains for the System VI 7.

Three classes of trains terminating in triads of the system: (1) 6 trains, figure 9; (2) 1
trains, figure 6; (3) 17 trains, figure 1.

One class of trains terminating in a cycle of period 72 : (4) 1 train, figure 204.

Group for the system VI y. — The sets of transitive elements are A; B C; Ot a^ a^ &i 5, 63
r, c, C3 di 6,2 d^; these with the trains separate the system into five nonpermutable subdivisions.
The group is generated by

s = {A) {B (7) (a, di 63 c, a^ d^ 6, fi a^ d^ h^ c^),
and is of order 12.

Trains for the System V4(31.

Nine classes of trains terminating in triads of the system: (1) 3 trains, figure 160; (2) 3
trains, figure 180; (3) 9 trains, figure 2; (4) 3 trains, figure 138; (5) 1 train, figure 139; (6) 3
trains, figure 3; (7) 3 trains, figure 27; (8) 3 trains, figure 164; (9) 7 trains, figure 1.

Group for the system V40^ ■ — The sets of transitive elements are A; B; C; a^ a^ a^; i^ 62 ^3^
c, C2 c^; di d^ d^; these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

s= (A) (B) iO (a, 02 a,) (6, h i,) {c, c^ c,) (d, d^ d,)
and is of order 3.

Trains for the System V4/32.

Seven classes of trains terminating in triads of the system: (1) 3 trains, figure 134; (2) 3
trains, figure 10; (3) 3 trains, figure 29; (4) 6 trains, figure 2; (5) 1 train, figure 95; (6) 3 trains,
figure 7; (7) 16 trains, figure 1.

Two classes of trains terminatmg in cycles of periods 9 and 6 respectively: (8) 1 train,
figure 191; (9) 3 trains, figure 190.

Group for the system V4P2. — The sets of transitive elements are A; B; C; a, Oj O3; &, &2 ^3/
c, C2 c^; (Z, (Z, d^; these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

s= (A) (B) (C) («! 0.3 a.3) (6, 62 63) (c, fj C3) ((?, (?2 J3)
and is of order 3.

Trains for the vSystem V47I.

Thirteen classes of trains terminating in triads of the system: (1) 3 trains, figure 107
(2) 3 trains, figure 42; (3) 3 trains, figure 100; (4) 3 trains, figure 8; (5) 3 trains, figure 36
(6) 3 trains, figure 41; (7) 3 trains, figure 78; (8) 3 trains, figure 150; (9) 3 trains, figure 97
(10) 3 trains, figure 117; (11) 1 train, figure 95; (12) 3 trains, figure 63; (13) 1 train, figure 1

Group for the system, V-j-yl . — The sets of transitive elements are A; B; C; a, a, a^; 6, b^ is,'
Ci C2 C3; di d^ d^; these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

s={A) (B) (0) (a, 03 a,) (b, K b,) (c. c, c,) (d, d, d,)
and is of order 3.

Trains for the Systems V472.

Ten classes of trains terminating in triads of the system : (1 ) 3 trams, figure 161 ; (2)3 trains,
figure 71; (3) 3 trains, figure 8; (4) 3 trains, figure 3; (5) 3 trains, figure 68; (6) 3 trains, fig-
ure 168; (7) 3 trains, figure 158; (8) 3 trains, figure 155; (9) 1 train, figure 140; (10) 10 trains,
figure 1.

Group for the system V4y2. — The sets of transitive elements are A; B; C; a, aj (^3! ^1 ^2 ^3/
c, Cj C2! ^1 <^2 ^3' these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

s={A) (5) (6*) (a, a^ a^) (6, b^ b^) (c^ c^ c^) (d^ d^ d^)
and is of order 3.

NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 35

Trains for the System V451.

Nine classes of trains terminating in triads of the system: (1) 3 trains, figure 128; (2) 3
trains, figure 170; (3) 3 trains, figure 6; (4) 3 trains, figure 172; (5) 3 trains, figure 154; (6) 1
train, figure 66; (7) 3 trains, figure 147; (8) 3 trains, figure 137; (9) 13 trains, figure 1.

Group for the system V451 . — The sets of transitive elements are A; B; C; ay a^ a^; &, h^ h^;
c, Cj c^; (/, d^ d^; these with the trains separate the system into 13 nonpermutable subdivisions.
The group is generated by

s={A) (B) (C) (a, a., a^) (6, i^ h) (<^i c^ Cj) (J, d^ d^)
and is of order 3.

Trains for the System VIlj/S.

Nine classes of trains terminating in triads of the system: (1) 2 trains, figure 176; (2) 2
trains, figure 2; (3) 2 trams, figure 128; (4) 4 trains, figure 142; (5) 4 trains, figure 152; (6) 4
trains, figure 62; (7) 4 trains, figure 6; (8) 4 trains, figure 107; (9) 9 trains, figure 1.

One class of trains terminating in a cycle of period 4: (10) 1 train, figure 182.

Group for the system VIIS- — Tlie sets of transitive elements are A; B C; Oj 6, a^ h^;
Cj 62 03 63; a^ 6, flj 65/ these with the trains separate the system into 11 nonpermutable subdi-
visions. The group is generated by

s= (A) (B C) (a, a^ b^ b^) (a^ 63 b^ a^) (a^ 65 b^ a^)
and is of order 4.

Trains for the System ¥12^0.

Eleven classes of trains terminating in triads of the system: (1) One train, figure 105;
(2) 1 train, figure 11; (3) 4 trains, figure 3; (4) 1 train, figure 149; (5) 2 trains, figure 8; (6)
4 trains, figure 2; (7) 2 trains, figure 181; (8) 2 trains, figure 6; (9) 1 train, figure 28; (10) 2
trains, figure 175; (11) 15 trains, figure 1.

One class of trains terminating in a cycle of period 24: (12) One train, figure 202.

Group for the System F/^^a. — The sets of transitive elements are A; B; C; a, &,; a^ h^;
ds ^3/ O'i iiJ ^5 ^5/ (^e K- These with the trains separate the system into 21 nonpermutable sub-
divisions. The group is generated by

s=(A) (B) (0) {a, b,) {a, b,) (a, b,) (a, &J (a, b,) (a, bj,

and is of order 2.

Trains for the System YI2^S.

Eighteen classes of trains terminating in triads of the system: (1) One train, figure 18
(2) 2 trains, figure 130; (3) 1 train, figure 132; (4) 2 trains, figure 133; (5) 2 trams, figure 153
(6) 1 train, figure 38; (7) 2 trains, figure 174; (8) 2 trains, figure 126; (9) 2 trains, figure 37
(10) 1 train, figure 8; (11) 2 trams, figure 62; (12) 2 trains, figure 120; (13) 3 trains, figure 2
(14) 2 trains, figure 6; (15) 2 trains, figure 156; (16) 1 train, figure 109; (17) 2 trains, figure
112; (18) 5 trains, figure 1. .

One class of trains terminating in a cycle of period 4: (19) One train, figiu-e 182.

Group for the System VI2^5. — The sets of transitive elements are A; B; C; o, b^; a^ b^; a^ b^
a^ bj flj 65/ Co \. These with the trains separate the system into 21 nonpermutable subdi-
visions. The group is generated by

s = (A) (BC) (a, 6,) (a.2 b^) (a, 6,) (a, b,) {a^ JJ (a, J,),

and is of order 2.

Trains for the System VI2<«.

Fifteen classes of trains terminating in triads of the system: (1) One train, figure 144;
(2) 1 train, figure 8; (3) 2 trains, figure 167; ^4) 2 trains, figure 139; (5) 1 train, figure 165;
(6) 2 trains, figure 157; (7) 4 trains, figure 2; (8) 2 trains, figure 62; (9) 1 train, figure 30; (10)

1 train, figure 6; (11) 2 trains, figure 115; (12) 1 train, figure 114; (13) 2 trains, figure 27;' (14)

2 trains, figure 166; (15) 11 trains, figure 1.

36 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

Two classes of trains te.rminatmg in cycles of periods 10 and 4, respectively: (16) Two
trains, figure 192; (17) 1 train, figure 1S2.

Group for the System VI2it. — The sets of transitive elements are^; B; C; a^ 6,; a^ \; a^ 63
a^ bj Oj Sj/ Oo &«. These with the trams separate the system into 21 nonpermutable subdi-
visions. The group is generated by

s={A) (B) (C) (a, 6.) (a, h,) (a, b,) (a, b,) (a, b,) (a, b,),

and is of order 2.

Trains for the System VlSja.

Seven classes of trains terminating in triads of the system: (1) Two trains, figure 143;
(2) 2 trains, figure 110; (3) 4 trains, figure 2; (4) 4 trains, figure 61; (5) 4 trains, figure 91; (6)
4 trains, figure 136; (7) 15 trains, figure 1.

Two classes of trains termmating in cycles of periods 18 and 20, respectively: (8) Two
trains, figure 193; (9) 1 train, figure 201.

Group for the System VlS^a. — The sets of transitive elements are A; B C; a^ b^ a^ h^; a^ 63
^e ^6/ c-i h C"o ^5- These with the trains separate the system into 11 nonpermutable subdi-
visions. The group is generated by

s={A) {B C) (a, a^ b^ b^) {a^ b^ a^ 63) K "s ^4 ^5))

and is of order 4.

Trains for the System VI337.

Six classes of trains terminating in triads of the system: (1) Four trains, figure 177; (2) 2 trains,
figure 79; (3) 4 trains, figure 173; (4) 6 trains, figure 27; (5) 2 trains, figure 92; (6) 17 trains,
figure 1.

One class of trains terminating in a cycle of period 4: (7) One train, figm-e 182.

Group for the System VlS^y. — The sets of transitive elements are A; B C; a^ &i a, \; a^ 63 a^ b^;
«4 ^4 «5 ^5> these with the trains separate the system into 1 1 nonpermutable subdivisions. The
group is generated by

s = {A) (B C) (a, Oj 6, b^) (a^ b^ b^ a^) (a^ a^ 6, 65),
and is of order 4.

Trains for the System VISjS.

Ten classes of trains terminating in triads of the system: (1) Two trains, figure 58; (2) 4 trains,
figure 87; (3) 4 trains, figure 141; (4) 4 trains, figure 76; (5) 4 trains, figm-e 151; (6) 4 trains,
figure 99; (7) 4 trains, figure 34; (8) 2 trains, figure 81; (9) 6 trains, figure 27; (10) 1 train,
figure 1.

One class of trains terminating in cycle of period 4: (11) One train, figm-e 182.

Group for the System VlS^h. — The sets of transitive elements arc A; B C; a, J, a^ b^; a^ b^ a^ b^;
«4 ^4 «5 ^5/ these with the trains separate the system into 1 1 nonpermutable subdivisions. The
group is generated by

s=(A) (B C) (a, a^ &, b^) (a^ b^ 63 aj (a^ a^ 64 65),

and is of order 4.

The trams show that 20 of the 71 systems obtained in Part 1 are new systems and the
remaining 51 systems are each congruent to some one of the 44 systems thus far derived. The
substitution which transforms each of these 51 systems into its congruent system is given below

T -.—TrTTi. —/'a b c d e 1 2345a/373 A

I,l=Vnby s^(4 7 b e I a d g 3 6 f 2 5 S c)

I, 2 anew syetem.

T— TTTA 1, _/a b c d e 1 2 i -i n a ft y 5 e\

I^IIAby s=(^ g 5 8 e c 6 I 2 7 b -l d / ^)

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CUMMINGS.

37

II, 1,=VA by

= ("i "a 1, 6i 62 ''3 fi '■2
Ml 1 i a b c e 5

II, I2 new system; III3 a new system.

II, 5=1 1 A by s^f"! "a ^3 61 ''2 ''3 c, fj

Vrf 4 8 a 6 c / 2
II, Ig new system; 1 11, new system.

«=/'«, O2 »3 f>

Va 4 / rf
s^{A a

\5 a
s~/A a

Vl a
s^fA a

Vl a

(•3 </.
7 /

'•3 rfi

8 3

fl ^2 ^3

9 2 6

)

II, 2=VII by

III, 1=IIIA by
III, 2=IIID by
III, 3=IA by
V, la=VA by

c d

d c

■J. e 8

'' / 9

9 f e

c d c f g

' 9 <^ /

« / 9

e / 9

e d
c d

c d

V, I7 a new system.
V, 15=1, 2 by s

V, 2a=IIIB by s:

V, 2^=VD by

z/A B C a^ a^ a, 6j tj
\a6crfl4e2

= fA B C 0, 02 03 61 62
Voj 02 83 rfj e, f, 62 6,

eAI 5 C a, 02 "3 ''i ^2
V«12c/(/38

1 7

(4 rf.

Ci f2 e.
9 3 6

>)

S:=fA B C 0, f(2 "3 &1 ^2

V(;86ca637
V, 27=IIIDby s~(A B C a, Oj 03 6; h^

Vdl5/e684
F, 2«=VBby s^(A B C a, Oj 03 &, b^

\g53cab76
V, 3a^V, la by s^ (A B C a, flo 03 6, 62

Vaj Cj 6[ 63 C3 rfj (4 02
F, 3/3=IIICby s^/^ B C a^ a^ ((3 6, 62

V6 1 e 5 8 7 2 4
F. 37=111 A by s=A4 B C a, 02 03 6, b^

^d I 6 € f g 2 i
V, 4^1, a new system; V, 4/3,, a new system.
V, 4/^2, a new system; V, 47,, a new system.
V, 472, a new system; V, 45,, a new system.
V, 5ai=IC by s=fA B C a, a.^ 03 6, 6,

U 8 g' '5 3 2 / c"
V, 5/J,=IB by s=/.4 B C a^ a^ 03 6, 63

V4 1 a 3 2 7 6 c
V, 50^IA by s^/^l B C Oj 02 03 6, 62

U 8 jr 3 2 5 c «
V, 57i=VIDby s=/.ri B C o, n, 83 6, 63

\2 6 3 3 8' 5 6 rf
V, 572=VIC by s=/A B C o, a^ 03 5, 6,

\5 4 g 8 6 3 a b
V, 55i=VIAby s=/A B C a^ a^ O3 5, b^

"3 w
3 7

63 Cj

63 03

63 Cl

6 5

63 c,

4 2

h Ci

2 3

63 c,

4 8

63 c,

62 a,

63 c,

3 /•
63 c,
3 6

5 6

^2 ''3

C3 f3

^2 f3

4 7

'"2 C3

1 5

7 6

C2 C3

1 2

C2 Cj

C2 f^a

7 8

6

4

6
8 4
5 6
2 5
rfi (/. d;
9

9'
I)
I)

I)
f)

d, d^ dA

^2 ^2 «2'

b g e y

d, d, d,\

d 9 f}

d, d, d,\

g c a /

d, d, d,\

d } c)

\ % 'd

di ^2 d\

a c b /

a c b

'd

5 g 7

1 d

VI, l,a=VA by s^/.l B C a^ a.^ a, a^ a

VI, 1,/J^IVA by s=/

VI, I2 =IIFby s=/A B C a^ 02 a^ a^ a^
\fad4321c

c b d 9~ 1 8 4
.4 B C Oi Oj Qj a^ a^
cabd9l25.

B C b, d\ b^ d^ 63

VI, l3a=iV, I7 by s=i/.4 B^ C a^ a^ a^ O4 a^

VI, I3/3, a new system.

VI, 2ia:^ElVA by s=/A B C a, o, 03 a^ a^

U a c d ^" 3 4 7
VI, 2,/3sIII(Jby s=/,4 B C ai a^ a^ a^ a^

\6ac56rfyl
VI, 2,S=IIIB by 8=/^ B C Ot a^ a, a^ a^

\bacedl26
VI, 2 a=IIA by s=/A B C a, Oj 03 04 a^

\/adb6S7b
VI, 230i=VC by « =/.4 B C at a„ 03 a, a^

\g f a d c' 6 5 2
VI, 24a, a new system.
VI, 247=11,1, by 8=/.! B a a, b, oj h "3

Vflj 62 '^ "3 "1 ''2 di 's

63

4

fa
6

f3

7

d,
a

d,
d

f)

63

e

8-

'i

'•3
6

i

9

f)

>

4

f2

6

C3

7

a

d^
d

i')

>

1

C2

7

C3
4

d,
a

d.
e

f.)

&3

7

C2
2

1^

d,
c

d-i
f

J)

>

'3

'■2
8

C3
6

d,
a

f

f')

°6

7

61

60

63
2

l^

I'

S-)

6^

6,

e

>

63
3

4

1.

s

?•)

"6

e

l^

62
6

^

64

7

I'

"8

K

62

63

K

K

''6\

^3

"i

fi

"2

<^i

"3

^3>'

5

>

62
e

&3

2

1

65

8

^)

I'

I'

^

;■

64

e

65
2

^)

7

>

63
3

4

5

l^)

«6

!7

','

^3=

1'

^4

1

65

>)

"6

4

^

63

1

64
3

7

^sO

'^

04

64

I'

i:

d.

»

38 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

VI, 2^7, a new system; VI, 2^1, a new system.

VI, 3, =VIA by s=/A B C a^ a^ a, a, 05 Og 6, h^ fr, 64 65 iA

Vc98125a66473(ic/
VI, 830, a new system; VI, 837, a new system.
VI, 838 , a new system; VI, 847, a new system.
VI, 34a=VII by s=/A B C a, a^ a, a, 05 o, fc, 63 63 5, 65 66\

U55rrfa47/2361ec8/
VII, 1, =IIIAbys=/l 23456T AaBbCcDd\

\abcdegfl234b687J
VII, 1, =IIIB by s=/l 234567^aB6 Cci)d\

Vo6crfe5r/1284568 7>'

VII, It =IIIC by 8=/l 234567 AaBbCcDd\

\abcdegfl2345eS7J
VII, 1. =IIIDbys=/'l 284567^aB6CcZ»<^

Va6crfe(7/1234568 7/
VII, 23 =IIEby s=/l 234567J[aB6CcDrf\

^ca642138576d/jf«/
VII, 2. =IVA by s=/l 2 34567^aB6CcZ)(A

Va(fe/s?c534785612/'

VII, 2, =IVBbys=/l 234567 AaBbCcDd\

\adegfbc21G5783 4/
VII, 4, =IIA by s=/l 234567^aB6Cc/)rf\

Uca«yd/3124857 6/
VII, 43 =IIFby s=/l 284567vlaB6CcDrf\

\6ca/d{?e2481856 7y
VII, 4^ =IIB by s=/l 284567^aB6CcX>cf\

\6ca/d(;e2431856 7/
VII, 45 =IIDby s=n 234567 AaBbCcDd\

ycbadfgeS56742l3J

VII, 4e =IICby s=/l 284567AaB6CcDd\

•V6ca/d(!^8124678 5/

VII, 5^ =VA by 8=/! 2 3 4567 AaBbCcDd\

\abcdgf€8521374 6/
VII, 5. =VCby ss/l 234567 AaBbCcDd\

Kb c a g f d e 1582634 7>'
VII, 5a =VB by s=n 2 34567 AaBbCcDd\

\>ibcgdefl25S647 3J

9 d e f
4 5 6'
a / g d

Trains foe Triad Systems on 15 Elements whose Group is of Order Unity.

VII, 5, =VDby s=n 2345 6' 7 AaBbCcDd\
\cbaf(ide7436285 1/

The trains for each of the 36 noncongruent groupless systems on 15 elements have been
determined. Tliese 36 systems furnish 449 distinct types, different from the trains of the sys-
tems with a group. Among these appear trains terminating in polygoms of 4,6, 11, 12, 13,
and 14 sides, respectively.

Hence the 80 noncongruent systems applied as transformers to the 455 triads on 15 ele-
ments, produce 665 distinct covariants or trains.

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CLJMMINGS.

at)

PLATE I- SYSTEM VI 3 Ay.

FIG I.

-H 1

Fig. 2 TlO. «. Fio. 1S2. F'°- "83.

A

A

A.

\

^ \

:i. ::::». -:^ ^^

Fig 2in.

^,

\

\ ^^.x.

H

:^^ •:^^:i»_l

no. 211.

no. 21.1.

FlO. 214.

^

PlQ. 206.

40

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV

PLATE II; SYSTEM V 4 a\.

H -H ^-

Fig. 1. Fio. 2. Fig. 3.

1

Fio. B.

Fig. 205.

Fio. 207.

<^\
^^^i_

H

_::^.

— I

Fig. 208.

^

\

Fig 209

Fio. 212.

^

H

\

^::^

Fio 216..

NO. 2.. TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 41

^_, ^W/

—i 1

F,o.l. F.0.-2. • Via. 3. F.o.4. 1"'°*

Fio 6. Fio. 7. Fio. 8.. Fio. 9.

— :^ -H tl^ _, ^ ^ ^

Fig 10, Fig. II. J.'io. i2. f:g. 13.

#Jh ^1-^ i^-ln ^"i

Fio. 14: Fio. 15. Fio. 16. Fic, u.

:^H :^J/_, ^-^u IjiL,

Fig. 18. fio. 19. Fig. 20. Flo. 21.

42 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol. xiv.

^^ ^^^H ^:^\I/h ^\^

FlQ. 22. Fig. 23. Fio. 24. FiO. 25.

:^^l/^

Flo. 26, Flo. 27. FlO. 28.

_rr-. , :^A^ 1

Fro. 29. FIO. 30, T'0. 3J.

Fio. 3Z. FiQ. 33. Fio. 34.

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 43

^^:^_, ^^/

Fi(j.3S. ■ Vio. 36. FlQ. 37.

Fio. 38. FiO. 39. Fw. 40.

-^::^::-._< ^•^A_^ :^A^^

Fio, U. Fig. 42. Fio. 43.

rio:44: F10.45. fig. 46.

44 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv

Fio. 47. Fra> 48. FlO. 49.

FiQ. S3.

^^^_, :^^^_J ^^^

Fio.50. Fl«.61. FlQ.52.

\

Fio. 54. Fio. 65.

:^^^H -^^-H -^^1-1

FiQ. 56. Pro. 57. Fio. 58.

NO. 2.] TillAD SYSTEMS— WHITE, COLE, CUMMINGS. 45

N

FlO. 65.

:^^\l_

FlQ. 59. FlO. CO. FlO. HL

Fig. 62. IlG.63. Fio. 64.

^^_H ^""^Ix-H ^4^^\'-

Fig 60. Fio. 67.

p.- so FlQ. 70.

FlQ. 68. FI0-.-B9.

46 MEMOIRS NATIONAL ACADEMY OF SCIENCES. (vol. xiv.

FlO. 71. FIO. 72. Flo. 73.

^A^_) ^^i^^H -^^1^

FlO. 74. FlO. 75.

Flo. 7tt.

FlO. 77. FlQ. 78. FlQ. 79.

^■rL^:i._i _'::::^.:^_i _r^^:^_4

FlO. 80. FlB.,» Fl0.a2^

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 47

Pig. 83. Fic. 84.

Fig. 85.

Fin. SB i?.^ n-» _ »

FiS Sa Fic 80. Fio. 9)

^ j _ ^Il^i j 2:^^ j

Fig 92. FiO. 93 KiO. M.

48 MEMOIRS NATIONiiX, AC^UiEMY OF SCIENCES. [Vol. xiv.

FlO. 115.

fia. 97. '•-'o- *S-

Fio. 99. FlO. 100.

i:x^ i^ _ :^ —j ^ S".i^ -^ ^ — 1

Fig. 101. Fig. 102.

No. 2.) TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 49

FiQ. 105.

^:^:^:^

FlO. 107.

Fig. 103. ^'O- !«*•

i=-.^.. _-if4

Fig. 106.

Fio.lOS.

\1

-^^^ I _^^^ H

FlO. 109. FlQ. 110.

54061°— 19 4

50 MEMOlilS NATlONiiL ACADEMY OF SCIENCES. (vol. xiv.

-iTr:::^ , _l::^^___j

Fig. Ill,

FJC. 113. Fio- 111-

^- N^

^^:^ \ :^^ ^^ 1

Frq.. 115,

FlQ. 116.

fiC !J7. FiC 118-

N°-2.' TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 51

FIO.U9. FIO. 120.

^_^V^_

FlO. 121. FlO. 122.

FIG. 123. ^«»- '«•

Flb.T2S. FlQ. I2«

52 MEMOmS NATIONAL ACADEMY OF SCIENCES. ivol. xiv.

no. 127. r^n- '2S.

Fio. 129. " Fig. 130.

f 10. 131. ^'° '^^■

Fjg. 133. Fig. 134.

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 53

^

:^_j ^^^^ I

Fig. 135. Fio. m.

Fig. 137. Fic. las.

Fig 139. Flc HP

FiG. 1«.

54 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

Fig. 143. Fi(. 144.

StS. 145. Fig. 146.

:^^5. j:-.j^^ 1 J^ \. ::^ ^ — 1

Fig. 147. Fig. 148.

FlO. 149. FlO' '50

NO. 2.] TlllAD SYSTEMS— WHITE, COLE, CUMMINGS. 55

^ -4

:^s^ ::iw "t^ __j

Fro. 151.

Fia.152.

^

^":---^:^_,

Fio. IS3.

\

no. 151.

56 MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivou xiv

FlO. 155.

^^ -::^ j

Fig. lie.

Fig. 157.

:^ ::i^ _ A j:^ :i^ _|

Fn».I68.

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 57

i^^::^^^ _ :-. ^

•^^ \

Fia. 169.

Via. 160.

__>^^:^_i

yio.161.

Fio; ax

58 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xn-.

FlO. 163.

FlO. IM.

fW. 165.

fio. lee.

No. 2.] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 59

-:i>- — ^^ I

Fig. 167.

Fig. 168.

■^^

FIO. 169.

no. 170.

60 ]VIEMOIltS NATIONAL ACADEMY OF SCIENCES. [Vul. xn-

^>--^ 0.x\

_A^ ^^A^^

FlO. 172.

Fig. 173.

FlQ. 174.

No- 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 61

I

FlO. 175,

IT ^— ::^_\.— =^\^ __j

Fio. J7a,

\

^^ A x:^^_|

Fio. I77i

ElO. 178.

62 MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivol. xiv.

-^i^A.

A.

^^ \ C^)

\ \ -^

C57

•-^ \ \ A^ \^ ^> "^ v^ •

Fig. 179. ^'^

(I), (2), (3) similar; (4), (S), (6) similar.

FlO. 180.

No. 2.1 TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 63

I

I
I

1/ /

\'^

1/

!/

I /
1 /
\1/
\l

V

y

1

64

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

IVOL. XIV.

FlO 1S2.

Fpo 133

Fio. 134.

W

ri5. isfi. M hexagon.

\

Fio IS6, "M bexason.

Fio. 1ST; H beMgon.

Jir i=^ Z5i» -_ -iA.

\
/

Fro. 188; M hexagon.

_::.:.=..___A

Flo. 189: H bexagon.

No. 2.]

TKIAD SYSTEMS— WHITE, COLE, CUMMINGS.

65

A\

\

^\

>5^

^:^

^A -.^^^^^

TlQ. 190.

Fio. 191; H polygon ot nine sides.
54061°— 19 5

/ /

Fio. 192.

66

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

A

^

-^i^^

^.

A_A^^X-_

FlQ. 193: ^4pQlygono^ eighteen sides.

Fio. 194; ^ polygon of efgbteen sides.

Fig. 195.

No. 2.J

TRIAD SYSTEMS— WHITE, COLE, CUMMINGS.

G7

FlO. 197.

Fio. IfiS

PiO. IDS.

Fig. 199.

68

MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivol. xiv.

FtO. 200.

Fio 2U1; H polygon ot iwenty sides.

\

^

Fio 202; H polygoa ol twenty-tour sides.

:^-

Fig. 203 H polygon of thirty eidcs.

•^.

^

riii. :::^j-=.

Fio. Wi A polygon ot seventy-lwo sides.

PART 3.

GROUPLESS TRIAD SYSTEMS ON 15 ELEMENTS.

By H. S. White and L. D. Cummings.

All noncongruent systems, A15, with a group having been determined in Part 1, there
arises next the qxicstion concerning the possible existence of triad systems on 15 elements with
the group identity. Systems whose group is identity, or gi'oupless systems, do not exist for
7, 9, or 13 elements. In a paper' ah-eady pubhshed Mr. White has proved the existence of
many groupless systems on 31 elements. An investigation given below in some detail has led
to the discovery of a considerable number of noncongruent systems on 15 elements with the
group identity. Every groupless system on 31 elements whose existence has thus far been
demonstrated contains one or more systems Ais and, therefore, is a headed system. On the
contrary, every groupless system on 15 elements Ls headless.

In any triad system the jiaire of elements are more or less interconnected or interlaced.
These interfacings may bo determined by applying to the system under consideration a modi-
fied form of the method - of examination by sequences and indices. The A 15 is exhibited in a
1 5 by 7 array. Each element heads one column ; below it are placed the seven dyads, which, with
the element at the head, constitute the triads of the system. Heretofore sequences and indices
have been derived from the three columns of a triad in any A15; the same process is now appHed
to every pair of columns and yields what may be called the two-column or contracted indices
for the system. Since the number of combinations of 15 columns, two at a time, is 105, this
number of pairs of columns must be examined unless the group for the system is known and
is different from identity. If the group contains an operator of order 7n, then in general m pairs
of colunms are examined simidtaneously. The process may be illustrated in its apphcation to
a system VII3/3, with a group of order 4 generated by <=(a) (be) (dSel) (jSgi) (1536). Pairs
of columns selected from the follomng table show every type of two-column or contracted index
that can occur in any system.

a

b

i

I

dc

if

at

a2

fg

eg

if

63

12

ac

c2

ce

34

13

g»

d5

56

24

IS

/s

78

57

37

gi

be

68

46

67

Pairs of
columns.

Index.

Contracted sequences.

ab
bl
da
db

25
S'
6
2,4

delgfid, 12/43/1, 66/87/5;
dfimpsid, 24lgiical2;
c2!15IMI37ISgllblc:
15/73/1, af/s8/64/2c/o.

The substitution t applied to the pair of colimins ah gives the pair ac with the same index
and similar sequences. If t is apphed to the pair of columns 61, the three pairs c5, l>2, c6 are
obtained with index 3". The analysis of the 105 pairs of columns shows that the contracted
indices 2'; 2, 4; 3-; 6 belong, respectively, to 2, 24, 4, and 75 pairs of columns.

1 White, H. S.: Transactions of the American Mathematical Society, vol. 14 (1913), pp. 13-19.

' Cumtnings, 1,. D.: Transactions of the American Mathematical Society, vol. 15 (1914), pp. 311-327.

70

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XTV.

Tlie new groupless systems are formed by interchanging duads of one column wnth those
of another column. For example, in the pair of columns ab, the duads de, fg, of column a,
may be exchanged with df, eg, of column h; such an interchange involving four elements, con-
tained only in two pairs in each cohimn, shall be designated as a quadrangular transformation.
The columns d, e, f, g must now be rewiitten in agreement with the new triads introduced
into the system, and the imdisturbed nine columns of VII318 with the reconstructed six col-
Tuuns form a new system 13. The four duads 12, 34, 56, 78 of column a might be interchanged
with tilt! fom- duads on the same elements in column b, forming an octagonal transformation,
but tliis is equivalent to the above quadrangular transformation followed by the interchange
of the elements a and b.

In the pair of columns bl the duads df, 68, 57 of column b may be interchanged with the
duads /8, 67, d5 of column 1; such an interchange, involving six elements appearing exchi-
sivel}' in three pairs in each of two cohimns, shall be designated as a Tiexagonal transformation.
New colunnis d,f, 6, 8, 5, 7 must next be constnicted; these eight reconstructed columns, with
the seven undistm-bed columns of Vllgi?, form a system 4. The appHcation of the second
hexagonal transformation in 61 is equivalent to an application of the first hexagonal trans-
formation followed by an interchange of the elements b and 1.

A transformation on 12 elements simply interchanges the two elements wliich head the
colunms. Therefore only the quadrangidar and the hexagonal transformations wliich exist
in a system require consideration.

By means of the operatore of the gi'oup of the system, the maximum number of non-
congruent transformations of each of the above types is determined — for example, in Wl^&
the eight hexagonal transformations reduce to one, and the 30 quadrangular transformations
to fom- noncongruent transformations.

Each of the noncongriient transformations is now applied to the system VII318, and the
sequences and indices are determined for the five transformed systems.

The 35 triads of the system 4, arranged in classes according to their indices, are shown
in the following table:

1<26

1<35

Wb

1'9

1»2'4

1«28

1»37

1>40

1»5»

157

012

2SS

(178
cg5

108

034

gU
dgS

cd2

624

613

1227

1230

1245

1,11

23 42

23M

5,7

6»

ade

die

afg
gV

f26
/'.•5
c/6
C38
rf37

drfS
6/8
607
beg
eel

(•48
d/1
?36

ef7

f35
/23

06c

The enumeration of the elements in the 17 classes shows that the sets of transitive ele-
ments are a; h; c; d; e; f; g; 1; 2; 3; 4; 5; 6; 7; 8; hence the group for the system is identity.
Therefore under this hexagonal transformation the system YII3P, with a group of order 4,
is changed into a system 4 with the group identity. The four quadrangidar transformations
apphed to VIlj/S jaeld four noncongruent systems. One of these is a new groupless system
13; the remaining three are the known systems Vl2^e, VI2,5, VI2^7, with groups of orders 2, 2,
and 6, respectively.

The headless, groupless system 4 which has been derived by a hexagonal transformation
from a headless system VI 1 3/3 with a group of order 4, may also be derived by a quadrangidar
transformation from the headed system IC with a group order 3. Hence a quadrangular
transformation may alter the number of systems A 7 in a A15 and change a headed system
into one without a head. Hexagonal transformations, on the contrary, leave unchanged the
number of systems A, in a A 15 and, therefore, always transform systems with or mthout
heads into systems with or without heads, respectively.

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CUMMINGS.

71

In further illustration of the productiveness of this method in generating new systems
from those already known, we exhibit the results of its application to a groupless system 27
previously obtained by this same method. The 35 triads of the system arranged in classes
according to their thi'ee-column indices are shown in the following table :

I'?! 5

1«234

1»9

IS 2" 4

1!28

1»37

1'5'

1236

647

/37

cfg

def

035

M5
157

aeg

a/I
(•34
ff45

aci
c25
bgS

1245

1,11

2' 6

V4'

2,10

3,9

5,7

6'

o78

/24

be2

/56

468

o66

dgd

126

bfS

bc7

ad2

238

ebS

bU
cdS
eel

c36
<il3
f67
1)27
gl8

The analysis of this system by two-column indices reveals the existence of 1; 24; 15; 65
pairs of columns with the indices 2'; 2, 4; 3^; 6, respectively. The group for this system is the
identity and, therefore, it is possible that the appUcation of all the quadrangular and of all the
he.xagoual transformations would yield 39 noncongruent systems. Since however a quadran-
gular transformation may lead back to a headed system, already completely determined, we
apply only the 15 hexagonal transformations. These generate the following 15 systems: 30, 33,
21, 18, Yl2,a, 28, 29, III,, 31, 9, 32, 17, 24, 17, V472; 2 of these are congruent, 3 are systems
with groups already determined in Part 1, while 11 are new groupless systems, which may be
shown to be noncongruent by a comparison either of their indices or of their distinctive sets
of trains.

The application of this method to a number of the systems given in Part 1 yielded the 33
noncongruent groupless systems tabulated below.

We make use of the notation (ai) c d e f to denote a quadrangular transformation which
occurs in the pair of columns a, b and which involves the four elements c, d, e, f. Similarly (ab)
c d e f g 1 is & hexagonal transformation in the columns a, b which involves the six elements
c,d,e,f,g,l.

The S5'stem 1 may be derived from the system IB by the application of the quadrangular
transformation (a 2) b d 5 3, and for the sake of brevity we shall write this in the form 1 = IB,
(a 2) bd 53.

The 33 new groupless systems are derived as follows:

l = IB, (a 2) bd53;

3 = IB, (b 8) c/5 2;

5 = IC, (aS) C€3 5;

7 = IC, (6 3) eSfif 6;

9 = IIC, (c3)(Z/7 6
11 = lie, (64) (Ze86
13 = 7/F, {b2)de7 5
15 = IID, (c 3) eg 8 5

2 = IB, 0)3) ci n;

4 = 10, (a2) c€ 4 1;

6 = IC, (a5)/!7l4;

8=y7<?, (6 3)/^ 1 7;
10 = lie, {a 2) 5 6dg;
12 = 7/5, (c5) e/4 1;
14 = 77(7, (c4) df8 5;
16 = 77, I3, (03 dj) Ci 62 61 ffj p, e^.

In the system 7, 2 we now replace the elements a, /3, y, 5, e, by/, g, 6, 7, 8, respectively.

17 = 7,2, (/I) 62 57;
19 = 17, (6l)a65f2/8
21 = 17, (6c)a6e3 14
23 = 17, (68)a6/5<'7
25 = 17, (6e)8 5/d;
27=17, (23)6(77/4e;
29 = 27, (e3)6 2a{?58;
31 = 27, (2 4)6fil63 8;
33 = 27, (a/)665378.

18=17, ((Z/) a2 4 7 1 3;
20=17, (c d) a i 7 b 5 2;
22 = 17, (6 5) ffl3 e 8/ 6;
24 = 17, (a6) c 4/1 7 8;
26 = 25, (c5) 63o4^/;
28 = 27, (c 5) o 4 /fir 3 6;
30 = 27, (a6) c4 1 /8 7;
32 = 27, {2 5) b d e 8 a 3;

72 MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivol. xiv.

The 77 noncongrueiit systems thus far derived are interconnected by quadrangular and by
hexagonal transformations, and in general a system is not united uniquely to another system
but is derivable from several systems by different transformations; for example, the system 7
possesses the folio wmg interconnections: 7 = I C, {b3)e8g6; 7=F4al, (dg) a3 ef 7 6;7 =11,
1, (a 1) & c 4 e 6 2.

After this point in the investigation many other systems which were transformed by this
process fiu-nished only repetitions of the 77 noncongruent systems thus far determined, showing
that the number of groupless systems was probably not much in excess of 33. An exhaustive
determination of all groupless systems by this empirical method requires that every new system,
as it appears, shall be subjected to each one of its possible quadrangular and hexagonal transfor-
mations. The enumeration of these transformations for the 14 new systems derived above
from the system 27 shows that a complete investigation even of these systems would necessitate
the application of more than 435 transformations. Hence while this empirical method for
generating new systems is productive, the amount of work involved in an exhaustive investi-
gation is prohibitive. Therefore it is evident that a new starting point and a new method are
requisite to insure a complete detei'mination of the groupless systems. This desideratum is
fully met in the following Part 4.

PART 4.

STRUCTURE AS DEFINED BY INTERLACINGS, HEADS, AND SEMIHEADS;
A COMPLETE CENSUS OF TRIAD SYSTEMS IN FIFTEEN ELEMENTS.

By F. N. Cole.

1. INTRODUCTION.
In forming triad systems in 15 letters, there are only four typical openings, viz:

1 2 3

1 4 5

1 6 7

1 89

1 10 11

1 12 13

1 14 15

I

2 4 6
2 4 6
2 4 6
2 4 6

2 5 7
2 5 7
2 5 8
2 7 8

2 8 10
2 8 10
2 7 9
2 9 10

2 9 12
2 9 11
2 10 12
2 U 12

2 11 14
2 12 14

2 11 11
2 13 14

2 13 15
2 13 15
2 13 15
2 5 15

II

Ill

IV

which, from the way in which the triads containing 1 are laced with those containing 2, may be
called the single tetrad, triple tetrad, hexad, and duodecad types, respectively. It turns out
in the present investigation that, witli a single exception, the tetrad tyije (single or triple) is
always present, so that in the final census only openings I and II need be considered. These
openings are then treated in sections 4, 5.

2. THE DUODECAD OPENING.

We show here that a triad system in the 15 letters can not be made up with duodecads
alone. To this end we note that opening IV above has the following group of 24 substitutions
which convert it into itseK :

(1 2) (4 5 15 14 13 12 11 10 9 8 7 6),|
(4 6) (5 7) (8 15) (9 14) (10 13) (11 12)/

If the triads with 1 and 2 (excluding 12 3) are denoted by a, h, c, d, e,/and a', h', c', d', e',f',
respectively, this group is equivalent to

{ iaf'fe'ed'dc'cb'ba'), (ai) {cf) (de) (b'f) (c'e') }

and suffices to interchange the accented and unaccented letters with preservation of order of
sequence, to move each set of letters in a cycle, and to reverse the order of each set.

If now the triads with 3 are laced through those with 1 and 2, in the duodecad manner in
each case, it may happen that these new triads (1) connect two successive ones of the 1 set or
2 set, or (2) do not exhibit such a sequence. It readily follows then, with the help of the group
above, that the only typical lacings are the following 14 :

dbcdef abdcfe ahefcd
abfced
abfdec

abcdfe
abcfed
abcefd
abcfde

abdfec
abdfce
abecfd
abedfc

acebfd

the first 13 presenting the sequence ab, and the last one no such sequence.

This last one, with no sequences, may be worked out in some detail as an example of the
method employed throughout this paper. We have to write down the triads with 3, lacing,
say, those with 1 in the order acebfd, and those with 2 without sequence. We can not use 3 4 8,
since this would give the sequence a' b' ; we must take 3 4 9, 3 5 8, or 3 5 9. These lead to the
four possibilities :

73

74 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

3 4 9

3 8 13

3 6 12

3 7 15

3 10 14

3

11

5

3 5 8

3 9 13

3 6 12

3 7 14

3 11 15

3

4

10

3 5 8

3 9 13

3 7 12

3 6 14

3 10 15

3

4

11

3 5 9

3 8 12

3 6 13

3 7 15

3 10 14

3

4

11

The substitution (123), with proper adjustment of the remaining numbers, converts the first of
these into the other tliree. The first may then be taken as typical. It has the following group
of six substitutions into itself: {(12 3) (4 8 10) (5 7 14) (6 13 11), (1 2) (11 13) (14 10) (9 15)
(5 8) (4 7)}. This suggests the formation of the triad 9 15 x. Here x can not be 1, 2, 3, 4, 5,
7, 8, 9, 10, 14, or 15, since 1 9, for example, has already been used in a triad; for % we must take
6, 11, 12, or 13. But 6, 11, and 13 are equivalent under the group of order 6 above. And the
remaining possibilities 9 15 6 and 9 15 12, if followed out, lead to tetrads or hexads.

Returning to the table A, we may note that in the fir&t 13 cases the 3-triad which joins
a and h must be 3 5 7, for 3 4 6 is at once excluded, and 3 4 7 and 3 5 6 involve tetrads of 1
and 4 and of 2 and 5, respectively. Starting with 3 5 7, and WTiting in the remaining 3-
triads, we find that the first, second, sixth, and eighth cases lead directly to tetrads or hexads.
The remaining cases prove to be partly equivalent to each other, and those which survive are
found on continuation to the 4-triads, etc., to involve tetrads or hexads.

3. THE HEXAD OPENING.

This has the following group of 144 substitutions into itself:
{(4 7 8) (5 6 9), (4 5) (6 8) (7 9);
(10 13 14) (11 12 15), (10 11) (12 14) (13 15);
(4 10) (5 11) (6 12) (7 13) (8 14) (9 15);
(1 2) (5 6) (7 8) (11 12) (13 14)}.

The triad 3 4 a; can not have x= 1, 2, 3, 4, 5, or 6, since these have already been used; nor can
x=7 or 8, smce these give tetrads of 4 and 1 or of 4 and 2. If the triad system is to have no
tetrads, we must take x = 9, 10, 11, 12, 13, 14, or 15, and of these the last six are equivalent
imder the group of order 144 above. Hence the only distinct tj^jes are 3 4 9 and 3 4 10.

Starting, then, with 3 4 9, we find for 3 5 i/, only y = 7 or 10, and note that 3 5 10 is
equivalent under the group above to 3 4 10, the case to be considered later. It turns out,
then, that we can have only

349 357 3 68 3 10 15 3 1113 3 12 14

and we note that this is invariant under the group of order 144 above. If we now write in
the 4-triads, we come at once to tetrads.

The case 3 4 10 leads to 3 9 11, 3 9 13, or 3 9 15. Following each of these out in detaU,
we encounter everjrwhere tetrads, with the single exception of the Ileffter system:

1 2 3

14 5 16 7

1 8 9

1 10 11

1 12 13

1 14 15

2 4 6 2 5 8

2 7 9

2 10 12

2 11 14

2 13 15

3 4 10 3 5 7

3 6 11

3 8 15

3 9 13

3 12 14

4 7 12

4 8 13

4 9 14

4 11 15

5 6 14

5 9 10

5 11 13

5 12 15

6 8 12

6 9 15

6 10 13

8 10 14

7 8 11

7 10 15

7 13 14

9 11 12

This, then.

is the only triad system in 15 letters that can

be constructed solely with hexads

and duodecads.

4. THE TRIPLE-TETRAD OPENING.

This has a group of 768 substitutions:

{(4 5) (6 7), (4 6) (5 7), (4 7) (5 6);
(8 9) (10 11), (8 10) (9 11), (8 11) (9 10);
(12 13) (14 15), (12 14) (13 15), (12 15) (13 14);
(4 8 12) (5 9 13) (6 10 14) (7 11 15),
(4 8) (5 9) (6 10) (7 11);
(1 2) (5 6) (9 10) (13 14)}.

NO. 2] TRIAD SYSTEMS— WHITE, COLE, CUMMINGS. 75

Wc find that the x of 4 3 a; must be 7, 8, ... , 15, and that 8, . . . , 15 are equivalent. The
typical cases are then 4 3 7 and 4 3 8. Of these 4 3 7 has a group of order 128, composed ol
those substitutions of the group of order 768 which leave 4 and 7 unchanged or interchange
them. With the aid of this group of order 128 we find that we may take as typical cases
3 5 6 or 3 5 8.

The case 3 4 7, 3 5 6 exhibits a triad system in the seven letters 1,2, 3,4,5, 6, 7, included
in the triad system in the 15 letters. We then speak of the latter as having a 7-head, or simply a
head. The triads with 3 are now found to be the six followmg sets:
347 356 38 11

3 8 12

3 9 10

3 12 15

3 13 14

3 9 12

3 10 15

3 13 14

3 9 13

3 10 14

3 11 15

3 10 15

3 11 14

3 9 14

3 10 13

3 11 15

3 10 15

3 11 13

Working out their continuations, wc find the 22 triad systems with triple tetrads and 7-head
exliibitcd ui the previous chapters of this memoir.

The case 3 4 7, 3 5 8 presents what one may perhaps be permitted to call a semihead.
The triads with 3 are here:

3 4 7 3 5 8 3 6 9 3 10 12 3 11 15 3 13 14

3 6 11 3 9 12 3 10 15 3 13 14

3 6 12 3 9 10 3 11 15 3 13 14

3 9 13 3 10 14 3 11 15

3 10 15 3 11 14

3 9 14 3 10 13 3 11 15

3 10 15 3 11 13

Continuing these, we find 12 triad systems with triple tetrads and semihead.

In the absence of cither head or semihead, the 3-triads form one of the following sets:
3 4 8 3 6 12 3 5 9 3 7 13 3 10 14 3 11 15

3 10 15 3 11 14

3 7 14 3 10 13 3 11 15

3 10 15 3 11 13

3 7 15 3 10 13 3 11 14

3 10 14 3 11 13

3 5 10 3 7 15 3 9 13 3 11 14

3 5 11 3 7 15 3 9 13 3 10 14

3 9 14 3 10 13

3 5 15 3 7 11 3 9 13 3 10 14

and these lead to 26 triad systems with triple tetrads but no head nor semihead.

5. ONLY SINGLE TETRABS.

The group of the single tetrad opening is of order 64 :
{(4 5) (6 7), (4 6) (5 7), (4 7) (5 6);
(8 11 15 12) (9 10 14 13), (8 9) (10 12) (11 13) (14 15);
(1 2) (5 6) (9 10) (11 12) (13 14)}.
This group also leaves the triad pair 3 4 7, 3 5 6 xmchanged, and the 7-head case has the follow-
ing sets of 3-triads:

347 356 38 11
loading to a single triad system with 7-head but no triple tetrads.

3 9 13

3 10 15

3 12 14

3 9 15

3 10 13

3 12 14

76 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vot. xiv

With a semihcad 3 4 7, 3 5 8 there are 17 typical sets of 3-triads:

3 4 7 3 5 8 3 6 9 3 10 12 3 11 15 3 13 14

3 10 13 3 11 15 3 12 14

3 10 14 3 11 13 3 12 15

3 10 15 3 11 13 3 12 14

3 6 11 3 9 10 3 12 15 3 13 14

3 9 13 3 10 15 3 12 14

3 9 14 3 10 13 3 12 15

3 9 15 3 10 13 3 12 14

3 6 13 3 9 10 3 11 15 3 12 14

3 9 11 3 10 15 3 12 14

3 9 14 3 10 12 3 11 15

3 10 15 3 11 12

3 6 15 3 9 10 3 11 12 3 13 14

3 11 13 3 12 14

3 9 11 3 10 13 3 12 14

3 9 13 3 10 14 3 11 12

3 9 14 3 10 13 3 11 12

These lead to only three triad systems with semihead but no head or triple tetrads.

There now remains only the case of no triple tetrads, heads, or semiheads. The first
3-triad may be taken as 3 4 8, and the opening set of 14 triads is unchanged by the substitution
s=(l 2) (5 6) (9 10) (11 12) (13 14) only. The triad 4 15 a; has x = 7, 9, 10, 11, or 12, arid under
s 9 is equivalent to 10 and 11 to 12. We have then three typical cases: 4 7 15, 4 9 15, 4 11 15.
For the first of these 8 15 a; gives x = 5 or 11, and the continuation is not unreasonably long.
But the cases 4 9 15 and 4 11 15 each subdivide into foiu" cases instead of two, making the total
labor five times that of the 4 7 15 case imless some method of compression could be devised.
Fortimatcly such a method was at hand.

The triad pairs 4 1 5, 4 2 6 and 7 1 6, 7 2 5 exhibit a tetrad. If, then, a triad system has
been constructed so far as to show its triads with 1, 2, 4, and 7, it may be possible by inter-
changmg the pairs 1, 2 and 4, 7 to throw this system into one already identified. For example,
arriving at the set

4 3 8 4 9 15 4 7 11 4 10 12 4 13 14
7 3 9 7 8 13 7 10 14 7 12 15

we find that the substitution (1 4) (2 7) (3 15 12 9 13 11) (8 14 10) throws this into a set with
the original 1-triads and 2-triads, but containing 3 4 8, 4 7, 15 and therefore coming under a
case already explored.

As another example, the set

4 3 8 4 11 15 4 7 10 4 9 13 4 12 14
7 3 11 7 8 13 7 9 14 7 12 15

is thrown by (1 7 2 4) (5 6) (3 9 11 12 15 13 10) into the case 4 3 8, 4 9, 15. In fact, the nearly
150 cases under 4 11 15 reduce to only 6 by this process.

In the end there are found to be 15 triad systems with no triple tetrads, head, or semihead.

6. SUMMARY.

The following table gives the number of triad systems of each type for 15 letters:

Triple tetrad and head 22

Triple tetrad, no head, but semihead 12

Triple tetrad, no head, or semihead 26

No triple tetrad, but head 1

No triple tetrad or head, but semihead 3

No triple tetrad, head, or semihead 15

No tetrad (Heffter's system) 1

Total number of types 80

No. 2.]

TKIAD SYSTEMS— WHITE, COLE, CUMMINGS.

77

Table of Triad Systems in Fifteen Elements,

I. TRIPLE TETRADS AND HEAD.

Systems 1-20.

Systems 1-20— Continued.

123 145 167 18

9 1 10 11 1 12 13 1

14 15

13.

4 8 13

4 9 15

4 10 12

4 11 14

24 6 257 2 8 10 2 9 11 2 12 14 2 13 15

5 8 15

5 9 14

5 10 13

5 11 12

6 8 14

6 9 10

6 11 13

6 12 15

1.

347 356 38 11

3 9 10

3 12 15

3 14

14

7 8 12

7 9 13

7 10 14

7 11 15

4 8 12

4 9 13

4 10 14

4 11

15

5 8 13

5 9 12

5 10 15

5 11

14

Group of order 8.

6 8 14

6 9 15

6 10 12

6 11

13

14.

5 8 14

5 9 10

5 11 13

5 12 15

7 8 15

7 9 14

7 10 13

7 11

12

6 8 15

6 9 14

0 10 13

6 11 12

Group of order 8!/2.

7 8 12

7 9 13

7 10 14

7 11 15

2.

5 8 13

6 8 15

7 8 14
Group of order 192.

5 9 12

6 9 14

7 9 15

5 10 15

6 10 13

7 10 12

5 11

6 11

7 11

14
12
13

15.

Group of order 12.

347 356 38 12

4 8 15

5 8 14

3 9 13

4 9 10

5 9 12

3 10 14

4 11 12

5 10 15

3 11 15

4 13 14

5 11 13

3.

5 8 14

5 9 15

5 10 12

5 11

13

6 8 11

« 9 14

6 10 13

6 12 15

6 8 15

6 9 14

6 10 13

6 11

12

7 8 13

7 9 15

7 10 12

7 11 14

7 8 13

7 9 12

7 10 15

7 11

14

Group of order 4.

Group of order 96.

16.

4 8 15

4 9 14

4 10 13

4 11 12

4.

5 8 14

5 9 12

5 10 15

5 11

13

5 8 11

5 9 10

5 12 15

5 13 14

6 8 15

6 9 14

6 10 13

6 11

12

6 8 14

6 9 15

6 10 12

6 11 13

7 8 13

7 9 15

7 10 12

7 11

14

7 8 13

7 9 12

7 10 15

7 11 14

Group of order 8.

Group of order 168.

6.

4 8 12

4 9 13

4 10 15

4 11

14

5 8 14

5 9 15

5 10 13

5 11

12

17.

,5 8 U

5 9 12

5 10 15

5 13 14

6 8 13

0 9 12

0 10 14

6 11

15

6 8 14

6 9 10

6 11 13

6 12 15

7 8 15

7 9 14

7 10 12

7 11

13

7 8 13

7 9 15

7 10 12

7 11 14

Group of order 32.

Group of order 24.

6.

5 8 14

5 9 15

5 10 13

5 11

12

18.

347 356 38 12

3 9 13

3 10 15

3 11 14

6 8 13

6 9 14

6 10 12

6 11

15

4 8 15

4 9 10

4 11 12

4 13 14

7 8 15

7 9 12

7 10 14

7 11

13

5 8 11

5 9 14

5 10 13

5 12 15

Croup of drder 24.

6 8 14

7 8 13

6 9 15

7 9 12

C 10 12
7 10 14

6 11 13

7 11 15

7.

4 8 12

4 9 14

4 10 15

4 11

13

5 8 15

5 9 13

5 10 12

5 11

14

Group of order 4.

6 8 13

6 9 15

6 10 14

6 11

12

19.

347 356 38 12

3 9 14

3 10 13

3 11 15

7 8 14

7 9 12

7 10 13

7 11

15

4 8 15

4 9 12

4 10 14

4 11 13

Group of order 288.

5 8 13

5 9 10

5 11 14

5 12 15

8.

347 356 38 11

3 9 12

3 10 15

3 13

14

6 8 11

6 9 15

6 10 12

6 13 14

4 8 13

4 9 10

4 11 14

4 12 15

7 8 14

7 9 13

7 10 15

7 11 12

5 8 15

5 9 14

5 10 13

5 11

12

Group of order 12.

6 8 14

6 9 15

6 10 12

6 11

13

20.

347 356 38 12

3 9 14

3 10 15

3 11 13

7 8 12

7 9 13

7 10 14

7 11

15

4 8 15

4 9 10

4 11 12

4 13 14

Group of order 4.

5 8 11

5 9 13

5 10 14

5 12 15

9.

5 8 15

5 9 14

5 10 12

5 11

13

6 8 14

6 9 12

6 10 13

6 11 15

6 8 14

6 9 15

6 10 13

6 11

12

7 8 13

7 9 15

7 10 12

7 11 14

7 8 12

7 9 13

7 10 14

7 11

15

Group of order 3.

Group of order 2.

Systems 21-22.

10.

5 8 15

5 9 14

5 10 13

5 11

12

12 3 14 5 16 7 1

8 9

1 10 11

1 12 13 1 14 15

6 8 14

6 9 13

6 10 12

6 11

15

2 4 6 25 7 2

8 10

2 9 12

2 11 14 2 13 15

7 8 12

7 9 15

7 10 14

7 11

13

3 4 7 3 5 6 3

8 11

3 9 13

3 10 15 3 12 14

Group of order 2.

4 8 12 4 9 14 4

10 13

4 11 15

11.

4 8 13

5 8 15

6 8 14

7 8 12
Group of order 2.

4 9 14

5 9 13

6 9 10

7 9 15

4 10 12

5 10 14

6 11 13

7 10 13

4 11

5 11

6 12

7 11

15
12
15
14

21.

5 8 13

6 8 14

7 8 15
Group of order 3.

5 9 15

6 9 10

7 9 11

5 10 14

6 11 13

7 10 12

5 11 12

6 12 15

7 13 14

12.

5 8 15

5 9 13

5 10 14

5 11

12

22.

5 8 15

5 9 11

5 10 12

5 13 14

6 8 12

0 9 15

6 10 13

6 11

14

6 8 13

6 9 15

6 10 14

6 11 12

7 8 14

7 9 10

7 11 13

7 12

15

7 8 14

7 9 10

7 11 13

7 12 15

Group of order 3.

Group of order 3.

78

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

II. TRIPLE TETRADS, NO HEAD, BUT SEMIHEAD.

Systems

1-12.

1

Systems

1-12— Continued

1 2 3

14 5 16 7 18

9 1 10 11 1 12 13 1 14 15 1

7.

4 8 15

4 9 12

4 10 14

4 11 13

2 4 6 2 5

7 2 8 10 2 9

11 2 12

14 2 13 15

6 6 11

5 9 10

S 12 15

5 13 14

6 8 14

6 9 15

6 10 13

8 11 12

1.

3 4 7 3 5 8

3 6 11

3 9 12

3 10 15

3 13 14

7 8 13

7 9 14

7 10 12

7 11 15

4 8 13

4 9 14

4 10 12

4 11 15

Group of order

3.

5 6 14

5 9 10

5 11 13

5 12 15

6 8 12

6 9 15

6 10 13

8 11 14

8.

3 4 7 3 5 8

3 6 12

3 9 14

3 10 13

3 11 15

7 8 15

7 9 13

7 10 14

7 11 12

4 8 14

4 9 10

4 11 13

4 12 15

No group.

5 6 11

5 9 12

5 10 15

5 13 14

2.

5 6 12

5 9 15

5 10 13

5 11 14

6 8 13

6 9 15

6 10 14

8 11 12

6 8 15

6 9 13

6 10 14

8 11 12

7 8 15

7 9 13

7 10 12

7 11 14

7 8 14

7 9 10

7 11 13

7 12 15

Group of order 2.

No group.

9.

4 8 14

4 9 12

4 10 15

4 11 13

3.

4 8 13

4 9 15

4 10 14

4 11 12

5 8 15

5 9 13

5 10 12

5 11 14

S 6 11

5 9 10

5 12 15

5 13 14

6 8 12

6 9 14

6 10 13

8 11 15

6 8 13

6 9 15

6 10 14

8 11 12

7 8 14

7 9 10

7 11 13

7 12 15

7 8 15

7 9 13

7 10 12

7 11 14

No group.

Group of order 4.

4.

3 4 7 3 5 8

3 6 12

3 9 10

3 11 15

3 13 14

10.

4 8 13

4 9 10

4 11 14

4 12 15

4 8 13

4 9 12

4 10 15

4 11 14

5 6 11

5 9 15

5 10 12

5 13 14

5 6 15

5 9 14

5 10 12

5 11 13

6 8 14

6 9 13

6 10 15

8 11 12

6 8 11

6 9 13

6 10 14

8 12 15

7 8 15

7 9 12

7 10 14

7 11 13

7 8 14

7 9 15

7 10 13

7 11 12

No group.

No group.

6.

3 4 7 3 5 8

3 6 12

3 9 13

3 10 14

3 U 15

11.

3 4 7 3 5 8

3 6 12

3 9 14

3 10 15

3 11 13

4 8 14

4 9 10

4 11 13

4 12 15

4 8 13

4 9 10

4 11 14

4 12 15

5 6 11

5 9 15

5 10 12

5 13 14

5 6 11

5 9 15

5 10 12

5 13 14

6 8 13

6 9 14

6 10 15

8 11 12

6 8 15

6 9 13

6 10 14

8 11 12

7 8 15

7 9 12

7 10 13

7 11 14

7 8 14

7 9 12

7 10 13

7 11 15

No group.

No group.

6.

3 4 7 3 5 8

3 6 12

3 9 13

3 10 15

3 11 14

12.

4 8 14

4 9 12

4 10 13

4 11 15

4 8 14

4 9 10

4 11 13

4 12 15

5 6 11

5 9 15

5 10 12

5 13 14

S 6 11

5 9 15

5 10 12

5 13 14

6 8 15

6 9 13

6 10 14

8 11 12

6 8 15

7 8 13

6 9 14

7 9 12

6 10 13

7 10 14

8 11 12
7 11 15

7 8 13

7 9 10

7 12 15

7 11 14

No group.

No group.

III. TRIPLE TETRADS, N

0 HEAD OR SEMIHEAD.

Systems 1-26.

Systems 1-26 — Continued.

1 2 3

14 5 16 7 18

9 1 10 11 1 12 13 1 14 15

8.

3 4 8 3 5 9

3 6 12

3 7 14

3 10 15

3 11 13

246 257 28 10 29

11 2 12

14 2 13 15

4 7 10

4 9 13

4 11 14

4 12 15

5 6 11

5 8 15

5 10 12

5 13 14

1.

3 4 8 3 5 9

3 6 12

3 7 13

3 10 14

3 11 15

6 8 14

6 9 15

6 10 13

8 11 12

4 7 14

4 9 12

4 10 15

4 11 13

7 8 13

7 9 12

7 11 15

9 10 14

5 6 15

5 8 14

5 10 13

5 11 12

Group of order

2.

6 8 13

6 9 10

6 11 14

8 12 15

9.

4 7 10

4 9 15

4 11 12

4 13 14

7 8 11

7 9 15

7 10 12

9 13 14

5 6 11

5 8 13

5 10 14

5 12 15

Group of order 3.

6 8 15

6 9 14

6 10 13

8 11 14

2.

3 4 8 3 59

3 6 12

3 7 13

3 10 15

3 11 14

7 8 12

7 9 13

7 11 15

9 10 12

4 7 14

4 9 12

4 10 13

4 11 15

Group of order 6.

5 6 15

5 8 13

5 10 14

5 11 12

10.

4 7 13

4 9 12

4 10 14

4 11 15

6 8 14

6 9 10

6 11 13

8 12 15

5 6 15

5 8 14

5 10 13

5 11 12

7 8 11

7 9 15

7 10 12

9 13 14

6 8 13

6 9 10

6 11 14

8 12 15

Group of order 4.

7 S 11

7 9 15

7 10 12

9 13 14

3.

4 7 10

4 9 14

4 11 13

4 12 15

Group of order 2.

5 6 11

5 8 15

5 10 12

5 13 14

11.

3 4 8 3 5 9

3 6 12

3 7 15

3 10 13

3 11 14

6 8 13

6 9 15

6 10 14

8 11 12

4 7 10

4 9 12

4 11 15

4 13 14

7 8 14

7 9 12

7 11 15

9 10 13

5 6 15

5 8 14

5 10 12

5 11 13

Group of order 12.

6 8 11

6 9 13

6 10 14

8 12 15

i.

348 3 5 9

3 6 12

4 7 10

3 7 14

4 9 13

3 10 13

4 11 14

3 11 15

4 12 15

No group.

7 8 13

7 9 14

7 11 12

9 10 15

5 6 13

5 8 15

5 10 14

5 11 12

12.

4 7 10

4 9 14

4 11 13

4 12 15

6 8 11

6 9 14

6 10 15

8 13 14

5 6 11

5 8 12

5 10 15

5 13 14

7 8 12

7 9 15

7 11 13

9 10 12

6 8 13

6 9 15

6 10 14

8 11 15

No group.

7 8 14

7 9 13

7 11 12

9 10 12

E.

4 7 10

4 9 14

4 11 13

4 12 15

No group.

5 6 11

5 8 12

5 10 15

5 13 14

13.

3 4 8 3 5 9

3 6 12

3 7 15

3 10 14

3 11 13

6 8 15

6 9 13

6 10 14

8 11 14

4 7 10

4 9 13

4 11 14

4 12 15

No group.

7 8 13

7 9 15

7 11 12

9 10 12

5 6 11

5 8 12

5 10 15

5 13 14

6.

4 7 13

4 9 12

4 10 15

4 11 14

6 8 14

6 9 15

6 10 13

8 11 15

5 6 15

5 8 13

5 10 14

5 11 12

7 8 13

7 9 14

7 11 12

9 10 12

6 8 14

6 9 10

6 11 13

8 12 15

No group.

7 8 11

7 9 15

7 10 12

9 13 14

14.

3 4 8 3 5 10

3 6 12

3 7 15

3 9 13

3 11 14

7,

No group.

4 7 11

4 9 12

4 10 15

4 13 14

4 7 9

4 10 14

4 11 13

4 12 15

5 6 10

5 8 14

5 11 13

5 12 15

5 6 11

5 8 12

5 9 15

5 13 14

6 8 15

6 9 13

6 11 14

8 11 12

6 8 13

6 9 14

6 10 15

8 11 15

No ETOUD.

7 8 13

7 9 15

7 10 12

9 10 14

No group.

7 8 14

7 10 13

7 11 12

9 10 12

No. 2.]

TEIAD SYSTEMS— WHITE, COLE, CUMMINGS.

79

Systems 1-26— Continued.

15.

4 7 14

4 9 15

4

10 12

4 11 13

5 6 13

5 8 14

5

9 12

5 11 15

6 8 11

6 9 14

6

10 15

8 12 15

7 8 13

7 9 10

7

11 12

10 13 14

No group.

la.

4 7 9

4 10 12

4

11 15

4 13 14

6 6 15

5 S 14

5

9 12

5 11 13

6 8 n

6 9 14

6 10 13

8 12 15

7 8 13

7 10 14

7

11 12

9 10 15

No group.

17. 3 4 8 3 5 U

3 6 12

3 7 15

3

9 13

3 10 14

4 7 10

4 9 14

4

11 13

4 12 15

5 6 9

5 8 15

5 10 12

5 13 14

6 8 14

0 10 13

6

11 15

8 11 12

7 8 13

7 9 12

7

11 14

9 10 15

No group.

18.

4 7 13

4 9 14

4

10 15

4 11 12

5 6 9

5 8 15

5

10 12

5 13 14

6 8 14

6 10 13

6

11 15

8 11 13

7 8 12

7 9 10

7

11 14

9 12 15

No group.

19.

4 7 12

4 9 14

4

10 15

4 11 13

S6 9

5 10 12

5

8 15

5 13 14

6 8 14

6 10 13

6

11 15

8 11 12

7 8 13

7 9 10

7

11 14

9 12 15

No group.

20.

4 7 12

4 9 15

4

10 13

4 11 14

5 6 10

5 8 15

5

9 12

5 13 14

6 8 13

6 9 14

6

11 15

8 11 12

7 8 14

7 9 10

7 11 13

10 12 15

No group.

Systems 1-26— rontimied.

21.

4 7 9

4

10 13

4

11 14

4 12 15

5 6 10

5

8 15

5

9 12

5 13 14

6 8 13

6

9 14

6

11 15

8 11 12

7 8 14

7 10 12

7

11 13

9 10 15

No group.

22.

3 4 8 3 5 11

3 6 12

3

7 15

3

9 14

3 10 13

4 7 14

4

9 13

4

10 15

4 11 12

5 6 9

5

8 15

5

10 12

5 13 14

6 8 13

6

10 14

6

11 15

8 11 14

7 8 12

7

9 10

7

11 13

9 12 15

No group.

23

4 7 10

4

9 13

4

11 14

4 12 15

56 9

S

8 IS

5

10 12

5 13 14

6 8 13

6 10 14

6

11 15

8 11 12

7 8 14

7

9 12

7

11 13

9 10 15

No group.

24,

4 7 12

4

9 13

4

10 15

4 11 14

56 9

5

8 15

5

10 12

5 13 14

6 8 13

6

10 14

6 11 15

8 11 12

7 8 14

7

9 10

7

11 13

9 12 15

No group.

25.

3 4 8 3 5 15

3 6 12

3

7 11

3

9 13

3 10 14

4 7 13

4

9 12

4

10 15

4 11 14

5 6 14

5

8 12

5

9 10

5 11 13

6 8 11

6

9 15

6 10 13

8 13 14

7 8 15

7

9 14

7 10 12

11 12 15

Group of order 3.

26.

4 7 14

4

9 15

4

11 13

4 10 12

5 6 13

5

8 12

5

9 10

5 11 14

6 8 11

6

9 14

6

10 15

84 3 U

7 8 15

7

9 12

7 10 13

11 12 15

No group.

IV. NO TRIPLE TETRADS, BUT HEAD.

One system

One system— Continued.

1. 1 2 3

1 4 5

1 6 7

1 8 9

1 10 11

1 12 13

1 14 15

5 8 15

5 9 13

5 10 14

5 11 12

2 4 6

2 5 7

2 8 10

2 9 12

2 11 14

2 13 15

6 8 14

6 9 10

6 11 13

6 12 15

3 4 7

3 5 6

3 8 11

4 8 12

3 9 15

4 9 11

3 10 13

4 10 15

3 12 14

4 13 14

Groupof order 21.

7 8 13

7 9 14

7 10 12

7 11 15

Systems 1-3.

12 3 14 5 16 7 IS

9 1 10 11 1 12 13 1 14 15

246 257 28 10 29 12 2 11

14 2 13 15

1 347 358 36 11

3 9 14

3 10 13

3 12 15

4 9 10

4 8 13

4 11 15

4 12 14

5 6 12

5 9 15

5 10 14

5 11 13

6 8 14

6 9 13

6 10 15

8 11 12

7 8 15

7 9 11

7 10 12

7 13 14

Group of order 3.

2. 347 358 36 11

3 9 15

3 10 13

3 12 14

4 8 14

4 9 10

4 11 13

4 12 15

V. NO TRIPLE TETRADS OR HEAD, BUT SEMIHEAD.
Systems 1-3— Continued.

5 6 12

5 9 13

5 10 14

5 11 15

6 8 13

6 9 14

6 10 15

8 11 12

7 8 15

7 9 11

7 10 12

7 13 14

Group of order 3.
3. 3 4 7 3 5 8 3 6 13 3 9 14 3 10 12 3 11 15

Group of order 3.

4 8 15

4 9 13

4 10 14

4 11 12

5 6 11

5 9 15

5 10 13

5 12 14

6 8 14

6 9 10

6 12 15

8 11 13

7 8 12

7 9 11

7 10 15

7 13 14

VI. NO TRIPLE TETRADS, HEAD, OR SEMIHEAD.

Systems 1-15.
123 14 5

16 7 18

9

1 10

11 1 12 13 1 14 15

2 4 6

2 5

7 2 8

10

2 9 12 2 11

14 2 13 15

1. 3 4 8 3 5 9

3 6 14

3

7 11

3 10 13

3 12 15

4 7 15

4

9 10

4 11 12

4 13 14

5 6 12

5

8 15

5 10 14

5 11 13

6 8 11

6

9 13

6 10 15

8 12 14

No group.

7 8 13

7

9 14

7 10 12

9 11 15

2. 3 4 8 3 5

12

3 6 14

3

7 11

3 9 13

3 10 15

4 7 15

4

9 10

4 11 12

4 13 14

5 8 9

5

8 15

5 10 14

5 11 13

6 8 11

6 10 13

6 12 15

8 12 14

No group.

7 8 13

7

9 14

7 10 12

8 11 15

3. 3 4 8 3 5 10

3 6 13

3

7 12

3 9 14

3 11 15

4 7 15

4

9 10

4 11 12

4 13 14

Systems 1-15— Continued.

No group.
4. 3 4 8 3 5 9

No group.

5. 3 4 8 3 5 10

No group.

5 6 11

5

8 15

5 9 13

5 12 14

0 8 12

6

9 15

6 10 14

8 11 13

7 8 14

7

9 11

7 10 13

10 12 15

3 6 11

3

7 14

3 10 13

3 12 15

4 7 15

4

9 10

4 11 12

4 13 14

5 6 12

5

8 15

5 10 14

5 11 13

6 8 13

6

9 14

6 10 15

8 12 14

7 8 11

7

9 13

7 10 12

9 11 15

3 6 12

3

7 13

3 11 15

3 9 14

4 7 15

4

9 10

4 11 12

4 13 14

5 6 11

5

8 15

5 9 13

5 12 14

6 8 14

6

9 15

6 10 13

8 11 13

7 8 12

7

9 11

7 10 14

10 12 15

80

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Systems 1-15 — Continued.

Systems 1-15— Continued.

e. 3 4 8 3 S 14

3 6 12

3

7 U

3 9 15

3 10 13

11. 3 4 8 3 5 J3

3 6 14

3

7 12

3 9 11

3 10 15

4 7 15

4

9 10

4 11 13

4 12 14

4 7 15

4

9 14

4 10 12

4 11 13

5 6 13

5

8 15

5 9 11

5 10 12

5 6 11

5

8 15

5 9 10

5 12 14

6 8 11

6

9 14

6 10 15

S 13 14

6 8 12

6

9 15

6 10 13

8 13 14

7 8 12

7

9 13

7 10 14

U 12 15

7 8 11

7

9 13

7 10 14

11 12 15

No group.

Group of order 3.

7. 3 4 8 3 5 13

3 6 11

3

7 12

3 9 14

3 10 15

12. 3 4 8 3 5 14

3 6 11

3

7 9

3 10 13

3 12 15

4 7 15

4

9 10

4 11 13

4 12 14

4 7 15

4

9 10

4 U 13

4 12 14

5 6 14

5

8 15

5 9 11

5 10 12

5 6 13

5

8 12

5 9 11

5 10 15

6 8 12

6

9 15

6 10 13

8 13 14

6 8 14

6

9 15

6 10 12

8 U 15

7 8 11

7

9 13

7 10 14

U 12 15

7 8 13

7 10 14

7 11 12

9 13 14

No group.

Group of order 5.

8. 3 4 8 3 5 14

3 6 13

3

7 12

3 9 11

3 10 15

13. 3 4 8 3 5 14

3 6 9

3

7 13

3 10 15

3 11 12

4 7 15

4

9 10

4 11 13

4 12 14

4 7 15

4

9 10

4 11 13

4 12 14

5 6 11

5

8 15

5 9 13

5 10 12

5 6 11

5

8 13

5 9 15

5 10 12

6 8 12

6

9 15

6 10 14

8 13 14

6 8 14

6 10 13

6 12 15

8 11 15

7 8 11

7

9 14

7 10 13

11 12 15

7 8 12

7

9 11

7 10 14

9 13 14

No group.

Group of order 3.

9. 3 4 8 3 5 13

3 6 10

3

7 12

3 9 14

3 11 15

14. 3 4 8 3 5 11

3 6 13

3

7 9

3 10 15

3 12 14

4 7 15

4

9 13

4 10 14

4 11 12

4 7 15

4

9 11

4 10 12

4 13 14

5 6 11

5

8 15

6 9 10

5 12 14

5 6 10

5

8 13

5 9 14

5 12 15

6 8 12

6

9 15

6 13 14

8 11 13

6 8 14

6

9 15

6 11 12

8 U 15

7 8 14

7

9 11

7 10 13

10 12 15

7 8 12

7 10 14

7 11 13

9 10 13

Oroupofor<ier4.

Group of order 4.

10. 3 4 8 3 5 11

3 6 12

3

7 14

3 10 15

3 9 13

15. 3 4 8 3 5 10

3 6 15

3

7 13

3 9 11

3 12 14

4 7 15

4

9 14

4 10 12

4 11 13

4 7 U

4

9 15

4 10 12

4 13 14

5 6 13

5

8 15

5 9 10

5 12 14

5 6 9

5

8 14

6 11 13

5 12 15

6 8 11

6

9 15

6 10 14

8 13 14

6 8 13

6 10 14

6 11 12

8 11 15

7 8 12

7

9 11

7 10 13

11 12 15

7 8 12

7

9 14

7 10 15

9 10 13

Oroupoforder4.

Group of order 36.

VII. NO TETRAD.-

HEFFTER'S HEADLESS SYSTEM.

123 14 5 167 18

9

1 10 1

1 12 13 1 14

15

5 6 14

5 9 10 5 11 13 5 12 15

246 258 27

9

2 10 12 2 11

14 2 13

15

6 8 12

6 9

15 6 10 13 8 10 14

3 4 10 3 5

7 3 6 11

3 8 15 3 9 13 3 12

14

7 8 11

7 10 15 7 13 14 9 11 12

4 7 12

4 8 13 4 9 14 4 11

15

Group of order 60.

PART 5.

SEQUENCES AND INDICES FOR ALL GROUPLESS TRIAD SYSTEMS ON

15 ELEMENTS.

By L. D. CuMMiNGS.

Tlio 80 systems described in this paper separate into the three distinct types:

(a) Twenty-three systems with a group and with a head.

(b) Twenty-one systems with a gi'oup and with no head.

(c) Thirty-six systems with no group and with no head.

The indices for type (a) and the Heffter system of type (b) have been fully discussed in an
earlier paper ' and therefore are omitted here. The indices for types (b) and (c) are given in

' Cummings, L. D. Transactions of the American Mathematical Society, vol. XV, No. 3, pp. 311-327.

the following tables 1 (b) and 1 (c), respectively. Now in a given system the index of a triad
is invariant under all the substitutions of the symmetric group ; two systems, therefore, which
differ either in their indices or in the niunber of triads eniunerated imder each index are cer-
tainly incongruent. Hence tables 1 (b) and 1 (c) show conclusively the noncongruency of these

56 systems.

Table 1 (b).

System.

eo

?<

?

10

eo

%

0

a

CO

00

en

fO

3j

Si

CC

M3

s

Si

«3

Si

Si

0

CO

C5
CO

00

tb

V47 1

3
2

1

3

4
2

1

3

3

8
3

3
2

3
4
5
3
3

6
4

"2

1

'3'
3

3

4

1

VI 3,5

4

VI24J

1

5
3
6
9

4
3
6

4

5

2

2
10
6

VI2«»

1 -

2
6
3

2

2

Vl2fr

1
1

Ill

6

7
1
14
4
7

1

3

3

3

1

Vial

4
3

3

6

3
9

3

3

1

V*y2

3

6

3

4

"4"

6

3
6

VI3ja

2

i

V-W2

3

1

3
2

3
2

18

6
3

6

8

6

3
2

6

1

VI2(e

1

1

Till

2

V4^1

6

3
4

9
8

10

3
8
12
S

4

Vlljfl

6
6

s

4

4

4

VK

12
11

4
7
2
6

4

12

5

5
4
3

VI3,T

2

4
3

8

4
"2

3

2

V«l

3

3
4
1

3

3
12

■ r

3

VI3rv

2

4

ni«

4

3

12

S

54061°— 19-

81

82

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table 1 (c).

System.

«

n

?.

CO

0

i

01

V

a

00

t>.

to

M

0

6
S
6
2
2
1
1

"5'
2
2
1
2
3
5
1

s>

"3'
"2

2
2

1
1
2

....

1
2

"i'

2
1

1
2
1
1
2

■4"

i
2
7
2
1
2
1

to

M

2

6
5
3
5
2
2
3

1

2

1
7
7
4
2
3
3
2
3
2
2
2
5
1
2
1
1
5
3
3
2
3
5
2

n

"

Si

1
....

to

^

Ol

0

oa

CO

00

s>

15

1

2
3
2
2

1

....

1
1

2

3
2

2

2
3
3
5
1
2
3
1
....

1
3

1
1
1
3
2
1
1
1
1

3
2
3

"2

2
2

"2'

■3'
1

7
2
5
5
3

2
2
2

1
3
3
3
1
4
2
1
1
3

"2"
1

1

"i'
1

1
1

4

5
1
1

\

11

4

2

5
4

S
6
3
3
5
6
S
8
8
2
4
5
5
S
g
8
4
3
9
7
5
5
7
7
4
4
5
4
6
9

"i"

14

1

■2'

1
1

"2

1
1

1
1

3
3
5
5
1
3
4
3
3
3
2
3
5
3
3
2

"2'

1
1
3

2

1
2
2

2

"2"

2

1
1

1
1

2

2
2

1

1

2
2

4

6

}

1

1

....

n

2
2
3

3

1

5

1

1

I
3

1

25

1
3
2
3
3
2
1
1
5
4
3
4
3
4
4
2
2
1
2
1
1

1
1

1

2
6
3
5
7
3
8
4
8
4
7
7
9
7

3
1
2
1
5
2
4
3
2
2
1
5

■'2'
1

1
3
2
5

1
3

2

3

1

27

1
1

"2'

3

2

..„

3
2
3

"2

1
3
3
1
3
1
1
2

"4'

1

20

n

^4 r Col ft!

5

29

1

'i'
"i'

1
1
2
2
1
2

?.

3
2
2
2

4

26

2
3

■5"
4
2
2

1

1

22

2
3
3

1
1
1
2
2

i
3

1

"2

1
3
1
1
1
2
3
1
3

1

21

"i'

1

32

1

2
2

31

3

3

10

3

30

1
2

1

1

18

2

2

2

The analysis by the method of sequences of the 80 noncongruent systems derived by Mr.
Cole in Part 4, shows that 77 of these systems are congruent either to systems derived in Part 1
by means of operators of their groups; or to systems derived from the former, in Part 3, by
means of quadi-angular or hexagonal transformations.

The names or numbers of these 77 pairs of equivalent systems and the substitution wliich
transforms each of Mr. Cole's systems into its equivalent system are given below.

I, l=IIIAby si

I, 2=IIIBby s;

I, 3=IIICby s;

I, 4=IIEby s\

I, 5=IVAby s:

I, 6=IIAby S-:

I, 7=VAby s;

I, 8=IVBby s;

I, 9=IICby s]

I, 10=1 ID by s;

I, ll=VIBby s;

I, 12=VID by si

I, 13=VC by s;
I, 14=VD by

-0

1

a

1 2
6 a

1 2

6
9
6
/
6

2
6

2
b

1 2

d /

2

d

3 4

6 d

3 4

e d

3 4

c d

3 4

6 c

9 /

5 6

1 4

5 6

? «

5 6

/ «

5 6

e /

5 6

e a

5
c

a f

3 4

rf /

3 4

c 6

3 4

c a

8 9

1 2

8 9

1 2

8 9

1 2

8 9

9
3

8 9
4 5

10 11
3 4

10 11
3 4

10 11
3 4

10 11

9 f

10 11

3 4

10 11

7 8

10 11

5 8

10 11

3 7

10 11

8 7

10 11
3 6

10 11

5 6

10 11
3 8

10 11

6 5

10 11

5 7

12

13

5

6

12 13

5

6

12

13

5

6

12

13

5

7

12

13

5

6

12

13

2

1

12

13

4

3

14 15

14 15

14 15
8 7

14 15

14 15

12 13

4 8

12 13

5 6

12 13

1 7

12 13

2 3

12 13

7 5

12 13

4 8

12 13

8 3

5)

?)

)

)

5)

14 15 \
4 3/

14 15 \
6 7/

14 15 \
& 2 )

14 15 \
2 1 )

14 15 "S
2 8/

14 15 \
4 i )

14 15 \
1 6 )

14 15 ^
1 3

14 15
6 2

I)

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CUMMINGS.

83

I,

15=IIF by

■Ki

2
f

3
a

4
c

5
9

C
6

7

8

1

9

(;

10 11 12 13 14 15 \
7 3 2 5 8 4/

I,

lG=IIIDby

-e

2
d

3
6

4
/

5
e

6

9

7
c

8

1

9
2

10 11 12 13 14 15 \
5 6 3 4 8 7/

I,

17=VB by

-0

2
e

3

d

4
!7

5

6
6

7
c

8
1

9
3

10 11 12 13 14 15 \
7 2 4 8 5 6/

I,

18=1IB by

-c

2
/

3
6

4

5
e

6
d

7
c

8
4

9

3

10 11 12 13 14 15 \
5 6 2 18 7/

I,

19=VIA by

<

2
d

3

4
6

5

6

/

7
c

8
5

9

8

10 11 12 13 14 15 \
12 4 3 7 6/

I,

20=VIC by

-G

2
c

3

e

4
d

5
a

6
/

7

8
4

9

8

10 11 12 13 14 15 \
2 3 6 5 1 7 y

I,

21=I(' by

-c

2
d

3
c

4
6

5

e

6

a

7
/

8
8

9

1

10 11 12 13 14 15 \
4 5 6 3 7 2/

I,

9.9—lB by

-a

2
c

3

e

4
5'

5
/

6
a

7
6

8
6

9
1

10 11 12 13 14 15 \
7 5 3 2 8 4 y

II>

1=10 by

-(;

2
d

3

1

4
/

5
3

6
c

7
7

8
6

9

8

10 11 12 13 14 15 N
f 4 jr a 2 5 /

n,

2=11 by

-c

2
/

3
5

4
1

5

2

6

7

7
8

8
d

9
c

10 11 12 13 14 15 \
9 6 € 6 4 3 /

n,

3= 9 by

-a

2
6

3
1

4
a

5
<7

6

5

7
2

8
8

9
4

10 11 12 13 14 15 N
6 c c 7 / 3 /

n,

4=15 by

-e

2
/

3
e

4

5
c

6
0

7
5

8
8

9

7

10 11 12 13 14 15 ^
2 13 4 6 5/

11,

5=13 by

-(}

2
a

3

d

4
£7

5

6
6

7
c

8
2

9
8

10 11 12 13 14 15 \
17 5 4 6 3/

n,

6=12 by

<

2
d

3
c

4
9

5

c

6

/

7
6

8
1

9
2

10 11 12 13 14 15 \
6 5 4 3 8 7/

II,

7=V47l by

•<i

2

3
63

4

5

6
C3

7
a,

8
a2

9
C

10 11 12 13 14 15 \
A d^ 61 rfs Ci 83/

II,

8=VI2,7 bys=(j^

2

3

05

4

5

6

7
C

8
03

9

02

10 11 12 13 14 15 \
A a., B 65 bi 6, /

II,

9=VI335 by

-c.

2

3

"2

4

"6

5

"5

6
64

7

8
6.

9

66

10 11 12 13 14 15 \
A a^C 62 *3 *ja

II,

10=14 by

-G

2
3

3
6

4
4

5
c

6
0

7
2

8

9

8

10 11 12 13 14 15 Y
^ e f 7 & d )

11,

11= 8 by

-G

2
d

3
6

4
e

5
/

6
9

7
c

8
7

9
G

10 11 12 13 14 15 \
4 3 12 5 8/

II,

12=25 by

-0

2
1

3
a

4
3

5
c

6

8

7
e

8
4

9
3

10 11 12 13 14 15 \
6 d 7 2 5 6 /

III,

l=V4a2 by

■<h

2
A

3
B

4

5

6
6,

7

8
62

9
d^

10 11 12 13 14 15 \
02 C2 03 C3 63 dj

III,

2=VIl3i3 by

-a

2
B

3

C

4

5
h

6

7

8
62

9

02

10 11 12 13 14 15 N
61 a, 63 03 64 aj

III,

3=Vl7 by

-a

2
B

3

C

4

5

6
6,

7

Ol

8
03

9
63

10 11 12 13 14 15 \
f3 d3 C2 *! 02 62/

III,

4=28 by

•=a

2
3

3

1

4
5

5
6

6
6

7
3

8

7

9
4

10 11 12 13 14 15 \
/ e 2 a 8 c /

III,

5= Gby

-G

2
1

3
8

4
3

5
6

6
c

7

8
e

9
6

10 11 12 13 14 15 \
2 7 5 / 4 a /

III,

6= 4 by

-G

2
8

3

1

4
d

5
c

6

4

7
5

8
6

9
3

10 11 12 13 14 15 \
b e 2 7 f a J

III,

7= 7 by

-G

2
8

3
1

4
5

5
4

6
c

7

8
/

9

a

10 11 12 13 14 15 \
2 7 3 « 6 6 /

III,

8=VI2^a by

-a

2
B

3
C

4
04

5
64

6

"3

7
63

8
6.

9

10 11 12 13 14 15 \

62 Oj fee 65 Be fls/

III,

9=IIla by

-a

2
6,

3

4
C3

5

6

7

^2

8

02

9

"3

10 11 12 13 14 15 \

«i di d., 62 «3 63/

III,

10=VI2,e by

-a

2

3
C

4

04

5
64

6

"3

7

&3

8

9

10 11 12 13 14 15 \

62 O2 fie *6 "e Oj/

III,

11 =a new groupless

system

(Reference n

umber 34.)

HI,

12=25 by

■<i

2

3

5

4

c

5
4

6

8

7

7

8
2

9
9

10 11 12 13 14 15\
a G c 3 / 1/

III,

13= 5 by

<

2

1

3

8

4

7

5
2

6
h

7
e

S
a

9
/

10 11 12 13 14 15\
4 5 6 3 d c/

84

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV

III

III

III

III

III

III

III

III

III
III
III

III

III

IV

V:

V
V

VI
VI

VI

VI

VI

VI

VI

VI

VI

VI

VI

VI

VI

VI

VII

14=21 by

<l
<\

-G
-G

15=20 by

16=19 by

17=26 by

18=32 by

in= 1 by

20= 3 by

21=27 by

22=22 by

23 =a new groupless

24= 2 by s=(j

25=V472 by s=(^^

26=31 by s=Q

]=IAby ..=(J

]=V4/32 by s=A

2=V461 by s=( ^

3=II4/J1 by s=A
Vi

1=23 by s=(^.

2=a new groupless

3=18 by s=Q

4=33 by

5=16 by

6=17 by

7=30 by

8=29 by

8 9 10 11 12 13 14 15\

/ rf 8 5 c 3 7 6/

8 9 10 11 12 13 14 15\

8 7 d A I f e gj

2 3 4 5 6 7

6 10^42

2 3 4 5 6 7

c 2 3 5 6 6

2 3 4 5 6 7

d 3 6 4 5 7

2 3 4 5 6 7

d 7 g c e 8

2 3 4 5 6 7

c 6 / 3 d 4

2 3 4 5 6 7

6 4 c 1 / 7

2 3 4 5 6 7

c 4 e 1 d 3

2 3 4 5 6 7

/ 6 c 2 jr 4

2 3 4 5 6 7

6 6 8 5 e d
jystem. (Reference number 35.)

2 3 4 5 6 7 8 9 10 11 12 13 14

34 cl/725da6

2 3 4 5 6 7 8 9 10 11 12 13

9 10 11 12 13

c / 8 a 6

9 10 11 12 13 14

/ 1 6 c 2 5

9 10 11 12 13

1 6 c ^ 2

9 10 11 12 13

5 d a 6 g

9 10 11 12 13 14 15\

6r 6 a 8 5 / 6^

9 10 11 12 13

a 7 1 d b

9 10 11 12 13

3 1 a 7 2

14 15\
2 gj

14 15\
5 aj

14 15\
a 8)

14 15\
8 e)

14 15\
/ h)

14 15\
e 8/

14 15\

c y)

C A bi a^ d^ c^ d, c^ b^ a^ 63

2 3 4 5 6 7 8 9 10 11 12 13
/65(f8ea3172c

2 3 4 5 6 7 8 9 10 11 12 13 14
/6acecf45631

2 3 4 5 6 7 8 9 10 11 12 13 14 15 \
«! flj C a^ Og $2 di B A d^ d^ b^ b^ b^J

2 3 4 5 6 7 8 9 10 11 12 13 14 15 \

<■, Oj C 03 Oi e^ di A B d^ rf, 63 6, 5,/

4 15\
g f)

13 14 15 \

"3 '^2 <^\'

14 15\
4 ?/
4 15\
7 2/

-G
-G

-e

-G

9=VI33c<bys=Q^
10=VI3,7bys=(^^
11=111, by s=Q^
12=12 by s=A,
13=1113 by s=(^^
14=VI337by«=Q^
15=1112 by s=Q
=VIIby .=(1

C 03 C2 O2 a, r3 rfj .(4 b^ d^B d^ 63 62

2 3 4 5 6 7 8 9 10 11 12 13 14 15
6672alc5'34de85

system. (Reference number 36.)

2 3 4 5 6 7 8 9 10 11 12 13 14 15
8c«/561a967243

2 3 4 5 6 7 8 9 10 11 12
535rc4266dc

2 3 4 5 6 7 8 9 10 11 12 13 14 15 \

C, C3 62 Qi (ij i\ O-l h "3 «3 Ci C2 «2 ''3/

2 3 4 5 6
64 65 04 ^ B

5> 3 4 5 6

}

D
I)

12 13 14 15\
a 1 8 7/

14 15 \

«2 bj
14 15\
6 3/

D

9

6,
7 8

9 10 11 12 13 14
a e 1 d

9 10 11 12 13 14 15\

d c 4 a 6 / 1/

9 10 11 12 13 14 15"^
a 6 g f c 8 c^

9 10 11 12 13 14 15 \

«! 61 C 62 fls °2 "6/

9 10 11 12 13 14 15>

'^

a, 05 62 Oa 04 64 65 02 65 6, 03 63 A CJ
23456789

C| 03 U2 ^2 ''3 ^3 ^2

2 3 4 5 6 7 8 9 10 11 12 13
la625768435rc

2 3 4 5 6 7 8 9 10 11 12 13 14

9 10 11 12 13 14 15 \
02 c/3 61 di Oi f, ej

14 15\

\ 15 \

'1 «3/

15)

2 05/

0

0

2 3 4 5 6 7 8 9 10 11 12 13 14

64 b^ b,B A 04 Qj ie C" 61 Oi a^ b^ u^/

2 3 4 5 6 7 8 9 10 11 12 13 14 15 \

03 C3 ^2 fli 'h di C3 c, 63 62 ("i ("z ''2 *i/
2 3 4 5 6 7 8 9 10 11 12 13 14 15>
/4137gr2c58rfe6 6>

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CCIMMINGS.

85

In Part 4 of this paper Mr. Colo disliiiKuishos four varieties of interlacing of duads in a
system, namely, tlio single tetrad or oktad, the tri])]o tcitrad, the liexad, and the dodekad.

Thi«e ty])es correspond to what wo have designated in Part 3 as two-colimin or contracted
indices. Tlie triple tetrad, the oktad, the hcxad, and the dodekad corresponding, respectively,
to the indices 2'; 2, 4; 3=; 6.

Since these four types of interlacing form the basis for Mr. Cole's derivation of tlie 80
systems, it seemed probable that the two-column indices might fui-nish a sufficient and unique
characterization for a triad system.

The two-column indices for each of the 80 systems were, therefore, determined and are
exhibited in the following Table 2 :

Table 2.

System.

ni.\..

IIIB..
lUC.
HID..
UE...
IV A..
IVB..
VD...
VA...
VC...
VB...
II A...
VID..
IID...
IIC...
IIB...
HF...

15

9

V,47l
VIB..
IC...
V,4al
VIC.

11

10

VI, 335

2"

2,4

105

0

57

48

49

24

49

0

29

60

29

60

25

36

25

36

21

36

21

36

21

12

15

66

13

57

13

54

13

54

11

42

11

42

11

36

9

33

7

36

5

54

5

45

5

24

4

48

4

45

4

42

4

42

System.

4

7

IB

VIA....

12

VI,2,J..

13

VI, 2,7.

6

3

1

VI, la^..

26

28

V,l>...

25

V,4t2..

14

8

5

27

35 rc<)lel

32

VI,2,a.
21

vi,'2i«;.'

2

23

2.4

32

6

4

27

7

67

4

24

8

69

3

42

7

53

3

42

4

56

3

36

7

59

3

33

6

63

3

33

5

64

3

21

6

75

2

30

10

63

2

27

13

63

2

24

9

70

2

24

4

75

2

21

15

67

2

21

13

69

2

12

12

79

2

10

30

63

36

16

52

36

9

59

33

8

63

27

9

68

24

15

65

24

12

68

24

12

68

21

20

63

21

10

73

21

8

75

21

7

76

System.

31

24

20

19

36 [Cole].
lA

V,4/31...

V::::::.

22

VI,3,Y..

II, l7

34 [Cole] .
V,4(32-..
V,4J1....
VI, 3w..

23

VI,3jT..

33

II, Ij

18

29

16

30

II, I3

VII

2»

2,4

3»

18

18

18

14

18

12

15

12

12

11

0

42

7

0

30

10

0

30

5

0

27

12

0

24

12

0

24

7

0

21

15

0

21

14

0

21

7

0

21

7

0

18

21

0

18

14

0

18

13

0

18

11

0

18

9

0

15

16

0

15

16

0

15

14

0

16

11

0

6

15

0

0

15

68
72
74
77
81
S6
65
70
66
69
74
69
70
77
77
66
73
74
76
78
74
74
76
79
84
90

This table shows that the two-column inchces suffice to estabUsh the noucongruency of
72 of the 80 systems, but fail to distinguish uniquely the remaining 8 systems. The systems
not uniquely determined by their two-colimm indices consist of the five pairs of headed sys-
tems IIC, IID; IIB, IIF; HE, IVA; IVB, VD; VA, VC; one pair wdtli a group but no head
V451, V4;82; and two pairs of grouplcss systems 18, 29; 32, 35 [Cole].

Perfect discrimination is possible by the use of a double entry table, 15 by 15, shoAving
not merely the number, but the exact distribution of triple tetrads, oktads, hexads, and
dodekads in each of the eight pairs of apparently duplicate sj'stems.

llie investigation for one of these pairs of apparently duplicate systems, IIF and IIB,
is given below in some detail.

The system IIF is arranged in a 15-by-7 alTa3^ Each element heads one column; below it
are placed the seven duads of elements that occur with it in triads of the system. The two-
column indices for the 105 pairs of columns are determined, and we find that the indices 2';
2, 4; 3-; 6 belong, respectively, to 11, 42, 0, and 52 pairs of columns.

To exhibit more evidently the types of intcrlacings existing amongst the duads in the 105
pairs of colunms, we now arrange the two-column indices for the system IIF in the following
15-by-15 array of Table 3.

86

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table 3.

[Vol. XIV.

a

b

c

d

c

/

g

1

2

3

4

5

6

7

8

2

2

2

4

2

4

6

6

6

0

6

6

6

6

b

2

2

4

4

4

4

6

4

6

4

6

6

6

6

2

2

4

4

4

4

6

6

0

6

4

6

6

4

d

2

4

4

4

2

4

4

6

4

6

6

4

4

6

4

4

4

4

4

2

6

6

6

6

4

6

6

4

/

g

2

4

4

2

4

4

6

6

6

6

6

6

6

6

4

4

4

4

2

4

6

4

6

4

6

6

6

6

1

6

8

6

4

6

6

6

4

2

4

4

6

6

4

2

6

4

6

6

6

6

4

4

4

2

6

4

4

6

3

6

6

6

4

6

6

6

2

4

4

4

fi

6

4

4

6

4

6

6

6

6

4

4

2

4

. .

6

4

4

6

5

6

6

4

6

4

6

6

4

6

4

6

4

4

2

6

6

fi

6

4

6

6

6

6

4

6

4

4

2

4

7

6

6

6

4

6

6

6

6

4

6

4

4

2

4

8

6

e

4

6

4

6

6

4

6

4

6

2

4

4

In this table the two-column indices 2'; 2, 4; 3^; 6 are replaced, for the sake of brevity, by
the single figures 2, 4, 3, 6, respectively. The figure placed at the intersection of any row with
any column shows the index for the pair of colimins formed from the two elements which lead
the row and head the column, respectively. For example, in Table 3, the indices for the pairs of
columns he, Id, hi are 2^; 2, 4; 6, respectively.

In the 15 by 7 rectangular array for the system IIF, the column headed by the element a
may be united with each of the columns headed by one of the remaining 14 elements to form 14
pairs of colunuis. The figures tabulated under the element a in table 3 shows that of these 14
pairs of columns there are 4 pairs with the index 2^; 2 pairs with the index 2, 4; and 8 pairs
with the index 6. Interpreted in terms of interlacings these two-column indices place in
evidence the fact that the duads of column a arc united with the duads in the remaining 14
columns by 4 triple tetrads, 2 oktads, and 8 dodekads. Similar results for each of the 15
elements are briefly summed up in the following Table 4, which we shaU designate as the table
of interlacings for the system IIF.

Table 4. — Interlacings for system TIF.

a

b

c

d

c

/

S

1

2

3

i

5

6

7

8

Index 23

Index 2, 4

4
2
8

2
6
6

2
6
6

2
8
4

1
7
6

2

4
8

1
7
6

1
5
8

1

6
7

1
5
8

1

6
7

1

6
7

1
5
8

1
5
8

1
6
7

Since only those elements whose duads are similarly interlaced through a system may
belong to the same set of transitive elements, Table 4 shows clearly that the possible sets of
transitive elements for the system IIF are a, he, d, eg, f, 1367, 2458. A fact in exact accord-
ance with the resiilts obtained previously in the examination of this system by the method of
trains and also by the method of the thi'oe-column indices.

The interlacings for the system IIB are exhibited in Table 5.

Table 5. — Interlacings for the system IIB.

a

b

c

d

e

/

9

1

2

3

4

5

6

7

8

Index 23

4

2

2

2

1

2

1

1

I

1

1

1

1

1

1

Index 2,4

2

6

6

6

9

6

5

5

6

5

6

6

5

5

6

8

6

6

6

4

6

8

8

7

8

7

7

8

8

7

A comparison of Table 4 mth Table 5 demonstrates conclusively the noncongruency of the
systems IIF and IIB. Table 4 shows that there is a column d in the system IIF the interlacings
of whose duads with the duads of other columns in IIF are represented by the numbers 2, 8,
and 4, corresponding, respectively, to two triple tetrads, eight oktads, and four dodekads. Table
5 shows no column in IIB with similar interlacings, therefore these two systems IIB and IIF
are certainly incongruent. The reader will observe other distinctive columns in these two
tables.

No. 2.]

TRIAD SYSTEMS— WHITE, COLE, CUMMINGS.

87

The tables of intcrlacings for the remaining seven apparently duplicate pairs of systems
are adjoined below and establish the noucongrucncy of the two systems in each pair.

Table of interlacings for the system IIC.

a

t>

c

i

e

/

9

1

2

3

4

5

6

7

S

Index 2'

Index 2, 4

Indoxe

4

6
4

2
6
6

2
8
4

4
8
2

I
7
6

2
8
4

1
9
4

1
6
7

1

8
5

1
6
7

1

8
5

1

8
5

2
6
6

2

K

1
8
5

Table of interlacings for the system IID.

a

b

c

d

e

/

9

1

2

3

4

5

6

7

8

Index 2>

Index 2, 4

Index 6

4
6
4

2
6
6

2

8
4

2
6
6

3
9
2

2

8
4

1
9
4

1
6
7

1

8
5

1
6
7

1

8
5

2

5
7

1

9
4

1
9
4

2
5

7

The system IID contains no column similar to the column d of IIC, therefore IIC and

III) are noncongruent.

Table of interlacings for the system HE.

a

b

c

d

e

/

S

1

2

3

4

5

6

7

8

Index 2>

Index 2, 4

Index6

8
6
0

4
10
0

6

4

8
2

3
9
2

4

8
2

3

9
2

2

8
4

4
6

4

2

8
4

4
6
4

3

9
2

4

S
2

4

8
2

3
9
2

Table of interlacings for the system IVA.

"

6

c

d

e

/

9

1 2

3

4

5

6

7

8

Index 2»

Index 2, 4

Index6

6
8
0

6
8
0

2
12
0

4
6
4

4
6

4

4
6
4

4
6
4

4
0
4

4
6
4

4
6
4

4
6
4

3
11
0

3
11
0

3
11
0

3
11
0

The system IVA contains no column similar to the column a of HE, hence HE and IVA

are noncongruent.

Table of interlacings for the system IVB.

"

b

c

d

«

/

9

1

2

3

4

5

6

7

8

Index 2»

Index 2, 4

Index6

'I
4

2
8
4

2
4

8

4
6
4

4
2

8

4
6
4

4
2
8

4
S
S

4
5
5

4
5
5

4

5
9

3
6
5

3
6
S

3
6
S

3
6
5

Table of interlacings for the system VD.

a

b

c

d

e

/

»

1

2

3

4

5

6

7

8

Index 2>

Index 2, 4

Index 6

2
8
4

2
8
4

2
8
4

3

7
4

7
3
4

■
3
7
4

3

7
4

3
6
5

3

«
5

4
6
4

4
6
4

3
6

5

4

6
4

4

6
4

3

6
5

No column in VD is similar to the column a of IVB, hence IVB and VD are noncongruent.

Table of interlacings for the system VA.

a

6

c

d

e

J

9

1

2

3

4

5

6

7

8

Index 2»

Index 2, 4

Index 6

2
12

0

2
12
0

2
12

0

3
3

S

3

I

3
3

8

3
3

8

3
3

8

3
3

8

3
3

8

3
3

8

3

3
3

8

3
3

8

3
3

8

88

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table of interlacings for the system VC.

[Vol. XTV.

a

b

c

d

e

/

g

1

2

3

4

5

6

7

8

Index 2>

Index 2, 4

Index 6

2
4
8

2

8
4

2

8
4

3
11
0

3
11
0

3

11
0

3

11

0

3

5
6

3

5
6

3

5
6

3

5
6

3
5
6

3
5
6

3

5
6

3

5
6

No column of VA agrees with any column in VC, therefore VA and VC are noncongruent.

Table of interlacings for the system V4&1.

a

b

c

d

e

/

ff

1

2

3

4

5

6

7

8

Index 2, 4

3

0

3

2

2

2

2

2

2

4

4

4

4

4

4

Index 33

0

1

1

1

1

1

2

2

2

1

1

1

0

0

0

Index 6

11

13

10

11

ir

11

10

10

10

9

9

9

10

10

10

Table of interlacings for the system V4^2.

a

b

c

d

e

/

?

1

2

3

4

5

6

7

S

Index 2, 4

Index 3'

IndexO

0

7
7

3
0
11

0
4
10

2
0
12

2
0
12

2
0
12

2

1
11

2

1

11

2

1

11

5
1
8

5

1
8

5

1
8

4
1
9

4
1
9

4
1
9

The two systems V431 and V4|82 show dissimilar columns a, and therefore are noncon-
gruent.

Table of interlacings for the system 18.

a

i>

c

d

e

/

S

1

2

3

4

5

6

7

8

Index 2, 4

Index 3=

IndexO

0

1
13

3
3

8

3

3

8

4

4
6

3

1
10

2

1
11

0
3
U

0
2
12

2

2

10

2
3
9

2

1

11

2
3

9

2
2
10

3
1

10

2
2
10

Table of interlacings for the system 29.

a

I>

c

d

e

/

ff

1

2

3

4

5

6

7

8

Index 2, 4

Index 3»

Indexfi

2

1

11

1

1

12

2

1

•11

3
2

9

2

4
8

2
2
10

2
0
12

1
3
10

1
3
10

2

4
8

5
3
6

3

3

S

0
3
11

3

1

10

1

1

12

The tables of interlacings for the systems 18 and 29 show manj^ dissimilar columns, there-
fore these systems are noncongruent.

Table of interlacings for the system 32.

a

I>

c

d

€

/

a

1

2

3

4

5

6

7

8

Index 2»

0

0

0

0

1

0

0

0

0

0

0

0

0

1

0

Index 2, 4

3

■1

5

3

3

£

5

2

3

3

4

3

0

2

3

Index 3»

2

1

1

2

3

2

0

1

2

4

2

1

3

0

0

9

9

8

9

7

7

0

u

9

1

8

10

11

11

U

Table of interlacings for the system 35 [^Cole^.

a

b

c

d

e

/

fl

1

2

3

4

5

6

7

8

Index 2'

Index 2, 4

Index 3^

0

4
1
9

0
3

4
7

0
3
2
9

0
4

1
9

0

4

1

9

0
3

1
10

0

1

1

12

1

3
0
10

1
2
3

8

0
2
2
10

0
3
5
6

0
4
0
10

0

2
2
10

0
4

1
9

0
5
1

S

Index 6

No. 2] TRIAD SYSTEMS— WIIITE, COLE, CUMMINGS. 89

The column e of 32 has no (hiplicato in 35; hence the two systems are noncongruent.

Wo have derived, then, iia Part 5 a new method of comparison for triad systems by means
of the two-column indices and the table of interlacings for the system.

Tlois method of comparison, since it naturally yields at least a partial, in some cases a com-
plete, separation into sets of transitive elements for the system, will also facihtate the deter-
mination of the group belonging to the system.

CONCLUSION.

Lookmg toward the census of triad systems in more than 15 elements, we have in the
foregoing memoir four modes of classification which woidd bo applicable to the construction
and comparison of systems. Of these wo venture to express the behef that the method of
indices will bo found most convenient for comparisons, whUe for construction there is no doubt
that a group, where one can be prescribed, is the most direct auxiliary. Any exhaustive
census, certainly for 31 or more elements, is out of the question in finite time; but systems
admittmg, for example, certain cychc groups are not numerous nor difficult of construction,
the method of indices showang very quickly their noncongruency. In the present state of the
theory the most desirable forward step would be a demonstration that some one of these methods
is (or is not) a sufficient means of provuig congruency for triad sj'stems of any nxmaber of elements
above 15.

WASHINGTON : QOVEENMEXT PRINTING OITICE : 1919

MEMOIRS

OF THE

NATIONAL ACADEMY OF SCIENCES

THIRD MEIVtOIR

WASHINGTON

GOVBRNMENT FEINTING OmCB

1922

NATIONAL ACADEIVEY OF SCIENCES.

Volume XIV.

THIRD THElVrOIR.

TABLES OF MINOR PLANETS DISCOVERED BY

JAMES C. WATSON.

PART II.

ON V. ZEIPEL'S THEORY OF THE PERTURBATIONS OF THE
MINOR PLANETS OF THE HECUBA GROUP.

BT

ARMIN O. LEUSCHNER, ANNA ESTELLE GLANCY, and
SOPHIA H. LEVY.

CONTENTS,

Page.

Preface 7

Introduction 9

I. Formuhc and fables for the FTeeuba group, areording to the theory of Bohlin-v. Zeipel.and an example of their

use 10

Determination of constant elements and of perturbations of the mean anomaly 10

Perturbations of the radius vector 20

Perturbations of the third coordinate 21

Check computation 22

Computation of the perturbations for the time ( 22

Comparison of the re^i.'sod with v. Zeipel 's original tables 27

Table A 28

Table B 30

Table C 31

Table D 34

Table E, 35

Table Ej 35

Table F 36

Table G 38

II. Tables for the determination of the perturbations of the Hecuba group of minor planets 41

Development of the differential equations for W and for the third coordinate 41

Integration of the differential equation for H-'. 78

Comparison of tables 120

Perturbations of the mean anomaly 121

Comparison of tables 134

Perturbations of the radius vector 137

Perturbations of the third coordinate 140

Comparison of tables 146

Constants of integration in no: and c 146

Comparison of tables 156

Eratain "Angeniiherte Jupiter-Storungen fur die iJecufta-Gruppe, " II. v. Zeipel 155

Erata in " Sur le D^veloppement des Perturbations Plan^taires," § 1-7 and Tables 1-XX, Karl Bohlin 157

5

PREFACE.

Part I of "Tables of Minor Planets Discovered by James C. Watson," containing tables
for 12 of the 22 Watson planets, was published in 1910 in the Memoirs of the National Academy
of Sciences, Volume X, Seventh Memoir, with a preface by Simon Ncwcomb, in which he
gives an account of the early history of the investigations of the perturbations of the Watson
planets under the auspices of the Board of Trustees of the Watson Fund.

In the introduction to Part I ' reference is made to the Watson planets of the Ilecuha
group, for wliich it was found necessary to construct special tables on the plan of Bolilin's
tables for the group 1/3. A comparison of these tables with similar tables by v. Zeipel remained
to be made before applying either of them to the development of perturbations of planets
of the Hecuba group. This comparison was completed in 1913 with the assistance of Miss A.
EsteUe Glancy and Miss Sophia H. Levy, with the results set forth in the following pages.

Publication of these results was delayed, partly because it seemed desirable to verify the
tables by application to a number of planets and partly on account of interruptions caused in
recent years by war conditions. Miss Glancy, in particular, had undertaken to test the accuracy
of our tables, which we had applied to v. Zeipel's example, (10) Hygiea, by further investi-
gations on this example after joining the Observatorio Nacional at Cordoba in 1913. This
test has now been completed with higlily satisfactory results. The tables have also been
successfully applied to the Watson planets of the Ilecuha group, including (175) AndroniacJie,
which, on account of unusually large perturbations and other imfavorable conditions, forms
so far the most striking example of the applicability of the Bohlin-v. Zeipel method and of our
revised tables for the Hecuba group.

The plan of work included conferences, in which Miss Glancy and Miss Levy took a leading
part, for the discussion of the Bohhn-v. Zeipel method, involving verification of all mathe-
matical developments and formulation of plans for the construction of tables, and, after the
appearance of v. Zeipel's tables, for the comparison of v. Zeipel's original, and our revised tables.
The niunerical work was carried out by Miss Glancy and Miss Levy, who have also contributed
very largely to the theoretical part of the work, and have prepared the principal details of the
manuscript.

To avoid confusion v. Zeipel's notation and method of procedure have been followed
throughout in completing our tables for the Hecuba group, which were well under way when
V. Zeipel's memoir appeared.

To aid computers in the use of the formulae and of the revised tables. Miss Glancy has
prepared detailed directions illustrated by an application to (10) Hygiea, the example first
chosen by v. Zeipel. These are contained in the first section of the present memoir.

Miss Glancy's contributions to this investigation and her work on {10) Hygiea were accepted
by the University of California in partial fulfillment of the requirements for the degree of
doctor of philosophy.

Miss Levy's contributions and her work on {175) A-ndromacTie were similarly accepted for
the same degree.

It seems highly desirable to make the tables for the development of the perturbations of
minor planets of the Hecuba group at once available to astronomers. They are therefore
pubhshed herewith, in advance of the perturbations and tables of the remaining Watson planets,
as Part II of "Tables of Minor Planets Discovered by James C. Watson." One or two parts,
which are to follow, will contain aU the numerical results for the perturbations and tables of
Watson planets not pubhshed in Part I (1910).

This memoir is presented in two sections. The first section, entitled "Formulae and Tables
for the Hecuba Group, according to the Theory of Bohhn-v. Zeipel, and an Example of their

• Pp. 20O-201.

8 PREFACE. [Vol. XIV.

Use," contains a collection of the formula to be used for any planet of the Hecuba group, the
general tables of the pertuibations which must be employed, and a more complete appUcation
of the formula; and the revised tables to the plane* (10) Hygiea, than v. Zeipel gives. The
second and more extensive section, entitled "Tables for the Determination of the Perturbations
of the Hecuba Groui> of Minor Planets," concerns the construction of the tables and their dis-
cussion with reference to the corresponding tables by v. Zeipel. It forms the preliminary part
of the in jestigation but is presented last as supplementary to the final results given in the
first section.

In the second section the tabular values which differ from the corresponding numbers in
V. Zeiper,s tables are placed in brackets. The general Tables XXXV, XXXVIII, XLIII,
LIV, LVi, LVii, LVI, LVII, of the second section, which, in order, are required to compute
the perturbations of any planet of the Hecuba group, are repeated without brackets at the
end of the first section as Tables A, B, C, D, Ej, Ej, F, G, so that the first section is complete
in itself for use in developing the perturbations of any planet of this group without the necessity
of reference to the second section.

A general account of the investigations of the perturbations of the Watson planets was
presented to the Academy on April 16, 1916, and is published in the "Proceedings of the
National Academy of Sciences," Volume 4, No. 12, March, 1919.

Washington, D. C, 1918, December.

Akmin O Leuschnee.

TABLES OF MINOR PLANETS DISCOVERED BY JAMES C. WATSON.

By Armin O. Leuschner, Anna Kstelle Glancy, and Sophia II. Levy.

INTRODUCTION.

Those planets whose mean daily motions are approximately 600" are classed with the
planet Hecuba, or, in the group for which

n'

n

^i(l-w)

where n' and n are the mean daily motions of Jupiter and the planet, respectively, and w is a
small quantity.

Among the minor planets discovered by James C. Watson there are several of this type.
In the course of the general program of deteVmining the perturbations of the Watson asteroids,
there arose the necessity of computing special tables for the Hecuba group in preparation for
the application of Bohlin's method to individual planets.

General tables for the group h were in the process of construction, under the direction of
Professor Leuschner,' according to the method of Bohlin,' when tables for this group were
published by H. v. Zeipel.' The computers. Dr. Sidney D. Townley and Miss Adelaide M. Hobe,
made a comparison of their tables with those of v. Zeipel and found certain discrepancies
Because of this fact the completion of the tables for the Hecuba group was deferred. These
discrepancies have been explained, as a result of a careful investigation, and the tables have
been completed by Miss A. Estelle Glancy and Miss Sophia H. Levy, under the direction of
Professor Leuschner.

In the completion of the tables, v. Zeipel's method and order of procedure have gener-
aJly been followed. There are numerous discrepancies between our tables and v. Zeipel's. As
far as possible, with the aid of the original manuscript, kindly forwarded by the author, we have
traced the source of these disagreements. In some of the more comphcated functions it was
not possible to do so, and these discrepancies remain unexplained. Our own results, however,
are substantiated by the employment of independent developments. Further, where we found
terms omitted which were of the same order as those which were included, we frequentlj'
extended the tables. In this connection, it is pertinent to remark that it is very difficult to set
up a consistent criterion for the omission of terms. With the exception of a few scattered
neghgible terms, our tables are published in full. They contain terms which may ordinarily be
omitted, yet their numerical magnitudes depend upon the elements of the particular planet
imder consideration, and their use is left to the comjiuter's judgment. Many of them are
incomplete, i. e., the tabulated coefficients do not necessarily include all the terms of a given
degree in the eccentricities or mutual inclination or of the small quantity w, which depends
upon the difference between the planet's and twice Jupiter's mean motion. In other words, the
coefficients may not contain all the terms of a given degree having the factors

IV, TjP, fj'l, f

which are defined on page 12. But, assuming certain numerical limits for the fundamental
auxiliary functions, the coefficients are of this magnitude. The value of the additional terms
will be shown best in an application of our tables to the same planet for which v. Zeipel computed
the perturbations.

Unless stated otherwise, the references to Bohlin refer to the French edition and are
designated by B. References to v. Zeipel are designated by Z.

' Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 200.

' Forraeln uud Tafein zur gruppenweisen Berechnung der allgemeinen Storungen benachbarter Planetcn (I'psala, ISPC).

Sur le Dfiveioppement des I'erturhations Plan^taires (Stoclcholm, 1002).
• Angenaherte Jupiterstorungen f ilr die Ueculia-Gnippe (St. Pfitersbourg, 1902).

9

I. FORMULAE AND TABLES FOR THE HECUBA GROUP, ACCORDING TO THE
THEORY OF BOHLIN-v. ZEIPEL, AND AN EXAMPLE OF THEIR USE.

DETERMINATION OF CONSTANT ELEMENTS AND OF PERTURBATIONS OF THE MEAN ANOMALY.

The planet (10) Hygiea was selected by v. Zeipel as an example of the use of his tables for
the group ^. We have used it as a preliminary example for the application of our own tables,
so as to provide further comparison of our tables with those of v. Zeipel.

This example is presented with the direct purpose of meeting the needs of the computer.
For this reason, no attempt is made to explain the significance of the functions involved, yet
their use will be less mechanical, if, in a general way, some of the essential principles under-
lying their development are understood. The theory of v. Zeipel is taken up in the second
section of this memoir.

The method proposed by v. Zeipel is a practical adaptation of Bohhn's method of com-
puting the perturbations by Jupiter upon planets whose mean motions bear nearly commen-
surable ratios to that of Jupiter. In particular, the formulae are derived for the planets of the
Hecuba group. Tracing the history of this method one step further back, Bohhn's method is a
modification of the theory of Hansen for the indeterminate case of nearly commensurable mean
motions. Or, concisely, in \'. Zeipel's own words, ''Die benutzte Methode kann einfach dadurch
charakterisirt werden, dass die Differentialgleichungen von Hansen mittels des Integrations-
verfahrens des Herrn K. Bohlin gelost worden sind."'

Certain principles of Hansen are fundamental to an understanding of some of the important
equations. Briefly, the perturbations are reckoned in the plane of the orbit and perpendicular
to it. In the plane of the orbit nbz signifies the displacement in the planet's mean anomaly
(52 is the perturbation in the time) ; v gives the disturbed radius vector through the relation

r = f(l +v)

u
and the displacement in the third coordinate is denoted by =. With Hansen's choice of ideal

COS »

coordinates, the fundamental analytical relations are:

£ — e sin £ = nt-\-c-\- nSz

f cos f=a (cos £ — e) (1)

r sin f=a ■>Jl —e^ sin £

r = f(l +1^)

dB = ^^.asm 1"
^ cos %

Jx = dp cos a (2)

Ay = dp cos h

Az = dp cos c

x = r sva.asva. {A' -\-f)-\-Ax

y = r sin b sin (B' +/) + Ay . (3)

2 = r sin c sin (C +f) +Az

where £,/, f are fictitiously disturbed coordinates, which, in connection with constant elements

u '
and the perturbations nhz, v, and = give the true position of the body. A', B', C, sin a, sin h,

sin c are the constants for the equator. The notation for the eccentric anomaly and the true
anomaly is v. Zeipel's; in Hansen's notation they would be written e,f.

1 Angenahertc Jupiterstorungen fiir die Hecuba-Grappe, p. 1.
10

No. 3.] MINOR PLANETS— LEUSCHNER, GL.A.NCY, LEVY. U

When Jupiter's mean motion and tliat of the planet are nearly commensurable, the inte-
gration of Hansen's difTerential equations becomes impracticable through the presence of large
integrating factors. The integrals are of the form:

sm

■('■-'^7

' \(in-i'n')t
' cos

•OO <t< +00
0<i'< +00

For the Tlecuba group the mean motion is approximately twice the mean motion of Jupiter.
Hence, for exact oommensurability,

7l' 1 1 •/ o • 1

j = 00 ; I = 2, a = 1 .

'n 2'

■(*-*¥)"

By introducing the exponential in place of the sine and cosine, the indeterminateness can
be removed, for if in — i''n' = 0, then (.V-i('''-i'n')(= j. This ia one of Bohlin's modifications.

For any given ])lanet the ratio is not exactly commensurable, and the developments are
originally made for the case of exact commensurabihty. They are then expressed, for a given
case, by Taylor's series in ascending powers of a small quantity w, which depends upon the
difference between the real ratio and exact commensurability. In addition to positive powers
of w there will occur negative powers. They are due to the following causes. An argument 0 is
introduced (see p. 13), from which the mean anomaly of Jupiter is eliminated through the intro-
duction of w. It is a necessary consequence of the form of the partial differential equations in

which -J- appears, that the integration of first-order terms shall contain w~^ and that higher

order terms shall contain other negative powers. Hence the integrals are series in both posi-
tive and negative powers of w.

In distinction to the method of Hansen the elements appear explicitly in the arguments
or as factors in the terms of the series.

An important feature of v. Zeipel's theory is his treatment of the constants of integration.
Since the method is essentially Hansen's, the constants of integration must be determined con-
sistently with that method. Given osculating elements, the constants of integration are deter-
mined by the condition that, at the date of osculation, (t = 0), the perturbations and their
velocities shall be equal to zero.

V. Zeipel ailopts osculating elements as his initial elements. With these elements and the
perturbations and their velocities at the date of osculation, he computes elements, designated
by the subscript unity, in which the constants of integration are absorbed. They are analogous
to Hansen's constant elements, i. e., the fundamental equations of Hansen are valid.

Our transformations of the elements differ from v. Zeipel's in two respects. First, the

constants in -., and in its velocity have not been introduced into the elements i, Si, but

cos ^

are treated in the usual Hansen manner. Second, v. Zeipel introduces certain terms in the

perturbations which have the same period as the planet (argument e), into the elements to

form mean elements. This ha^> not been done.

The general tables, XXXV, XXX\TII, XLIII, LIV, LVi, LVii, LVI, LVH, which are
required in computing the perturbations, are given at the conclusion of the formulae. The
formulae for any planet of the group ^ are given completely, and they are supplemented by
numerical values for the planet (10) Hygiea.

The references to v. Zeipel's paper arc indicated briefly by Z, followed by the number of
the page.

The osculating elements of the planet are taken from Z 139; the elements for Jupiter are
taken from Astronomical Papers of the I'nited States Nautical Almanac Office, Vol. VII, p. 23.

(10) Hy^iea.

jch,

1851, Sept.

17.0, Ber. M. T.

OSCULATING

ELEMENTS.

«o

= 634?S50

= 0

? 176347

"Po

= 5° 46/28

= 5

?7713

fo

= 227 46.61

= 227.

7768

S2o

= 287 37.19

= 287.

6198

'^0

= 300 9.42

= 300.

1570

%

= 3 47.14

= 3.

7857

12 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

Jupiter.
Epoch, 1851, Sept. 17.0, Ber. M. T.

ME.A.N ELEMENTS.

n'= 299fl284= 0?0S30912
ip'= 2°45.'95= 2?7658
Tz'= 11 54.45= 11.9075
ft'= 98 55.97= 98.9328
co' = 272 58.48 = 272.9747
i'= 1 18.70= 1.3117
Co = 126 59.81 = 126.9968 c'=199 57.70=199.9617

Mean equinox and ecliptic, 1850.0.
Epoch, 1851, Sept. 16.96279 Gr. M. T.

The following notes in regard to these elements are of importance:

Jupiter's elements were first taken from Z 139. They were used only in the equations
nimibered (1). In these equations either set of elements may be used with sufficient accuracy.
In fact, it is not necessary to know Jupiter's elements as accurately as those of the planet, for
they appear only in the arguments of the perturbations. We have adopted Hill's values of
the elements and Newcomb's value of the mass of Jupiter. The tables of the perturbations
are based, however, on Bessel's value for m'. To correct the perturbations for the improved
value, it is only necessary to multiply them by 1.0005, and this is done in the formulae which
follow.

The original epoch of Jupiter's elements was 1850.0 Gr. M. T. It was changed by the

formula c' = 148° 1.'97 +n'< (4)

The elements of Hygiea are very good osculating elements, computed by Zech. They
include perturbations by Jupiter, Saturn, and Mars and are based on five oppositions. The
reference for these elements is doubtful, for in Astronomische Nachrichten 39, 347, the elements
given by Zech are not identically the same, although the differences are very small. The values
given by v. Zeipel were probably taken from Zech's manuscript, to which he had access. They
may, therefore, contain some later corrections.

The auxiliary quantities *, *, J are first computed by the formulae :
sin ^ J sin ^ (* + 'l>)=sin -^ (Q.o- SI') sin ^ (ig+i')

sin 2 J cos 2 ('!' + *) =cos 2 (.Sl^-^') sin ^ (%-*')

cos 2 "/sin 2 ('*'-*) =sin ^ iQ,o-Q,') cos -^ do+i') (5)

cos 2 e/ cos 2 ('*'-*) =cos 2 (9,0- ^') t'os 2 (io-i')

rv, 1. sin^_sin'l> sinfSJp— S^')
^'^^'^^'- sini„~sini'~ sin J
Then follow

n. = ^„-ft„-*; Vo = J; n'=;r'-fi'-M'; ,'=|'-

y2=sin^ '2 cos^ k 'Po cos^ o ^'' ''='*^" "^ ^'^'^^ 2'''' ^^■^

j„=n„-n'; j„ = n„+n'

Wo=^—

No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 13

and the arguments for the date of osculation:

fl, = ^fo - 9' '^vhere g' = c' + [n'dz'] ; [n'dz'] = (9.5215) sin 1 15?326, (7)

where the coefficient in parentheses is logarithmic in degrees.

eo-^-osin £„ = r„: r = ^ £„ + ^„ + Jo (8)

J„ = 215?8f)79
i'„= 28.9289

(?„ = 223.23.34 (a)

^c^-ic' + ln'dz']) =223.2445 (b)

logw'„= 8.76072 ^ Co- c' = 223.5448 (c)

See footnote.'
e, = 131?3236; r=145?0746

(10) Ilygiea

*=186?4792

n„=.302?3984

* = 357. 7586

n'= 86.5.305

J= 5.0856

log jj„= 8.70139

logf= 7.29275

Iog,'= 8.38238

log I = 8.94739

With these initial quantities all the arguments and factors in Table LVI or F are computed.
The required function, w — Wg, is computed by successive approximations, the first approximation
being

In the first trial the smallest terms and the last digit may be omitted; the second trial should
be accurate; a third trial, if necessary, will require only corrections to the largest terms.
The mean motion n is then given by

2n'

(10) Hygiea.
The three successive trials for w give

w-w„

+ 0.00388

w= +0.061208

+ 0.003541

logw= 8.78681

+ 0.003568

n = 637f2633

Designating by Cand S series to be computed next from Table LVTI orG, it is evident bv
inspection of Table L\1I that

C cos <1> + S sin <l> = Ic cos (^ + X) = Ic cos X cos ip—Ic sin X sin ^
from which

C=i:c cos X; 5= -Jcsin X (10)

' Three numorlcal values for the argument 9, are given. According to the theory (sec footnote, Part 2, p. 147), (a) Is rigid; (6) Is rigid ivithin the
accuracy of the developments by v. Zeipcl; (c) is an approximation which v. Zclpel used and which is used here. The value (6) Is preferable.

In equation (4), [n'hz'] — +0°.31 U and is the complete perturbation of Jupiter by Saturn taken from Hill; in all other parts of the computstlon
n'it'\ is only the long period term used by v. Zelpel.

14

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

To make the order of computation evident, the successive steps for a group of terms for
Hygiea are given.

X

c

X

-5

+ C

-5r+fi9„+6J„

+ 0.4

o

111.10

+ 0.4

- 0.1

-4r+69„+6J„

4- 1.9

256. 18

- 1.8

- 0.5

-3r+6fl„+6J„

+ 4,6

41.25

+ 3.0

+ 3.5

-2r+6'*„ + 6Jo

+ 6.8

186. 33

- 0.7

- 6.8

- r+efi„+6Jo

+21.5

331.40

-10.3

+18.8

6(?o+6i„

-63.0

116. 48

-56.4

+28.1

r+6flo+6i„

- 4.0

261. 55

+ 4.0

+ 0.6

2r+6fl„+6J„

- 3.1

46.63

- 2.2

- 2.1

3r+6e„+6J„

- 1.9

191. 70

+ 0.4

+ 1.9

The second column contains the sum of the niunerical coefficients multiphed by their respec-
tive factors w'rjP-q'i'f-'. The columns —S and + C contain the required terms from this group in
the table. They can be computed at the same time if a traverse table is used.'

From S and C the elements tz and ip can be computed by the formulae:

e sin {it — 7ta)==S cos v'u

e cos (;: — ;:„) =^0 + '^cos Vo (H)

e = sin <p

In place of -q^, J^, !„ the following are used hereafter:

(12)

"'1

i = J„+(;r-;T„)

I-

= i'(,+ (;:-ffo)

S'=+1215f0
C= +2191.1
A = 218?8882

(10) Hygiea.

;r-;r„=+3?0203
r = 230. 7971
S= 31?9492

log 7, = 8.74517

log sin.p = 9.04620

vj = 6?3858

There remains one more element to determine, namely, c, but the computation must be

deferred until vs'e know the perturbation ndz at < = 0. (See equation (1), page 10 or page 16.)

The fictitiously disturbed eccentric anomaly at the time t = 0 denoted by £„ is determined

through the relations :

£o-eo sin fo = C(, (13)

where £„ is calculated with the aid of Astrand's table ' ;

ifi'Kvo-To) =y J-^" fgho> 'i'i^i = -\/r+i ^si^'^o-^)

(14)

£o=131?3236

(10) Hygiea.

r„ = 3?2968

£, = 127?6064

£,-esinf, = 122?5578

The perturbation ndz i.s computed as follows:

The function 1 +<i!>(t?) is computed from Table XXXVIII or B. The coefficients are mul-
tiplied by their respective factors, the trigonometric functions of the arguments are expanded,

and the coefficients of . * ii? are collected, (j is the numerical coefficient of t>).
sm

' Memoirs of the National Academy of Sciences, Vol. X, Seventh Memoir, p. 218.
'Hulfstafcln zur leichten und genauen AuflSsung des Kepler'schen Problems (Leipiig, 1890).

No. 3.1

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

15

UO) Hygiea.

1 +(/>(t?) = (1 -0.008064) {1 -0.0r)5937 sin 2i? + 0.017170 cos 2(?

+ 0.01(i057 sin 4i? + 0.012244 cos 4t?

+ 0.000905 sin (iiJ- 0.005081 cos Gt?+

+ ((>-:?„) ( + 0.000007 -0.000490 sin 2i? -0.001206 cos 2t>

-0.000361 sin 4t? + 0.000409 cos 4t?+ )}

where the coefBcients are in radians, and d„ is the value of i? at < = 0.

Let 1 + <7 be the nontrigonometrical term in 1 + 0(d), take it out as a common factor, and
denote the numerical coefTicients by A^, B^, A^, Bt, A^, B^, 6„, a^, b^, a^, h^, respectively.
With these coefficients the following are computed:

1 1

K=

1—w sin 1'

S,= E

C,= K

- 1 {A,A, + B,B,) +Ia, (A,' + 5/) + |a,j

-5,+|(^A-5^J -\bm2'+b,') +lbj^

\a, + ^^{A^,-B^,)-^^A,{ZB,>-A,')^
-\b, -^^(A,B,+B,AJ - jr.B.iZA'' - 5,')]

s^'-

w

C^ — — K ^a^

.w

S/=Kj (b,+AA-B,a,) (15)

.w

C,'=-K^(a, + A,a, + B,b,)
C,"^K^ {b,-Ajb,-B,a,)

There are check formulae for these quantities in Z 134, equation (153), (161')- In equa-
tion (153) there is a misprint; in equation (161') there are two misprints. The errors and their
corrections are noted in the list of errata which accompanies the second section of this paper.

A part of the long period terms in iidz, denoted by [n8z\, is expressed by

[ndz\ = S:, sin 2i;;+Cj cos 2!^ + S^ sin 4,^+C, cos 4c + 5e sin 6C+Ce cos 6^+ . . . .

+ ^(C - Co) (5/ sin 2C + <7,' cos 2^ + S,' sin 4^ + C/ cos 4c +

) +

(I)"-

Co)'^o'' +

(16)

{10) Hygiea.

l+(P(t?) = (1-0.008064) {1-0.056384 sin 2«> + 0.017308 cos 2«? + 0.016186 sin 4ty
+ 0.012342 cos 4i? + 0.000912 sin 6t>-0.005122 cos 6r?+ . . .
+ (<? - t>o) ( + 0.000007 - 0.000494 sin 2^ - 0.001276 cos 2t> - 0.000364 sin U

+ 0.000412 cos 4t> +

) +

. }

6o= +0.000007

A,=

+ 0.017308

Cj= -0.000494

B,=

-0.056384

Jj= -0.001276

^.=

+ 0.012342

a^= -0.000364

B.=

+ 0.016186

J«= +0.000412

Ai =

-0.005122

B,=

+ 0.000912

16 ]MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

Unit of A^, etc., is one radian

[7i&].= (3.59592) sin 2C +|(C-Co) [(0.933„) sin L'c
+ (4.09785) cos 2^ + (0.521) cos 2^

+ (3.0783) sin4C +(0.085) sin 4C

+ (3.2230„) cos 4C + (0.005) cos 4^

+ (2.4390„) sin 6C + ]

+ (1.494„) cos ec +

+

in which the coefficients are logarithmic in seconds of arc. For this planet it is not necessary

to include C\".

In equation (IG) let

iS„ = Ic cos K (7„ =^ sin Z .^_>

5'„ = - it' sin Z' C'n = ¥ cos K' ^ '

Then

[ndz\^nc sin (nc+ii0+ |(C-Co)^^'' cos {nC+K')+ (18)

The argument ^ is given by the relation :

and ^0 is the value ol t^ &t t = Q, in which, [n'dz'], the long period term between Jupiter and

Saturn is :

[n'dz'] = (9.5215) sin{ (9.58539) T+ 1 15?32C} (20)

where the numerical coefficients are logarithmic in degrees, and T is measured from the date of
osculation in Julian years.

The complete expression for the long period term in ndz is :

[ndz] = [n8zl^^^^j^^^lls-in'3z']) (21)

It is important to remark that, in equations (19), (21), the eccentric anomaly is computed

by the usual formula,

s — e sin s = c + nt + ndz (1 )

m which the multiples of 2;r must be retained, for s is used here as if it were the time. Since
ndz is unknown, the computation is by successive approximations.

UO) Hygiea.

M2],= (4.11837) sin (2^+ 72?5246)
+ (3.3130) sin (4^ + 305. 627)
+ (2.442) sin (6^+186.48)

+

+ |(C- Co) {(0.963) cos (2C+ 68.83)

+ (0.199) cos (4c + 309.75)+ ....}+....
in which the coefficients are logarithmic in seconds of arc.

The argimient ^ in (ndz -[ndz]), the short period part of ndz, is given by

^='L^[nd2], + c (22)

and the function itself is computed from Table XXXV or A.

No. 3.] MINOR PL^VNETS— LEUSCIiNER, GLANCY, LEVY. 17

The numerical coefficients in Table XXXV or A are niultijilied by their respective factors

and the terms are then coUectod in the form

nd2-[ndz] = ICl'^^ ii^e + j^ + U-lJ) (23)

By expanding the trigonometric functions, the known pai t of the argument, namely,

U-lI

is incorporated in the coefficients, and the terms are collected in the form:

ndz - [ndz] = la sin x + ~b fos x + (^^ - ''o) (^(^' si" X + -Sb' cos x)

+ (t? -.?„)= (Ja" sin x + 2-i" cos x) (24)

+

where
Let

Then

X^^i'^s+i^ (25)

a = 7c cos K b =k sin K

a' =-l'' sin IC h' =¥ cos K' (26)

a"= k" cos K" b" = Jc" sin K"

ndz-[ndz] = Ik sin (x+K) + (i}-d,) Ik' cos (x+ A'' )

+ (i? - ^,)'IJc" sin (x + A"') +

(27)

The tabulation of n8z — [ndz] for (10) Hygiea is given on page 27.

Finally, the complete perturbation in the mean anomaly is:

Tidz = [ndz] + {n3z - [ndz] ) (28)

It is now possible to determine c by successive approximations from equations (20), (19),

(IS), (21), (22), (27), (28).

From equation (1), which holds for any time t,

c = £i — e sin £, — ndz

t = o (29)

As a first approximation

ndz = 0 c = s, — esinjj

Introducing this value of c in equation (19), a first approximation for ndz is made. For < = 0,

(:-Co)=0 (30)

(,?-«?„) =0

Substituting the value of ndz in equation (29), and computing a new value of c, the procesa of
solution by trials is repeated until a satisfactory agreement is reached.
110379°— 22 2

18

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

(10) Hygiea.
Below is the last approximation for the constant c.

[Vol. XIV.

(See tabulation of rwz— |n^;] on pg. 27.)

x=i

,^J^

x+A'

log sin (x+i')

isin (x+A')

Approx. (102

+0?6124

f,— e sin £,

122. 5578

•5£+ 0

285?683

323?619

9. 7732„

— 282"

Approx. c, equ. (1),

p. 10

121. 9454

ie+30

9.443

291.021

9. 9701„

- 680

^-i^c,p.l3

57. 240

j£ + 5l?

93. 203

258. 21

9. 991„

- 260

'-'% C,p.l2

217. 278

-i£+ <?
-}^+3,S

158. 077
241. 837

183. 00
335. 37

8. 719„

9. 620„

- 6

- 17

e

127. 606

135. 14

9.848

+ 25"

1 ^„ p. 13

+3. 9053

£ + 2<J

211. 366

288. 414

9. 9772„

-3403

£+4;?

295. 126

256. 179

9. 9872„

- 723

[n'Si'], equ. (20), p

16

+0. 3003

£+6.3

18. 886

223. 38

9. 837„

- 168

(9. 99572)^1 £,-K

«\-'])

+3. 5697

£+8<J

- £+2,5

102. 646
316. 154

186.8
14.11

9. 07:-l„
9.387

— 5
+ 23

- £ + 4,?

39. 914

129. 91

9.885

+ 3

(8.3192„)0f,-[n'

52'])

-0. 0752

!£+3.J
f£+5a

137. 049
220. 809

74.51
39.53

9. 984
9.804

+ 121

+ 44

f£ + 7,>

304. 569

3.25

8.754

+ 1

f, equ. (19), p. 16
2C

220. 848

2£+2.J

338. 972

236. 180

9. 9195„

- 86

81. 696

2£+4.5

62. 732

209. 15

9. 688„

- 19

163. 392

2£+6.>

146. 492

183.0

8.72„

- 1

245. 088

4£+5^

348. 415

2.4

8.62

0

4£ + 7,?

72. 175

327. 9

9.72„

- 4

2c+ 72?525, p. 16
4^+305. 627

154. 221

109. 019

+ 21 7" -5654"

6C+186. 48

71.57

nSz — lnoz]

f - 5437"
1- 1?5103

log sin
log sin

9. 6384
9. 9756

(8.3192„)(|£,-KoV])

- 0.0752

log sin

9. 9771

[nSz],

+ 2. 1994

ndz, equ. (21)

+ 0. 6139

+ 5712"

C=Ci

121. 9439

+ 1944

+ 262

(6. 8050„)r,
c„ p. 19

- 0.0778
121. 8661

[,idz]i

/ +7918"
1 +2? 1994

l-w

(9. 67154)[n32],

+1. 032

2 '•i

+57. 240

0, equ. (22), p. 16

221?880

1 — «'
2 '•. ^'

217. 278

2<>

83. 760

3!>

305. 640

(9. 6715)[)i52],
,?„, equ. (22)

+ 1.032
221. 880

4>>

167. 520

5i>

29. 400

6!>

251. 280

7.>

113. 160

8,?

335. 040

i£, p. 14

63. 803

t

127. 606

If

191. 409

2£

255. 212

If

319. 015

Collecting the elements, and adopting a change of notation, introduced at this point by
V. Zeipel, namely, the addition of the subscript unity to the elements just now determined,

n, = 637f2633 =
^5,= C?3858
;r, = 230.7971
c, = 121.9439

0?17701758

No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 19

Th&se elements are constants; they differ from constant osculating elements only by the
constants of integration in nSz and v. They are to be used in the same manner as Hansen uses
constant osculating elements.

It is possible, in a similar manner, to absorb the constants of integration in the third coordi-
nate in the elements •?„ and J2„, but this transformation will be omitted.

It is a convenience to the computer to have n, and <•, transformed to mean elements. The
last term in equation (21) increases in magnitude, progressively with the time. The computa-
tion of this terra of large magnitude may be avoided by modifications of the elements n, and c,.

The method of transformation can be clearly shown from the example {10) Hygiea,

By equation (1)

(6.80497„)£ + (8.3192) [n'8z'] = {(>.H0497„) c,-0.''40(37 t- 14f(3 sin e

+ (G.80-i97„)n82+ {H.3192)[n'Sz'] ^^^

It is evident from equations (1), (21), and (23) that the first tei'm on the right-hand side
of equation (32) may be combined with the mean anomaly at the epoch to form a mean mean
anomaly, given by ^^ ^ ^_ + (fi.80497„)c,

Furthermore, the second term on the right-hand side of equation (32) may be combined
with nt in equation (1). A mean mean motion is thereby introduced, which is given by

n, = 7i 1 - 0.'4067 = G36f8566

Again, the third term on the right-hand side of equation (32) may be combined with a
term in {Ti32 — [ndz]) which has the argument s. In the construction of {nS2 — [ndz]) there
occurred the terms ^ 3^,3 ^j^ ^ ^ ^,q cos e = (1 .545) sin (e + 7?53)

The addition of — 14f6 sin e from equation (32) gives

-|-20f2 sin s-h4r6 cos £=(1.320) sin (£+12?74)

These two values for the argument x = e are tabulated in the body of the table given on p. 27.

Further, since it is intended to improve the perturbations by the use of Xewcomb's value
for the mass of Jupiter, ndz must be multipUed by the factor 1.00050. The combination of the
correction for the mass of Jupiter with the term of the same form in equation (32) gives

(-F 0.00050 -0.00064)r!.52= -0.00014 ndz

This correction is the la-st step in the determination of ndz, since it depends upon the pertur-
bation itself.

Without change of notation for ndz, the collected results are:

£ — e sin s = c2 + ndz + n2t (33)

where

ndz = [noz]i + {ndz-[ndz]) -O.OOOli ndz+ (8.319) [n'dz'] (34)

It must be remembered that [ndz]i and (ndz — [ndz]) are numerically different from their
original values, but there should be no confusion if this transformation is not made before the
con-stant c has been determined.

The constant elements are now:

Epoch and Osculation, 1851, Sept. 17.0, Ber. M. T.
Tij = 636.''8566 = 0? 17690461

Cs =

121?8661

^1 =

6.3858

^1 =

230.7971

ft,=

^287.6198

io =

■ 3.7857

20 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

uinox and ecliptic,

18

50.0

1 ^

e, = sin <p^

P2=aj(l-f,')

log P = 9.98741

log e, = 9.04620

log ^2 = 0.49191

log n,' = 8.49548
log 0.3'= 1.49193

log 0.2 = 0.49731

Certain other transformations of the elements which v. Zeipel makes are omitted. Those
terms of the perturbatioas which have the argument £ have the same period as the planet and
can, therefore, be absorbed in the elements. It would be necessary to set up formulae for this
transformation to mean elements, and it is not profitable to do so.

PERTURBATIONS OF THE RADIUS VECTOR.

The perturbations in the radius vector are computed in a manner similar to that for
{nhz — [n^z\). In Table XLIII the numerical coefficients are multiplied by their respective
factors w*, jjp, r)'i, j^, the terms are collected, the kno\vn parts of the arguments are incorporated
in the coefficients, and the terms are grouped in the form:

v= la sinx + -^& COSX+- • • •

+ (!?-!?„) {Ja' sinx+^6' cosx+- • • •} (35)

+ (!?-.>„)^{2'a"sinx + 2'6"cosx+- • • •} + • • • •
Let

o = — ^ sin K b =Jc cos K

a' = k' cos K' ¥ =¥ sin K' (36)

0"= -V sin K" h" = l-" cos K"

Then

v^Ikco3(x+I0 + (^- ^0)^^' sin (x + K') + (t? - «?„)='i'/<.-" cos (x + A"') + • • • • (37)

and to correct the perturbation for the use of the improved value of the mass, v should be
multiphed by 1.00050.

If the mean motion n^ is adopted, the constant in v must be corrected by

2 n^-n, _1

3 rij sm 1

This correction of the constant in v permits the use of the relation

n2V/ = P

in the computation of a geocentric place; without this correction it would be necessary to use
the relation

in the determination of the parameter p. In the computation of the eccentric anomaly it
is permissible to use either ?ij or tIj, for the difference is taken up in the modification of ndz,
but the theory of Hansen demands the use of constant elements. Hence, strictly speaking,
71, must be used in computing a geocentric place. The modification of the constant in v renders
the employment of n^ equivalent to the use of ti,.

(10) Hygiea.

2 71,-71., 1 2 054067 1 _ ^^,0

3 71, sin 1" 3 637.''3 sin 1" ' ''^

The constant in Table XLIII or Cis +47f6. Therefore, the new constant is:

+ 47.'6 - 87.^8 =-40f2 = (1.604) cos 1S0?00

where the coefficient is logarithmic in seconds of arc.
The perturbation is tabulated on page 27.

No. 3.1 MINOll PLANETS— LEUSCIINER, GLiVNCY, LEVY. 21

PERTURBATIONS OF THE THIRD COORDINATE.

The perturbations of the third coordinate are derived from Tables LIV, LVi, LVn or D,
E„ Ej. The first of tiiese is of the same form a-s tlio tables for (ndz — ljiSz]) and v. After mak-
ing analogous transformations and multiplying by the factor i cos i, (i is defined by equation (6)),

t cos il Up.g iffj'i sin A = lie ^in (x + K) (39)

Both Table LVi or E, and Table LVu or Ej lead to a single numerical quantity, since all
the factors and arguments are known constants.
The perturbation u is given by

u = I cos i [I Up.q r;P)j'« siu A + rt^t { A'l (cos £ — ej + K^ sin s} +c\ (cos £ — €,)+ c^ sin e] (40)

in which c,, c^, the constants of integration, have not been determined.
The constants i\ and c^ are determined by Hansen's conditions:

w = 01
^w^J = 0 (41)

ds

Substituting these relations and equation (39) in equation (40), the determination of c, and
c, is given by the solution of

d'Y
C, (cos s — e,) + 6j sin s= —Ik sin (x + A") ; C, sin j — C, cos £ = lJc j- cos (x + K) (42)

(43)

where C, = i cos i.Cj

Cj = t cos i.c,
and

where

de~l+i(A,' + B,')2V'^ 2 d^ ) ^^*^

A double notation is used here, for cos i is the cosine of the inclination of the orbit, and ^ is

the numerical coefficient of £ in the argument x, but this should cause no confusion.
Dividing and multiplying the factor

by 365.25

. I cos i-rij ™ -._,

t cos %-nji= , T (45)

where Tis the interval in Juhan years, measured from the date of osculation.
It is evident that

. , . Ci (cos £ — «,) + C^ sin e

can be incorporated m

lie sin (x + K)

in the same manner as similar terms were treated in {7id2 — [ndz\).
For symmetry of form, let

I cos i-n^t{Ki (cos £ — ei) + Ajsin E)=Ik' cos ix+ K') (46)

Finally, then, without change of notation,

w = Jt sin (x + Z) + Tlk' cos (x + K') (47)

in which the constants of integration are absorbed in the first term. The perturbation u is
tabulated on page 27.

The perturbations in the heliocentric; coordinates are computed from equations (3) The
signs of cos a, cos h, cos c are determined as follows:

cosa>Oif 0<S2 <180°
cos 6 <0 if -00°<Q<+90°
cos 6 < 0 in any case, if £ >i
cos c > 0 if sin i cos Ji < cos %
cos c>0 in any case if ■i<45°

22 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

(10) Hygiea.

t

= 0

^[ndz\ = [(.4A19m

cos (2^0

+ 72?5246)

+ (3.9150)

cos (4Co

+ 305?627)

+ (3.220)

cos (6Co + lSG?4S)]sin I"

log

2 1+UA,'+B,')-^-^^^^^

^ = 0.0285

log

^;^-9.67154

^ = 1+0.0.

!85/

Ii:sm(x+K)=- 70.'5

Ji^cos (x+K)=+lQlf6

<7, = + SS.'Q
(7j=+120.»8

From Table LIV, multiplied by i cos i we have three terms in

Ik sin (x+ K)=- 4.''2 - l.^O sin e + 2f7 cos s
which, added to

(7, (cos £-€,) + (72 sin £ =+ 1 20.''8 sin j + 35:'9 (cos £ - ej

gives for two terms in lie sin (x+ K)

-T.'S + llS.'gsin £ + 38^6 cos £=(0.89) sin 270?0+ (2.0970) sin (£ + 17?99)

CHECK COMPUTATION.

After the elements have been determined and the final tabulation of the perturbations is
ready, the following checks should be performed, even if the computation has been duplicated.

t = 0

(?o = i(£i-«'isin£,)-^'

1 — w
g' = c' + [n'dz'] Oo = d„+ —^ {ndz - [ndz] ) - ijw sm £

where the necessary quantities are to be taken from the last approximation for c.
Secondly, the heliocentric coordinates

x — Jx, y — Ay, z — Jz

for < = 0 must check when computed by the usual formulae for two body motion and osculating
elements, and when computed with the final set of elements and the corresponding perturba-
tions, 71^2 and V, taken from the final tabulation.

The final tabulation of the perturbation in the third coordinate is checked by the test

t = 0 ; M = 0

COMPUTATION OF THE PERTURBATIONS FOR THE TIME t.

It is well to emphasize here the distinction between the elements n^ and c, and the elements
7^2 and Cj in their relation to the perturbations. Let 7)52, denote the perturbation in the mean
anomaly computed according to equations (20), (19), (IS), (21), (22), (27), (28), and let 71^22
signify the pertiu-bation computed according to equations (20), (19), (18), (22), the final tabu-
lation of {ndz — [ndz]), and an equation analogous to (34). (It must be remembered that equa-
tion (34) is for {10) Hygiea only. The numerical coefficients are determined for each ]>lanet
individually.)

Before the determination of c there can be no confusion, for there is but one way to com-
pute the perturbation ndz. Later, when both c, and c^ are given, the computation may be per-
formed in either manner. The latter method is, of course, adoj>ted. The question then arises,
what values of £ and c are to be used in equation (19) ?

No. 3.)

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

23

Clearly, there is only one value of s, for

£ — c, sin s = Cj + n,< + ndzi = c, + n^t + ndz^

and both n8z and e must be found by trials. Further, since the intioduction of n^ and c, arises
merely from a transfer of certain terms in the perturbation, the argument of the perturbation
is independent of this transformation. Therefore c, is the constant in eijuation (19).

For any time t the order of computation is: equation (33), neglecting ndz, (20), (19),
(18), (22), final tabulation of iidz — indz], and the equation analogous to (34). Since the per-
turbations are large, the argument s is not sufficiently accurate when ndz is neglected. It Ls,
therefore, always necessary to make a second approximation for ndz. In the first trial the small
terms may be omitted.

(10) nygiea.
Perturbations ndz, v, u, for 1873, Sept. 20.4491, Ber. M. T.

log ei (degrees)

0. 80432

log sin

9. 9387„

log 2

8. 48578

log sin
log sin

9. 9918„
9. 703„

1 1 2

'°e 57.30 •«,

9. 75»il

217?278

Co

220?848

log COS

n.74i„

Y=i — :+jO

221?880
121?8661

log COS

log C-Co)

9.419
1. 6337

/V *} ' J

,2

i£+ ^

315?350

i"g^i:-:o>

1. 3898

i£+3,>
ie+5,?

119. 524
283. 698

1873

log -(C-Co) COS

2
log ^,(C-Co) COS

1. 131„

-i£+ .?
-is+Si*

208. 824
12. 998

Ber. M. T.

Sept

' 20. 4491

0.809

t

+

803M4491

£+2t>

106. 526
270. 700

nU

+

1422?2156

11405"

£+4lJ

74. 874

r-,+n.,t

{_

1544. 0817
104.0817+1440°

2017
140

£ + 6l>
£-|-8l>

239. 048
43. 222

ndz

» 3. 666
100. 416

124
+10

- £ + 2.>

- £ + 4<>

57. 648
221. 822

(

f
I

M06. 526+1440°
1546. 526

log aSz\ (sees)

13676"
4. 13596„

|£+ iJ
i£ + 3,)

fj+5^

f£ + 7l>

61. 876
226. 050

30. 224
194 398

10g£

3. 18935

log h5z], (degrees)

0. 57966„

^V *• ^J\fKJ

1 "W

loggs

1. 67513

{nSz\

-3°. 7989

-!£+ .?

102. 298

IB

+

47?329

2«
2£+2>>

213. 052
17. 226

l-'-Kaz']

+

47?053

log (9.6715) \nSz\

0. 2512„

2£+4.>
2£+6i>

181. 400
345. 574

log (|£-[n'az'])

log (9.99572) (|s-[«'oV])

1. 67259

(9.6715) \n5i\

-1?783

5£ + 5,>

136. 750

4£+7.?

300. 924

1. 66831

<?

2629087

(9.99572)('|«-[n'52'n

+

46?592

H-f}^

40?207

^
V

263?870

0

262?087

2:

167?740

2t)

164. 174

4:

335. 480

3^

66. 261

6:

143. 220

40
5tJ

328. 348
230. 435

C-Co

43. 022

6.?
7»

132. 522
34. 609

2c+ 72°5246

240?265

8*

296. 696

4:+305. 627

281. 107

(')

6C+186. 48

329. 70

£

53°. 263
106. 526

2c+ 68?83

236. 57

!-'

159. 789

4^+309. 75

285. 23

2£

1^

213. 052
266. 315

I CoiT. for aberr.

• From previous approx.

• From Astrand'3 table.

* Scecq. (1), page 16.

24

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

(10) Hygiea.
Perturbations n32, v, u, for 1873, Sept. 20.4491, Ber. M. T.-

Continued.

\-i-^+j3

n52— [nor]

"

u

i )

x+ZiT

log sin

k sin U+K)

x+K

log cos

(x+K)

k cos (x+K)

X+K

log sin
(x+K)

1
;:sinCx+A')

0 0
0 2
0 4

0 6

1 1
1 3

1 5
-1 1
-1 3

2 0
2 2
2 4
2 6

2 8
-2 2
-2 4

3 1
3 3
3 5

3 7
-3 1

4 0
4 2
4 4

4 6

5 5
5 7

0

353. 286
41. 102
88.71
233. 74
106. 53
119. 27
347. 748
35. 927
83.53
127.4
115. 61
311. 82

163. 51
208. 94
253. 08

274. 434
327. 82
22.1
150.8
196.6

9. 0679„
9. 8178
0.000
9. 907„
9.982
9.941
9. 3268„
9. 7686
9.997
9.90
9.955
9. 872„

9.453
9. 685„
9. 981„

9. 9987„
9. 726„
9.58
9.69
9.46„

- 56"
+ 479"

+ 265

- 85
+ 41

+ 18

- 761
+ 437

+ 243
+ 35
+ 84

- 3

+ 36

- 34

- 13

- 104

- 21
+ 5

+ 5

_ 2

O

180. 00

58. 608

98. 841

146. 28

173. 425

221. 824

268. 86

192. 38

111.41

300. 02

167. 726

21 5. 194

263. 148

309. 92

88.6

349. 03

276. 65

341. 94

28.42

54.02

136. 98

156. 79

198. 96

330. 79

16.85

0. 000„
9 7167
9. 1867„
9. 920„
9. 9971„
9. 8723„

8. 298S„

9. 9898„
9. 562„
9.699

9. 9900„

9. 9123„

9. 0767„

9.807

8.39

9. 992

9.064

9.978

9.944

9.769

9. 864„

9. 963„
9. 976„
9.941
9.981

- 40"
+375"

- ?,3

- 52

- 137

- 193
_ 2

- 3

- 13
+ 3

-1852

- 329

- 15
+ 27

0
+ 6
+ 1
+ 85
+ 34
+ 1

- 2

- 9

- 4
+ 10

+ 8

o

270. 00
296. 33
238

80.40
110. 78
156. 04
122. 28
172. 10
124. 52

79.94
104. 99
150. 50

0.51

225. 83
257. 49
278. 56

4.98
348.8
40.38
85.1
92.97

0.000„
9. 952„
9. 928„

9.994
9.971
9.609
9.927
9.138
9.916
9. 993
9.985
9.692

7.948

9. 856„
9. 990„
9. 995„

8. 939

9. 288„
9.811
9.998
9.999

- 8"
-12

0

+ 11"

+ 14
+ 3
+ 11
+ 1
+103
+ 59
+ 25
+ 5

+ 1

- 5

- 3

- 1

0

0

+ 1
+ 1
+ 1

+1648" -1079"

+550" -2684"

+236" -29"

<<>-i>.)'orT

X+K'

log COS

'.X+K')

k- cos (x+^')

x+K'

log sin
(x+K')

fc' sin (<+*'')

(x+K')

log cos

(x+K-)

k- cos (x+K')

0 0
0 2
0 4
2 0
2 2
2 4
-2 2
4 0
4 2
4 4

o

292. 53
4.7
41.3
123. 85
219.90
104.1
154.82

9.583
0.00
9.88
9. 746„
9. 885„
9.39„
9. 957„

+ 371"
+ 2
+ 6

- 2"

- 20

- 1

- 1

o

270. 00
232. 94
281. 51
292. 573
351. 93
41.29
305. 02

104. 23
147

0.000„

9. 902„
9. 991„
9. 965„
9. 147„
9.819
9. 913„

9.986
9.736

- 6"

- 8

- 7

- 447

0
+ 3"

- 1

+ 4
+ 1

180. 00
47.67

0. 000„
9.828

0
+ 6"

+ 379" - 24"

+ 8" - 469"

+ 6"

(•^-■».)'

x+K"

log sin

(x+K")

Jc"sin(x+^")

x+K"

log cos

(x+K")

I:"cos(x-^A-")

2 0
4 0

o

296. 23
227. 15

9. 953„
9. 865„

- 3"

- 1

O

112. 79

47

9. 588„
9.834

- 1"
0

- 4"

- 1"

' For perturbation u use factor T.

No. 8.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

(10) Hygica.
Pertiirb.m-ion's n3z, v, u, for 1873, Skpt. 20.4191, Ber. M. T.— Continued.

25

log(t>-J„)rad.

9. 8462

log(,?-,>o)'

9. 692

logT

1. 3426

log_iL^sin 1"

5.183

"cos I

nSz

V

u

2h sin ix+K)

+ 569"

n cos (x+A')

- 2134"

2k sin (x+.^)

+ 207"

Ik' cos (x + A'O

+ 355

Ji-' sin (x+ATO

- 461"

I¥ cos (x+A")

+ 6

Ji^sinCx + A'")

4

Ik" cos ix+K")

1"

log JF cos ix+K')

2.5502

log Ik' Bin (x+K')

2. 664„

log Ik' cos (x+A"')

0.778

log(0-Oo)Ik'coa{x+K')

2.396

log(i)-Oo)Ik'Bin(x+K')

2. 510„

logT. 2i-'cos(x+A-')

2.121

{O-Oo) 2 k' cos ix+K')

+ 249"

{0-Oo)2k'sm{x+Kn

- 324"

T. Ik' cos (x+A-')

+ 132"

losn-"Bm(x+K")

0. 602„

V

- 2458"

u

+ 339"

Ioe{0-Oofliy'ein(x+K")

0.294„

log V (sees)

3. 3906„

logM

2.530

(^-■Jo)'^

2"

log V (rad)

8. 0762„

log (Iff

7.713

log (1+1/)

9. 99480

log cos a
log cos 6

8. 798„

9. 619„

ndz-[ndz]

+ 816"
+0°. 2267

log cos c

9.958

(8.3192) [n'52']

+0°. 0058

log Jx

6.511„

[nSzl

-3°. 7989

iogJy

7. 332„

n8z

- 3.5664

logJz

7. 671 1

-0.00014 nSz

+ 5

nSz

- 3°. 566

ill

Jz

-0. 00032
-0. 00215
+0. 00469

The computation of the geocentric place on page 26 is analogous to the usual method for
two body motion, the fundamental equations being (1), (2), (3). A complete set of formulae
and an example of the computation is also given in Memoirs of the National Academy of
Sciences, Vol. X, Seventh Memoir, p. 215.

Constants for the Equator.

A' yearly var.

B' yearly var.

C yearly var.

log sin a log cos a

log sin b log cos b

log sin c log cos c

IS.50.0
1900.0
1930.0

3209833+0901399
321. 532+0. 01399
322.232+0.01399

2299182+0901404
229. 885+0. 0H05
230.587+0.01105

23.S9657+0901310
239.312+0.01308
239. 965+0. 01306

9.99914 8.799o

9.99914 8.797„

9.99915 a 795a

9.95S84 9.619„
9. 9586S 9.619o
9.95853 9.620„

9.62355 9.958
9.62423 9.958
9.62490 9.958

26

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

(10) Rygiea.
Comparison, Observation — Computation, 1873, Sept. 20.4491, Ber. M. T.

[Vol. XIV.

1873

X

+3. 0709

Ber. M. T.

Sept. 20.4491

X

-1.00281

r._,+n^(

104?0817

Jx

-0. 00032

nUz

- 3. 5660

«

+2. 0678

M=c^+n^t-\-nSz

100. 5157

y

-0. 89314

dM°

0?4843

r

+0. 03260

dM'

- 29/06

Jy

-0.00215

dp'

+ 3/15

'■/

-0. 86269

dv
dtp

+ 1. 8124

dip

+ 68

2

-0. 19677

+ 8

z

+0.01415

^ 20
I. dtp'

+ 5/73

r

+0. 00469
-0. 17793

d(v-M)ldM

- 0. 0674

'UD^.dM°

+ 8

S.dM'

+ 1/94

log p cos 3 cos a

0. 31551

v-M

4- 12° 0/14

COS a

9. 96515

v-M,

/+ 12° 7/81
1 + 12°. 1302

sin a

9. 58550„

log p cos 3 sin a

9. 935S6„

f=v,

112°. 6459

log tg a

9. 62035„
f337° 21' 14"
\ 22" 29°' 24'. 9

log cos/

9. 58550„

a

log f, COS J

8. 63170„

Red to Tnie a

+ 1.5

log (1+e, cos/)

9. 98099

True a

22" 29" 26'. 4

logr

0. 51092

Obs. a (A. N. 2029)

22" 29" 07". 1

log (1+..)

9. 99480

logr

0. 50572

log p cos 3

0. 35036

A'

321?1548

cos 3

9. 99864

B'

229. 5058

sin 3

8. 89852„

C

238. 9584

log p sin 3

9. 25025„

log tg 3

8. 89989„

A'+f

73. 8007

3

-4° 32' 26"

B'+7

342. 1517

Red to True 3

+6"

C'+f

351. 6043

True a

-4° 32' 20"

Obs. 5 (A. N. 2029)

-4° 33' 27"

log sin a

9. 99914

log sin (A'+f)

9. 98240

log I

0. 48726

logp

0. 35172

log sin 6

9. 95877

log sin (B'+f)

9. 48643„

logy

9. 95092„

(0-C)

ja 008 3

-19? 3

log sin c

9. 62387

J3

-1' 7"

logsin(C'4-/)

9. 16438„

log J

9. 29397„

Given a series of observations well distributed around the orbit and extending over as long
an interval as is available, the elements can be corrected by the method of least squares.

For this purpose the formulae by Bauschinger ^ are convenient. The equations of condi-
tion are set up for the residuals in the plane of the orbit and perpendicular to the plane, as seen
from the earth. This resolution of the residuals is convenient because it keeps the same reso-
lution into components as is used in the theory of Hansen.

It is to be noticed that the elements to be used in computing the differential coefficients
are the finally adopted constant elements referred to the equator by the proper transformation.

The value of r to be used is

r = r(l +y)
except in the equation

sin/ (Hansen's notation)

sm £ =

ajVl-

■Tafel lur Berechnun? der wahron .\nomalie, Vorodentlichun^en des Rechen-Instituts der Koius;licheii Sternwarte zu Berlin No. 1.

' Ober das Problem der Bahnverbe-sserimg, Veroflentlichungen des Koniglichen Astronomischen Rechen-Instituts zu Berlin, No. 2!, Berlin,

loai.

No. 3.]

MINOR PI^iNETS— LEUSCHNER, GLANCY, LEVY.

27

The use of i,/, r and constant elements is equivalent to the use of osculating elements for the
given date of observation.

(10) Hygiea

UnJtott-l"

x-ij+jo

nj2-

[TIJZI

V

«

i

J

logi:

r

log*

K

log*

0.89

K

0

0

•

1. 604

9

180. 00

270. 00

0

2

2. 8570

254. 434

1. 118

132. 16

0

4

2. 33G1

130. 493

8.25

270

0

G

1.800

13.76

n,'iz-[nSz]= Ik Bin (x+S^

1

2. 6771

37. 936

2. 1397

218. 075

1.057

125. 05

+{d-d„)Ik'coa{x+K')

3

2. 8627

281. 578

2.4l;',5

102. 300

1. 161

351. 26

+(>J-.5o)=i'i-"8m(x+^'0

5

2. 4238

• 165.01

1.965

345. 16

0.930

232. 34

_]_

1

2.022

24.92

0.55

343. 56

1.119

273. 46

— 1

3

1.628

93. 53

1.543

98.41

0.981

159. 10

o

0

/ [1. 545]'
\ 1. 320

|7. 53]'

v=Ik C08 ix+S^

12.74

0.711

193. 49

2.097

17.99

+ (d-Oa)Ik'Biii{x+K')

2

2

3.5546

77. 048

3. 2776

257. 026

1.777

169. 24

+ (,>-^„)2i'ib"C0B(x+r")

2

4

2. 8719

321. 053

2.6054

140. 320

1.412

30.12

2

6

2.389

204. 49

2. 1033

24. 100

1.034

271. 45

2

8

1.64

84.2

1.62

266. 70

-2

9

1.970

57.96

1.27

31.0

1.824

302. 86

u^Ikmn(x+K)

-2

4

0.602

90.00

0.80

127. 21

+ TIk' COB ix+K')

3

1

0.90

214. 77

0.826

163. 95

3

3

2.100

297. 46

1.95

115. 89

0.446

31.44

Where T is expressed in Julian years
from date of osculation.

3

5

1.841

178. 72

1.583

358. 20

0.171

248. 34

3

7

1.12

58.68

0.34

219. 62

-3

1

0.42

34.68

0.673

262. 68

£

4

0

0.00

135. 7

x=i^-+jO where in e the multiples of

4

2

2. 0170

257. 208

0.270

23.15

2w must be retained.

4

4

1.589

146. 42

0.97

335. 39

9.91

263.7

!?o=221.811

4

6

1.14

36.5

0.66

213.39

9.73

107. 40

5

5

1.038

14.0

1.062

194. 04

5

7

0.88

255.7

0.94

75.93

(>>-t>o) or T

logi'

K'

log*'

K-'

log*'

K'

0

0

•

0.799

•

270.00

9.690

9

180. 00

0

2

1.021

68.77

0

4

0.86

313. 16

2

0

2. 9862

186.00

2. 6850

186. 047

0.957

301. 14

2

2

0.18

94

0.12

81.23

2

4

0.88

326. 4

0.60

326. 42

-2

2

0.60

66.20

0.11

247. 37

4

0

1.414

6.85

4

2

0.68

86.9

0.580

87.00

4

4

0.11

833. 42

0.09

326

(■>-<»«

•

logi"

K"

logi"

K"

2

0

0.58

«

189. 70

0.26

9

6.26

4

0

9.91

14.10

9.6

194

COMPARISON OF THE REVISED WITH V. ZEIPEL'S ORIGINAL TABLES.

It was originally planned to conclude the example with a least squares solution of the orbit
on the basis of the observations used by v. Zeipel for the same purpose, and to test conclusively
the relative value of the revised and v. Zeipel's original tables b}' representing recent observa-
tions with both sets of elements and tables.

In the course of the computation doubt arose regarding the accuracy of some of the
observations selected by v. Zeipel, which led us to reject them and substitute other observa-

> In the determination of the constant c use quantities in brackets.

28

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. xrv.

tions. This substitution produced an unfavorable distribution of the observed places in the
orbit and the resulting least .scjuarcs solution was not satisfactory.

In the meantime, pending a resumption of the least squares solution on the basis of a more
favorable distribution of observed places/ the following conclusions may be drawn regarding
the revised and v. Zeipel's original tables:

1. V. Zeipel's tables have been slightly improved by the correction of some numerical errors.

2. A moderate further improvement has been accomphshed by an extension of the tables
in so far as seemed practicable without a more exhaustive and unwarranted study of the prac-
tical convergence of the auxiliary series, by including certain terms of higher order and degree.

With reference to the correction of the orbit and the representation of observations by a
least squares solution, it should be observed that

(1) A symmetrical distribution of the observed positions in the orbit is essential to coun-
teract the effect of neglected perturbations of higher order and degree and of major planets
other than Jupiter. For the Hecuba Group, in general, the mean motions of the minor planets
may be nearly commensurable with those of Saturn, Mars, or the Earth in the ratios 3/2, 3/1,
or 3/5.

(2) However accurate the initial osculating elements may be, comparatively large residuals
may remain on account of neglected perturbations.

Logaritlimic.

Table A (XXXV).
nSz—[nSz]

Unit-l"

Sin

W-'

w->

IC-l

w

w

w'

vv'

Jf+ <J

4. 1570

4. 8741„

h'-+ >5+ J

2. 7684„

3. 3827

3. 7172„

-)'

i-'+ .5+ J

4. 0056„

4. 7686

^"

i£+ <?+ J

4. 0766„

4. 8295

J'

i£+ <>+ i

4. 1365

4. 8738„

W

i.'+ 0+2A

3. 3345

4. 5162„

i

i£+3i?+2J

4. 2240„

4. 9611

5. 6685„

V

iE+3i>+3J

4. 0671

4. 8483„

5. 5636

Y'

i£+5<>+3J

5. 0926„

6.0018

vv'

if+5(?+4J

5. 2325

6.1714„

f

i.'+5.?+5J

4. 7675„

5. 7344

j'

i£+5i)+4J-i'

3. 8050„

4. 7998

i

-i£4- {^

3.3112

3. 8350,,

4. 1355

V

-\c+ .?+ A

3. 2065„

3. 7910

4. 0833„

n"

-i£ + 3<>+ J

3. .5338

4. 6236„

W

-i£+3<?+2J

4. 0879

5. 0382

■j'

-i£+3.?+3J

3. 6012„

4. 5318„

P

-iE+3.>+2J-2'

3. 2074

4. 1925„

ij

t

9. 868„

0. 5689

2.922

3. 4600„

3. 3670

,'

t+ J

9.482

0. 2533„

2. 673.,,

3. 2959

3. 1772„

iY

t+2d+ J

0. 746„

1.384

3. 2927„

4. 14906

4. 6990„

t+20+2J

9. 788„

2. 47560

3. 10847„

3. 4540

3. 3960„

v'

e+2<5+2J

0.645

1. 342„

2. 305„

3. 6179„

4. 4018

V'

t+2d+2J

0. 32G

1. 119„

2. 935„

3. 3017„

4. 39206

f

e+20+2J

3. 4276„

4. 23764

4. 76933„

rin'

e+2{f+SJ

0.28„

1.102

3. 1738

3. 5449„

3. 8446„

W'

£+4:7+2J

3. 6004

4. 27485

r,'

£+4.>+3i

9.057

0. 692„

3. 10161

3. 9302„

4. 52415

4. 78162„

,v

£+4^+3J

4. 0519„

3. 7975

,'3

£+4^+3J

4. 1385„

4. 6961

JW

c+ia+3J

4. 2431„

5. 1290

ij

e+40+iJ

9. 500„

0.522

2. 9351„

3. 8035

4. 41616„

4. 63017

^'

£+4iJ+4J

3. 7714

4. 2108„

^l'"

£+4,5+44

4. 4165

5. 0931„

Pi,

£+4^+44

4. 1524

5. 0661„

%v

£+4,5+5J

4. 0588„

4. 8136

■ Since 1913, when the revision of the tables was concluded, Miss Glancv has continued the problem of (10) Ilijgka independently at the Ohser\a-
torio Nacional, C6rdoba, with the following highly satisfactory results, which substantiate further the increased accuracy of the revised tables
(1) The original osculatin<; elements and the revised tables resulted in a greatly improved representation of the selected observations (is^9-lssi)
over the representation ojjtained with the original tables. (2) After the correction of the original osculating elements by least squares solution
(a) on the basis of v. Zeipel's tables and residuals, (b) on the basis of the residuals resulting from the revised tables, the representation ol the
selected observations was equally satisfactory; but 3 later observations, taken in 1910, 1914, and 1917, are represented tar better by the revised
tables and correspondins elements than by the original tables and corresponding elements, (cf. Astronomical Journal, Vol. 32, p. 27, No. 748,
January 1919) A. O. Lcuschncr.

No 3] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

Logarithmic. Tablh A ^XXXV)— Continued.

29

Unit-

sin

1

«;-»

w-J

w"

to

ic

^',' ,

e+4.)+3J-i'

3. 2322„

4. 2342

f v'

e+4i?+4J-J

2.744„

3. 09C2

<

£+6.)+4J

0.28

0.64„

3. 8027

4. 77998„

5. 52852

Vr/

£+6>?4-5J

0. 59r>„

1.070

3. 9374„

4. 94342

5. 70347„

■f

£ + fit? J- 6 J

0. 255

0,8„

3. 4684

4. 50125„

5. 27451

f

£+6,?+5J-i'

8.8

9.3„

2. 415

3. 4823„

4. 2931

'

E+8.J+5J

4. 55G4

5. 49!t'»„

'C

£+8,J+6J

4. 8668„

5. 8410

O^'.

E+8.7+7J

4.6990

5. 7030„

'',

I £+8>?+8J

4. 0531„

5. 0844

?. '

E+8-7+6J-J

3. 5829

4. G352„

/',

£+8J+7J-J

3. 3768„

4. 4540

<'

- £ + 2<>

0. G06

1. 422„

3. 2132

3. 6657„

3. 9260

'/'

- £ + 2,?+ J

0. 791„

1.690

3. 3777„

3. 8866

4. 72168

■,'

, - £+2,J+2J

0.418

1. 365„

2.894

3. 461';^

3. 8078

J"

- c+2,?+ A -I

9,34

0.28„

2.938

3. 4714„

3. 7862

'

- £+4.?+ ^

3. 5208

4. 07255

■ni'

- £+4.5+2J

3. 49G5„

4. 59582

,,'

- £+4,?+3J

3. 2416

4. 54fi7„

i'

- c+id+AA

2. 430„

3. 9848

J'^''

- £+4,J+2J-2

3. 5496

4. ]n8.52„

i'n

- £+4,?+3J-J

3. 3247„

4, 05994

r,r/

j£+3i5+2J

3. 6731

4. 0029„

It+ZO+ZJ

2. 3528

3. 2475„

3.9005

s

U+-iO+ZJ

3. 6181„

4 2122

f

4£+3,9+3J

3. 4072„

4! 4000

r.r,'

|£+3,3+4J

3. 5244

4. 4012„

'/'

4£+5i)+4J

3. 3533

4. 4231„

5. 2725

' .,

i£+5,5+.5J

3. 1780„

4. 2730

5. 1359„

'^

4£+7,J+5J

4. 2775

5. 4708„

','

4S+7.5 + 6J

4. 4051„

5. 6177

n'

|f+7>?+7J

3. 9296

5. 1605„

^ ,

2£+2>J+2J

9.486

2, 1744„

2.708

2. 889„

2. 599„

<

2s + 2.' +3 J

1. 948„

2.501

2. 516n

^r,'

25 + 4i?+3J

8.8„

0. 561

2. 789„

3. 5813

4. 1074„

2£+4.>+4z(

8.90„

9.599

1.711

2. 5795„

3. 1726

'',

2s+4(?4-4J

9.2

0.34„

2.618

3. 4962„

4. 0890

V

2£+0,J+5J

9. 819„

0.5840

2. 7821

3. 7794„

4. 51865

>/

2.-+61J+6J

9. 653

0. 4645„

2. 5979„

3. 6265

4. 38424,

^-+5i?+5J

1. 2340

2. 1166„

2. 7076

i'

|j+7<?+6J

2. 3679

3. 3518„

4. 0587

>;

. |£+7<>+7J
(.)-.)„) cos

2. 1758„

3. 1926

3. 9204„

'i.

£

0. 1021„

0.728

2. 8978„

3. 4504

3. 7168„

',,

£

].377„

2.346

3. 8211„

4. 6762

.,' '

£

1. 941„

2. 815

4. 4076„

5. 1971

•"' ,

£

1.364

2. 220„

4.4076

5. 1971 „

5. 7086

?'

£+ i

9.658

0. 774„

2. 7836

3. 3840„

3. 6946

"' ^J

£+ J

1. 863

2. 755„

4.2546

5. 08i4„

,'3

£+ J

1.844

2. 642„

4. 1953

4. 9770„

P V

£+ i

1. 170„

2. 049

4. 3715„

5. 1770

5. C9?5„

•,"»:'

£+ 2J

1. 742„

2.574

4. 0203_

4.8466

JP ,'

£+ i'

0.716

1.65„

4.0809

4. 8829„

5.4008

Pn

£+ J+i'

1.00„

1.89

4. 3427„

5. 0837

5. 5553„

7''/'

- £+ ./

1.562

2.455„

3. 9535

4. 7803„

'\

2£

9.801

0.43„

2. 5842

3. 1493„

3.4158

>!>?'

2£+ J

(<>-.>„)» sin

9. 357„

0.473

2.4548„

3. 0830

3. 3936„

"i

£

9.56„

0.42

1'

1

£-r J

9.43

0. 32„

nSz-

-[ndz]=IW^Pij'9j^C^ sin ATg.+(0 -0„)I n'rjPj/gftC^ cos Arg.
where C,, C,, Cj represent the respective coefficients.

-{d -Oa)'Iu^TiPr/gftCt sin Arg.

30

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Logarithmic.

Tablk B (XXXViri).

Unlt=l radian.

Cos

W-'

U7-»

W-'

W-*

«j-»

IC-l

w

to

w'

1.5

3. 909„

4.960

6. 6748„

7. 2764

7.540„

7.31

f

2.0

4. G44„

5.160

6.150

8. 048„

8.838

8. 655„

8. 100„

n''

1.9

3.41„

4.75„

6.509

8. 2077„

8.994

8. 919„

P

2. 83„

5.146

6. 299„

7.994

8. 740„

8.656

\r,'

i

2.34„

4.446

4.57

6. 728„

8. 4022

9. 1999„

9.0854

8.079

rin"

20

1. r,

2.6„

5. 744

6. 535„

8. 3811

9. 1031„

9. 0128

v'

2iJ+ i

0.8„

3. 068

5. 2988

7. 2212„

7. 3772

8. 0372

8. 764„

8.668

vW

2^+ A

2.32„

3.30

5. 886„

6.718

8. 5059„

9. 2804

9. 2017„

','3

20+ 4

5. 301„

6.149

8. 2302„

9. 0154

8. 938„

f V

20+ A

8. 5592

9. 3245„

9. 2428

ri'

20+2J

2.48

3. 40„

5.422

6. 292„

7.476

8. 664„

8.636

ij

20+2J

1.22

2. 94„

5. 1206„

7. 6416

7. 9638„

7. 083„

8.645

8. 582„

,,«

20+2A

1.9

3.0„

5.442

6. 328„

8. 0915„

8. 630„

8.742

J'v

20+2A

8. 590-4„

9. 3489

9. 8024„

9. 6532

fr,'

20+3A

2.04„

3.00

4.98„

5.89

8. 0326

8. 1973„

7. 69

j'v

20+ A-I

4.51

5.42„

8. 1011

8. 873„

8.792

rv

20+2A--I

4.04„

5.00

6.89„

8.182

8. 158„

ri"

A0+2A

2.66„

2.7

6. 1031

8. 4188„

8. .5297

6.0

7.90„

vn'

A9+ZA

2.72

4.369

6. 2526„

8. 5594

8. 7988„

7.94„

8.287

8.210

r,'

iO+iA

2.20„

4. 624„

5.824

8. 0924„

8. 4333

7.24

7.74„

8. 044„

r-

AO+iA-2

1.5„

2.45

4.68

7. 1747„

7.301

8.111

8. 127„

l"

6i?4-3J

5. 301„

6.149

9. 1294„

9. 7728

9. 6609„

,,'2

&0+AA

5.92

6.74„

9. 4432

0. 14644„

0. 05077

^w

6i?+5i

2.0„

3.0

5. 93„

6.79

9. 2774„

0. 03298

9. 9494„

v'

6^+6J

2.0

3.0„

5.420

6. 292„

8.634

9. 4351„

9. 3608

P ri'

&0+iA-S

4.04„

5.00

8. 272„

9. 1028

9. 0334„

Pr,

6d+hA-I
{0—0 a) sin

4.51

5.42„

8. 0554

8. 926„

8.864

n^i'

J

2.60„

4.71

5.94„

6. 507„

6.606

ri'

20+ A

1.36

2.48

4.49

5. 255„

5.51

5.25„

V

20+2A

1.82„

2.42

4.64„

5.350

5.51„

5.16

v"

A0+2A

2.34

3.00

5.392

6. 179„

6. 528„

6.665

vv'

AO+ZA

2.89„

3.46

5. 702„

6.467

6.851

6. 979„

n'

iO+iA
(0-0„y cos

2.66

3. 459„

5.357

6. 127„

6. 530„

6.653

r)'

2.08„

2.08

5. 546„

5.546

r

2.54

2.54„

5. 396„

5.396

nn'

A

2.5„

2.5

5.776

1

5. 776„

m.'3

m'3

m'3, m'^

m", m''

m'% m'

m'\ m'

m'2, m'

m'2, m'

■m", w

*(t5)=i'tt'».i)P.T)'9.j^.C, cos Krg. + [0-Os,)Iu-».rf.ri'<I.j'i.C^ sin \T:g. + (0-0aYIwK-qP.j)'1.j'^'.C3 cos Arg.
where C,, Cj, C3 represent the respective coeflicients.

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

Table C (XLIII).

31

Logarithmic.

Unit-1"

Cos

w~^

«'-«

w-y-

t£>«

w

U)»

8.72

9.88„

1. 6349

2. ]070„

2. 2333

7'

9.80

0. 212„

2.759

3. 4922„

r

8.9

9.23

2.937

3. 6295„

j'

2. 937„

3. 6295

vV

J

9. (;a„

9.78

3. 1],S6„

3. 8440

,,'»

2.?

0. 55()„

1. 204

3.211]„

3. 7970

v'

2.?+ J

0. 504„

2. 3472

2. 456„

2. 686„

3. 4735

vV

26+ J

0.997

1.711„

3. 6559

4. 3103„

r,"

20+ J

0. 438

1. 220„

3. 3654

4. 0763„

P V

2d+ J

3. 6975„

4. 3810

7

2^ + 2J

0.438

2. 952„

3. 2529

3. 0689„

3. 3979„

V'

2.5+2J

0. 732„

1.497

3. 2410„

4. 0643

vv"

20 +2 J

0. 772„

1.589

3. 4136

4. 0723

fr,

2,5+2J

3.9048

4. 5649„

4. 9303

vW

2,J+3J

0.505

1. 344„

3. 4757„

2.783

fr,

26+ J -I

9.33„

0.15

2. 93a„

3. 5830

f V

2,>+2J-i'

9.20

0.10„

2. 0251

3. 2961„

n"

4,J+2J

8.9

1. 2819„

3. 5514

3. 6173„

3. 8147

vv'

4J+3J

9.75„

1.5024

3. 7885„

4. 1394

4.3110„

n'

4.J+4J

9.98

1. 1342„

3. 4007

3. 9091„

4. 1480

f

46+3J-I

9.64„

2.305

2. 542„

2. 749„

v"

6,? +3 J

0.438

1. 220„

4. 2675

4. 7993„

,,'»

6^+4J

1. 125„

1.862

4. 6479„

5. 2324

vY

6,?+5J

1.198

1. 947„

4. 5397

5. 1768„

^'

6^+6J

0. 732„

1.508

3. 9457„

4. 6328

f n'

M+4J-I

9.20

0.10„

3. 4099

4. 1710„

fv

6d+bJ-I

9.70„

0.56

3. 2C01„

4. 0542

riri'

is+ 0

3. 4878„

4. 1106

i^+ •■>+ J

8.3„

2. 210G

2. 7179„

2.919

J'

is+ ■5+ J

3. 5709„

4. 2261

ri'

i-'+ 6+ J

3. 4507

4. 1296„

V

i-'+ ^+ J

3. 5100

4. 1837„

V l'

it+ .J+2J

2. 579„

3. 9270

n'

Jf+3iJ+2J

0. OS

3. 6873

4. 1471„

4. 7839

T)

i£4-3.>+3J

'J. 5

3. 5727„

4. 1511

4. 7545„

v"

i£+5.>+3J

4. 5568

5. 1414„

vn'

i£+5,?+4J

4. 7261„

5. 4067

f

i£+5i?+5J

4. 2862

5. 0418„

f

i!+b6+4J-I

3. 2570

4.0005„

r/

-i^+ 6

1. 086„

2.7090

3. 3467„

3. 7098

V

-i£+ <>+ J

0.88

2. 1967„

3. 0952

3. 5836„

v"

-i£+3l>+ J

2.514

4, 1049„

vY

-j£+3.?+2J

4. 0853

3. 9122

1'

-if+3<J+3i

3. 8341„

3.8118

p

-i€+36+2J-I

2.416

3. 6926„

1

c

9.62

0.58„

2. 143„

2.682

2. 9151„

/

e+ J

9.04„

9.9

2.061

2. 666„

2. 9477

rir,'

c+2d+ A

0.444

1. 1661„

3. 0588

3. 8035„

4. 2554

£+2i>+2J

9.487

2. 1744„

2. 7280

2. 972„

2.976

r^'

C+2.J+2J

0. 344„

1. 1143

2. 692„

3. 5334

4. 0772„

1"

f+2iJ+2J

0. 025„

0.828

2.634

3. 0726

4. 0416„

f

£+2i>+2J

3. 1265

3. 8806„

4. 3473

n^'

t+26+'AA

9.98

0. 811„

2. 873„

3. 1697

3. 5856

,,'».

1+46+24

1.105

1.89„

2.864

4. 3477„

l'

e+46+ZA

8.8„

0.398

2.8000„

3. 5327

4. 0065„

4. 3207

^\

£4-4>)+3J

1. 260„

2.083

3. 0931

4. 4160

,'3

£+4^+3i

3. 8375

4. 044fi„

P i

S+46+ZA

0.267

1.15„

3. 9421

4. 6972„

T)

c+46+44

9.19

0. 248„

2. 6356

3. 4317„

3.9469

4. 2558„

v'

£+4^+4.4

0.774

1.66„

3. 0934„

3. 786fi„

vv"

£+4,J+4J

4.1154„

4.5547

Pv

£+4,>+4J

0. 455„

1.32

3. 8518„

4. 6436

riW

£+4,J+5J

3. 7579

4. 3244„

Pri

£+4,>+3J-J

3. 0030

3. 8869„

32

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table C (XLIII)— Continued.

Logarithmic.

[Vol. XIV.

Unit- 1"

Cos

K,J

w->

w-'

V)'

w

w

J- r/

£+'t-3+4J-i'

2. 4425

1.85„

,'2

£+0>3 + 4J

n.98„

0.480

3. 501 G„

4.3723

4. 9952„

vn'

£+Gi?+5J

0.296

0. 823„

3. 6369

4. 5582„

5. 2093

f

£+Gi?+6i

a 95„

0. 538

3. 1685„

4. 1334

4. 8131„

P

£+6i3+5J-i'

8.5„

9.15

2. 114„

3. 0881

3. 7886„

,'=

£: + 8l?+5J

4. 2554„

4. 9349

riV'

E+8i?+6J

1.320

2. 152„

4. 5657

5. 3010„

iw

£;+8i3+7J

1. 228„

2. 093

4. 3995„

5. 1827

v'

E+Si) + 8J

0.648

1.54„

3. 7543

4. 5812„

P V

£-|-8i?+(;j-i'

3. 2818„

4. 1442

w

£+8^+7J-2

3. 0763

3. 9759„

Y'

- £ + 2.?

0.305

1. 1007„

2.912

3. 4958„

3. 8151

vY

- £+2J+ A

0. 490„

1. 3330

3.0166„

3. 7273

4. 3119

n'

- £+2^+2J

0.117

0. 982„

2. 288„

3. 2375„

3. 7892

f

- £+2,3+ J-S

9.04

9.96„

2.636

3. 281 7„

3. 6568

n"

- £+4,3+ i

3. 2197

3. 9650„

rir,''

- £+4j+2J

1. 146„

1.89

3. 0204

4. 2441

vY

- £+4,3 +3J

1.005

1.78„

3. 5247„

4. 0012„

f

- £+4,3 +4 J

0. 290„

1.15

3. 1793

2. 982

f v'

- £+4,3+2J-i-

3. 2486

4. 05S5„

h

- £+4,3+3J-i'

9.98„

0.8

2. 957„

3. 8580

fj

|£+ ,3+ A

9.0

2. 3363

3. 0704„

3.5111

yf

|£+ ,3+2i

9.5

1.500

2. 3585

3. 1842„

rin'

|£+3i3+2J

2.779

3. 7820„

*£+3,3+3J

9.28

2. 1614„

3. 0257

3. 6491„

n'

|£+3,?+3J

1.32

2.966

ri"

|£+3,?+3J

3. 3450

4. 1111„

P

|£+3,3+3J

3. 2309

4. 1965„

vY

|£+3,3+4J

3. 2994„

4. 1520

,'

|£+5i)+4J

1.017

3. 1617„

4. 1967

5. 0160n

Tj

|£+5,3+5J

0. 88„

2. 9688

4. 0380„

4. 8781

Y'

|£+7,3+5J

4. 0855„

5. 2422

rin'

|£+7,3+6J

4. 1991

5. 3823„

n'

|£+7,3+7J

3. 7114„

4. 9188

P

|£+7,?+6J-i'

2. 615„

3. 8317

'/'!'

-*£+ i*

3. 2411

3. 7872„

rr-

-^£+ ,3+ A

2. 819„

3.4476

P

-le+ 0- I

2. 9181„

3. 4813

t

2e

2. 364„

3. 0737

^r/

2£+ A

2.624

3. 3489„

v"

2£+ 2i

2. 207„

2. 97S

P

2£+ A+I

2. 620„

3. 2765

^

2£+2,3+2J

9.8„

1. 63

2. 362„

2.873

V

2£+2,3+3J

9.5

1.796

2. 303„

2. 1007

riv'

2£+4,3+3J

1.92„

2.700

2£+4,?+4J

8.7

8.8

1. 5802„

2.4158

2. 9867„

r,'

2£+4,3+4J

2.330

3. 1764„

l"

2£+4,3+4J

3. 1079

3. 9008„

P

2£+4i)+4J

2. 736

3. 6809„

nri'

2£+4,5+5J

2. 9881„

3. 8425

r,'

2£+6,3+5J

9.64

0.53

2. 652„

3. 6204

4. 3270„

I)

2£+6i(+6J

9.48„

0.36„

2. 4419

3.4512„

4. 1S92

%"

2£+8,3+6J

3. 6135„

4. 6784

7 >)'

2£+8,3+7J

3. 7124

4. 8075„

^'

2£+8,3+8J

3. 2109„

4. 3338

P

2£+8,3+7J-2'

2. 068„

3. 2092

|£+5i)+5J

9.3„

1. 140„

2. 0056

O f^79^

v'

|£+7,3+6J

0.5„

2. 2749„

3. 2377

3. 9184„

^

4£+7,3+7i

0.3

2.0512

3. 0565„

3. 7710

^£+7i>+7J

8.1

0.4J„

1.346

1. 959„

(t3-,3o)sin

'!'/'

A

9.66

0. 810„

2. 7559

3. 3840„

3. 6946

r/

20+ A

9.79„

0.54

n

20 +2 A

9.92

0.63„

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

Tablk C (XLII I)— Continued.

33

Logarithmic.

Unlt-1"

(i>-i>o) sin

K-J

«...

W-l

w>

W

w'

T

t

9. 801„

0.425

2. 5970„

3. 1493

3. 4158„

i'

I

1. 075„

2.045

3. 5201„

4. 3751

U"

£

l.G-!0„

2.514

4, 1066„

4. 8961

A

£

1.0G3

1. 916„

4. 1066

4. 8961„

5. 4076

1,'

£+ ^

9. 3()

0. 471„

2. 4824

3. 0830„

3. 3936

fv'

£+ ^

1. 565

2. 4o6„

3. 9671

4. 7890„

¥'

£+ J

1. 543

2. 341„

3. 8942

4. 67fi0„

P V

£+ d

0. 87„

1.75

4. 0705„

4. 8759

5. 3965„

W

e+ 2J

1.441„

2.273

3. 7192„

4. 5456

P V

£+ i'

0.42

1.36„

3. 7799

4. 5819„

5. 0998

Pr,

£+ -i + i"

0. 695„

1.585

4.0417„

4. 7827

5. 2543„

ij

£+4t5+4J

9.59„

0.45

V

e+4.5+3J

9.46

0.34„

I?

2£+2tJ+2J

9.45

0. 11„

%'

2«+2<?+3J

9.32„

0.04

nW

- £+ J
(t>-l5„)»C08

1. 255„

2.149

3. 6240„

4. 4615

r.

e

9.25

0. 117„

\'

«+ i

9.12„

0.02

m''

m'»

m'\ m'

m'

m'

m'

v=Jtu'r)Pr)'gj-tCi C03 Arg. + (.>-i)„)Ju;»i}P)j'?/'C2 sin Arg.+(i?— t>o)'2w»ijP»)''y** C, coa Arg.
where C,, Cj, C3 represent the respective coefficients.
110379°— 22 3

34

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table D (LIV).
Logarithmic. lUp.gfjPYi Bin Alg UnIt-1".

[Vol. XIV.

Bin

tc-1

U'O

U)

1}

- J-W

3. 0621„

3. 7258

,'

-n'

2. 8235

3. 5.528„

2e+ J-n'

2. 2831

2. 8483„

I)

4fl+3J-n'

1,705

3. 1591„

3. 9166„

\'

4e+2j-n'

3. 2462

3. 8608

n

ie+ e -n'

3. 2112„

3.8544

','

i£+ e+ J-n'

2. 5875

3. 4153„

i£+30+2J-n'

2. 2787

2. 6304„

Y

i£+5(9+3J-n'

3. 3(L55

3. 5865„

V

i£+5(?+4J-n'

3. 0779„

3. 3972

■n

-i£- fl-2i-n'

3. 1158„

3. 7378

i

-if- e- j-n'

3. 1493

3. 7544„

-ie+ « -n'

2. 3242

3. 0060„

i

-j£+30+ ^-n'

3. 3863

4. 1833„

V

-i£+3fl+2J-n'

3. 3532„

4. 1452

Tj

£+2e+ ^-n'

2.6364

3. 3704„

3. 8423

\'

£+2fl+2J-n'

1.423„

2.706

3. 4014„

E+4fl+3J-n'

1. 4042„

2. 1720

2. 6339„

y{

£+6e+4J-n'

2. 3306„

3. 1922

3. 7582„

5

£+60+5J-n'

2. 1137

3. 0138„

3. 6101

•n

- t-2e-ZA--a'

2. 7175

3. 4858„

3. 9484

\'

- f-20-2i-n'

2. 7756„

3. 5070

3. 9456„

- £ - A-W

1. 6810

2. 2463„

i

- £+29 -n'

2. 8125

3. 4427„

3. 7846

V

- £+20+ J-n'

2. 9121„

3. 4958

3. 8338„

1?

^£+30+2J-n'

2. 6058

3. 5312„

Y

4£+39+3J-n'

1.760

1.82„

f£+5fl+4j-n'

1. 7510„

2. 8113

Y

f£+76l+5J-n'

2. 9120„

4.0813

1}

-4£-3l9-4J-n'

2. 8673

3. 8458„

','

-|£-39-3J-n'

2. 9620„

3. 9124

-|£- e-2J-n'

2. 0569„

2. 7932

v'

-f£+ 9- J-n'

2. 9275„

3. 4708

V

-f£+ e -n'

2. 9702

3. 5487„

V

2£+49+3J-n'

1.640

2.7B1„

Y

2£+4e+4j-n'

1.617

2. 340„

2£+60+5i-n'

1. 206„

2. 2110

■n

-2£-40-6J-n'

2. 4012

3. 3634„

\'

-2£-4fl-4J-n'

2. 5241„

3. 4544

-2£-2fl-3i-n'

1. 5290„

2. 3210

i

-2£ -2A--a'

2. 3174„

3. 0558

V

-2£ - j-n'

2. 3514

3. 0737„

7n'

u

( COS t

=JUp.gi)Pri"l Bin Arg.+v{-S'i(co8 £-«,)+ZaBiii£|+c,(co8£-«,)+Cj

sin c.

No. 8.)

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.
Table E, (LV,).

logarithmic.

Logarithmic.

Kt=Iw'rjPTj'gj'^ COB Arg.
Table E, (LVn).

Unlt-1».

Cos

vl>

to

u>

■ni

i-ii'
zf+n'
i+ii'
J+n'
J+n'
2J+n'

2. 9180„

1. 9821

2. 8035
3.5175

3. 1764„
3. 4580„

3. 7732
2. 5473„

3. 7182„

4. 3017„

3. 9772

4. 2668

2. 8138

to'

Unit-l"

Sin

vf

w

a,i

j-n'
n'
J+n'
J+n'
J+n'
2j+n'
J+n'

2. 9180

3. 7799
1. 9821„
3. 7744„
3. 5175„
3. 4580
3. 1764

3, 7732„

4. 5819„

2. 5473
4.5420
4. 3017
4. 2668„

3. 9772„

2. 8138„

m'

K^^IW^p -q'qpt sin Arg.

" .=2 Up.qTjPjj'g ain Arg.+nK^,(co8 £-e)+i'j sin el+c,(cos £-«)+<;, sin*
i COs t y, )

35

36

MEMOIRS NATIONAL AC.IDEMY OF SCIENCES.

[Vol. XIV.

Logarithmic.

T.^BLE F (LVI)

w-w„

Unit— 1 radian.

Cos

IC-J

W-!

lit-i

M-O

w

Ufl

4.360

5. 1966„

5. 7767

r

4.766

6. 6599

7. 3732„

7. 7492

2r

4. 446

7, 1194

7. 7.572„

8. 0553

sr

4.412

6. 8442

7. 5458„

7.9060

AF

4.434

6, 5883

7. 3450„

7.7602

5r

6. 3437

7. 1490„

7.6136

ir

5.875

6. 7632„

7. 3134

Vo

-5r+2(?o+2Jo

6. 5090

6. 6325;,

7. 4746„

-4r+2flo+2Jo

4.161„

6.169

7. 0658

7. 8fi98„

-3r+20o+2Jo

3.19

6. 882] „

7. 6078

7. 9974

-2r+25„+2J„

3.52

7. 098fi„

7. 6970

7. 9394„

- r+25o+2Jo

5. 1420

6.359

7. 0722„

7. 4480

2»o+2Jo

4.379

7. 635.5„

8. 2144

8. 4125„

r+2(?„+2Jo

4. 856„

8. 0894„

8. 9548

9. 5668„

2r+25o+2Jo

4. 92„

7. 8150„

8. 6561

9. 2006„

3r+2/?„+2Jo

5. 5174„

7. 6056„

8. 46.50

9. 0111„

4r+2«o+2Jo

5. 4248„

7.4]28„

8. 2958

8. 8561„

5r+2fl„+2Jo

7. 22.54„

8. 1426

8. 7346„

7r+29o+2Jo

6. 8746„

7. 8484

8. 4936„

v'

-5r+29„+ Jo

6. 8776„

7. 5604

7. 842.5„

-4r+2{)o+ Jo

4.582

6. 881.5„

7. 4536

7. 5238L;,

-3r+29„+ Jo

4.674

6. 6271„

6. 7816

7. 3174

-2r+20o+ Jo

4.99

6. 7985

7. 4732„

7. 7966

- r+29o+ Jo

5. 4623„

2flo+ Jo

4. 605„

7. 1987

7. 8314„

8. 1061

r+2(?o+ Jo

5. 0056

8. 2964

9. 1086„

9. 6833
9. 3296

2r+2(?o+ Jo

4.38

8. 0434

8. 831 6„

3r+29„+ Jo

5. 6251

7. 84.58

8. 6564„

9. 1558

4r+2«o+ Jo

5. 5812

7. G603

8. 5030„

9. 0248

5r+29o+ Jo

7. 4778

8. 3544„

8. 9050

7r+25o+ Jo

7. 1130

8. 0545„

8. 6668

V

r

sr

4r

4.664

4.71

5.83

7. 8102
7. 7520„
7. 6172„
7. 7135„

8. 6250„
8. 1242
6. 6043„
8. 2308

Vo'

-4r+4&„+4Jo
-3r+49o+4Jo
-2r+4»„+4Jo
- r+49o+4Jo

7. 1862
7. 1804
6.817
8. 4680„

7. 9072^
7. 8679„

7. 45G„

8. 8822

4eo+4Jo

4.666

5. 807„

8. 0913

8. 8270„

9. 2073

r+4eo+4Jo

8. 7850

9. 8236„

2r+45o+4Jo

8. 5144

9. 4910„

3r+49o+4Jo

8. 3274

9. 3006„

4r+49o+4Jo

8. 1627

9. 1494„

5r+4e„+4Jo

8. 0050

9. 0105„

voY

-4r+49o+3Jo
-Sr +400+3 Jo
-2r+45o+3Jo
- r+40o+3Jo

7. 354„

7. 5708„

8. 8838

8. 1083
8. 2084

9.054^

400 +3 Jo

4. 516„

6. 20S4

8. 5565„

9. 2180

9. 5174„

r+40o+3Jo

9. 2783„

0. 2833

2r+40o+3Jo

9. 0241„

9. 9635

3r+40o+3Jo

8. 8480„

9. 7850

4r+40o+3Jo

8. 691fi„

9. 6434

5r+40o+3Jo

8. 5401„

a 5128

77l'2

771'-

m'\ 777'

m'^ 77/

771'

m'

No. 3.]

MINOR PL-INETS— LEUSCHNER, GLANCY, LEVY.

37

Logarithmic.

Table F (LVI)— Continued.

IV — w„

Unit— 1 radian.

Cos

U'-5

U>-2

»■-'

w'

w

w>

VoY

-4r + Jo

-2r f j„
- r \- Jo

7.7640
7. 4203

7. 8104„

8. 0479„

7. 8364„

8. 3915
8. 6268
8. 8018

+ Jo

4. 518„

5. 886„

5.70„

r + j„

7. 1339

7. 8.500

2r + Jo

7. 8421

8. 4293„

3r + Jo

7. 9669

8, 6796„

4r + Jo

7. 9760

8. 7576„

V'

-4r+4fl„+2J„
-3r+40„+2Jo
_2r+40„+2Jo
- r+4(7„+2Jo

6.9002
7. 1638

8. 1860„

7. 6938„

7. 8502„

8. 4016

4fl„+2Jo

3. 7G

6. 0608„

8. 4157

8. 9760„

9. 1661

r+40„+2Jo

9. 1714

0. ]382„

2r+4(?„+2Jo

8. 9358

9. 8333„

3r+4flo+2Jo

8. 7718

9. 6681 „

4r+4^o+2Jo

8. 6236

9. 5372„

ri"

2r
zr

3.76

5. 7516

4.7

7. 8677

7. S610„

8. 1026„
8. 1538„

8. 6727„
8. 2228
8. 7296
8. 8728

P

r

■2r

4r

7. 941 8„
7. 9312„
7. 7920„
7. 639„

8. 7337
8. 7154
8. 6154
8.5001

P

-4r+49o+3J„-i-„
-3r+4eo+3J„-i'o

-2r+4e„+3Jo-i'o

- r+4<)o+3Jo-i'o

7.446
7. 1858

7. 6176,

8. 1156„
7. 8677„

7. 9693

4fl„+3Jo-Jo

4. 804„

7.168

7. 9368„

8. 3724

r+4e„+3Jo-i'o

7. 7887

8. 8492„

2r+4So+3Jo--ro

7.448

8. 453L„

3r+4(?o+3Jo-Jo

7. 19,76

8. 2026„

4r+40o+3Jo-i"o

6.978

7. 9y|i3„

"Jo'

29„+2Jo

5. 418„

6.292

7. 4754„

8. 6636

ri.'

6fl„+6Jo

5. 418„

6.292

8. 6H28„

9. 4351

VoW .

2*0+ ^D

5. 8S5

6. 719„

8. 5059

9. 2804„

ri.W

2eo+3J(,

4. y74

5 896„

8. 0326„

&i 1975

r^oW

e^o+oJo

5.935

6. 780„

9. 2774

0. 0330„

Vo rr

2e„

5.744„

6.535

8. 3811^

9. 1030

n.r

2eo+2Jo

5.441„

6.327

8. 0917

8. 6300

6^0+4 Jo

5. 'JJ9«

6.744

9. 4432,

0. 1404

n"

2*0+ ^0

5. 301

6. m„

6. 14^

8. 2302

9. 0152„

<'

6e„+3Jo

5.301

9. 1294

9. 7729„

Pno

2fio+2Jo

8. 5904

9. 3492„

9. 8022

Pr-io

2^0+ ^0-^0

4. 502„

5.41

8. 1011„

8. 8726

Pno

60o+5Jo-J:o

4. 502„

5.41

8. 0.564„

8. 9263

P l'

2e„+ J„

8. 55=t2„

9. 3245

P r,'

20„+2j„-J„

4.057

5.021„

6.887

8. 1804„

P r,'

60o+4J„-2'„

4.057

5 021„

8. 2718

9. 1021„

w—Wo=ICw'TiPi^'9pt COS Arg.
■where C represents the coefficient.

38

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Logarithmic.

■Oo

Vo

Vo'

rio'

Cos

4,-i.r +2e^+2ia
vi-3r+2eo+2Jo
<p-2r +2e„+2ia

4,- r+2do+2Jo
^ +29o+2J„
4,+ r+29o+2Jo
4,+2r+2d^+2i„
^!.+3r+20„+2J<,

v!.+4r+2eo+2J„

^+5r+2eo+2J„

4,-hr+ie^+Ada

^_4r+40„+4Jo
v!r-3r+4e„+4io
^(,_2r+40o+4J„
^i- r+4eo+4J„

V!r +4fl„ + 4Jo

4+ r+4e„+4Jo
i!r+2r+4e„+4J„

^+3r+4fl„+4Jo
v'.+4r+40o+4i„

v!'-4r

i-sr
cj-2r

i- r

<p

'!'+ r
4'+2r

4>-5r+49o+5Jo

4:-4r+4.eo+3Jo

4,-3r+4do+3Jo

•i>-2r+4eo+sdo
4>- r+45„+3Jo

,{> +49o+3io

<!>+ r+45„+3Jo

v!r+2r+4e„+3i„
v!.+3r+4eo+34o

<p-zr

i'-2r
4- r

4'

4+ r
4,+2r

•P+ir

+ ^0
+ ^0
+ ^0
+ ^0
+ ^0

+ ^0
+ ^0
+ ^0
+ -^0

V'— 5r+65„+6J„

4.4-4r+69o+6Jo
i^-3r+60o+6Jo
V'— 2r+6eo+6Jo
V'— r+69o+6Jo

^J+ r+6fl„+6J„
(J+2r+69o+6zlo

9J-5r+25o+2J„
v!.-4r+20„+2Jo
0-3r+20o+2Jo
v!.-2r+2So+2J„

v!— r+2eo+2J„

0 +29„+2J„

,!.+ r+2flo+2Jo
0+2r+2i9„+2Jo

Table G (LVII).
jS sin ^+C cos ip

9.199

8. 76„

W-2

9.196

9. 04„

9. 140
9. 274„
9. 137„

0. 158

y. yoyi

0.344„

9. 013„

9. 885„
9.009

1. 1109„

1.017
9.45

8.81

9.009

9.318

9.207
711
1712„
230„
220„
724„
494„
100„

9. 771 „
0. OG't^
0. 3185„

0. 497„

1. 0286„

2. 6172
0. 7226
0.669
0. 9435
0. 5122

9. 814„
0. 0434„
0. 354]„
0. 362„
0. 4164„
0. 1436„
0. 3102„
9.918
9.465
9.20„

9.476
9.781
9.811
0. 3489
0. 9511
2. 7932„
9. 961„

0. 491„

1. 0464„
0. 678„

9.848
0. 0792
0. 3941
0. 248
9.901
0. 8518
0. 1664
9. 76„
9. 38„

3. 1673„

2. 689„

1. 082„
1. 2314„
0.931
1. 6478

1. 9.50

2. .5678
2. 3541„
1. 9114„
1. 5:572„
1. 2.544„
1. 018„

042„

723„

1626„

7787„

2379„

2511„

1702

7877

5117

2732

1. 925

2. 0527
2.145
2. 1351
2. 3504„

2.497
1. 9006„

0. mn

1. 406„

1.327
1.447

2. 1070

2. 5095

3. 3599
3. 3085
3. 3609„
2. 9943„
2. 7293„
2. 4992„

2. 0766„
2. 1609„
2.]57„
2. 0455

2.5^4

1.836
2. 1633
2. 1064

1. 9892

2. 3144

2. 9538

3. 3102
3. 4970
3. 9455
3. 9296
3. 9144„
3. 5594„
3. 3121„

1657

1255

234

576

1995

4822

2480

1612

5710

5493

604„

1070,1

3426„

565n

1493

6867

3831

1315

9034

1.
1.
1.

2.

2.
2.
3.
2.
2,
2
1,

1.868
2. 3515

2. 6961

3. 0649
3. 1223
3. 4930
4. 1580„
3. 7083„
3. 4261„
3. 2042„

2. 634„
2. 6896„
2. 675„

2. 3850„

3. 0929

3. 1875„
1.0453
2. 5218„
1.729

1. 889„

2. 1506„
2. 6309„
2. 9557„

2. 7758

3. 4526„

4. 3114
3. 8728
3. 6067
3. 3946

2.712
2. 6968
2.491

2. 78p8^

3. 2539„

2.44^
2. 6170„
2. 7194„
2. 6870„

2. 9730„

3. 3785„
3. 5843„
3. 8423„

3. 7269„

4. 3377«

5. 0372
4. 5942
4. 3236

2. 7221„

2. 8004„

3. 1304^
3. 3804„
3. 8325n
3. 9938„
3. 2839„
3. 8424„

Unlt-l"

w>

1. 612„
0. 989,„
1.916

2. 2333
2. 3713

7107„
1657„

862871
633871

4248„

2. 357„
2. 6814„

2. 921 4„

3. 0993„
3. 9385„

4. 9365
4. 3605
4. 0450

984

9432

744

4864„

5397„

5978
8834

3. 3564

2299
5419
8j608
0952
9726

0691„

4922„
1945„

2. 9697„
2, 7976„
51

2380
6434

3029„

2433

9212

No. 3.1

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

39

Logarithmic.

'^ABLE G (LVII)— Continued.
S Bin ^+ C COB (p

Unit- 1"

Cos

ip-s

uu

icJ

7/)0

w

w'

•Jo'

v!l-5r-20„-2Jo
4,-Ar-20„-2ia
v!.-3r-29o-2J„

2. 700„

2, 817„
2. 9247„

3. 5481
3. 6251
3. 6905

^J-2r-29o-2J„

9.59„

3. 024 1„

3. 7470

,J- r-20„-2J„

3. 13e4„

3. 8346

ij, -29„-2Jo

0. 117

0.95„

2. 297„

2. 785()„

3. 0614

,J+ r-2e, -2Jo

2. 8942„

3. 5604

^,+2r-2(J„-2Jo

2. 297„

3. U29

In V'

V'-4r+6flo+5J„
^-3r+69o+5Jo
Vr-2r+G0o+5Jo
,J- r+69„+5Jo

2. 4885„

2. 97G„

3. 054 1„

3. 9514„

4. 3903„

3. 1691
3. 5500

3. 8829

4. 1032
4. 0037„

^ +60„+5Jo

0.295

1. 360

3. 6364

4. 330 1„

4. 00f;2

Vi4- r+6fl„+5J„

4. 4005

5. 4960„

^+2r+6e„+5Jo

4. 0582

5. 0(;i2„

v!.+3r+6flo+5J„

3. 8204

4. 8027„

ijol'

^-5r+20o+^o
0-4r+29o+:lo

^_3r+2e„+jo

0--2r+29o+^o

•

2. 426„
2. 399„
2. 410„

2. 701„

3. 2842„

3. 0684
3. 0310
3. 1305
3. 4602
3. 8558

^ +29o+Jo

0.444

1. 188„

3. 0569

3, 7266„

4. 1122

4,+ r+2fl„+Jo

2. 8541

3. 5823„

^+2r+2e„+io

3. 2]91„

3. 7635

^ol'

v!.-5r-2eo-^
v!r-4r-2eo-^o

4,-zr-2e,-\

3. 1551
3. 2454
3. 3100

3. 9530„

3. 9948„

4. 0023„

^_2r-25o-J„

9.93

3. 3277

3. 9401„

^_ r-2fl„-j„

3. 1976

3. 4598„

V^ -2flo-Jo

0.490„

1.324

3. 0145„

3. 7326

4. 2787

^+ r-29„-io

3. 3632

3. 9402„

,!.+2r-2flo-4

2. 7792

3. 5224„

'Jo'?'

9'.-5r+2eo+3Jo
Vi-4r+20o+3io
^-3r+2eo+3Jo
^j_2r+2eo+3J„
V^- r+20o+3Jo

2. 2738„
2. 116„
2. 5858„
2. 809„
2. 650„

2. 847

3. 0290
3. 3787
3. 5429
3. 7297

4, +2eo+3J„

9.98

0.60„

2.873„

2,685

3. 7980

4,+ r+2(?„+3Jo

3. 5126„

4. 2856

v!.+2r+2eo+3J„

9.46„

3. 3438„

4. 1208

ri"

;i-5r+69„+4io
0-4r+6eo+4J„
<5-3r+6eo+4i„
^_2r+6fl„+4Jo
v!— r+69„+4J„

1. 9950
2.6112
3. 0556

3. 7934

4. 2260

2. 7422„
3. 1949„

3. 5583,„

3. 7947„

4. 4064

V^ +69o+4J„

9.98„

0.76„

3. 5017„

4. 1098

4. 3552„

^+ r+60„+4J„

4. 2852„

5. 3521

vJ+2r+69o+4Jo

3. 9567„

4. 9249

r,"

^,_5r+20„+2Jo
^J-4r+2fl„+2J„
v!.-3r+25„+2Jo
v!.-2r+25o+2Jo

4- r+2o„+2Jo

2. 5018
2.453
2. 4799

2. 9375

3. 2833

3. 0963„
3. 0935„
3. 2779„
3. G294„
3. 8982„

^ +29o+2Jo

0. 025„

0.60

2.634

3. 2781

4. 0439„

^+ r+29„+2j„

3. 5607

4. 2381„

(/r+2r+29o+2Jo

3. 4629

4. 1704„

v"

4,-6r-28„
4, -if -200
^-3r-20„

^-2r-2eo
^- r-29„

3. 0090„
3. 0676„
3. 0764„
2.958^
3. 1140

3. 7477
3. 7445
3. 6664
3.3121

4. 0201;,

1^ -2ffo

0.305

1. 127„

2.912

3. 5491„

3. 9085

4,+ r-29o

3. 0396„

3. 6320

v!i+2r-2ff„

2. 4706„

3. 2330

40

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table

G (LVII)— Continued.

Logarithmic.

S ein}j/-\-C cob4

Unit-l".

Cos

UH*

10-1

to-"

U't

u>

w*

f

v''-5r+6fl„+5J„-Jo

v'.-3r+6e„+5Jo-i'o

^J.-2r+6e„+5Jo-^o

4— r+effo+sJo-i'o

2.006
2.335
2.544
2.718
2.970

2. 7505„

2. 981„

3. 1436„
3. 2445„
2. 9116„

9^ +6eo+5J„-J„

8.6„

9.7

2. 114„

2.^23

3. 40G7„

v!.+ r+60o+5i„-2o

2. 7948„

3. 9420

9!.+2r+69„+5i(o-i'o

2. 3824„

3. 4488

P

^I— 5r+25„+2J„
^;,_4r+2(?o+2Jo
^-3/^+2fl„+2Jo
vJ-2r+20„+2^„

9,6

1. 9,16„

2. 5178„

2.387
2.911
3. 3047

2. 938„

3. 6294

(J- r+2e„+2Jo

3. 3406„

3. 9,3.30

^ +20<,+2J„

0. 5910

3. 1266

3. 802 1„

4. 1894

4>+ r+29o+2Jo

3. 4070

4. 3178„

sH-2r+2eo+2J„

3. 0472

3. 9308„

P

^(,_5r_2(9„-^„+v„

0. 732„

1. 085

^_4r-20„-^+i'„

«

0.35

1. 895„

4,-zr-2e,-A,+i,

1.463

2.614B„

4,-2r-20„-j,+i,

2.064

3. 0255„

4'- r-2e,-A,+2,

2,6816

3. G280„

4 .-20„-i„+v„

9.04

o.n„

2.636

3. 3284„

3. 7399

4+ r-20„-Jo+j„

3. 0572„

3. 6430

vi+2r-2(?o-J„+i'o

2. 9121„

3. 5491

w

V''+ 4eo+4J„

0.775

1. 65„

3. 1052„

3. 0342„

4- 40o-4Jo

0. 29„

1.10

3. 1888

3. 6104„

9^+ 85o+8J„

0.65

1.54„

3, 7520

4. 5812„

rioY

4+ 4e„+5J„

3. 7577

4. 3244„

^J+ 49o+3z(o

1. 280„

2.081

3. 1240

4. 1388

,!.- 4eo-3z(o

1.005

1. 77„

3. 5356„

3. 3.560

9!.+ 8fl„+74„

1. 228„

2.093

4. 3980„

5. 1827

vr

^+ 40„+4^o

4. 1155„

4. 5547

4+ 40o+2J„

1.106

1.88„

2.831

4. 1803„

S';- 4e„-2io

1. 146„

1.88

3. 0422

4.0180

V^+ 80„+6J„

1.321

2. 152„

4. 5658

5. 3010„

,'3

3. 8375

3. 2197

4. 2553„

4. 0446„

3. 9650„

4. 9349

i' -Jo

^J+ 4(?e+3Jo-i'o

3.0024

3. 8634„

4- 40„-3J„+i-„

9.98„

0.8

2. 956„

3. 8331

^!.+ 80 +7i„-J„

3. 0757

3. 9759„

4+ 40o+4i„

0.46„

1.32

3. 8514„

4. 6436

/ n'

^J+ 4/9„+4J„-J„
^J- 4(?„-2J„+i-o
^J+ 8e„+6z(„-J„

2.442
3. 2486
3. 2818„

1. 846„
4. 0585„
4. 1441

^+ 45„+3J„

0.27

1.15„

3. 9421

4. 6972„

S Bin ^f'+C cos 4'='ZCu'S-qP-q'<lpt COS Arg.

where C represents the coefficient.

II. TABLES FOR THE DETERMINATION OF THE PERTURBATIONS OF THE

HECUBA GROUP OF MINOR PLANETS.

DEVELOPMENT OF THE DIFFERENTIAL EQUATIONS FOR W AND FOR THE THIRD COORDINATE.

It would be futile to attempt to give a brief but comprehensive outline of the fundamental
developments in the theory of Bohlin-v. Zeipcl which would assist the reader to an imderstanding
of the construction of the tables. In broad outlines, the problem is the integration of Hansen's

differential equations for vZz, v, and -•> by means of the method developed by Bohlin and

COS V

according to the modifications introduced by v. Zeipel for purposes of numerical computation.
The first division of the problem is the development of functions of the partial derivatives of
the perturbative function; the second division of the problem is the integration of the Hansen
equations in the form of infinite series.

For the theory the reader is referred to the original works of Hansen', Bohlin^, and v. Zeipel'.
As indicated in the introduction to the first section, unless otherwise stated, the references to
Bohlin refer to the French edition and arc designated by B; references to v. Zeipel are desig-
nated by Z. Although dupHcation of material which can be found in either reference is to be
avoided, our experience in attempting to reproduce v. Zeipel's tables led us to fill in certain
gaps which are troublesome to the reader and the computer.

The first section of v. Zeipel's theory is concerned with an independent development of
Hansen's differential equations for ndz and v and a repetition of the differential equation for

-t and the introduction of Bohlin's argument 0. In passing, it is well to emphasize two

COS %

facts: First, the variables e and /"used throughout the theory are analogous to Hansen's s andy-
the dash is unnecessary, for the physically real values do not appear. Second, the constant
elements a, e,Tz, c, Q,,i are neither osculating nor mean elements; they are defined in the section
on constants of integration.

The perturbative function and its partial derivatives are developed in Fourier's series, in
which the arguments depend upon the relative positions of the disturbed and disturbing
bodies and in which the coefficients are infinite series in ascending powers of the eccentricities
and the inclination of tlie orbits. The coefficients in the latter are elliptic integrals depending
upon the ratio of the semi-major axes.

Since these elliptic integrals are functions of the ratio of the semi-major axes, or of the
mean daily motions, they can be developed in Taylor's series, in which the given function and
its successive partial derivatives are expressed for exact conamensurabiht}' and the series pro-
ceeds according to a small quantity w, defined by w=l— 2//, where /i is the ratio of Jupiter's

mean motion to that of the planet and where fi differs but little from ^ • These elliptic integrals

enter the coefficients in all of the subsequent trigonometric series. Hence all the coefficients are
series in w. With some exceptions the terms in w°, w, and v/ have been used. The develop-
ment of all functions in powers of vj is the essential principle underlying the group method of
determining perturbations.

Tlie following pages contain the tables which are, in general, parallel to those of v. Zeipel.
At the end of sections 2, 3, 4, 5 there are brief written comparisons. To facilitate comparisons

' Auseinandersetzunc einer zweckmassiRen Methoilo zur Bereehnung der absoluten Storungen dcr klcinen Planeten.

' Formeln und Tafeln zur gruppenweisen Bereehnung der allgemeincn Storungen bcnachbarter I'lanetcn. Nova Acta Reg. Soc. So. UpsalienslJ,
Scr. Ill, Band XVII, U'96.

Bur Ic Di'vcloppcmcnt dcs Perturbations Plan^taires. Application aui Petites Planfttes. Stockholm, 1902.
••■VngeniilieneJupiterstoningenfurdie Hecuba-Gruppe. St. PStersbourg, 1902.

41

42 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

with V. Zeipel's tables, those numerical quantities which are in disagreement are inclosed in
brackets. There are also certain mathematical developments useful to the reader. These
relations are sometimes taken from v. Zeipel and sometimes supplement his text.

Certain simple functions of the elliptic integrals Tj""", defined by Z 19, eqa. (73), (74), (7.5),
are tabulated in Table I (cf. Z 23).

Tables II-IVw^ (cf. Z 2G-32), giving the partial derivatives of the perturbative function,
are computed according to Z 24, eq. (77), by means of Table I and B 184, Tables XVI-XVIII
and B (Ger.) 182, Tables XII-XIV.

The eUmination of Jwpiter's mean anomaly from the argument gives Z 25, eq. (78), in
which the coefficients are derived from Table II-IV vy^ by the formulae given in .S 61. These
coefficients are tabulated in Tables V-VII w' (cf. Z 33-39).

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

43

2

o

n

o

9. 73559
9. 22161

8. 88734
8. 60172

CD OS
OS d
OS £N

oo'

0. 62442„
0. 21732„
9. 96912„

co^^^

CI CO
CO CO
t- CO
t^ CO

t—i ^

COCO

os^
in b-

r-. 00
r^O

b-
CO

ci

o>

nHCs) r- CO

OS TJ* ^ 00
OS O) OS CO

COr-t

1— t CD

Or-;

e c c

t^ ino c
o CO ino

t^ O -HO

o o CD in

00 -^ rHOS

O' O' O OS

O OS

•— ' -^

CO h-
OS O
rJc4

■^ CO in
b- CD in

CO in CO
OS c--: OS

CIO CO

.-; r-Jo

s
s

ci

00

CD

Tp COCO 00

eo(M Mcooo

CO CD C-1 CO CD
r-H CO coo t^

O OS oi oi CO

oo in

O CO
COCO
r-l r—i

r-i r-i

see

00 CO 00 e e

CO 00 CD -^ CO
OS T' CO •** t-

co OS in -rr* -**

OS in CO rH OS
O O O O" OS

CD CI

CO CO
CO o
O CI

ci ci

r-l Crs 00

CO 00 in

TfH rH O

OS

ci

t^

CO

W CO CO '^

i-H CM GO ai Oi

rH ^D CO t-H Tf

rH 00 'T »0 CO
TJ* CO lOd 05

O OS OS O) CO

O CO OS

CO cs in

CO CD r-(

coco CO

r-i r-i r-i

C^

<7S
CO
CO
CO

CM

e c e c e

CC O O CM r^

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n

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1. 898161„
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2. 283141 „
2. 283141

2. 153770„
2. 153770

2.006171„
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1. 847875„
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1. 68250„
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1.5]210„
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2. 522558
2. 522558
2. 522558„
2. 522558„

2. 083079
2. 937464
2. 683079„
2. 937464„

2. 528880
2. 958044
2. 528880„
2. 958044„

2. 366191
2. 92477(>
2. 366191„
2. 924776„

2. 197675
2. 858077
2. 197G75„
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2. 02490
2. 76886
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1. 84887

2. 66352

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2. 812563

3. 024413„
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2. 462558

3. 261391

2. 794447„

3. 076598„
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3. 246209

2. 523495„

3. Or)8902„
1. 856833„
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2. ]97fi75„

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1. 76164„

2. 9,1803„
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2. 09513„
2. 89541

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3. 49405„
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3. 70912

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2. 94112

2.51524
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2. 84832
2. 84832„

2. 78195
2. 78195„

2. 69286
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2, 58759
2. 58759„

2. 47019
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i?,.o[n + l.-n+]
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3. 45486„
3. 62946„
3. 45468
3. 62946

3. 34235„
3. 66471„
3. 34235
3. 66471

3. 21906„
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3. 21906
3. 66409

3. 08741„
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3. 08741
3. 63529

2. 94903„

3. 58453„

2. 94903

3. 58453

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3. 69598
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3. 53278
3. 76576

2. 99523„

3. 91658„

3. 33224
3. 78484

2. 24789„

3. 90661„

3. 08741
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2. 51136

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2. 77380

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3. 51583n
3. 51583

4. 60272
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3. 76747„

lia-o

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3. 1148„
3. U48

3. 0520„
3. 0520

2. 9231

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4. 0409„

4. 0961

4. 0774„

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3. 8736„
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4. ]593„
3. 6562

4.3090

54

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

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MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

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62

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table VII.

[Vol. XIV.
Unit=l".

R„,o(n.-n+l)+ff'

Iio.o(n.-n-l)-n'

i?,.„(n+l.-n+l)+,r'
i?,.„(n-l.-n+l)+>r'
RUn+l.-n-l)-,/
Ry.o(n-l.-n-l)-n'

R„.Jn.-n+2)+n'
J?o.,(n.-n)+!r'
fio.i(n.-n)-^
Ro.iin.-n-2)-T/

RUn.-n+D+Tz'
R,.,in-2.-n+l)+n'

i?,.,(n-l.-n+2)+ff'
" ■ ■ " -n)+7r'
.-n)+7/

K,.,n+1
/?,.,(n-l
i?,.,(n-l

;j,.,(n-i

B<,.,(n.-n+l)+^
iJo.j(n.-n+l)-T'

i?„.o(n-l

i?o.o(n-l
i?o-o(n+l
iJo-o(n-l

—n)—a+7:'
-n)+d+7z'
-n+2)-d+7:'
— n)+<7— ff'
— n)— 5— jt'

79.10

+ 79. 10

+ 372. 6

+ 293. 5

- 293. 5

- 372. 6

- 649. 5

- 333. 1
+ 333. 1
+ 649. 5

+ 2012

+ 2012

- 2012

- 2012

- 533
+ 533

+ 533
533

191. 93
+ 191. 93

+ 482. 0
+ 865. 9

- 290.1

- 1057.8

- 1057.8

- 1057.8
+ 290.1
+ 1825.5

3119
2330

6583
+ 1656

- 142. 48
+ 142. 48

+ 266.7

+ 979. 2

- 124. 2

- 1121.6

- 622. 9

- 1192.9
+ 53.0
+ 1762,8

+ 5118

+ 873

- 101. 43
+ 101. 43

+ 130. 9

+ 942. 4

- 29.5

- 1043.8

- 333. 8

- 1145.3

- 71.9
+ 1551.0

- 70. 45
+ 70. 45

+ 52.0

+ 826.9

+ 18.5

- 897. 4

- 157. 6

- 1003.0

- 124. 1
+ 1284.8

48.14

+ 48. 14

+ 9.6

+ 683. 6

+ 38.5

- 731. 7

- 57.8

- 828. 0

- 134. 8
+ 1020. 6

6

- 32.52
+ 32. 52

- 10.7
+542.1
+ 43.2
-574. 6

-5.6
-655. 9
-124.5
+786. 0

iJ„.„fn.-n+n+;r'
i?o.o("--n-l)-T'

i?,.o(n+l.-n+l)+7r'
i?,.o(n-l.-n+l)+;r'
i?,.„(n+l.-n-l)-;/
R,.o(n-l.-n-l)-n^

/?o.,(n.-n+2)+„'
Ro.An.-n)+7:'
Ra.i(n.—n)—7:'
R^.M--n-2)-j:'

/?,.„(n.-n+l)+^
Bj.o(n-2.-n+l)+s'

+

327.5
327.5

•R...(n-1
i?,.,(n+l
if,.,(n-l
i?,.,(n+l
i?,.,(n-l

-n+2)+n'

-n)+;r'

-n)+n'

-n)-it'

-n)-rt'

Ro.^{n.-n+\)-\-7z'
R,.,{n.-n-\-\)-v/

i?o.o(n-l.-n)-a+!r'

i?„.o(n-l
i?o.on+l

-n+2)-5+;r'

—n)-\-a—7c'

-n)-S—,^

1950
1622
+ 1622
+ 1950

+ 3096
+ 1786

- 1786

- 3096

-13734
-13734
+ 13734
+13734

+ 3280

- 3280

- 3280
+ 3280

+ 705. 2

- 705. 2

- 2850

- 4260
+ 2145
+ 4966

+ 4966
+ 4966

- 2145

- 7786

+23018
+15418

+

605.
605.

1897
4923
1292
5529

+ 493.0

+ 386. 9

493. 0 - 386. 9

+

+
+

+ 3410

+ 5831

- 989

- 8252

-3,3562

1163
5107
670
5600

+
+

+ 2149
+ 6093

- 177

- 8065

+40061
-11862

- 5854

295.
295.

- 643

- 4898
+ 256
+ 5285

+ 1223
+ 5866
+ 325

- 7413

- 299

- 4432

+ 4
+ 4727

+ 594

+ 5318

+ 587

- 6499

/?o.o(n.-n+l)+T'
i?„.„(n.-n-l)-;r'

i?,.o(n+l.-n+l)+;r'
fi,.o(n-l.-n+l)+^
ie,.„(n+l.-n-l)-!r'
J?,.o(n-l.-n-l)-jr'

R„.An.-n+2)+n'
Ra,An.—n)-\-K'
Ra.i{n.—n)—n'
R„.i{n.-n-2)-n'

- 1303
+ 1303

- 7080
-12290

- 1127
+ 1127

+13600

+ 838

-14465

- 7475
+ 7475

-14430
+ 4532

+20370

No. 8.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 63

With these tables wo compute terms of the first order in the mass in Hansen's differential
equations for the function W and the perturbation in the third coordinate. See Z 7, eq. (33)
and Z 8, eq. (39). The first order parts of the equations are expressed in Z 41, eqs. (82), (83),
in the form of trigonometric series, in which the coefficients are computed from the formulae
given in B 67. These coefficients comprise Tables VIII-XIVw^ (cf. Z, 42-48).

Table XV (cf. Z 50, eq. (88)) is an auxiliary table of the same type of construction, which
is employed in the computation of terms of the second order in the mass in the differential
equation for W (cf . Z 53) .

64

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

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+ 1 + 1
s s s s

1 1 1 1
s s s s

tl-

r-J p-i rH l-H i-^

1 + 1 + 1

7+7+7

1 1

+ 1 + 1

-S-S.-S-S-

1 1 1 1

1."

rH r-i

1 +

o o

9 9 9?

-S-S-

£-£.£-£.-£

^s.

S^^£,£,

^~o^

9 9 9 9

-S"S

^-S.

tejtqtutq

(M M M N

ttjCq

tijaftqtQtJ'

o o o o 9

tlftl?

(N (M C^ (M

(MC<I<M(M

t:fte

tq tq

(71 at

)}0B^

Wo. 3.3

MINOR PLANETS— LEUSCHNER, GLANCY. LEVY.

75

+ +

1 +

+ 1

I I

+ 1 I

iJL
++

R R
I I

8 e S 8

15

V V +

t: ts'o

+ + I

JL++

S R R
I I I

I I

iJL

R^R
I I

1 + I + I

o o o o o

o. o. c- o c

WW

+ + 1 1

fc a

k u

+ 1

++I 1

+. s 1

+ 1

R R R S
1 1 1 1

+±il

S R
1 1

+ 1 + 1

R R R K

MM

s s

S S R R

R R S R

c o

c o o o

sfsf

W (M (M C^

,.7l JOJDBJ

76

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIT

■-1 o

+

+ +

to CO

I +

+

no i© t--

C CO c: 00

.— ( ■^ I— I CD

+ + I I +

C5 t-

c^ o

iC oo

+

I + + I

+

d 05 C5 CC
W O (MM

+ + I 1 +

+

I I + +

-^ O O lO

+ +

I +

•J

pa
■<

Eh

+

oc cc c^i ■^ t*

-H rr (M CO 00
C^ CO CD M CO

I + +

CO

Tf 00

coo

t- QO

r- to

.-I -^

X.

t- 00

M

CC (^

•* CO

M* CO

t— 1

PHIO

05

IN

(MO

;d

,-^ U3

^ rj

+ + I I +

I +

I I

+ I

+

+ I

++

I I

++ I I

IN

+ +

+ 11 +
c = c s
I l_ 1^ 1^

,— I r-i r-4 I— i

+ 1 + 1

c c o o
o c o

05 oqoo oooq CQO5O!} tj^cc'yi'yi tC't>30o cooo^c^

'. + I
s c s

<^

Odcc.

+ 1

s s

I I

— _c o

Cq C<;,

(/; On 0;^ 0^ Cri On Cq

rrt JOJOEjI

No. 3.1

MINOR PLiVNETS— LEUSCHNER, GLANCY, LEVY.

77

g:5

t^ CO

+

+ 1

1 +

(—1
00

O 00

lO CO
CD so

+ + 1

+

X! O t^ O 'M

4- + I I +

lO UO OS CO o

c^j I— ( o as

+ + 1 1 +

Oi CO
CO rp
(N CD

+ I I ++

I— t CO C^ lO

+ I I ++

s s
I I

+ 1 + 1

a o o o
o o o o

GOCqOQGQ

+ 1

c s

I I

s e

Oq &3 C>2 Cc 0;;

J n JojOBj

78 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xrp.

INTEGRATION OF THE DIFFERENTIAL EQUATION FOR TF.

With the exception of Tables LVI and LVII all the following tables are concerned with
the integration of functions whose coefficients can be derived, more or less directly, from the
preceding tables. The terms of first order in the mass, before and after integration, are of the

^Cp.,(n+r.-n + s),P,v{cii

A+s-tJ)
A-e + (p

where Cp.g = C„.p., + C,.p.,- w + C^.p.g-uj' -\ (see Z 25)

and A = [n + r-i{n-s)]£+{n-s)d+in+i' W

In the argument A the factor n is always a positive integer; the factors r, s, i, and i' are
positive and negative integers. Evidently, the factor of s is i-^ where Jc is any positive integer,

and the arguments in a series are I^^r-s-^- Within the extent of Bohhn's tables all of the
coefficients can be written in symbolic form from B 188, XVII, XVIII. In the notation for
the coefficients the particular values of r and s are given, and there remains to be foimd only

the positive value of n, if there is one, for each multiple of -^ •

The following tables present, in skeleton form, any series of the given type. There are
properly two tables, one for perturbations in the plane of the orbit, and the other for perturba-
tions perpendicular to the same. The headings J and I are defined by

j=n-n'
i' = n + n'

Considering first the tables referring to the plane of the orbit, omitting for the moment

the arguments bearing the subscripts ±5 or ±a, the argument A for any term is read from a

Ice
main heading ± -^ and the first two columns under this heading. The tabulated numbers are

the respective factors of d, J, and I. The degree of the factors in the eccentricities is indicated
in the subscripts p-qin the symbol for the coefficient. Further, when j^ = 1

i n+i'n'=n(n-n')=ni.

Hence the coefficient of i is also the number n in the proper table of the numerical values of
the coefficients. For instance, in the function T^ (Z 41, eq. 82) we have for one term

F,.o(n-i.-n));sin (e + 40 + 4J)

where F, taken from Table VIII, is numerically

F,.„(n- 1. -Ti)„^= - 1514" +5780"w-8976"'U^.

Adding e - </> to the argument and taking the coefficients from Table IX, we have also in the

function T, G,.,{n-l.-n)r,^^T) sin {2e-4> + 4d + 'iJ)

where <?i.o(w- l.-«)«=4= +452"- 1475"w+ 1451' V.

In this manner the series is built up.

The coefficients having subscripts ±d and ±a belong to terms depending upon the mutual
inclination of the orbit planes. They differ from the preceding type of terms in three ways.
In the fiirst place the subscript signifies the addition of ± J and ±2 to the argimient, respec-
tively. Evidently, if ±i is added to the argument, the factor of J is not n but7i±l, from
which we determine n. Lastly, these terms contain the factor f, i. e., within the extent of
our tables the exponent t is not greater than unity.

For the tables referring to functions which concern the perturbations in the third coordi-
nate the same explanations hold, with the exception that the additional subscript ±7t' signifies
the addition of ±11' to the argument.

These tables, in connection with the proper tables of numerical coefficients, enable the
computer to write a complete series by inspection or segregate any term of given degree and
given argument.

No. 8.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

79

-9--S-

1 +

+ 1

,^^^

-f^

fi

h

•eS

o

(T)

^ '^

5

X s:

"o

a X

»

>-3 «0

2 +

a

^ f

CL(

1

4)

n

+

r

s

o

^

o

N

+

^

■^

t-

uao>

(OQO

«)

r-

iOO>

lOCT.

^N

1

1

•«

+

tN

--H

■n

»C

Mr*

-f (©

— iCOi

CltO'^'OO

eomr*

ClXiOiO

iCi

fcr^.

MI*

-HlOC»

^■OiOOJ

^tOC31

— CitOiO

1

^■^

1

'n

-

o

o

*

-

-

-

+

f-l

"7

'n

«

-HiO

Cl'^I'

CO t-

o -^ci to

^eoio

OOCOCO

X

CO

— o

-.U7

COf*

— eccot*

1

— cor*
1

— I* CO CO

1

1

N

■n

-

O

CO

O CI

-"

1

-^ CO

1

X

"^

*"*

1

+

<-»

7

"n

-

«

OM

— lo

CJO-ff"

— CO

Tf — —

<£>

-

«

«M

-ilo

^ — lO

-o

■ O-H —

(•)

+

"n

«

-*■ 00

*or-

«

CO

-J- 00

■voo

1

•-»

7

7

-1

o

C» — O

-

c

»

o

CJ CI CI

CI

0

+

^i

--

---

---

1

^

C)'«

MiO

c -^ X

— lOCO t-

CI-^O

— I*-*--!- CI too cot*irtO) -^oto-j-x coior-

CO lO 0 --0 CJ CO

CI « ->: CO ic CO »o

<t

-r

CM tC

<N<C

o -»•«

O -J- -»" OC

C T- K

ox-r-*- ci'-co CI (o too CJO<0--0 0 ci--=o

C) 0 0 -O CI to

ci to c n '-0 CI to

77

1

^-1

1

1

'^

o

CI

--

c

— . O-'J'— -co OC) —

CO 0

0 w —
1

<»

o

CI

c.

CI

CJ o-ro-«r o-r •?•

T C'

0 -I' 0

••

^i

7

---

---

cs

o -?■

.-.CO

ClO

e^j^tft

OCJ -T-

iCMci o-ifoo — .locor* ciOtpckc — coio

— eor^'fO'V

©•.fCO— CO — "

<to

w

O"^

©■«•

wo

CIMCO

CIMCO
1

fOC»CJ OTfOO OTf-Tf-oO OCO'-J-OC O-rX

0 -f X -r 0 -»•

O'ff'XO-rO^

=-1

7

77

77

■n

o

C)

-

O-l-

.-* CO

oc*

CO O M-C CO — iO O (NO** —CO

— 10 Cl CI

0 -^ — — —
1

■»

o

n

C)

O-r

O -I"

Ol-

-»■ O C) to C) W •© CJ CI CI <0 CI to

1

CI CO CI CI

0 to CJ CI C)

1

1 l_

+7

+7

S B
1 1

■S5

9 ■?

cJ 7c»

+ '. 1

++77

esse

'. '. '. I

+7+7

•SSS5

CJ M

+^ 1

C K C

1 1 1

C e a
V^

o o o

+ 11 + ^^^ ^+^ 1 +.^^ 1 1 ^^^
'. '. '. '. '. '. '. i '. 1 '. '. '. '.'.'. 1 1 i

^^^^ — — CO , ri 1" ='• — — — —— 1 Y 1

esse Kes cess eccec ess

r ?

+7'fT + T

b ^'b
1 + 1

XX\t\+X

c e K e R e e
1 1 1 1 1 1 i

M7M.7

e e e e s c

+77+777

R B e e R e K

80

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

'2
t~

0

-a

■3
»

a

03

2

1

"^1

7

•^

0

0 (N

'-

0

~

1 +

+ 1

.SS
1

*

'^

oH CO

!

n

CO

n

+

^i

7

"7

■^

■M^

0 "^ CI 0

w^nmifi

Pl«50 fXi

en -1 m T^ ta m t~~

0 CJ f CI f a

•O CI ct

-"^■'-'

«.

M>0

1

—.CO ror-

1

— lO -. 1-- a>

— — u^ — 0 0 Ol

ro — ir: — . .c 3v
1

0--

— 3> u- 0

7

7

1

^

■n

0

0 01

'^

« o*«j<

— — CO

0 0 c»

^

n 0

15

—

— «

1

CO

— f-tO

— -. L-i

- -0

^

10 ^

t-l

7

7

-1

OCl

;•) Or*.

OH— CO

<z>-v cgto

— CO CO — lO

0 CI 0 Cl f

CO 0

1-0 w c^

«

1

r--H^

— f-HlO

— CO COI--

I

7 '^

— ro — ro r-
1 1

n —
1

r^nn

=?

'<

7

^

^

0

0

1

M

c^

"

.0

>

c-1

+

k,

--

"7

-1

M !-•;

— L-! in r--.

cif-rto

cor- — ioo>

OfWtC>C)?Of«)

— coiOcoor-

00 CO CO

C'l 30 .-^ '-

PQ

+
s
1

+

*

(M-J

Of -*■ 3G

Of -*cc

C-l (O ci <c> 0

0 <M 0 « CI 15 « 0

I

Di OHO ci !0 0
1

CI X> CI C^I

1

c-i 0 -0 -.o

b-f

7

7

0

1

0)
(1h

—

1

1

"^

0

— M

0 0 CJ

"^

0

1

«

c-i

01

0 <-!"

0 0 -«<

f

0

I- 0

+

N

7

"7

1

^i^

CO — 10

0 CI c-1 f

— 10 cor-

c* 0 f 0 f M «

— CO — CO 0

f — —

O-.5C0M

■»

O"**

<NC»1«

1

Of fQO

OOf 0 ffGO

0 f 0 •»■ 30

f 00

000 'i- -s.

=-H

T

7

-l

-

-H CO

00 CI

CO — <o

0 0 ClO-1'

— —CO

<N —
1

-I--.-.

X

CI

0 -r

00-*

CI MO

i

(M 0«

CI (N
t

«(NN

;.^

+ 1

77

++77

s c s e
L '. '. '.

+7+7

K p e e

+zAi

e e s s
1 1 1 1

His

+. + 1 ^ 1
+++7JL7

a-- a a-ra
'.J'. '.il.

e e = s s K

+ +. . * . X^

+++ilA 1 1

ecKCKses
'. '. '. '. '. '. '. '.

+7+7+7+7
SSSSC.SSS

eo^Q— Z^^

++ 1 + 1 1
p s; s P ff C
1 1 1 1 M
K s- e c: C 5=

"k ^k

+++ 1

+XX+
a a a a
1. 1. 1. 1.

+7+7

a a a a

1 1 + 1

iiii

a a a s
1. 1. 1. 1.

+7+7

0 0 0 •»

000000

0090

No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 81

Our problem is now the integration of the partial diiTerential equations Z 7, eq. (33), Z 8,
eqs. (37) and (39), and Z 9, eq. (47').

In the trigonometric series to be integrated the argument is a function of d, e, (p, J, I.
The last two are constants. According to the principles of Hansen, ([/ occurs outside the opera-
tion. Numerically, however, it is ec^ual to e. The argument 6 contains s implicitly. See Z 9,
eq. (43). Hence we must, in general, write

F(b, 6)
and

di ds dd ds

In order to set up the partial differential equations from the total derivative, the following
notation is introduced:

F(£, d) = [F{e, 6)] + F{e, 6) - [F{t, 0)]

where [F{s, d)] signifies that part of the function which is independent of s. Again, since b has
the period of the planet, there can be no secular terms in s (with the exception of the function d),
i. e..

On the other hand, the argument d varies much more slowly, and there may be secular terms
in 0. Hence

D

m

^ Mo

and 0 may occur outside the sign of integration.

Owing to the presence of the required function in the differential equation, the integrations
must be performed rank by rank where rank is defined as follows:

In the course of the developments there arise negative powers of w. Since w is a small
quantity, these factors increase the numerical value of the terms, or, in other words, they lower
the order. Therefore, it is better to define order in terms of both the disturbing mass m' and w.
For this purpose v. Zeipel makes the assumption that both w and ■^jm' are quantities of the
first order. Order so defined is called " rank," and the word "order" is reserved as usual for

7/1 '"

the powers of m'. The factors — j- are arranged according to rank in Z 53.

Any function is then written in the form

F{,,d) = F,{,,d) + F,{e,d) + F,{,,e)+

where the subscript denotes the term of lowest rank, for F,- (=, 6) contains terms of more than
one rank since each coefficient is itself a Taylor's series in w. In assigning rank it is to be noted
that the coefficients in all the preceding tables contain the factor m' implicitly. The implicit
mass factor is indicated at the foot of each table which follows.

On the basis of the foregoing principles, the differential equation for W,

dWbW bW dd^
de bs bd de

expressed in Z 52, eq. (91), is broken up into four equations, Z 53, eqs. (95, — QS^), according
to rank, and before integration they are again subdivided according to parts which contain s
and parts which are independent of s. The total derivative is then in the form of eight equivalent
equations, and the integration can be performed in the following order:

W,; W,-[W,]; [FJ; W,-[W,]; etc.

It is possible to avoid the computation of T^, as v. Zeipel did, by the introduction of some
auxihary functions, but we found it preferable to tabulate them.

Employing Table XVa, and by inspection of Tables "VIII, IX, X, XI, T, is written directly.
( T has no terms of first rank.)
110379^—22 6

82

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

I Vol. XIV.

Table XVc.

Unit-l"

Sin

- t+i>

£+ 25+24
2£-^;'+2fl+2il

2e+ 4e+4J

3i-4>+4e+4J

e+<l)+4e+4J

2£+</.+6(9+6J

29+2J

c-4>+2e+2J

- £+^+2fl+2J

2e->/>
9

c+ iO+iJ
2f-i/'+4fl+4J

^+4e+4j

2t+ 29+2J

3t-(p+26+2J

e+^.+20+2J

2e+ 6fl+6i

3£-0+69+6J

e+<p+M+6J

2t+</>+ie+iJ

2t+<l>+Se+8J

29+ J
£-v''+29+ J

- !+(i>+2e+ J

£+ 4

2£-lJ+ J

<J,+ A

t+ 4e+3i

2£-s!'+4e+.3J
v'.+49+3J

2£+ 2fl+3i

3£-^&+29+3i

£+V'+2e+3i

2£+ 69+5J

3£-^«'+6(9+5J

£+9!r+6e+5J

2f+v^+40+5J

2£+^+8e+7il

- £+^

45+4J

- f+^+4(9+4J
£+ 2fl+2J

2£-^!r + 20 + 2.^

<p+2e-\-2ji

+

43.1
43.1

+ 271. 5

- 67. 1

- 294.9

+ 159.9

- 45.7

- 167.4

- 81.7

-1180
+ 273
+ 1496

- 173

- 211

+ 384

-1514
+ 452
+ 1679

- 6

- 83
+ 136

-1149
+ 360
+1227

- 102

+ 7.50

+ 318
+ 222

- 646

+ 130
+ 112

- 285

+2279

- 580
-2460

- 314

+ 127
+ 291

+ 1887

- 542
-1974

+ 390

-1263

+ 568

- 568

+6716
-2114
-7960

-;- 128
+ 535

- 978

- 128.0
+ 128.0

- 636.6
+ 108.2
+ 740.6

- .593.3
+ 122. 1
+ 637.4

+ 418.9

+2962

- 179

-4265

+ 512
+ 899
-1410

+5780
-1475
-6656

+ 408
+ 262

- 878

+5902
-1734
-6415

+ 112

-4900

-1081
-1012

+2452

- 484

- .565
+ 1211

-7160
+1410
+8138

+ 702

- 399

- 537

-8417
+2221
+9002

-1556

+7397

- 3106
+ 3106

- 26627
+ 6488
+ 33462

- 3166

- 2505
+ 7431

+

171
171

+
+

526

32

734

+

869
174
984

-

907

- 2935

- 1170
+ 5572

+

684
1921
2605

- 8976
+ 1451
+ 11172

+

1307

564

2285

-12820
+ 3301
+14400

+ 1552
+ 2227

- 4296

+ 808
+ 1393

- 2475

+ 8896

- 520
-11342

- 90
+ 598

- 478

+ 15550

- 3377
-17350

+ 44700

23105

No. 8.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

Table XVc — Conlinued.

83

■i\

Unlt-l"

Sin

UJ«

w

UI«

>»'

«+ 6(?+6J
2j-^J+60+6i

+ 7969

- 2624

- 8819

- 41736
+ 12577
+ 47347

-111337

V'

-^+2fl+2J
-2e+^J+2fl+24

+ 2246

- 396

- 3596

- 6168

- 1494
+ 12561

+ 9351

■»'

2c

+ 423
+ 357
- 780

- 1797

- 2207
+ 4005

1'

2f+ 49+4i

3£-v!r+4e+4J

£+vJ+4e+4J

- 1783
+ 924
+ 1220

+ 3946

- 3327

- 1026

v'

2£+ 8e+8J
£+vf'+8fl+8J

+ 6749

- 2247

- 7252

- 44127
+ 14052
+ 48051

vY

£-iJ+ 4

- 285

- 1004
+ 1574

+ 1210
+ 5771
- 8192

- 2475

vY

49+3i

e-0+49+3J

- e+vJ+4fl+34

-17218
+ 4253
+20345

+ 56961

- 8340

- 73031

- 79400

vv'

«+ 2(9+ i

2£-^+2(?+ A

il>+2e+ A

- 1429

- 523
+ 2280

+ 6138
+ 3792
- 11302

+ 28347

vv'

£+ 29+34

2£-v(i+2fl+3i

^+29+3i

+ 1725

- 1003

- 1492

- 3054
+ 3753
+ 677

+ 13097

riv'

e+ 6e+5J

2£-^+6/?+5J

vi+6ff+54

-23773
+ 7038
+25974

+108605
- 28427
-122380

+251019

■^v

- e+ 29+ J

-0+29+ 4

-2£+v:.+29+ i

- 965

- 2068
+ 3785

+ 3533
+ 10582
- 16928

+ 39011

vv'

2£+ A

3£-0+ i

- 820

- 470
+ 1488

+ 3797
+ 3185
- 7870

VY

2£+ 49+34

34-^5+49+34
€+^(•+49+34

+ 1815

- 1181

- 853

- 1190

+ 3807

- 3161

IV'

2f+ 49+54

3£-^5+49+54
£+^5+49+54

+ 4294

- 1571

- 4414

- 17092
+ 6629
+ 17198

nY

2£+ 89+74

3e_^+89+74

£+v!'+89+74

-21544
+ 6700
+22868

+126397
- 37167
-136294

n"

£-Vi

- £+s!.

+ 866
- 866

- 4261
+ 4261

v"

49+24

£-^5+49+24

- £+v5+49+24

+ 10682
- 1815
-12428

- 28347
+ 474
+ 37322

+ 32120

r

£+ 29+24

2£-^5+29+24

^•+29+24

- 1498
+ 1136

+ 861

+ 450
- 4394
+ 3794

- 22127

m'

84

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

XVc—

Continued.

Unit=l"

Sin

Ujo

1

XV

K-»

V'

£+ 6fl+4J

2e-^J+6fl+4J

v'/+69+4J

+ 17790
- 4675
-19046

- 69.344
+ 15200
+ 77260

-13.59.54

n"

+ 1634
- 1634

- 7081
+ 7081

+ 16199

1

2£+ 2J

3£-sI'+ 2J

£+(S+ 2i

+ 328
+ 154
- 591

- 1710

- 1141
+ 3420

ri''

2£+ 4fl+4J
3£-v!'+49+4J

- 5879
+ 2032
+ 5807

+ 19019

- 7361

- 17D9S

^"

2£+ so+ej
3£-</'+85+6J

+17340
- 5018
-18102

- r.')064
+ 24266
+ 95320

P

- 866
+ 866

+ 4260
- 4260

f

49+3J-J

£-^H-4e+3J-i'

- €+4>+ie+3A-I

+ 609
+ 232
- 1044

- 2958

- 1656
+ 5600

+ 6763

■ 3'

1

1

£+ 2ff+2J

2£-^f'+20+2J

i;r+29+2J

- 1760

- 331
+ 2677

+ 7189
+ 3096
- 12681

+ 30930

P

£+ 60+5J-J

+ 578
+ 10
- 780

- 3543

- 299
+ 5023

- 15302

P

-s!'+2fl+J-J

+ 866
- 866

- 4260
+ 4260

+ 10988

P

2£+ J+i'

+ 1152
+ 98
- 1634

- 4231

- 1440
+ 7081

f

2£+ 4y+4z(

3£-^+4tf4-4J

£+V?.+4»+4J

- 1795
+ 164
+ 2229

+ 9459

17

- 12595

P

2£+ 80+7J-i"

3£-(/'+8(?+7J-i'

£+</'+89+7J-i'

+ 392

- 40

- 482

- 2914
+ 194
+ 3G91

1

|£-0+ 0+ 'i

+ 47.1
+ 27.5
- 90.4

- 149. 3

- 111.4
+ 310. 5

+ 185
+ 207
- 455

;

!£+ 3(?+3J

^'-^Ir + SO + S^

i£+v''+3fl+3J

+ 216. 1

- 58.9

- 229. 3

- 655. 2
+ 150. 7
+ 722. 8

+ 749

- 93

- 905

^£-^+5fl+5i
fr + ^ + Sfl+SJ

+ 113.8

- 33.5

- 118. 2

- 499. 2
+ 137. 7
+ 527. 9

+ 892

- 213

- 977

'

i£+ 79+7J
?£-v''+75+7J
4£+v''+70+7J

+ 54.1

- 16.5

- 55.7

- 310. 2
+ 91.3
+ 322. 3

+ 757

- 209

- 801

?£+ 9fl+9J
5£4-v''+99+9J

+ 24.5

7.6

- 25. 1

- 173. 5
+ 52. 7
+ 178. 5

4- 537

- 157

- 559

v/

No. 3.]

MNOR PLANETS— LEUSCHNER, GLANCY. LEVY.

85

XVc — Continued.

Cnit=l'

Sin

w°

w

w«

V

i.'+ 30+3J

- 1497
+ 419
+1729

+ 4732

- 966

- 5826

- 59C3
+ 131
+ 8424

V

-f.+v'.+ 0+ J

- 114

- 224
+ 43G

+ 385
+ 1006
- 1720

- 548

- 2186
+ 3220

V

- 208

- 55
+ 314

+ 781
+ 349
- 1316

- 1315

- 1020
+ 2041

V

'.£+ 50+5J

-13GG
+ 420
+1480

+ G114

- 1711

- 6793

-11303

+ 2548
+ 13254

^

ii+ 35+3J
|£+(^+30+3J

+ 108

- 85

- 19

- 20
+ 256

- 329

- 847

— 3'j5
+ lo'-O

V

ie+ 70+7J

- 922
+ 292
+ 975

+ 5348

- 1618

- 5728

-13320
+ 3092
+14002

V

5f4- 59 f 5J

?£-iJ+53+5J

. 4r+;H5fl+.5J

+ 172

- 74

- 133

- 594
+ 298
+ 402

+ 491
- 470

<o

V

- 541

+ 174
+ 564

+ 3856

- 1205

- 4055

-120!::2
+ 3C0S
+12887

i

if +30+2J
f£-si+39+2J

+2041
- 431
-2290

- 5080
+ 499
+ 6274

+ 4928

- 1020

- 7597

n'

+ 384
- 384

- 1410
+ 1410

+ 2G05
- 2G05

i

{s + 0+2J

|£-Si+ 0 + 2J

- 131
+ lOG
+ 69

+ 12
- 3C6
+ 350

+ 717
+ 772
- 1728

r,'

If +50+4J
if -^+50+4 J
if+vi+50+4J

+2169
- 596
-2295

- 8241
+ 1980
+ 9008

+12080
- 2086
-14S23

'/'

|f +3fl+4J
^c-+^J+3<'+4J

- 389
+ 135
+ 383

+ 1251

- 479

- 1189

- 1212
+ 687
+ 930

r/

if +7/?+GJ
5f-sJ+7fl+6J
ff+(/.+79+Gi

+1550
- 457
-1609

- 7940
+ 2211
+ 8376

+ 17170
- 4245
-18050

r,'

Jf +5fl+6i
|f-v!'+5tf+GJ

- 349
+ 113
+ 352

+ 1665

- 543

- 1G78

- 3127
+ 1052
+ 3117

i

if +90+8J
ff-(^+90+8J
|f+sJ+9<'+8J

+ 937

- 286

- 963

- 6044
+ 1784
+ 6274

+16950
- 4724
-178S0

1'

if + 0+ J

|f-v''+ 0+ ^

-if+v''+ 0+ J

+ 757
+ 514
-1583

- 3272

- 3044
+ 8214

n'

is +bd+bJ

i'.-^+bO+bJ

-4f+^+50+5J

+7767
-2522
-8820

-35692
+ 10085
+42033

m'

86

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

IVol. XIV.

Table XVc— Continued.
To

rnit = l"

Sin

■e

■n r

')')'

vY

V V

n V

w

vr

vv

5 9'

-is +39+34
if-^!'+3fl+3J

?£ +30+3J
5£-v''+3fl+3i
^£+v''+39+3J

|£ +70+74
it-4'+7e+7A
is + 4+70+7J

-Is + B+ J
-h—l'+ S+ 4
-|s+v'.+ 0+ J

\e +9fl+9J
^E-v''+9fl+94
|e+v''+90+9J

is + 0

is + 0+24

|s-v^+ e+2j

-is + ^!r+ 0+24

is +50+44

|s-v''+50+44

-is+i!.+50+44

-is +30+24

is-9'.+30+24

-|s+9!'+30+24

4s +30+24
^£-9!r+30+24
i£+v'.+30+24

|s +30+44
4s-;J+30+44
is+iJ+30+44

|s +70+64
4S-VO+70+64
is+v!.+70+64

-!s + 0
-is-0+ 0
-4s+v'.+ 0

4s +90+84
|s-v!'+90+84
fs+v!'+90+84

is + e+ A

ic-4,+ 0+ 4

-i£+v!'+ «+ ^

is +50+34

4s-i!'+50+34

-is+v''+50+34

-is +30+ 4

is-0+3O+ 4

-|s+v!'+30+ 4

?£ - 0+ A
\s-4- 0+ 4
i^+v'-

+ 4758

- 1369

- 6128

+

+

882
732
177

0+ 4

+ 7549

- 2504

- 8212

- 32

+ 784

- 1031

+ 5780

- 1929

- 6154

- 768

- 1156
+ 1924

+ 209

- 771
+ 446

-218G9
+ 6125
+24564

-10182
+ 1576
+13528

+ 384

- 939
+ 715

+ 3250

- 1333

- 3256

-23414
+ 7146
+25145

+ 768

- 2308
+ 1540

-18847
+ 5935
+19837

+ 761
+ 906

- 1920

+15303

- 3577
-16828

+ 1582
+ 1300

- 3410

+ 451
+ 494

- 1096

- 15945
+ 2307
+ 22836

- 280

- 2580
+ 3816

- 44427
+ 13864
+ 49192

+ 220

- 4194
+ 5051

- 41583
+ 13412
+ 44735

+ 2821
+ 6406

- 9227

+ 1404
+ 3774

- 5879

+ 85960

- 19358
-101260

+ 27638
+ 2276

- 43309

+ 3626
+ 3382

- 8550

- 10017
+ 4996
+ 9153

+122108

- 34410
-133985

- 2821
+ 10637

- 7816

+122928

- 37138
-130949

- 3333

- 5387
+ 9831

- 49954
+ 7957
+ 58649

- 5765

- 6572
+ 14260

- 1890

- 2861.
+ 5381

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

87

Table XVc — Continued.

Unlt-l"

sin

w°

w

w'

Y'

fs +30+3J

- 3918

+ 87G0

4£-(J+30+3J

+ 1588

- 5289

4£+v!iH-39+3J

4- 3637

- 6391

i"

\t ■\-^JQ^-hi

+ 18292

- 83098

^f-v!i+7»+5i

- 5104

4 20825

i£+^!.+7e+5J

- 19286

4 89973

i'

i.- + <?+ 4

- 902

+ 3781

- 988

4 5721

4- 2191

- 10762

;■"

if +59+4J-i"

+ 634

- 3482

f£-v!i+5(?+4J-i"

+ 87

836

-i£+v''+50+4J-i-

- 933

+ 5479

j'

-is +39+2J-i-

+ 428

- 1816

i£-^;.+3fl+2J-j

+ 480

- 2805

-5£+^+39+2J-i"

- 1050

4 5226

P

ft +39+3J

- 1916

+ 8929

^£-^+30+3J

+ 2

4- 1220

i£+0+3O+3J

+ 2553

- 13126

P

55 +76+6^-^

4- 488

- 3307

f£-^;'+7(/+6J-i"

27

4- 23

i£+v:;+7e+6J-j

- 623

4- 4387

3'

-\, + 5 -i-

- 475

+ 1965

-i-'-»''+ 6 -i"

+ 1141

- 5536

-|£+^+ » -i"

- 508

+ 2916

?

., + fl+2^+i'

+ 1282

- 5447

7j_^+ e+oj+v

90

384

ij+^!>+ &+2J+J

- 1620

+ 7647

?

|£ +5fl+5J

- 1544

+ 9111

5E-^J+59+5i

+ 222

735

i£+^!'+5e+5J

+ 1838

- 11413

?

4£ +9(?+8J-i'

+ 304

- 2460

is-vi+9fl+8J-J

42

4 266

j£+V'+9i9+8J-i'

- 364

+ 3013

i'

2e+2J

- 1955

4- 14862

6fl+6il

- 35276

+ 189348

^

+ 3312

- 23724

^J+49+44

- 5097

- 4328

-i+49+4i

+ 6177

- 16310

^+8fl+8J

+ 45199

- 304998

,','

2fl+ J

+ 6733

- 33547

2e+3J

- 3730

4- 1693

69+5i

+142854

- 673242

•^ + i

- 9270

4- 61512

-'!> + J

4- 4207

- 28940

<l>+ie+zj

+ 5323

+ 55061

-4,+4d+ZJ

- 13730

+ 9080

J,+49+5J

4- 22898

- 84425

•;>+se+74

-200024

4-1218446

1,,"

2d

- 3268

+ 14164

2e+2J

4- 3445

+ 15177

6e+4J

-190467

+ 772593

'p

4- 12782

- 78712

<!' +2J

+ 5239

- 35125

>l>+i0+2J

4- 2712

- 60586

-^+40+2J

4- 4409

+ 41693

!ii+4g+4j

- 52183

+ 143461

i/,+80+U

4-294332

-1600036

m'

88

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XVc — Continued.

Unit=l"

Sin

w

w

u'

i'

2e+ J

+ 3479

- 17883

ee+3J

+ 83314

- 283500

4> + ^

- 7839

-1- 47423

-^+46+ J

+ 6634

- 36904

0+4e+3i

+ 27512

- 44330

v!l+8S+5i

-144023

+ 688658

fv

25+2i

+ 10709

- 50725

26+ J-J

- 1732

+ 8521

6e+5J-I

- 7799

+ 50227

<!>

- 12782

+ 78712

<P + J+i'

+ 11006

- G0629

ip+ie+sj-i

+ 4022

- 29208

-4:+4e+3J~2

- 3616

+ 27235

<p+4e+4J

- 28408

+ 176052

4I+S0+7J-2

+ 9526

- 75678

P V

26+ J

- 7475

+ 36068

26+2J-I

+ 159

- 2967

ed+4j-i

+ 11564

- 66719

4, + J

+ 11762

- 75153

4' +^

- 6024

+ 33182

-•P+46+2J-I

+ 7090

- 45771

4'+46+ZJ

+ 35006

- 199168

<li+46+4J-I

+ 1108

281

ip+se+ej-i

- 15308

+ 111481

m'

An inspection of the preceding table, which is t}"pical of all the trigonometric series under
consideration, shows readily that any function of this type is of the form

II-' sin K' + Iksin {K±iP)==IJc' sin K' + lie sin K cos <!; + II' cos K sin (J)
or

II-' cos K' + Ilc cos iE±tl>)=Il-' cos K' + Ilc cos 5: cos </>+ II- sin Ksin ^

or, more briefly, a + & cos ^ + c sin 4>

wherea, 6, c are trigonometric series and can be ^vl■itten by inspection from the tabulated function.
Hence, in v. Zeipel's notation (Z 54, eq. 96),

Ti = Xf+Yi cos 4> + Zi sin (p
and the integral may be written

Wi==Xi+yi cos ^ + Zi sin ([>

The functions T and W are to be used in this form in solving equations (95).
Considering only first order in the mass in T

T, = X^ + Y^ cos ^ + Z2 sin ^
X.^^IJc' sin K'; Y^ = I1c sin E; Z,= ±I]c cos E
or, Xj is the part of Tj which is independent of ^, Y^ is a trigonometric sine series having the
same numerical coefHcients as the part of T^ which contains (p in the argument, but in which
(p is omitted from the argtmaent, and Z, is the corresponding cosine series.

Considering the first two of the eqs. (95), the first one states that IFjisnot a function of £

a^o"''' 01"' W, - [ FJ = 0 ; If, = [ FJ.

Making use of this fact in the second, F, can be obtained from (dB^). (See Z 54.) Introducing
the auxiliary functions ^-j and w,, defined by (99) and (101), the difTerential equation for F, is
replaced by the ec^uivalent differential equations, (100) and (102), for ^, and u^.
The series r yj — 71 [ YJ

a"^' [Y,]cos,p + [ZJsin4>

can be written by inspection from T^, or, better, the integration itself can be performed in part
at the same time.

where

No. 3.] MINOR PLANETS— LEUSCHNER, GLiVNCY, LEVY. 89

The f miction <^i is given by Z 59, eq. (103), or,

From the table of T",, page 82, it is not difficult to Avrite immediately

\[X,]-r,[Y,])dO

3
T

r

The terms of higher order must be obtaincil by the usual method for the mechanical multi-
plication of series. A logarithmic multiplication is the most direct.

In each term in the expression for 0i the terms of lowest rank must be of the first rank.

Recalling the tabulation of factors in Z 53, w, — . — j, — ^, etc., are all of first rank. But the

coefficient for a given argument consists of three terms in ascending powers of w. Hence
(^, — w, within the limits of the given tabulation for T^, is of rank 1, 2, 3 for each order in the
mass. Table X\T:, giving cpi — w, is tabulated with double headings. The three subheadings
indicate the expansion of the coefficients in a Taj^lor's series and the main headings give the
factors in the development of the radical in Z 59, eq. (103).

Having found <^i, its reciprocal, 4>i~\ inclusive of first order in the mass, is given by

'l>r-' = i-^f ([X,]-ri[Y,])dO

The second term is the negative of the first three columns of Table XYI multiplied by itr^.

The product of 2<l>~'^ and that part of T^ which contains cp gives -^, and integration with respect

to 6 gives «!, tabulated in XVIII. The function Uj is of first and higher rank because the factor
<^,~' is of rank minus one and T^ is of second rank.

From Table XM^II j/i can be read by inspection, and j^i/j added to Table X\T gives 2,,
tabulated in Table X\ai. The function IF, is the sum of Tables XVII and XVIII.

In the integration those terms whose arguments are independent of d are of the nature of
constants. In accordance with the condition that there may be secular terms in 6, the integral
contains such terms as the following:

O-Jc sin ((P + J).
As the constant of integration

do-ksin {(f' + J)

is added. Hence the integral contains terms such as

(O-Oo) t sin ((l> + J,

where ^0 is the value of 0 for the time t = 0.

In passing, it should be noted that, in order that the expansion of Z 59, eq. (103), shall
represent the function, wo must have

{[X,]-r,[Y,])dO

< 1

and this condition should be tested for a given planet before applying this method of determining
the perturbations.

To the computer the extent of auxiliary tables, the arrangement of series in logarithms or
natural numbers, in seconds of arc or radians, inclusive or exclusive of numerical factors, and
foresight in combining operations — all these are of the greatest importance. But considerations
of this kind would carry the reader into complicated details which are best left to the com-
puter's own judgment.

On the other hand, general considerations about the extent of the pubhshed tables are of
importance in the discussion of the accuracy of the final tables. Yet, for a given limit of
accuracy, it is so difficult to determine, for each tabic, the highest powers of vi', w, t), r/, and f
that little or nothing is said about it in connection with individual tables, but the discussion
is reserved imtil later.

90

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

tVol. XIV.

Table XVI.

<^, — Jl'=I,— lj!/l = [(l— « C08 £)ir,]

Unlt—4th decimal of a radian.

Ui-l

w->

uu

w>

w

w«

W

w

w'

U)«

w

1,^

-0. 0460

+0. 231

-0.52

%"

-0. 0060

+0. 040

-0. 127

w

d

+0. 0331

-0. 195

+0.53

1?

20+2J

+ 42.889

- 107. 72

+ 106.7

,'

20+ i

- 15. 427

+ 52. 39

- 75.2

f

49+4/J

- 122.10

+ 484. 1

- 813

-0. 0460

+0. 231

-0.52

rir,'

40+34

+ 357.75

- 1183.5

+1650

+0. 0331

-0. 195

+0.53

i'

49+2J

- 258.93

+ 687. 2

- 779

-0. 0060

+0. 040

-0. 127

j'

4fl4-3J-i'

- 14. 75

+ 71.7

- 164

f

20+2J

+ 28.2

- 433

+0. 262

-1.70

+0. 0003

-0. 0022

f

60+6J

+ 428

- 2295

+0. 262

-1.70

+0. 0001

-0. 0008

iw

20+ A

- 316.1

+ 1592

-0. 767

+4.46

-0. 00021

+0. 0018

vW

29+3J

+ 108.5

- 49

-0. 094

+0.69

-0.00011

+0. 0009

W

6e+5J

-1889

+ 8902

-0.86

+5.2

-0. 0001

+0. 001

vv"

26

+ 237. 6

- 1030

+0. 555

-2.87

+0. 00004

-0. 0002

v¥'

2d +2 A

- 125.3

- 552

+0. 276

-1.85

+0. 00008

-0. 0009

,^

65 +4 J

+2770

-11237

+0.83

-4.7

,"

25+ i

- 168.7

+ 867

-0. 200

+1.21

i"

65+3J

-1346

+ 4581

-0. 200

+1.21

fv

29+2J

- 389.4

+ 1846

-4498

fi

25+ J-J

+ 126.0

- 620

+0. 032

-0.23

fi

65+5J-J

+ 113

- 731

+0. 032

-0.23

P Y

25+ A

+ 362.4

- 1749

f V

20+2A-I

- 7.7

+ 144

-0. Oil

+0.09

P ,'

65+44-2

- 187

+ 1078

-0. Oil

+0.1

m'

m'^

m"

Table XVII.

Xi

Unit=l".

W-'

ly— 3

to'

w

w«

tfO

w

w'

71

25+24

+ 1179.6

- 2963

+ 2935

,'

25+ A

- 318. 2

+ 1081

- 1552

;:

- 0.95

+ 4.8

45+44

- 3358

+ 13313

- 22356

- 1.27

+ 6.4

?

4

+ 0.68

- 4,0

45+34

+ 8609

- 28481

+ 39702

+ 0.79

-4.7

v"

- 0.12

+ 0.8

^

45+24

- 5341

+ 14175

- 16063

- 0.12

+ 0.8

j'

45+34 -!■

- 304

+ 1479

- 3383

f

25+24

+ 1955

- 14861

+ 7.2

- 46.6

v'

65+64

+11758

- 63112

+ 7.2

- 46.6

w

25+ A

- 6732

+ 33547

-15.2

+ 88.0

iw

25+34

+ 3730

- 1691

- 3.8

+ 27.9

iw

65+54

-47616

+224423

-21.7

+130. 0

1^"

25

+ 3267

- 14165

+ 7.4

- 37.8

r,r,"

25+24

- 3446

- 15176

+ 7.8

- 52.5

vY'

65+44

+63489

-257533

+19,0

-108.1

j'v

25+ A-I

+ 1733

- 8522

+ 0.4

- 3.1

A

60+'=,J-I

+ 2599

- 16744

+ 0.7

- 5,2

fr,

25+2J

-10709

+ 50748

-123705

\"

25+ J

- 3479

+ 17880

- 4.1

+ 24.9

,'3

65+34

-27772

+ 94500

- 4.1

+ 24.9

P Y

25+2J-J

- 159

+ 2966

- 0.2

+ 1.9

? ,'

65+44 -J

- 3855

+ 22240

- 0.2

+ 1.9

}' Y

25+ J
(5-5„)sin

+ 7475

- 36070

V Y

4

- 570

+ 2421

- 4950

- 0.45

+ 2,7

-7.2

m'

m'^

No. 8.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

91

Table XVIII.
Hi=y, COS (p+Zi sin (p

Unit-l"

Ul-l

MT-J

Cos

U'«

w

w'

u,«

w

w«

',-

vJ+49+3J

+ 294. 89
- 831). 5

+ 1229.8

- 740. 6
+ 3328

- 4069

+ 734
- 5586
+ 5671

- 0. 316
+ 0. 114

+ 1.59
- 0.67

- 3.6

+ 1.8

i

^+2fl+2J

+ 390
+ 978
+ 2940

+ 1494

- 7431

- 15782

- 9351
+ 23105
[+ 37112]

- 2.62
+ 4.42
+ 1.80

+ 16.8

- 28.4

- 11.7

V,'

-^^.+20+ J
v!'+2d+3-/

+ 2068
+ 1492

- 2280

- 8(i58

- 10582

- 677
+ 11302
+ 40793

- 39010

- 13098

- 28348

- 83730

+ 6.18

- 1.91

- 5.57

- 3.95

- 36.9
+ 13.6
+ 32.8
+ 23.6

p

-^i+2e
^!'+2(?+2J

- 1634

- 861
+ 6349

+ 7081

- 3794

- 25753

- 16199
+ 22127
+ 45318

- 4.04
+ 2.12
+ 1.90

+ 21.4

- 14.4

- 10.8

1

^5+2e+2J

- 866
+ 260

- 2677

+ 4260
- 1674
+ 12681

- 10988
+ 5101

- 30930

- 0.22
+ 0.07

+ 1.6
- 0.5

■I

^i+40+4J
-v!.+40+4J

+ 2549
- 3089
-1J300

+ 2164
+ 8155
+ 76250

-11. 9]
+ 3.9]
- 8.9

+ 89
- 25
+ 70

vi+45+5J

v!^+40+3J

-v!'+4e+3J

vi+80+7J

-11449
- 2661
+ 6865
+50005

+ 42212

- 27530

- 4540
-304611

+ 1.9
[+36. 4]

-20.3
[+33. 8]

- 23

[-241
+118
-248

v!'+4(?+4i

v!'+40+2J

-v!'+4tf+2J

0+8fl+6J

+26091

- 1356

- 2204
-73583

- 71730
+ 30293

- 20846
+400009

-10.1
[-25.5
+28.0
-41.9

+ 83
+ 153
-153'

+284

P

v!'+4fl+3J
-0+4<?+ J

-13756
- 3317
+36006

+ 22165
+ 18452
-172164

+10.1
-12.4
+16.6

- 65
+ 64
-104

v!'+4«+3J-i'
-v!'+4(?+3J-i'

vJ+4e+4J

- 2011
+ 1808

- 2381
+14204

+ 14604

- 13617
+ 18919

- 88026

- 1.9
+ 5.7

+ 14
[- 14]

+ 10
[- 42]

vJ+40+4J-J

-Vi+4(?+2J-i'
vi+80+6J-J
<;'+48+3J

(fl-fl„) sin

- 554

- 3545
+ 3827
-17503

+ 140
+ 22886
- 27870
+ 99584

+ 0.5

- 1.8

+ 1.3

[- 3. 7]

- 4
+ 14

- 11
[+ 28]

"J

<i'

+ 767. 72

- 2820. 9

+ 5210

+ 1.265

- 6.35

+14.3

'j'

4> + J

- 569. 95

+ 2421.1

- 4950

- 0.455

+ 2.69

- 7.2

i'

V^

+ 6624

- 47448

+23.8

[-221. 9]

v^ + J

[-185401

+ 8414

[+123024]
- 57880

-73.4
+36.0

+572. 4
-282. 2

^^ +2J

+10478
+25564

- 70250
-157424

+55.2
+87.3

-374. 8
-652. 8

,"

<!> + J

-15678

+ 94846

-69.9

+438. 6

+22012
-25564

-121268

+ 157424

+359162
[-511232]

+ 9.9
-23.1

- 77.0
+ 165.0

-12048
+23524

+ 76364
-150306

-251640
+498328

- 5.2
+ 14.8

+ 45.8
-112.0

m'

771'^

92 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv

After the determination of Wj, the function W, — [TF2] is obtained from the solution of
Z 53, eq. (95j). The integral may be written as in Z 63, eqs. (10.5), (106), or, quite as simply,
as follows:

F, - [ Fj = F/ -f r ( r, - [ r,])c7£

K'= - rj|(l-fi cos b){w+ TT;)-i[(l-c COS e){w+ Fj]W> de

The function F' is given in Table XIX.

Anticipating some later developments, for which we shall need

[(l-gcosc)lF!
the function _^

[(l-ecos£)ir/]
is tabulated in Table XX.

The determination of [ IF,] °^ay be accomplished according to Z 65, eq. (108) —Z 67, eq. (116),.
or in the manner outhned below, which we regard as preferable.

Repeating Z 65, eq. (107),

<^/^^ + fe-W. + W4j = j^<^.+|[(T^t-| E,)(W,+|H,)(l-e cos^)]

— (1 — e cos e)\{w

^+W^)^(,jiT,-[T,])de + ^j(T,-[TJ)dJ^ + 2[T,]

in which all the known parts are contained on the right-hand side, the development of equivalent
equations proceeds in a manner analogous to that for W^.
Writing

T3 = X3 + Y3 cos (/i + Z3 sin (p

and introducing

<^2 = [^2]-'?[y=]+w'

and equating parts independent of tp, coefficients of cos ^ and coefficients of sin (J), the three-
equivalent equations arc:

d[x.] dx^ ^ fZ.r, , sr,, V ™ 1 „ V w , 1 - M'^^ii

-[(1 -e cos e){w+ W\)^gj(X,-[X^)de'j-^il-e cos e) f (T,-[T,])d^ll^ + 2lX,]

^d[y.] , ^dv, ^(^Vi^^Vf, Yir 1 - V fif ^ 1 « M^-Vi

*'^+'^^rf^ = ^"^'W + 4L^^-^'^°^^\^'-3-A^^' + 9"0J^

-[(1-e cos e){w+ W\)^J{Y,-[Y,])de'j-^{l-e cos e)J {T,-[T,])dsJ^ +2[Y^

-[(1-e cos e) {w+ ]]\)^,j{Z,-[Z,])de'j-^{l-e cos e) j {T,~[T.^)de'^ + -2[Z,]^
Multiplying the second of these by tj and subtracting from the lust:

-[(1 -c cos o(w+ iTo^|J{A',-,r,-[X3-,rj}j£]
-[(i-e cos £)J(7;-[7;])(?£]'^+L-[A'3-,r3]-

No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 93

Midtiplying the second by cos 4', the third by sin (/- and adding:
0,^(«,-M>u.) =-</., + |[(l-e cos b)(^W,-\e^W, + \e^J]^ + 2{[Y,] cos ^ + [Z,] sin ^0

— (1 — « cos b){w+ l^i)^ I { ^'j cos 4' + Z.i sin (,'' — [I'j cos ^'' + ^2 sin ipDde

-[(1 -f cos e)j{T,-[T,])chJ^

in which

■"2 = [2/3] cos ^ + [2,] sin (;{'

and [A'J, [Y^], [Z^] are read by inspection from T,, which is to be determined as follows:
If Z 50, eqs. (89), (90), are written in the form

[1 — cos (/-co)] = 4 1 — 27J cos £ — cos (s — ^) +)} cos (2s — (/-) +ri cos (/-+

cos^ <p a^ cos^ (p

^ T + 3"^^ 2 [l-co3(/-c.i)]| = 3 + 14 f-8Ticoss + 2f cos 2 s-2 cos (s-^'')

cos-" (p[ a- a' cos' 9>

— 8 ]j^ cos (s — ^) + 2?) cos (2s — 4') + 2rj cos ^ +

then TgF and T',, given by Z 49, eqs. (84), (85), in connection with Z 50, eq. (87), are given by

T-^= — Tj — 4{1— 2?) cos £ — cos (s — 0) + tj cos (2s — tl') + ij cos 9!- + }

ISp. q{n + r. -n + s)rjPTj'^j-' sin ^

T', = {3 + 14ij' - 8)j cos £ + 2i}' cos 2s - 2 cos (s - ^) - Stj^ cos (s - ^'0 + 2)} cos (2s - ^) + 2jj cos ^

+ }ISj,. ,{n + r.-n + s)7jPri'9i-( sinA-^H-iv)^

and r, (Table XVIIIa) is computed by Z 53, eq. (94), in which

E = x + 2riii, Et = x, + 2riyt
The function

<t>2-'u<' = [X:,]-Tj[y2]

is tabulated in Table XXI; the function

Wj = [.V2] cos 4' + [^2] sin (p
is tabulated in Table XXII.

From the latter [j/j] can be read by inspection, and 7j[;/j] added to the former gives [zj].

Finally, (Table XXIIa),

f TT,] = [Xj] + [:/,] cos 4' + [22] sin 4

94

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XVIIIa.

T,.

Dnit=l"

sin

?('-*

ti-i

tc«

to

TO»

U'O

w

v>'

-e+'p

+0. 339

- 2.01

e +20+2J

2c-il>+20+2J

4,+20+2J

-0. 375
-0. 137
+0. 498

+ 2. 403
+ 0. 847
- 3.223

+ 7.72

2$ +4e+4J

-0. 438

+0. 429

+ 2. 234
- 2.338

2£+v''+C0+6J

+0. 361

- 2.372

V

20+2J

c-^+20+2J

- e+^+2e+2J

-0. 00047

+0. 0036

-0. 0123

+2. 199
+0. 286
-3. 294

-14.58
- 3.85
+23. 82

+ 33. 34

n

-0. 00038

+0. 0035

-0. 0136

-2. 811
-0. 688

+12. 20
+ 1.67

- 16.51

V

n

£ +4(9+4J
v''+49+4J

-0. 00015

+0. 0013

-0. 0048

+0. 432
-4. 536

- 2.58
+35. 80

- 95.79

V

£+^'.+2(?+2J

+ 1.017

- 6.333

n

£+v'.+60+6J

-3. 219

+22. 43

I

26+ J
£-v'.+20+ J

- £+v''+2e+ J

+0. 00017

-0. 0014

+0. 0055

-2. 520
-1.253
+4. 372

+14. 78
+ 10. 20
-28. 56

- 31. 95

?

£ + ^

+0. 00014

-0. 0014

+0. 0060

-0. 404

+ 1.188

+ 4.06
-11. 30

+ 34.07

?

+0. 00005

-0. 0005

+0. 0021

-0. 224
+6. 480

+ 1.53
-47. 37

+120. 37

?'

t+<l>+20+3J

+0. 214

- 1.66

n'

(B-do) COB

+5. 977

-36. 82

20+2J

i-<{>+2e+2J

- e+<;>+20+2J

-0. 00188

+0. 0143

-0. 0489

-1.141
+0. 235
+1.12

+ 7.14

- 1.12

- 7.39

- 20.54

9

4'

-0. 00059

+0. 0051

-0. 0189

-0. 357

+ 2.62

- 8.20

9

t +4(?+4i/

+0. 00155

-0.0141

+0. 0540

-0. 975
+0. 939

+ 7.39
- 7.27

+ 23.43

V

t-\-4:-\-2e-\-2A

-1.12

+ 7.39

t

20+ J
e-4>+20+ J

- £+v:.+20+ .1

+0. 00068

-0. 0058

+0. 0222

+0. 847
-0.17
-0. 828

- 5.79
+ 0. 93
+ 5.96

+ 18.12

v'

•p + ^

+0. 00021

-0. 0020

+0. 0085

+0. 265

- 2.10

+ 7.15

t-

£ +4e+3J

v!'+4(?+3J

-0. 00056

+0. 005C

-0. 0239

+0. 724
-0. 697

- 5.90
+ 5.80

- 20.35

->'

e+il>+2e+3J

+0. 828

- 5.96

m"

VI"

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

95

Table XIX.

Unlt-l"

w'

w-t

u>-*

Cos

w'

w

w'

w'

w

ttj

w'

to

-<l>+i

+0. 2108

- 1.059

+2. 379

<l>+e+ie+iJ

-0. 2108

+ 1.059

-2. 379

v

e

+0. 843

- 4.236

V

c+iB+4J

-0. 843

+ 4. 236

1)

-4:+e-20-2J

- 294. 9

+ 740.6

-733. 9

-1.200

+ 7. 772

7

-lp+s+20+2J

-0. 875

+ 5. 583

V

!p+c+20+2J

+ 294. 9

- 740.6

+733. 9

+0. 274

- 1.697

n

s+c+sg+64

+ 1.800

-11.658

V

-4,+2e

-0. 105

+ 0. 529

n

4>+2£+iO+iJ

+0. 105

- 0.529

r,'

£ + J

-0. 227

+ 1.344

r>'

£+4e+3J

+0. 227

- 1.344

i

-•!>+ e-2e- J

+1. 758

-10.233

i

-,/>+ £+29+ 4

+1. 083

- 6.493

i

^+ £+2fl+3J

-0. 204

+ 1.377

r,'

41+ e+Gd+bJ

-2. 637

+15. 350

f

-4'+ f

+ 384

-1410

ri'

-4+ s-id-4J

+1679

-6666

v'

£+2fl+2J

+ 1180

-2963

f

- £+2fl+2J

-1180

+2963

v'

<!>+ £

- 384

+ 1410

v'

4,+ £+40+4J

-1679

+6656

riv'

-4,+ £ - J

- 285

+1210

vY

-4>+ £-4fl-3J

-2460

+8138

vv'

£+25+ 4

- 318

+ 1081

rin'

£-29- 4

+ 318

-1081

VI'

4'+ c +4

+ 285

-1210

riY

4>+ e+4e+S4
{6 -do) sin

+2460

-8138

V

-4>+ e-28-24

+0.549

- 3.40

+0.00090

-0. 0068

v

4)+ €+26+24

-0.549

+ 3.40

-0. 00090

+0. 0068

r)'

-4>+ 1-26- 4

-0. 407

+ 2.75

-0. 00032

+0. 0027

Y

4+ s+26+34

+0. 407

- 2.75

+0. 00032 -0. 0027

m'

771"

77l'3

96

MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

Table XX.
[(1 — e cos £) W'2] Unit- 4th decimal of a radian.

Cos

w'

M-!

u-<

U'O

w

a- J

jfO

w

u,i

u-i

V

+0. 01022

-0. 0513

+0. 115

v'

+ IS. 0

- 6S

+0. 187

-1.76

+0. 00043

-0. 0039

ri"

+0. 296

-2.46

+0. 00020

-0. 0020

f

-0. 186

+1.34

-4.8

vv'

J

- 13.8

+ 59

-0. 529

+4.34

-0. 00075

+0. 0065

1

29+2J

- 14. 29

+ 35.9

- 36

-0. 1006

+0. 647

-1.91

-0. 000055

+0. 00041

V

26+ A

+0. 1377

-0. 811

+2.19

+0, 000020

-0. 00017

v'

49+4J

+ 81.4

- 323

+0. 477

-3.64

vn'

40+3J

- 119.2

+ 395

-1.295

+9.36

v"

40+2J

+0. 921

-6.09

f

40+3J-i"
(f)-e„) sin

+0. 036

-0.32

1

25+2J

-0. 0266

+0. 165

-0.49

-0. 000044

+0. 00033

v'

23+ A

+0.0198

-0. 134

+0.43

+0. 000016

-0.00013

^'

46+44

+0. 151

-1.16

+0. 00031

-0. 0027

n v'

46+3J

-0. 334

+2.47

-0. 00052

+0. 0045

r,"

40+2J

+0. 165

-1.24

+0. 00015

-0.0014

m'

7n'2

m"

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

97

-a

a

I

CI CO CO

■^ CO

I I -f

^ CO

+ I +

(M O

■X" O rH

1 + 1

+ 1+ I ++ I ++

F-l CI CO -^
C^ lO -* 00 t- <M
■^ <M O <M d O

i-i .— t CO r-^ lO c:

I ++7 I +

lO CO iC
O Oi CO
O 1^ t-

+ 1 -f

(M O Oi lO r- CD lO I— .(N

C^C-ICOiO-^-^C^'COW

O O O 1— ' ^ "^ '**' Oi O

I + I + 1 I + I I

CO Q ^ "^r I— t^^

Oi O O CO CT. O

TT -"T CO lO CO CO

O O C^ lO O d'

+ 1 1 + 1 I

X

n

3
I

OOOOOO'JrH'o'

+ I + I ++ I ++

CO (M CO CO O t -
Ol O^ O <M O O
r^ lO ■n" u^ o r^

O O CO I- O CO

o o o o o o

I ++ 1 -t

0)0 CO
QO --H TIH
CO CO (M

ooo

1 + i

O i-H CO
C-J t^ oo
lO !>. CJ
OOO

ooo

+ 1 +

oo

+ 1

ss

oo

I +

I I +

•r t^ c^j CO

O ■^ t- C)

^O ^-^Q

sssi

o o o o

I +++

C5 o ^ r- -— t 00

Ci 1* P-* rH O O

O O oo O CO

800-^0 0
ooooo

o o o o o o

+ 1 I + I I

a t^ ^

Tj" [^ ■^

1-1 rtO

ooo

I + I

00 -^ c-l
f-O o
CO C^l t^

ooo

++ 1

CO -V C^ CO
O CO CO CI
-— ' Q C-t O

+ 111

rH Q O ^ O CO

ooo — p O

O O OO o o

I ++ I ++

rr 00 O

t^ 00 -^
rH —.O

ooo
ooo

+ 1 +

c^ c^ -^ T r

? s.

+ I

<M C) -^ -^

B* B" t

(T- W K-

B* C- B-

B- B*
B* B-

110379*^—22-

98

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

•a

-3.

a

CQ

(— ^

»-H

^
X

+

a

ij

a

<

H

.^

in

OO

o>

s

+

1^

CO

o

<x>

8

s

1

g

-^

t>i

a

CD

+

OO

c<i

CO-*

o

M

o'o'

o

3

.-1 1^

1 +

+

Oi — ^^-.

rt« O

lOOJO

coco C^ CD

(MO

O CO OO

-^ Oi i-H t^

CO rH

1

S

-.J^ CO M

CO ifio ■'J'

c<i w

C<1 CO

CO

+ 1 +

+ 1 J_^+

+ 1

Oi i — ,

(M M

coo Cs

i« rf 050

r^ ic

^ O^iO

fH 00 -^ OO

lOCO

S

CD i>- irt

O t^ f?^ "^

eO(M

O c-i -^

f-i o' CO o'

<oo

1 +J^

+++ 1

1 +

05

»o CO c^

O coo 05

O CO

O -^ CO

■* If:; c^ c^j

•^ <M

t.

OOO

(MO (MO

OO

a

OOO

0 0 0(0

■do

++ 1

1 ++ 1

1 +

O

CO

t^ CO CO

O •-< t^ CD

N w

i-l (M 00

(M -^ t- CO

O CO

1

s

8S§

CD 1— < lO O

ooo5

o8

S

OOO

oddo

d(d

1 1 +

±' 1+

+ 1

0-iC-iC}

^ 00 OO 00

00 '^

C-1 CO OS

t^ lO CD CO

^ CD

CO r-H coo

^o

§oS

8888

88

OOO

(iOiOO

do

++ 1

1 ++ 1

1 +

5

t

i

95

^ ^ ^ 0

^^■n o -^

'M ^ CO -g

Tf CO "

+++ i

+++ ?

+

^ <s <a <-P

C-l ^ -^ ,

<o <c _?

^ Tf "^

+++ .=1

+ + (1,

-^-3>.-S- &

-S--S.-3--3- S- -3--S.

V

V V

(=-

P- K-

pr-

o

p- p-

K-

No. 3.1

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.
Table XXIIo.

99

In the construction of Tables XXI and XXII it is necessary to compute

/<

Unlt-l"

Cos

uu

wo

U)«

w

to"

a*

w

4'+20+2J

- 0,614

+ 4.059

-10.3

■0

20 +2 J
(j'+40+4J

- 4.255
+ 2. 791

+27. 89
-23. 39

- 271.5
+ 167.4

+ 636.6
- 637.4

?

28+ J

<;'+4e+3J

+ 5. 444
- 4.558

-31. 91
+33. 80

1

40+4J
^''+20+2J
<;'+GO+ej

-^'+2e+2J

+ 0.11

+ 14. 90

+1514
+1360
-1227
- 273

-5780
-3387
+6415
+ 179

J

40+3J

4-+2e+ J

4'+20+3J
d>+6d+5J

-4'+2e+ J

+ 0.13
-44.62

-2279

- 646

- 291
+ 1974

- 222

+7160
+2452
+ 536
-9002
+ 1012

45+2J

- 0.06
+30. 53

}'

4e+3J-i:

+ 0.34

n

2e+2j

v''+4(?+4J

- 1.64
+ 1.014
+ 0. 782

+ 10. 18
+ 8.43
- 5.96

I

2d+ J

4' + ^

(/■+4«4-3J

+ 1.22
+ 3. 249
- 0. 579

- 8.26
-30. 12
+ 4.74

v'

45+4J

+ 7.81

J

49+3J

+ 3.25
-16. 10

I

v"

4(9+ 2J
(e-e„)»cos

+ 7.64

V

>!>

- 0. 356

+ 2.62

Y

<!> + ^

+ 0. 266

- 2.10

,

m'»

ir

i'

as one factor of a product, but the more complete tabulation is best arranged as follows. This
function gives all of the terms of the first order i n the mass in Wj — [ W^]. Let

W,

V' = jiT,-[T,])ds

wo MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

and denote first order terms in TF3 — [W3] and W^ — [TFJ by W3" and W^", respectively.
Then because of the similarity in the equations for these functions of successive ranks, the
sum

can be computed by Z 70, eqs. (117), (118), (119). The coefficients F, G, B are tabulated in
Tables XXIII, XXIV, XXV. The mass factor m' is, of course, implicitly contained in the
tables.

Ehminating the distinction between (p and s, the function is

W,"+W,"+W,"

in which the coefficients Apg, determined by Z 71, eq. (121), are tabulated in Table XXVI.
The coefficients A^g in the function

(1-f cos«) {W,"+W,"+W/')

are computed by Z 71, eq. (123) and are tabulated in Table XXVIL
By means of Table XXVII we readily compute

[(l-fcos£) iW," + W," + W,")]
tabulated in Table XX\1II.

Proceeding now to the determination of

[(l-<'C0S£)ll^]

(from which we shall subtract [(1 — e cos) W^"], already included in Table XXV ill), we have
by Z 53, eq. (95)

^^W,-[W,])^{T,-m) il-e cos e)iw+W,)^'-l'^^'[i^-e cos s){w,-w-W^)
-|(l-eco3e)(W.-iSi) (W. + i^.)- [d-e coss){w+ F,)^']
-[(l-ecos£)(F,-wW;)]-|[(l-ecos£)(F,-i5,)(F. + |^.)]}

in which all quantities are known. The integration gives W^ — [ IFJ.

Having computed W^ - [ TF3], [ Wj,] can be obtained from Z 53, eq. (95).

[(1 -£ cos £)(w+ F,)]^^V[(1 -e cos £)[ W3]]^' = 2[r,]-[(1 -e cos £)( F,-wF,)^']

+ [1 - e cos e)w W,t^ -[{1-e cos e) (w + W,)^g{W, - [ W,])] - [(1 - e cos .) (TT^- [iF,])]^'

The function [TJ, computed from Z 53, eq. (94), is tabulated in Table XXVIIIff.

In a manner similar to the development of equations for W, and f W^], the right-hand side
of this equation, when computed, can be segregated into portions independent of 4>, terms
multiplied by cos <l>, and terms multiplied by sin (p. It is of the form

A + B cos ip+Csin (J)

where A, B, C are too complicated to be written analytically, but can be written by inspection
after the computation has been performed.

The equation can then be written in the three following equivalent equations:

^.%3+(^3— ')^ = ^

in which we define

No. 3.1 MINOR PI^NETS— LEUSCHNER, GLANCY, LEVY. 101

From the first two equations we compute

Let

'U'3 = [lh] COS (l> + [zj sin 4>.

Then from the second and the third equations

■"3 = I </>r' I B cos <I>+Csm4i- (03 - 11^)^^ \dd

.duy]

~de.

By inspection of w, the function [7/3] can be written, and r^ln^ added to [x^ — Ti[y^ gives [a;,].
Finally,

[ ^3] = fe] + M cos ^ + [Sj] sin 4>
and

[(1-€C0S£)F3]

is readily computed from TF,, which is tabiJated in Table XXIT^II?*.

But this function contains [(1 — e cos e) W^"], already included in Table XX\T^ri. By Z 69

[(1 -e cos t) 17,"]= — ^[(l-ecosO r[(l-e cos ^)^'-[(l -e cos O^'jjtZe]

Subtracting Table XXVUIc from [(1 — « cos e) W^ we have

[(l-ecos£) (iF,-F,")]
which is tabulated in Table XXIX,

102

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

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No. 8.1

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

103

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104

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV

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No. 3.1

MINOR PLANETS— LEUSCHNER, GLANCY. LEVY.

Table XXVIII.
[(1-e cos s){W,"+W/'+W/')] Uait-lth decimal of a radian.

109

Cos

w»

w

a'

- 4. 1829

+ 12. 406

- 16.58

l'

- 68.61

+ 347. 9

\"

- 83. 96

+ 413. 1

?

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Il'

J

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rt

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[- 165. 57]

r+255.8]

r,'

28+ J

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/

48+ 4J

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vv'

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h

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+ 405. 5

P

48+ iJ-I

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+ 273. 1

m'

Table XXVIIIa.
2[T,]

Umt=- 4th decimal of a radian.

«,-!

u>»

Sin

U!«

«•

w'

W

,l>+29+2J

-0. 00005

+0. 00073

-0. 0682

+0. 4056

1

28+2d

4'
sJ+49+4J

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t

28+ J

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+8. 472

m'^

77!

n

110

MEMOIRS NATIONAL ACAD-EMY OF SCIENCES.

[Vol. xiv:.

Table XXVIII6.
W,

Unit—4th decimal of a radian.

Cos

u'->

t/t-i

w

U)0

w

w'

w

ws

w'

a i

- f+v''

-0. 00004

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-0. 0032

-0.0005

-0. 4803

t +29+2J

2£-v''+20 + 2J

v''+20+2J

0. 00000
+0.00001

+0. 00004
-0. 00023

+0. 0237
+0. OO.-iO
+0. 0726

-0. 15101
-0. 0318
-0. 4.507

+ 13. 16
- 0.81

- 30. 86
+ 1.31

2f +40+4J

+0. 00004

-0. 00038

+0. 0153
+0. 0181

-0, 0770
-0.0814

+ 3.88
- 16.23

- 14. 38
+ 61.80 ,

2c+<l>+eO+6J

-0. 0088

+0, 0576

- 3.0

+ 15.7

V

20+24

e-,l)+20+2J

- £+v'-+20+2J

+0. 5242
+0. 1384
-0, 0508

-3, 3539
-0. 7747
+0. 4060

+ 13. 16
+ 14. 86
+ 58. 25

- 30. 86

- 11. 30 '

- 170. 9

V

£

+0. 0749

-0. 2385

f

V

1

c +4d+4J

il>+4e+4J

+0.1723
-0. 5378

-0. 8380
+4. 082

-154.6
- 16.23

+ 589. 2
+ 61.80

V

e+^+2e+2J

-0. 1801

+1. 1267

- 7.71

- 6. 66

V

e+4f+ed+ej

-0. 3275

+2. 032

+178. 4

- 933.0

i

26+ J

i-4>+20+ J

- £ + 4^ + 20+ J

-0. 6099
-0. 1412
+0. 1220

+3. 634
+0. 6843
-0. 8554

+ 10. 77
- 31. 34

- 49. 04
+ 118. 9

r,'

+ 4

+0. 0524

-0. 4182

i'

t +40+3J
<l>+40+SJ

-0. 0411
+0. 7660

+0. 2314
-5. 430

+221.0

- 694.3

i

e+(P+20+3J

+0. 0460

-0. 3060

+ 14. 12

- 26. 01

n'

c+4,+Gd+bJ
(e-Bo) sin

+0. 3718

-2. 0745

-287. 1

+1309. 3

■0
V

20+2J

€-4>+20+2J

- €+il'+29+2J

+0. 0490
+0. 0144
-0. 0542

-0. 2949
-0. 0705
+0. 3582

^

V

4'

+0. 5810

-4. 7017

c +49+4J

-0. 0618
-0. 0453

+0. 4650
+0. 3389

1

')

£+^■+25+2^

-0. 0010

+0. 0290

t

2(9+ J

£-(/. + 2<?+ i

- £+4'+2e+ J

-0. 0364
-0. 0107
+0. 0402

+0. 2398
+0. 0585
-0. 2889

v'

4, + J

-0. 7670

+5. 4890

n'

e +4fl+3J
il>+40+-iA

+0. 0459
+0. 0336

-0. 3715
-0. 2709

n'

e+4,+2e+U

+0. 0007

-0. 0220

m's

ni'2

m'

No. 8.)

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

Taiile XXVI He.
[(l-cco8£)TrV']

111

Unit— 4tb decimal of a radian.

Cos

u>

Vfi

a

\'

25+2J
20+ J

+60. 76
-20. 57

-152. 6
+ 69.8

■m/

Table XXIX.

[(l-«C08£)(r3-¥V')]

Unl^4th decimal of a radian.

Cos

w->

«7-l

u

u-a

w

Mil

w

„.

to

+0. 00038

-0. 0032
+0. 5106
-0. 6292

+0. 0092
-0. 0069

-0.0005
-3. 0290
+3. 463

-0. 0072
+0. 0094

+13. 16

-30.9

20+2J
20+ J

(6 -do) sin

2fl+2J
20+ J

-0. 00004

m"

m"

m'

These developments cover the function W within the extent of our tables. This does not
mean that W is always inclusive of all these terms, but that these terms occur in one or more of
the tables. With the exception of [(1 — e cos e) W], which contains W,— W,", Wis to be under-
stood to mean ^_ y^^ ^ '^y^, ^ [ ^y^-^ ^ ( ^^,, + w," + W,")

^^^ w= w,+ F,'+nig + (tF/'+ w,"+w,").

The ascending powers of w, -q, -q' , f are selected independently in each function.

To avoid along series which is analogous in construction to T^, the function W^" + TF," + W/'
is not tabulated. The sum of this function and Tables XVII, XVIII, XIX, XXIIa gives W.
Since W is so long and we only need W, it is not tabulated. The function

IS given m Table XXIXa.

It is convenient to collect here [(1 — e cos e) If], which is required later. The function is
given by the sum of Tables XM^, XX, XXI, XXVIII, and XXIX, and is tabulated in Table
XXIX6.

Wc shall also need the function S

H = x + 2,y = E.+H/ + [S,] + (S,"+H3"+E/')

Evidently E can be written by inspection if Wis tabulated. If the double headings are retained in
the construction of E the mass- factors and ranks are explicit as in the construction of W. If W
is not given, we can write by inspection E, (previously required in the computation), Ej' and
[Sj] from TF,, TF/, and [W^], respectively. The remainder, namely, ^2" + ^3" + 'ZJ' , can
be written from F/'+ F,"+ F/', i. e., by inspection of Tables XXIH, XXIV, XXV. The

function 5^ S is given in Table XXIXc.

112

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. xrv.

a

CO

CO

CO CO r'S

c^' o c^i

^^

+ 1 I

Oi C. Oi

lO lO lO '^ ^ O =3

OOO CJ r- CO t-- ':c

^rrr-< ^coc^j.-ij-H OT^^nHio -pasn ajaAi uranio:> siq:^ ui gnua; aaiSap puooas xX
I ++ -f 1 I I I I ++++

C^ CD !M

cido"
+ I I

O CO -TO u^ t-
COCO'ft-O COC-IC

CO t^ lO CI CO CJ C-J c

' O CO
> C^ CO

Tj<C0"^iCO<MiC0C oocoo-— 'O-h;

CO CO CO c^

COOCOOr— ( OOO^OCl

I ++++ + I I I I ++ I I + I I I

I I +++ I ++ I + ++ I 1

o

I

+

O i-H tT .-H C^

I I ++ 1

o O I':) t— t CO ^

++ I I I +

•— I (— 1 1— I ^
© o" oi r-i

I I++

C) lO o:

b- lO ^ 00 :d

O "^ Oi Ol CO

iOO t^ CO lO

c: O O CO cr.

i-H Ca

CO r-1 lO lO

lO r-l 1— 1 r^

I I + +++ II I I I I +

n

■<

iC >-l CO iC-t C^ C3 C^ t^ CO lO CO F— I

lO t--. Tj" CO CO r-t -^ O 00 CO Oi -^ (M

(MV/3CO t--0)<M-^r— 1 Tft-COOiM

—I I— t ■^ t- I— ( CO O

<M-<t*'^COCsaiOOOi
O CT> CO lO lO lO CO C-l

cococo-^cioscoo
■^ lO o: c-i CO CO lO

IM l-H O r-l O

coirao-^dcccit-io-rf
iccoo r-<Mcooii— it-ci
t^i— icvir^t^cocoi^»— 'C^

iMiO-^OSiiOCOCOCOOOO
I— ICOCO lO'^ (— If- <X)

t-HOO CO
CI O ^ CJ
ira t-- CO 05

CO C- (M CO

++ I +111+ ++++ I + I ++ II + I I + I I I I + I I + + I ++

(M CO t^ ■-•

CO CO C^

CO t^ o

COOO-(t*OiM (M CO Ol t^ <-!
Ot-OOOit- i—COirat-rH
CO C^ r- I— I t^ (M r-i O r- 1 (M

C0OC0C005C0C0C0

ICCOCOCOCOCO.— (CJ

t-. CD (M CD -^ CO CD

O iC CO (M Tf

COCOOOfMi-ICOt^COrHOS

osc^tNdtMoocoasosM

(MnHOf-HOiOCOOt^CO

11+ I +++ I +111+

I + I '^++ I + +7+ 1 + 1 I ++ I

coco <:DO
t^ lO 00 c^

r^ CO C^ i-l

1 + I I

(>4 CO

00 aJ

rH N

+

+ I

CO CO O CO
lO t~- "^ OS

CO 05 OS CO
CO (M

I +++

CIO C^ 00 00
O CO 05 lO CO

CO C^ "^ -^o o

CO <M r-. 00 C--)

+ I + i +

i-< rH a>
TT CO ^

CO CO CO
O CO

1 I +

(M-^r C^ -^tMCD COCOlC "^(NCOC^ tTCO CO COiO COvOC- (NCJ'^

++ + +++ + +++ ++++ ++ +++++ +++ +++

CCS lli

Qi

en '^ ^

CB «C <5i -IS CC '3S,

is <i5 qs

(M -^

(M

Tf CJ CD

C^l '^ (M CO

'tT CI CD C-) '^00

'q- CI CI CD O -tj- -»- CO

■^ CM CO

++

+ + +

++++

+++ + +

4- + + + + + + +

++

•u W U W <4| tu

W

C^ C^

IM (N

CM C4CJ

CI CJ CI CI

No. 3.]

MINOR PLANET&-LEUSCHNER, GLANCY, LEVY

113

aiqcX JO uopoiujBnoo em nj

00

TT in

00 ^

rt IM

00 t^

CO

O lO
CI

+ 1

rjr-.

Ort'

«o

+

1 +

1 +

+ 1

oo

+ 1

CO iC t- iC

CO 00 ^o r-

^ nHO r-l
COC^O '^O
.-1 tM ^

i-H005Tj*"»rcOOiTf

fMiOOcor-cocolO

COC^i— lOOS'^i— ICO

I + I I + I + I I +

^ CO ■^ -^ c^ CO CO r- t^ lO h- o

COl^TfO coo Q UO C^ C^ CO O

CJ CO lO t— I t^ C^ CO (M oo O CO CO

CO CO --^ i-l Tf C^ CO r-l rH

++ I + +++ I ++ I +

■^ t--0 CO
O t- CO CO

CO CO (M 00

I I + 1

C4 ■*»• CD

++

CJ "*r 00

++++

I I I + I

"^ T^ ■^ -^ -"q ■n

CO CJ O T}« 1^

+++4 ++

^ ^ ^ ^ ^ ^

1 103790—22-

114

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XXIXa — Continued.
W.

T-'nit-l".

Cos

w-i

U)->

w«

w

u"

w'

w

«.•«

v'

- £+4e+4J
£+8e+8i

+ 2549
- 3089
-11300

+ 2164
+ 8155
+ 76250

-11.9

+ 3.9
-8.9

+ 89
- 25
+ 70

riW

£+4(9+5J

£+49+3J

- £+4e+3i

£+8fl+7J

-11449
- 2661
+ 6865
+50005

+ 42212

- 27530

- 4540
-304611

+ 1.9
+36.4
-20.3

+33.8

- 23
-241
+118
-248

vY'

£+4e+2J
- £+4e+2J

+26091

- 1356

- 2204
-73583

- 71730
+ 30293

- 20846
+400009

-10.1
-25.5
+28.0
-41.9

+ 83
+ 153
-153

+284

v"

£+4»+3i
- e+iO+ J

-13756
- 3317
+36006

+ 22165
+ 18452
-172164

+10.1
-12.4
+ 16.6

- 65
+ 64
-104

Pv

e+4d+3J-S

- €+4e+zj-£

t+S0+7J-2

£+ie+4j

- 2011
+ 1808

- 2381
+14204

+ 14604

- 13617
+ 18919

- 88026

- 1.9
+ 1.9

- 1.1

+ 5 7

+ 14

- 14
+ 10

- 42

P i

e+4e+4J-i:
- e-\-49+2J-I
£+»0+6J-£
£+4(?+3J

(e-(9„)8in

- 554

- 3545
+ 3827
-17503

+ 140
+ 22886
- 27870
+ 99584

+ 0,5
- 1.8
+ 1.3
-3.7

- 4
+ 14

- 11

+ 28

V

2e+2J

€+4e+4J
2i+2e+2J

+ 767. 7

- 2820. 9

+ 5210

+ 1.265

- 2.19

- 5.34
+ 0.78

- 0.55

+13.6
+22.7
- 6.0
+ 3.4

Y

20+ J

£+ J

e+4e+3J

2£+2(?+3J

- 570. 0

+ 2421. 1

- 4950

- 0. 455

+ 1.63
+ 5.94
- 0.58
+ 0.41

-11.0
-37.3
+ 4.8
-2.8

v'

4d+4J

£+2e+2J

- £+29+2J

2£+4(?+4J

+ 10.93

- 2.19

- 1.92
+ 3.12

46+34

£

- 570. 0
+ 6624

+ 2421. 1
- 47448

- 4950

- 0.455

+23.8

+ 5.94
- 23.00
-221. 9

-7.2

iW

£+ 4

- £+ J

-18540
+ 8414

+ 123024
- 57880

-73.4
+36.0

+572. 4
-282. 2

nr/'

£+ 2J

+25564
+10478

-157424
- 70250

+87.3
+55. 2

-652. 8
-374. 8

Y'

£+ -^

-15678

+ 94846

-69.9

+438. 6

j'v

£+ 4+i'

-25564
+22012

+157424
-121258

-511232
+359162

-23.1
+ 9.9

+165. 0
- 77.0

P rf

£+ J
£+ i'

(fl-e„)'cos

+23524
-12048

-150306
+ 76364

+498328
-251640

+ 14.8
-5.2

-112.0
+ 45 8

\'

£

£+ 4

- 0. 356
+ 0. 26i>

+ 2. 623
- 2. 100

m'

m'"

No. 3.J

MINOR PLANETS— LEUSCJINER, GLANCY, LEVY.

115

Table XXIXa

— Continued

W

Dnit-l"

Cos

UIO

'X

w«

i£+ fl+ i

- 293. 4

+ 913. 6

- 1400. 1

f£+3fl+3J

+ 338. 1

- 2315

+ 9277

4£+5(?+5J

+ 42.9

- 284. 3

+ 948. 2

^f+70+7J

+ 10.6

79. 2

+ 288. 5

IJ

i£+3fl+3J

+ 6172. 8

- 20580

+ 86649

-i£+ fi+ J

+ 511.2

- 2834

+ 7746

f£+ e+ 4

- 467. 9

+ 2336

- 6259

^£+5fl+5J

- 2217. 1

+ 23971

-157308

4£+30+3J

5.8

+ 539

- 3713

|£+7e+7J

- 364. 3

+ 3259

- 15083

yf

i£+3(?+2J

- 8375.5

+ 20591

- 95913

-i£+ 6

- 1023.4

+ 4443

- 10251

f£+ e+2j

- 92. 3

- 444

+ 3212

|£+5fl+4J

+ 3383.4

- 34097

+214736

|£+3e+4i

- 138. 6

+ 608

- 1089

4£+7e+6J

+ 583. 3

- 4805

+ 20748

ri''

i£+ e+ J

- 5022

+ 24269

i£+5e+5J

-31492

+ 154466

-l£+3e+3J

+ 8169

- 18309

5£+3^+3J

- 59

+ 7449

f£+7fl+7J

+12392

-182737

-f£+ »+ i

+ 1133

- 5174

|£+9e+9J

+ 2342

- 25879

iv'

i£+ 0

+ 6163

- 26311

i£+ (? + 2i

+ 988

- 15732

i£+59+4i

+88784

-367666

-i£+30+2i

-14498

- 3083

|£+3(9+2J

- 1309

5

f£+3fl+4J

+ 4878

- 31947

|£+7fl+6J

-37540

+526187

-!£+ e

- 3487

+ 12764

^£+9e+8J

- 7382

+ 77025

V^

i£+ e+ i

- 5966

+ 27801

i£+5fl+3J

-61877

+192684

-i£+30+ 4

- 1709

+ 26144

\t- e+ J

+ 1693

- 6306

|£+3fl+3i

- 5297

+ 28649

|£+79+5J

+28418

-377278

?

i£+ e+ ^

+ 6846

- 30642

i£+5fi+4J-i-

- 3191

+ 15690

-i£+3fl+2J-i'

- 806

+ 10210

^£+3i!?+3J

- 3829

+ 33852

f£+70+6J-J

+ 932

- 14562

-^£+ e -2

+ 1762

- 6460

m'

116

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

a

3

•a

X

•J
n

■<

\h

o>

lO

00

<a

(N

O

5D

•^

CO

COrH

deo

rH

CO

00

C4 t~t

CM CO

3

I

+

c^^
1

++

1 1

CO

o

Tf

o

rH

■^

■^

r-* iO

i-H

c-i

^

W

oi (M

irfO

CO CO

CO OS

&

c->

t^

t- -^

lO 1— 1

>-* t^

,-i o

(M lO

CO o>

tT CO

■«>rH

1

+

+

+

+ 1

1 +

+ 1

1 +

00

O

C<1

CO

CO

t^

CO

in

C^

Oi

■^

o

e^

CO

O CO

(M(35

OSU5

r-i CO

i»

CS

CO

o

^

■*

csi

o5

d CO

d -H

c6c<i

aid

com

rHOiO

O CMO

t-O

O fO

0> Tf O

r-i

r-i

cx5

3

t^

1— 1

>ra oi

<M CO

00 05

t^rH

CO o:

o> -^-O

CO 00

CI T

-rf ^OO

(M

rH >n

f-H

-^ (M

o a>

O lOCl

00 iO

CO CO t^

t- rH O

IM

rH 00

1-^ r^
1 1 1

•^

(— t

rH r-i

1

1

+

1 +

+ 1

1 +

+ +

1 1

+ 1 +

++

1 + 1

1 ++

1

+

+

X5

^

^

CO

"*

<N

10X5

r^

o

o

»o

o

F-«

M iM

CO (N

t'.QO

-•** 03

CO

"^

CO

o

OJ

■^

CCC^

O0O5

I0 05

eooj

(M

r-1 tn

CO CO

t~

O 'J*

TJH t^

CO

Ift

t-

•^

o

iC

00

M rH

^r-i

c4 -rt*

f-i irt

CO 00

CO 00 Oi

t-^ lO O

oo' CO

CD Oi CO

CM t>^ t^

d

o"

CO

^

■*

1— t

CO

00

t-

rH

(M W

^ooo

CO <N t>

s?

CM OO

CO OO

rH

CO

<M

■^

COrH 00

Cl rH t^

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CO rn

CM

r^

1

+

'

1 1

++

1 1

+ 1

+ +

1 +7

+ 1 +

1 1

+ 1 +

+ 1 1

+

1

1

o

CO

r-

00

if^

o

b-

1— 1

CO -^

t^ C-1

CO rH

CO (M

o

o

s

t^

o

C^ OS

.-< CO

CO t

00 lO

oo

CO 05

t^ UT)

i-l r-i

CO

CO

05 a>

o

oc

o

c^

tT

-•l^ (M

t^ --r

CO -^

rHO

t^ t^

■>!< CO CM

OO 00 t^

CM CM

CI

CM

oo

r-i

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o

o

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d --i

d CO

d ci

dd

r-i r-i

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C-i r-i .*

r-i rH

d

d

dd

d

o

d

+

1

+

+ +

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+ +

1 +

1 1

+ + +

1 1 1

+ +

1

1

+ +

1

+

+

■Tf

rH

t^

CM

CO

o

■^

COO

t-co

OS CO

CO

CO

»o

a>

CO t^

\a CO

05 Tj^

(M (M

t^ ■«•

iC CO

88

CI

C-l

rH rH

r-i

00

lO^

■* CO

88

COCO

CO o; CO

lO t^ CO

CO

CO

rH r-i

8

8

o

H

o

o

o

oo

oo

o

(M (M

l^ O 00

in CMOD

CM CM

o

o

O O

o

b

o

1

d
+

d
1

do
1 1

do
+ +

dd
1 1

d

+

dd

++

odd

1 1 1

ddd

+ + +

dd
1 i

d
+

d

+

dd

1 1

d

+

d
1

d
1

ta

o

CO

CO

o

la

(M

f—i r-H

iO cc

fM t^

CO

t^

CO

o

00 <-*

tP Q

rH O

■^ <M

CN GO

00 o

TT a>

o

r-i

§

CM O

r-H Q rH

o o

o

T

s

8

SS

8 o

88

888

o o
o o

8

8

8

:t

d
1

d

+

dd

1 1

dd

+ 1

d

dd

1 1

ddd
+ + +

d d
1 1

d

d
+

d
+

o

r-( r-*

-^ 00 -^

r-l lO

50rH

CM rH ,-4

o o

S O

oo

888

1

o o

OOO

o o

dd

odd

ddd

dd

+ +

1 1 1

+ + +

1 1

g

1

1

1

1 1

'S

-n

-^

-n

-<q-<i

■n

1

■n'n

-^■^^

^■^

^ ^

1 1

1 1

■^

■^

<M

'<r

CO

M

CO

CM CO

COiO

C-J rp

CO

CM xn

CM •»

=5S CM

■<i<

+

+

+

+

+

+

+ +

+ + +

+ +

++

+ + +

+ + +

1 +

+

+

CD

■^

CB

Qi

t

<i:s

QS <CS

<^ 1i 1>

<3i CO ccb

<^ <C)

OS QiCa

<IS CC) -^i

^ "5^

•3^

^

^^^

(M

-v

-'f

■»<

'^

(M CD

<M (M CO

CM (M CO

CM CO

CMIMCO

CM CM CD

vS-CM

CM

■<i«

N

M

V

P-

^f=-

V

V

V

V

P-

K-

R-

e^

f=-

^-^

"p

p-

(■1

^-,

P-

(3~

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

117

g

V

g

■*

lo

oo

if^

f^ 00

Tl

iO

c:

CO

OilN

o

o

l:^

r-

g

irf t--^

ai

r-H*

i—J

^*

1

1 1

+

+

1

+

t^

to

M

o

CO

Ci CO

■^

00

^H

"'J'

«o

o

CO

CO

C-l

s

ood

-^'

o'

o

o'

n"

s

1 +

1

1

+

1

o

f-i

CD

O CD

■■o

(M

t^

00

M

Q C30

ro

>n

t--

C^l

V

5o

U5

o

o

o

p

O rH

o

o

o

o'

'■

1 1

+

+

1

+

s

i-hO)

■^

OS

t^

CO

om

•^

■^

r-

S T»<

■tf

.—I

.— 1

M

So

8

s

8

o

V

c o

O

o

o

o

-

1 +

1

1

+

1

6

t— 1

CO

CO

00

00

r^

00

cp :d

■^

8 o

*L

So

o

o

o

o

o'o

o

o"

o"

o

c

1 1

+

+

1

+

n

s

n

s

o

w

■^■^

•^

'n

CO

N

s

+

+

1

<:s

^

<5>

•«■

TT

M

«

V

B-

ST*

«*

r ■

^t=-

E"

c-

o

118

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XXIXc.

1„
3"

Unlt=l"

Cos

W-'

u'~*

urn

w

w'

w>

U)0

w

u'

S+20+2J

- 90.5

+ 302. 7

- 478. 2

2£+49+4J

- 26.6

+ 125. 5

- 270. 3

1

29+2J

+ 589. 8

- 1571.9

+ 1680

- 1.82
+ 0. 42

+ 12. 00
- 2. 10

£+4e+4J

+ 616

- 3638

+11175

- 0.42

+ 2.10

2J+20+2J

+ 23

- 161

+ 439

2e+6e+6J

+ 219

- 1451

+ 4616

i

26+ J

- 106. 1

+ 3G0

- 517

+ 1.81

-10. 64

£+ ^

- 43

+ 161

- 269

- 0.08

+ 0.45

e+id+SJ

- 760

+ 3906

-10778

+ 0.08

- 0.45

2£+2e+3J

+ 52

- 143

+ 87

2S+69+5J

- 314

+ 1874

- 5403

y?

-0. 317

+ 1.63

49+4J

-1679

+ 7272

-13527

-0. 633

+10. 02

£+29 +2 J

+ 274

- 63

- 1.20

£+69+6i

- 3474

+29267

+ 3. GO

- £+29+24

+ 1156

- 2171

- 2.40

2£

- 0.21

2£+49+4J

+ 180

+ 113

+ 0.21

2E+89+8J

- 1375

+11897

ni

J

+0. 227

- 1.30

49+3J

+3690

-12966

+19401

+0. 340

-19. 92

£+26+ J

+ 222

- 1234

+ 1.96

£+29+3J

- 769

+ 2197

+ 0.01

£+69+5J

+ 9240

-70866

- 7.25

- £+29+ d

- 646

+ 1806

+ 5.27

2£+ J

+ 99

- 444

+ 0.04

2E+49+3J

- 109

- 922

- 0.04

2£+49+5J

- 846

+ 4256

2£+89+74

+ 4012

-31827

v"

-0. 039

+ 0.24

i6+2J

-1780

+ 4725

- 5354

-0. 039

+10. 42

£+29+24

+ 499

- 649

- 0.32

£+69+4J

- 5930

+40905

+ 2.86

-£+29

- 2.54

2£+ 2J

- 55

+ 285

2£+49+44

+ 980

- 4150

2£+89+6J

- 2890

+20791

f

49+3i-2'

£+29+24

£+69+54-2

- £+20+ J-S

2£+ J+i-

2£+49+44

2£+89+74-J

- 101

+ 493
+ 587

- 193

- 192
+ 298

- 65

- 1128

- 2983
+ 1759

+ 705

- 1876
+ 616

+ 0.11

+ 0.14
- 0.14

m'

m'^

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY. LEVY.

119

Table XXIXc — Continued.

1-

3"

Unlt=l"

Cos

w

w

w'

i£+ «+ J

- 31.4

+ 131.0

- 255

^£+30+3J

- 48.0

+ 193. 6

- 360

^s+5fl + 5J

- 15. 2

+ 81.7

- 201

^£+-tf+7J

5.2

+ 34.7

- 107

7

i£+39+3J

+ 1304

- 8173

+30282

-i£+ e+ A

- 196

+ 506

- 598

!s+ 0+ ^

+ 34

- 146

+ 292

|;+5fl+5J

+ 356

- 2212

+ 6781

4£+39+3i

+ 1

67

+ 294

i£+7fl+7J

+ 138

- 999

+ 3437

1'

i£+3fl+2i

- 1361

+ 7468

-25691

|£+ fl+2J

+ 29

12

- 155

|£+59+4J

- 482

4- 2635

- 7209

|£+39+4J

+ 52

- 197

+ 280

4£+7fl+6J

- 207

+ 1348

- 4176

1'

i£+ 0+ ^

- 625

+ 3058

i£+59+5J

- 7151

+ 70387

-i£+3fl+3J

+ 5478

- 5874

*j+3(?+3J

+ 18

+ 1924

4S+70+7J

- 2111

+ 17665

-f£+ e+ J

+ 187

- 590

f£+99+9J

- 771

+ 6931

IV

i£+ fl+2J

- 231

- 1142

i£+59+4J

+17640

-159928

-i£+3(5+2J

- 9842

+ 1346

|£+3e+4J

- 892

+ 3699

f£+30+2J

+ 106

- 2494

i£+70+6J

+ 5918

- 45149

-!£+ 0

4£+99+8J

+ 2513

- 20914

l"

i£+ 9+ i

- 507

+ 2729

i£+5e+3J

-10202

+ 84314

-i£ + 3fl+ J

+ 1055

- 678

f£- 9+ J

- 100

+ 387

|£+3fl+3J

+ 871

- 2817

4£+79+5J

- 4065

+ 27951

p

i£+ 9+ J

+ 601

- 3122

i£+5fl+4J-i-

- 423

+ 4435

-i£+3fl+2J-Z

+ 285

- 356

f£+3fl+3J

+ 426

- 2410

4E4-79+6J-J'

- 108

+ 988

-|£+ e -I

- 106

+ 402

m'

120

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XXIXc— Continued.
3"

Umt=l"

rn«

U)-l

UI-'

,„0

w

w'

ufi

w

Ul2

f

2e+2J

60+6J

+ 1568
+ 5879

- 8912

- 31559

+3.6
+3.6

-23.3
-23.3

nW

26+ J
20+3J
60+5J

- 2385
+ 2238
-21644

+ 11662
- 1015
+102003

-4. 7
-2.6
-9.9

+26.6
+ 18.4
+59.1

Vrt"

29
2e+2J

ee+ij

- 1723
+25396

- 7588
-103013

-0.2
+4.0
+7.6

+ 1.7
-27.1
-43.2

V'

26+ J
66+3J

- 1160

- 9257

+ 5960
+ 31500

-1.4
-1.4

+ 8.3
+ 8.3

fv

ee+5J-x

26+2i

+ 1040
- 5354

- 6697
+ 25370

-61855

+0.3

- 2.1

/ V

26+ J

26+2J-I

66+iJ-J

(e-flo)Bin

+ 2492

- 53

- 1285

- 12023
+ 989
+ 7413

-0.1
-0.1

+ 0.6
+ 0.6

V

25+24

- 0.55

+3.40

i

26+ J

+ 0.41

-2.74

r,'

49+4J

J+29+2J

-£+20+24

- 3.12

- 1.10

- 1.10

vv'

J
49+34

£+25+ J
e+26+ZJ

-£ + 25+ J

- 569. 95

+ 2421.1

- 4950

+0. 45

+ 5.94
- 5.75
+ 0.20
+ 0.82
+ 1.01

v"

45+24
£+25+24
-£+25

(5-5„)»ros

+ 2.55

- 0. 15

- 0.15

vY

4

- 0.26

rP

+ 0.20

m'

m'2

COMPARISON OF TABLES.

As a computer would discover in constructing tables, and as will be evident from an appli-
cation of the method to a planet, the coefficients in Table II and others of the same form are
given with unnecessary accuracy. Although so many digits would never be required, except
in a much more exhaustive development, they are given, for completeness, as they resulted
from computation.

In all the tables whose constructions involve the multiplication of trigonometric series, the
errors are difficult or impossible to determine. Although v. Zeipel's manuscript, which the
author generously furnished for comparison, is of assistance, the computations are not entirely
parallel, and comparison is not always possible. Many of the computations are so long and

No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 121

complicated that the origin of certain discrepancies is obscure. Aside from possible errors of
calculation, differences are duo to the independent adoption of the highest powers of m', w, ri, tj', f,
and the number of arguments in a given series or product of series. In most cases our series
are more complete than v. Zeipel's. Whether or not the extension of the tables increases the
accuracy of the result remains to be seen from future applications of the theory-
Tables II-XV. — The discrepancies seem to be due to v. Zeipel's errors of calculation and to
their subsequent effects. The larger number of these errors have been traced in the manuscript.
Tables XVI, XVII check satisfactorily.

Table X\^II. — The bracketed quantities in the first three columns are in error through
previous discrepancies. We did not discover the som'ce of the general disagreement in terms
of the tliird degree, second order in the mass. These terms do not affect v. Zeipel's subsequent
tables, since they are of order higher than have been included.
Tables XIX, XX agree satisfactorily.

Table XXI. — The discrepancies are numerous and their origin is obscure because of the
very long computation involved. In addition to performing a complete duphcate computation,
the terms of first degree and a part of the computation of second degree terms were checked
by the solution of the differential equation in the form given in Z 64. With the exception of
three or four single instances, the discrepancies occur in two groups, having the following
probable explanations. The neglect of the term

in Z 6.5, eq. (109), by v. Zeipel accounts for one group of differences. The other group can be

fairly well explained by an error in the addition of second order terms in +:s <^i to c^, — -^<^,.

Assuming that for these terms he added — w<i>, and, correcting his table, three discrepancies are
removed and two others are improved.

Table XXII. — Considering the disagreements in Table XXI, Table XXII checks satis-
factorily.

Table XXIII-XX"\T^I. — These tables, like II-XV, are simple in construction, and the
discrepancies are due to errors of calculation, or they are the result of previous ones, with the
exception that some quantities have different numerical values because they are more inclusive.
The latter have been indicated by ( ).

Table XXVIII. — The discrepancies arise from the quantities in parentheses in Table XX\T!I.
The omission of the term depending upon the inclination is justifiable in view of its magnitude.

Table XXIX. — The discrepancies are niunerous and striking, but, since v. Zeipel does not give
the formulae of computation, they remain unexplained. The remark is made (Z 77), '"Die
Berechnung der Funktion [(1 — e cos s) ( IF3— W3")], welche eine sehr komphcirte war, wird hier
nicht im Einzelnen mitgetheilt." For this reason the development of the formulae which we
used has been included and the auxiliary functions 2[rj, W3, [(1— e cos e) W/'] have been
tabulated. The differences are not serious because of the high rank of the function. Our
table is deficient in certain terms whose computation would be long and the omission of which
is justifiable in view of their magnitude.

PERTURBATIONS OF THE MEAN ANOMALY.

For clearness some of v. Zeipel's developments will be amplified and repeated in an order
which we found more convenient.

The determination of the disturbed mean anomaly is accomplished with the integration
of Z 9, eq. (47), (which implicitly contains Z 8, eq. (38)). By Z 9, eq. (43),

0 = i{£-e sin £)-g' = ig-g'

The differential equation is repeated in Z 78, eq. (124), in which is emphasized the fact that

dW
the arguments are functions of both £ and ^, as is the case for —j-'

122 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv.

If we observe the character of d as it is expressed in the definition and recall that we have
admitted trigonometric terms in 6, mxdtiplied by t, it is evident that this argument, which is
a function of the disturbed positions of the planet and Jupiter, is not periodic, but varies con-
tinuously with the time. In the foregoing equation g and g' can not be regarded as angles
which are always less than 3(30°. 0 contains, therefore, a nontrigonometric secular part in s
and a periodic part in 0 and e.

If we write

d — [d] contams the secular term in c as well as periodic terms. The segregation of terms of
different type can be made explicit by the introduction of

d = ^ + 0,{^,s)+d^{&,i)+d^{d,s)+ Z78, eq. (125)

where t? is a function of £ and 5„ 62, 6^ • ■ ■ are the periodic parts of 6 — [6], i. e., they are
entirely trigonometric functions of s. This covers the condition that Oi can not include trigo-
nometric secular terms in s. By definition of § and (?<

y

^' = {F(0,e) - [F{0,e)]) - j^(n'8z' - [n'Sz'])

where [n'dz'] is the long period term between Jupiter and Saturn,
The derivative of (125) is

dd^dd dddd_
de hi d^ds

/d6 be dd, VA ,^^.,^^^4.^^3, Y&

~VdI+d£+d7+ ; + V d^+d^+d^ + Jde

Expanding F{d,s), eq. (124) in a Taylor's series in ascending powers of ^<and making the above

substitution iov-j^' (124) becomes (126), in which

F{d,^) = \{l-eco5i)[w+{\-w)W-\{\-ui)(W-\s^(^W+\E^+----^; 6 = ^

From the Taylor's series -^ is written m (127). This is the differential equation for 1?, the

right-hand side of which can be computed.

Substituting ;5^ in (126) and equating fimctions of equal rank, we have the differential

equations (128,-1283) for di, which can be integrated in succession.

Before integration we convert eqs. (128) into differential equations for ndz as follows:

Let

nhz = (ndz — [ndz]) + [ndz]

= ndzi+ ndz^ + ndz^-] [-[ndz] Z 88, eq. (144),

where ndzt is not only a function of fiist and higher orders in m', in which the lowest rank is i,
but is entirely trigonometric or periodic. Then

Z 9, eq.(46) gives n5z-[nSz] = j^f-;^|(?,(j?,s) +<?2(i?,£) +^3(i?,£) + +wrj sin s+ (n'dz' -[n'dz'])j

and

M2] = y^{.>-|e + [7j,'a2'] + <-'-A(c} Z 88, eq. (145),

where it is to be noticed that [ndz], unlike [ TF], is not free from terms in s. Subdividing the first
of these two equations according to rank, we have Z 79, eqs. (130), in which — n'dz' + [n'dz'] can
be neglected.

No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 123

Differentiating eqs. (130) partially with respect to e, substituting in eqs. (128), evaluating
the right-hand sides of eqs. (12S), we have eqs. (131, — I3I3), in which the superscript indicates
that only terms of first order in the mass are included, and where the argument «? replaces
the argument 0.

For purposes of calculation, the integrations arc arranged as follows:

In _ _ „ _. _

F= lf,+ IK' + [ irj + ( F/' + W," + W,")

consider first only W/' + W/'+W/' in the integration of eqs. (131). The integrations will
concern only part of the terms of first order in iidz^ + ndz^ + iidz.j. It is shown by v. Zeipel that
the integration for all three ranks can be performed conveniently at the same time. Let this
part of the function be indicated by enclosing it in ( ). The integral

(n52,(") + (na^^C)) + (n.523<")

which is a trigonometric series, is given by Z 80, eq. (135), in which the coeflficientsT.p.g are
defined by (130) and are easily derived from Table XX\T!I . The coefficients Lp.g are tabulated
in Table XXX.

The remaining terms of rank one which are of first order only, namely, ndz^^''— {ndz^'^'>), are
given by the first of Z 81, eqs. (137), in which TF,, TF,, [W^, can be written by inspection
from Tables XVII, XVIII, XIX, XXIIo. The fmiction 'is tabulated in Table XXXI.

The remaining terms of first order in ndz^_ and ndz^ are given by the sum of Z 82, eqs. (139)
and (140). The function

71^2/') - (nfeO) +n523("- (n^Sjti')
is given in Table XXXII.

These developments complete ndz <" within the limits of the tables, and we next consider
71^2 '^'. We shall limit ourselves to functions in which the lowest rank is the first or second.
Consequently, Tidz^ contributes nothing.

m'-
Any function of second order in the mass and first rank must contain the factor — r and in
•^ _ w^

the given F {d, e) this factor occurs only in TF/^'. We have, therefore, by Z 80, eq. (131,), for

one part of n.dz^''\ indicated by parentheses,

(7)^2,(2)) = r { (1 - e cos £) F,<^' - [(1 - e cos e) F/'']}d£

This function is tabulated in Table XXXIII.

124

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV..

a

X
X

a

■<

CO

lO iC

-^ iC

CD

o

o

o o

M -^

O ^ CD

CO a; CM lO

CD CO O

CM O -* iC

CO

lo r^

n Oi

1— 1

C4 CO

C5 (M 13

C3! •«<
CO

y—t

00

C^ lO

+

+ 1

1 +

! +

1 ++ 1

+ 1 +

+ + 1 1

1

1 +

+ 7

g

to eo

a a

t^

o

o c-i

CO CD

O CO ^

CO O O C5

=5 -< CO

-<*- CM t^ O

CO

1^ — H

lO c^

Tf* CO CO

CD CO iC t^

CO

(N

C<» CO

CO CM lO

.c o

CI r-t

lO

CO 00

CO ^

CO lO rH t^

o

C5

— H CD

CO

C<J

CM 05

CM

r—l

lO CO CO

r-i

+

+ 1

1 4-

1 +

1 ++ 1

+ 1 +

+ + 1 1

1

1 +

+ 1

1 + 1

+ 11 +

"

CO lO

lO CO

OO

1—1

o ^

O n-1

O CM o

Tt* C-1 '^ lO

C3 CO N

t^ CO CM Oi

c^

CO -H

T?< rH

CM '* C^

O CO 00 rH

lO

CO w

CO CO <M

f-( CO "^

lO rH rH

1— t

o

lO n

CM rH

^r OO "^ CO

00

CM o

CM W

t~

lO

CO CD
CO
CO

CO CO »o
CI t-'

CO

+

+ I

1 +

1 +

1 ++ 1

+ I +

+ + 1 1

1

1 +

+ 1

1 + 1

+ 11 +

CO Ci

t-( (M

CO

oi

o d

rH lO

O CO ■*

CM O O CM

CO Cl CO

C3 O -^ O

U5

t- CO

CO b-

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CM CI t^

■^

rt o

i-H O

tP CO C^

r-i r^ CO

rH lO CI -^

CI

lO

CO CI

CI ^

CO »0 to rH

1— 1

lO

-^ CO

CO CO

rH

»o

o

rH r-

r- OS r^

CO

r^

c^

t—i

1—1

CO Cl CO
OS

+

4- !

1 +

1 +

1 ++ 1

+ 1 +

+ + 1 1

1

1 +

+ 1

+ 1

+ 11 +

QO

00 (M

Oi «

CO

O CD

00 CO

1— 1 lO lO

fH b- O lO

■— < -^ CO

O rH o »-<

M*

-»< C]

CO CO

lO OO O

rrs M- ^ CO
rH Cm '^ CD

53

o=

rt CO

t^ CO

CO 1—1 1—1

CM ^ lO

c^ (M Tt" o:

lO

CO

C) '^

en ■*

o

CO

O)

CO

t^ '"JH

CO lO

CD

C^l

rH CM

OS 1-1

^ rH CO

lO

CI

lO

rr

CM

CD

CD "^
1-H

+ 1 + +

+

1 +

1 +

1 + 1

+ + 1 1

+ 1 +

+ + 1 1

1

+ +

+ t

+ 1 +

1— 1

lO t^

>o iC

00

t-^

CO CD"

C^ lO

(N «= O

CO -rf .— ( CI

I>- CO lO

CO C^l w -»

o

CD C^

CO 00

00 00 ■*

rH CO rH O

1— 1

CO O

C^ lO

^ o ^

CM -— t TT

CO CO CO -^

CO

'«*' Hi

02 -H

c^ •»•

rH CO CO CO

CO

CO »o

Tf< f-1 Q

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l« (N

T-l

l>

•-t o

CO -a"

CO O CM

«-H

CO

rH CO

CO

CD

00 (N

O CM CD

lO

*"*

f-H

f-H

- a

+

1 +

1 +

1 + 1

+ + 1 +

+ 1 +

+ 1 1 1

1

+ +

+ 1

+ 1 +

1 1 + 1

S

CO CO

tT CD

00

<— i

CO iC

i6 iri

<N (N

,-C iC CO -v

CO CM CO

iO CM r- OS

ai

O C5

in ^

C^ 0?

1^ cFo C-)

iC

C<I CO

lO lO

t^

CM rp 05 O

CO r- lO

CD CO t^ CO

t^

O C75

I!~ OJ

CD

C^ OO .-H CO

1— i

c^

OS

CO O CO

r- '<5*

CO rH OS

CO

C^ Tt<

c^ -v

t-H

" Q O

•V

(N

CO .-H O

■— 1

CD

Oi

CO

1^

lO CO Tf
C^ rt 00

+

1 +

1 +

1 1

+ + 1 +

+ 1 1

+ 111

I

+ 1

+ 1

1 +

1 J^+ 1

CO

CO

lO CD

^ CD

lO

i6

t^ oi

CO ci

CD ^ C3

r^ t^ M*

CO 1— 1 rp

rH O lO

CO

.-J CO

t^ CO

CO OS CD

IC C<1 CD

N

CO CO

CO CO

(M ■9' CO

in CO ifs

CM lO CM

CO

C<1 C4

1-t o

CO OS CM

00 rH M

n

w

1—1 CO

Oi

lO g

o <->

ic r*

rH -xr I>

r*

O lO

CO r-

CM O

rH CO

1— i

■^ CO

CO CO

1-^

rt o

rf

CD -^

CO t^

1— 1

CM

r-i

ti

rH

i-{ CO

+

1 +

1 I

+ 1 1

+ 1 +

+ 1 1

+ 1 +

1

+ 1

+ +

1 + 1

+ + 1

''t^

lO CI

rH

,—..—,

t^

CD

O ^ CO

■^ O 1— 1 '^

00 o

t- CM ira CI

i6

r^

lO CO

STo ^

05 00 CD -^

CO

CO

CD vC

lO r^ r-

CD CD .—1 -^

00 CO

CO rH rH lO

t^

00

t^ CO

CO 00 .-<

C>) C^ CO O

M

t^

lO t*

<— 1 t^ CM

I-H 0>

CM CD CO lO

o

CM TT

OI lO (N

03 00 o C^

w

CD

T— (

C^ -^ CM

t— 1

^ T-H CD

r-i

~~'

5

— ' lO CO

!M O Oi

CO rH O

1

+

1 1

+ 1 +

1 + + +

+ +

+ + 1 +

+

1

1 +

1 +^

+ 11 +

CD

CD

iO CO

t^

CO

"^

CD

CD CO

lO

CO O lO

CO cs t^

Ol O CO

CM CO r-H

CO

TT O

CD

C* OS CO

C<1 t^ rH

00

f— 1 o

CO

lO CO ^

CD »0 C5

CI CO rH

rH CO O

1-^

00 CO

CO

t^ CO lO

t- -H C4

iC

rH CD

(N .-< -^

CO lO t^

f-t CI th

iO -^ CI

Tp

O -H

l~

'<P CD rH

O rH ■»

rH

O

CM .-1

rH rH CO

rH

N

--I CD

rH CO rH

(M •«> CO

i

+ 1

!

+ 1 1

1 + 1

+ 1 +

+ + +

+

+ +

+

1 + +

+ J_ +

'^ '^

CO CD

iri ic

CO CO

CO CO

1— t CO CO '-H

OC CO

r^ t- o o

I> t~~

o o

CM CJ

OS CM CI OS

c^ c^

(M CI

TT -V

CI t- t-- CI

c o

CO' CO t^ r^

CO CO

C^l CM

»0 lO

CI CI C) CM

o

N cs

o o
.— ( 1— (

CO I— * <— 1 oc

-^r '^

lO lO lO lO

CO CO

c~ t-

CO CO

CO CO

CO TT TT CO

CI r- t^ CM

CO CO

+ 1

1 +

+ 1

1 1 + +

1 +

+ 1 + 1

1 +

+ 1

I +

+ + 1 1

6 e '3 'o

+ X + X

,^ ,—t ,—{ r^

I-H rH rH I—I

^^ ^^

^-v .-^

^^ ,-^

+ + 1 1

,— N ,'~~.

+ 11 +

,'~^ /-^

^

^-^ ,— >.

+ + 1 1

s e

»-t 1— 1

s s

s s s s

CM M

s s s s

s s

1—t t-H

S S

s s s s

c

_^^

1 1

+ 1

1 ^ 1

'. '. '. '.

+ ^ 1

1 1 1 1

^

1 1

+ 1

1 ^ 1

1 1 1 1

e

s s

■ fi

S ff ?^

g

s s

• s

nH I— 1

IM 1 IM

— 1 1— 1 .— 1 T— 1

,—^r-i,—^,-^

f-^ 1—i

CM , CJ

!

+ 1

1 1

+ 1. 1

+ 1 + 1

1 1 1

+ 1 + 1

i

+ 1

t 1

+ '. 1

+ 1 + 1

^

^-S-

s e

-£•-£.-£

s s s e

^-S-S

-£■£ ~£-S

^

-E^

s c

-£.-£.£.

-E-S-ES

^

^~9 ^

^ ^

9 o o

^~^ i i i

N W N

o o o o

o

o o

^^

^~9 ^~9 ^

T^ ^ T

M°

w"m"

W°M°

i'4'i4'i4'

iKJ'lkj'lKJI^'

1^=14=14

iKqlkflKjl^

M°

WM"

Ml^

!*4'w''[^

li^WM'-T

.71

JOIOBJ

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

125

oo

r-t

+

t^ CO «

(M r~ Tr CO

C-I CO

lO CJ

V O t* r-^

CO

r-H l>

fT t^

■»• o

CO .-1

00

f-H -f

f-l t^

I-H

1 4- 1

1 1 + +

+

+ 1

1 +

lO O lO

00 U3 iO lO

CO

r-{ CD

!>• 00

O^ M 00

CO ^ C^) rt

lo

o:

O "(J*

-> o

t> f-f OJ

CO

CI CO

(N «

CO

o

1 +T

1 1 + +

+

+ 1

1 +

CO CO -^

00 t^ 00 t^

00

''f lo

1-H rH

<-H iO CS

O (N rH <M

o

CO C^l

OS iC

rH <M CJ rp

CO

M ^

CO t^

(N

00

lO

<N

I + 1

1 1 + +

+

1 1

I +

irt OS lO

W ?P O lO

t^ Lft Q CO

r-H ^ ^ Oi

b, ^

Oi b-

CO CO Oi

(M

C) CJ

05 -^

1-t t^ --I

C<J

I-H C-l

CO -^

"<9* Oi

1— 1

■V

?o

.— (

CI

I-H

i + 1

1 1 + +

4-

1 1

1 +

«0 00 UD

00 OS CO Tt<

o

O oo'

lO o

lO OO C^

t^ t^ lO lO

Oi CO

o ;=H

1-H <M t^

C^ I-H t^ -^

lo

CO t^

iO O

b- iC

t^ W

CD

Ol

CO

c^

l—<

1 + 1

1 + + +

+

1 J_^

I +

:D iC CO

CO O O 00

lO t- cfe o

oo

t^ U5

>-i \a

Oi O (>■

OO

CJ l>-

O Oi

r-H CO OV

'V O lO 05

■^

CJ CD

CO CI

<— 1 t--

CO I-H 05

i-H

p-i CI

c~-

r-i irt

CO

TT

00

•"•

1-<

1 + +

1 + + +

+

1 +

1 +

»— 1 Oi "Tl"

^ iC 00

CI

00 00

00 rH

O CN CO

lO t-^ t^

Id

o o

O 00

CO CD rH

c^ as 1-H

Qi

CO ■^ 1

N rr

CO i«

t^

irt CD

O 1

<M g

^

CO

f— 1

1 + +

1 + 1

+

1 +

1 I

to 5C

CO 00 CO Q
t^ t- .— 1 00

Ci

t^

CI ^ (

03 C^

00

Oi

CO b-

O CO

<M lO <M <-<

'T

CO

Cl r-^

(-H

f-H ifS lO CD

.— I (-H CO

C-1

O CD

CO

1 1

1 1 + 1

1

+

4- 1

oo :d O

(M ^ 00

■^

p-H TT

-^

C3 CO CO

■V 00 CI

O r-*

c

CO lO o

(M -T CO

CD r^

i3

CO t^ CJ

C-) -< r^

iC

CO CI

CD -^

1 + 1

1 1 1

1

! 7

1

C: T.

00 oo o o

Tp -fj-

r-4 1—)

iC iC

in to o o

CO CO

CO CO

i-H 1— 1

CD 05 CO CO

00 CO

(— ( i-H

1-H f-H

CO CO

+ 1

1 + 1 +

+ 1

1 +

O 0 "o "o

+ 1 + 1

rH f-H 1— 1 I-H

+ 11 +

--V ,— ^

^-^ .^^

y~^ ,— V

(M C4

e s s s

e K

1— ( i—i

+ _ 1

1 1 1 1

_^

1 1

+ 1

see
1 1 1

+ 7+7

T

I-H 1— 1

+ 1

s s
1 1

3.~E-£.

S.SS~£-

^

^^

a e

M « M

o o o <=

9

o o

— ^

o o o

o o o o

o

. o o

W M W

M l-J W M

M

i4"[4*

MW

s" j<nJM

3

03

©

a
o

•a

CI

13

3

ifi

126

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIT

Table XXXI.

n52,(')

-(n5z,('))

UDit=l"

Sin

!i-'

«»

w

w'

£4-2.J+2J

+ 294. 89

- 740. 6

+ 734

V

£+4.J+4J

- 839. 5

+ 3495

- 6224

2e4-2tJ+2i

- 147. 4

+ 517

737

n'

€ + J

£+4i>+3J

+ 1229.8

- 4069

+ 5671

f

- £+2t>+2i

+ 784

(- 3570)

(+ 10522)

£+2i>+2i

- 202

(- 1657)

(+ 13183)

E+6.J+6J

+ 2940

(- 17009)

[(+ 43527)]

2£+4>9+4J

+ 41.5

(- 2587)

(+ 6440)

2£

- 192

+ 705

riY

- £ + 2.?+ J

- 2386

(+ 11567)

(+ 37527)

£+2iJ+3J

+ 1492

(- 968)

(- 12562)

£ + 2^+ J

- 1962

(+ 9257)

(- 23263)

£+6t5+5J

- 8658

(+ 42767)

(- 92732)

2£+4J+3J

- 615

(+ 3264)

( - 6905)

2£ + J

+ 142

- 605

V»

- £ + 2l>

+ 1634

- 7081

+ 16199

£+2i>+2J

- 861

- 3794

+ 22127

£+6t5+4J

+ 6349

- 25753

+ 45318

P

- £+2i>+ i-2

+ 866

- 4260

+ 10988

£+6t>+5J-2

+ 260

- 1674

+ 5101

£+2^+2J

- 2677

+ 12681

- 30930

f

£+4,?+4J

+ 5907

- 11149

- £+4.?+4J

- 269

+ 5158

£+8^+8j

-11300

+ 76249

YV'

£+4(9+5J

-11449

+ 42212

£+4<?+3J

-11270

+ 951

- £+4,?+3J

+ 1744

- 23941

£+8t9 + 7J

+50005

-304611

7) ,'2

£+4i3+4i

+26091

- 71730

£+4t5+2J

+ 3985

+ 16118

- £+4,J+2i

- 3137

+ 35021

£+8.?+6J

-73583

+400009

,"

£+4i>+3i

-13756

+ 22165

- £ + 4^+ J

+ 3317

- 18452

C+M+5J

+36006

-172164

fri

£+4.9+34-2

- 1707

+ 13125

- £+4<? + 3J-2

- 2112

+ 15096

£+8i)+7J-2

- 2381

+ 18919

£+4.J+4J

+14204

- 88026

j» ,'

£+4t5+4J-S

- 554

+ 140

- £+4.J+2J-2

+ 3545

- 22885

£+8,>+64-2

+ 3827

- 27870

£+4.?+3J

-17503

+ 99584

+(,>-,9o)cOB

;)

£

- 767. 7

+ 2821

- 5210

ij'

£+ 4

+ 570. 0

- 2421

+ 4960

yi'

2£

- 384

+ 1410

- 2605

riY

2£+ J

+ 285

- 1211

+ 2475

f

£

- 6624

+ 47448

T?Tl'

£+ 4

[+17970]

[-120603]

1 1

- £+ J

+ 8984

- 60301

,,"

£+2i

-10478

+ 702.50

c

-25564

+ 157424

>)"

£+ 4

+ 15678

- 94846

Pn

£+ J +2

-22012

+ 121258

-359162

£

+26565

-157424

[+511232]

f 1'

£ +i'

+12048

- 76364

+251640

£+ J

-23524

+ 150306

-498328

1

1

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

127

Table XXXII.

+nSz.,(')-{niz,('))

Unlt-l

Sin

»•

ir

£+2,?4-2J

- 294. 0

4- 1036

V

£+4i3+4i
2£+2.5+2J

+ 384

+ 1679

74

- 1410

- 10348

4- 149

>)'

+ ^
£+4t?+3J

- 285

- 2460

4- 12U
4- 13057

v'

- S+2.3+2J

£+2i? + 2J

£+6t(+6J

2e+4.?+4J

2£

- 101

- 978

- 8820
+ 424

[+ 96]

- 883
4- 6459

(4- 77487)

- 1332
[- 352]

vY

- £+2^4- A
e+2^ + 3J

£ + 2^+ i

£+6^+54
2;+4^+3J
2£ + J

- 2068

- 1492
+ 228D
+ 25974

- 615
[- 71]

+ 8418
4- 2460
- 12618

(-206223)
4- 1420

[4- 303]

ri"

- £ + 2<?

£ + 2<?+2J
£+6^+4i

+ 1634
+ 861
- 19047

- 5447

4- 2933

[4-134400]

f

- £+2i>+ J -2
£+6^+5i-S
£+2<>+2J

+ 866
- 780

4- 2677

- 3394
+ 7362

- 15358

v'

£+4.J+4.if

- £+4tJ+4i

£+8^+8J

- 5098
+ 4499
+ 45200

r,W

£+4.J+5J

£+4,>+3J

- £+4i>+3J

£+8<5+7J

+ 22898
+ 5322
- 11270
-200020

Vri"

£+4,5 +4i

£+4t?+24

- £+4,>+2J

£+8,>+6J

- 52182
4- 2712
+ 4408
+294332

v"

£+4,J+3J

- £+4.?+ 4

t+M+h4

4- 27512
4- 6634
-144024

Pv

£+4,?+3J-2
- £+4,?+3^-2
£+8.3 + 7J-2
£+4,>+4J

4- 4022

- 3616
4- 9524

- 28408

P V

£+4,?+4J-2
- £+4<J+2J-2
£+8,?4-6J-2
£+4.)+3J

4- 1108
4- 7090
- 15308
4- 35006

7)

i'

The coefficients in parentheses differ from v. Zeipel's values because they contain additional
terms. See p. 134.

128 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

The remaining terms in the differential equation for ndzy" are, by eq. (143),

|^{n52 w- (7i52 W)} = (1 _e COS £)[lf/(^) + n^p)-|(F/')-i =•/')¥ tfi(') + ^=-.('))j

-[(1-e cos e)[F/(')+r^F)-|(^FW-i^/'))(F,'» + ^^,('))n

aU the terms of which are of the second order whose lowest rank is the second. They therefore

contain the factor — ^ •
vr

To obtain ti^s/^' it is necessary to return to eqs. (124)-(130) and make developments for

terms of the second order similar to those for first order. The resulting differential equation is:

(1-e COS£)r_^_,„ ,„^. „n,Ail?(

f'NQ ''1 ' ' ^V-*- t> WO C y #V M/ OXXJ.C B, o

,ndz^(2) ^ ^' «^"^^' |^^2,(" - (n52,(')) }^, IF/" - (1 - e COS £)>) w sim.^. F/^'

_ra__e^os_£) ^^^^^(,) _ (^^^^(,)) } 1 17^(1) -d-e cos £)j? m; sins^^ F/^'l
-^[(1 -e cos 0 F.("]^{7i52.- (nfe/-))} -|^(n5z,«)-

The sum of the last two equations, when integrated, gives the terms of second order having

Factor — ~

ur

Ivticallv that

the factor — j- It has been shown by v. Zeipel through computation and we have shown ana-

and

Therefore,

d
de

F/(^'+^{n52/')-(naz,('>)}^F/" = )j w sins^^ F/^)
[{l-e cos e) W')]^ {n52i<"- (ri,fe/»))} +w^(7i(52i<=') =0-

{w5z.(=) - (71^2/=') +n32,^'^} = (1 - c cos £) j[ F,](=" - 1/ F/') - 1-^/")( l^V" + -J.= /")

-[(1-. cos ^)jn^F-|( W')_|^,o))(^F.(')+is/'))j]

The integral is tabulated in Table XXXIV.
Summarizing, we have included first order terms in

ndzi + ndz^ + ndz^

given by tables XXX, XXXI, XXXII and second order terms in

ndZi + ndz^

given by Tables XXXIII and XXXIV. The addition of Tables XXX-XXXTV gives the short
period terms in ndz, or, the function

ndz — \_nSz\
which is tabulated in Table XXXV.

Returning now to the differential equation for ■d, the evaluation of F (i?, s) and its derivatives

in Z 78, eq. (127) gives Z 91, eq. (146). The variable does not appear; -j- is a function of 6 alone

Therefore the function is of long period. The integration is one step in the determination of
\ndz\, the long period terms in the perturbations of tlie mean anomaly.
'(1 — e cos £) W\ is tabulated in Table XXIX6.

The function
The function

{\—e cos c

is given in Table XXXVI.

i)( IF- ^S'Y F+ \^S\ computed from Tables XXIXo. and XXIXc,

So. Z.]

MINOR PL.\NETS— LEUSCHNER, GL^VNCY, LEVY.

129

The function (1 -e cos e){0^ + 02 + 0,) -^ is computed as follows:
First, d-^ = Oi + 0, + 0,

is given by Z 93, eq. (150) i)y means of Table XXXV, and -^rj- is readily written b^' inspection

ou

of Table XXIXa. Performing the indicated multiplications and retaining ordy the terms which
are independent of s, we have the required function as tabulated in Table XXXVTI.

By eq. (140), the sum of Tables XXIX6, XXX\7, and XXXMI, multiplied by the factor

i^, gives 0W, tabulated in Table XXX\^II.

Table XXXIII.

(n52,(2))

Unit-l"

Sin

«•-»

»«

«•

«)'

1

e+4t>+4J

- 0. 316

+ 1.59

-3.6

>}'

£+4,3+3J

+ 0. 114

- 0.67

+ 1.8

v'

£+2.J+2J

£+6^+6J

2:+4i.+4J

+ 2.62
+ 4.42
+ 1.80
+ 0.16

- 16.8

- 23.4

- 11.7

- 0.8

+ 1.8

■0 1'

- £ + 2,?+ J

£+2.?+3J

£+2,54- i

£+6i>+5J

2£+4»+3J

- 6.18

- 1.90

- 5.57

- 3.95

- O.OG

+ 36. 9
+ 13.6
+ 32.8
+ 23. 6
+ 0.3

- 0.9

n"

- £ + 2l>

£+2^+2^
S+6.3+4J

+ 4.04
+ 2.12
+ 1.90

- 21.4

- 14.4

- 10.8

P

- £+2.?+ J-2
£+6i>+5J-2

+ (l>-l>o)c08

+ 0.22
+ 0.07

- 1.6

- 0.5

n

£

- 1.265

+ 6.35

-14.3

i'

£+ A

+ 0. 455

- 2.69

+ 7.2

i'

It

+ 0.63

-3.2

+ 7.2

V 11

2£+ J

- 0.23

+ 1.3

- 3.6

V

£

-23.8

+222

,»,'

- £+ J

+72.9
+36.5

-569
-285

■J-?'"

£ + 2J
£

-55.2
-87.3

+375
+653

r/»

£+ J

+69.9

-439

)\

£+ J + i-

-9.9
+23.1

+ 77
-166

J' rt'

+ 5.2
-14.8

- 45
+112

m"

110379°— 22-

130

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XXXIV.

Unit=I"

Sin

KI-'

a?"

w

w

- 0. 614

- 0.079

+ 4.06
+ 0,40

-10.3

')

£+4.?+4J
2E+2J+2J
2£+6.?+6J

- 0.74
+ 1. 74
+ 0.31
+ 0.45

+ 3.7
-IS. 1
- 2.0
-2.9

v'

£+4^+3i
2£+Ci?+5J

+ 0.30

- 4.26

- 0.66

- 1.8
+32.0
+ 3.8

n'

- £+2^+2i
£+2<J+2i

2£+4i?4-4J
2£+8i>+8J

-6.4
+ 6.4
+ 5.1

- 1.4

- 2.2

■^i

- £+2;5+ A

£+2<?+3J

£+2^+ A

£+6i3+5J

2£+4^+3J

2£4-4i?+5J

2£+8^+7J

+12. 0

- 0.9
-8.5
-11.8
+ 3.4

- 0.8
+ 6.5

n"

- £+2<?

£+2.?+2i

£ + 6!?+4i

- 5.1

+ 1.3
+ 6.4

J'

- £+2,?+ i-2
£+6i?+5J-S
2£+40l+4i

+ (,5-<?„)cos

- 0.3
+ 0.3
+ 1.4

n

£

£+4iJ+4J
2£+2^+2i

- 1.02

- 0.78
+ 0.41

- 8.4
+ 6.0

- 2.5

Y

£ + A

£+4.?+3J

2£+2<>+3i

- 3.25
+ 0.58

- 0.31

+30.1
- 4.8
+ 2.1

f

- £+2^+2J
£+2<?+2i
2e
2£+4J+4^

+ 3.6
+ 1.1
+ 0.5
- 0.8

vv'

- £+2<>+ A
2£+ A

-3.4
+ 1.6

V

£

- 0.36

+ 2.6

■n'

£+ A

+ 0.27

- 2.1

to'2

No. 3.]

Logarithmic.

MINOR PI^\NETS— LEUSCHNER, GLANCY, LEVY.

Tai.le XXXV.
n6z — [nSz]

131

Unit=l

Sin

«)-«

ra-2

u'-'

„>0

w

tt»

vY

h+ 0

4. 1570

4. 8741„

h+ ^+ i

2. 7684„

3. 3827

3. 7172„

n'

h+ .5+ i

4. 0056„

4. 7686

l"

h'-+ .5+ 4

4. 0760„

4. 8295

r-

4-'+ «+ d

4. 1365

4. 87.38„

rin'

h+ ''+2J

3.3345

4. 5162„

r,'

A-+3.3+2J

4. 2240„

4. 961 1

5, 668.5„

1

*r4-3i?+3J

4.0671

4. 8483„

5. 563G

n"

i;+5.3 + 3J

5. 0926„

6.0018

ir,'

i£+6t3+4J

5. 2325

0.]714„

r,'

hz+bO+bA

4. 7675„

6. 7344

r-

iE+5>9+4J-J'

3. 8050„

4. 7998

n'

-if+ <>

3. 3112

3. 8350„

4. 1355

V

-¥-+ .5+ A

3. 2065„

3. 7910

4. 0833„

v''

-ij+3i?+ J

3. 5338

4. 6236„

Vr,'

-i.-+3^+2J

4. 0879

5. 0382

v'

-i£+3t5+3J

3. 6012„

4. 5318„

f

-ij+3!?+2J-i'

3.2074 ■

4. 1925„

1

c

9. 868„

0. 5689

2.922

3. 4600„

3. 3670

n'

e+ ^

9.482

0. 2533„

2. 673„

3. 2959

3. 1772„

rii

£+2tS+ J

0. 746„

[1. 384]

3. 2927„

[4. 14906]

[4. 6990„]

£+2.?+2J

9. 788„

2. 47560

3. 1084 7„

3. 4540

[3. 3960„]

ri'

£+2,3+2i

0. 645

[1.342„]

2. 305„

[3. 6179„]

[4.4018]

i^

S4-2.5+2J

0.326

1. 119„

2. 935„

3. 3017„

4. 39206]

P

£+2^+2J

3. 4276„

4. 23764

4. 76933„

^i

£+2>'>+3J

0.28„

1.102

3.1738

[3. 5449„]

[3. 8446„]

in"

£+4-5 +2J

3. 6004

4. 27485

l'

£+4i»+3J

9.057

0. 692„

3. 10161

3. 9302„

4. 52415

[4. 78162„]

vW

£+4i( + 3J

4. 0519„

3. 7975

n"

£+4>?4-3J

4. 1385„

4. 6961

P V

£+4.3 +3J

4. 2431„

5. 1290

1

£+4i3+4J

9. 500„

0.522

2. 9351„

3. 8035

4. 41616„

4. 63017

n'

£+4i)+4J

3. 7714

4. 2108„

iV"

£+4;3+4J

4. 4165

5. 0931„

Pi

£+4(3+4J

4. 1524

5. 0661„

vW

£+4t3+5J

4. 0588„

4. 8136

Pri

£+4,J4-3i-J

3. 2322„

4. 2342

P V

£+413+4^-^

2. 744„

[3. 0962]

r,"

£+6,3+4J

0.28

[0. f,4„]

3. 8027

4. 77998„

[5. 52852]

vY

£+6i3+5J

0. 596„

1. 070]

3. 9374„

[4. 94342]

5. 70347„]

v'

£+6>J+6J

0.255

[0, 8„]

3. 46S4

4. .50125„]

5. 27451]

P

£+6>3+5i-Z

8.8

9.3„]

2.415

3. 4823„

4. 29311

i'

£+8i3+5J

4. 5564

5. 4999„

nr,'^

E+8.3+6J

4. 8668„

5. 8416

fn'

£+8.3+7J

4. 6990

5. 7030„

f

£+8,3+8J

4. 0531„

5. 0844

P 1'

£4-S.?+6i-i'

3. 5829

4. 6352„

Pri

£+8,3+7J-J

3. 3768„

4. 4540

1"

- £+2^

0.606

1. 422«]

3. 2132

3. 6657„

3. 9260

vY

- £+2.3+ A

0.791„

1. 690]

3. 3777„

3. 8866

4. 72168

v'

- £+2.3+2J

0.418

1. 36.5„1

2.894

[3. 4616„]

3. 8078

p

- £+2.3+ J-J

9.34

0. 28„ "

2.938

3. 4714„

3. 7862

r)"

- £+4.3+ A

3. 5208

4. 07255

'?'?"

- £+4.3+2J

3. 4965„

4. 59582

riW

- E+4.3+3J

3. 2416

4. 5467„

f

- S+4.3+4J

2. 430„

3. 9848

P r,'

- £+4.3+2J-J-

3. 5496

4. 19852„

Pr,

- £+4i>+3J-i'

3. 3247„

4. 05994

132

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Logarithmic.

Table XXXV— Continued.

n5: — \ndz]

Umt=l".

Sin

M)-'

W-2

w-i

w'

1"

w

'!')'

*£+3.?+2J

3. 6731

4. 0029„

4S+3J+3J

2. 3528

3. 247.5„

3. 9005

v'

4£+3.?+3J

3. 6181„

4 21 '^2

f

f£+3-?+3J

3. 4072„

4 4000

vY

f£ + 3!? + 4J

3. 5244

4. 4012„

r{

^£+5i?+4J

3. 3533

4.4231„

5. 2725

■Q

^;+5.54-5J

3. 1780„

4. 2730

5. 1359„

\"

4J+7.5+5J

4. 2775

5. 4708„

^i

|r+7i?+6J

4. 40.51„

5. 6177

f

f£ + 7!?+7J

3. 9296

5. 1605„

T)

2£+2iJ+2J

9.486

2. 1744„

2. 708

[2. 889„]

2. 599„

\'

2E+2.5+3J

1. 946„

2.501

2.516„

vY

2e+4t?+3J

8.8„

[0. 5611

2. 789„

[3. 5813]

[4. 1074^]

2c-+4i?+4J

8.90„

9.599

1.711

2. 5795^

3. 1726

f

2£+4i?+4J

9.2

[0. 34„]

2.618

[3. 4962„]

[4. 0890]

Y

2£+6t?4-5J

9. 819„

0. 5840

2. 7821

3. 7794„

4. 51865

^

2£+6,?4-6J

9.653

0. 4645„

2. 5979„

3. 6265

4. 38424„

4£+5^+5J

1.2340

2. 1166„

2. 7076

i

4J+7.J+6J

2. 3679

3. 3518„

4. 0.587

V

4-c+7>?+74

(l> — i3o) COS

2. 1758„

3. 1926

3. 9204„

jj

£

0. 1021„

0.728

2. 8978„

3. 4504

3. 7168„

ri'

e

1. 377„

[2. 3461

3. 8211„

4. 6762

nri"

e

1. 941„

2.815

4. 4076„

5. 1971

A

£

1. 364-

2. 220„

4. 4076

5. 1971„

5. 7086

,'

£+ ^

9.658

0.774„

2. 7836

3. 3840„

3. 6946

,Y

£+ 4

1.863

2. 755„

4. 2546

5. 0814„

^

e+ ^

1.844

2.642„ ■

4. 1953

4. 9770„

j^ }

£+ ^

1. 170„

2.049

4. 3715„

5. 1770

5. 6975„

vY-

£+ 2J

1. 742„

2.574

4. 0203„

4.8466

fV

£+ ^

0.716

1.65„

4. 0809

4. 8829„

5. 4008

w

£+ 4+^

1.00„

1.89

4. 3427„

5. 0837

5. 5553n

,w

- £+ 4

1.562

2.455„

3. 9535

4. 7803„

f

2e

9.801

0.43„

2. 5842

3. 1493„

3. 4158

?»'

2£+ 4
(,>-.?(,)» sin

9. 357„

0. 473

2. 4548„

3. 0830

3. 3936„

I)

s

9.56„

0.42

^/

£+ ■i

9.43

0.32„

n5z-[n52] = J'u)*i)Pi)'?f<C, ain l\.Tg.-\-(9 -»a)Sw»TiPTi'1ftC2 cos Arg.+(«»-tfo)'i"«^'j''')'»f* Cj ain Arg.
where C^, Cj, C3, represent the respective coefncienta.

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY

Table XXXVI.
-|-[(1 -« cos £) (W- J- E ) (W+^ H)]

133

Unit™ 4th decimal of a radian.

Cos

tr-1

W-'

W-'

«>-'

«'»

w

+0. 000032

-0. 0080

+0. 04,93

- 0. 176

+0.52

>)'

-0. 00028

+0. 0037

-0. 133

+ 1.10

-8.8

ri"

-0. 00014

+0. 0026

-0. 095

+ 1.27

-14.4

P

-0. 0003

+0. 139

-1.20

+ 5.9

vv'

J

+0, 00047

-0. 0070

+0. 252

-2.51

+22.8

n

2i>+2J

+0. 000017

-0. 00042

+0. 0437

-0. 366

+ 2.10

i

2tJ+ J

-0.000000

+0. 00045

-0. 0639

+0. 508

- 2.79

r?

4i?+4J

+0. 00004

+0. OOOU

-0. 194

+ 1.64

-11.4

vv'

4^+3J

-0. 00012

-0.0012

+0. 372

-3.59

+32. 2

ri"

4^+2J

+0. OOOU

+0. 0003

-0. 252

+2. 40

-19.8

P

4<J+3J-2
+ (!? — (>o'l sin

+0. 00001

-0.0001

+0. 032

-0.19

vv'

J

-0. 00004

+0.010

- 0.08

1?

2(>+2i

+0. 000066

-0. 00060

+0. 0399

-0. 275

+ 0.94

v'

•20+ J

-0. 000024

+0. 00047

-0. 0296

+0. 221

- 0.81

/

4,?+4J

-0. 00023

+0. 0028

-0.114

+ 1.02

-4.7

^v

4.)+3J

+0. 00039

-0. 0053

+0. 251

-2.20

+ 9.9

n"

4tf+2J

(t>-.Jo)'<'08

-0.00011

+0. 0024

-0. 124

+ 1.11

- 5.1

f

-0.00017

+0. 0014

-0. 052

+0.38

- 1.4

iv'

.i

+0. 00019

-0. 0021

+0. 077

-0.61

+ 2.4

r,"

-0.00005

+0. 0008

-0. 029

+0.24

- 1.0

m'3

m'^

m", m'-

r-/-

m'2

m"

Table XXXVII.

r(0-.5)(l-ecos£)|j']

Unit=4th decimal of a radian.

Cos

ii'-<

IP-'

UI-2

w-i

u-«

w

w'

+0. 000042

-0.01071

+ 0. 0883

- 0. 402

+ 1.31

-3.9

r,'

-0, 00043

+0. 0056

-0. 189

+ 2. 73

- 51.3

+ 299

ri"

-0. 00021

+0. 0048

-0. 296

+ 4.47

- 59.8

+ 416

f

-0. 0004

+0. 186

- 2.00

+ 11.7

- 40

nn'

A

+0. 00076

-0.0110

+0. 530

- 7.59

+ 104.2

- 682

V

2.'+2J

+0. 000055

-0. 00086

+0. 1005

- 1. 153

+ 21.86

- 81.5

+217

r)'

2>5+ A

-0. 000020

+0. 00090

-0. 1377

+ 1.463

- 9.50

+ 44.2

-176

ri'

4.)+4J

-0. 00031

+0.0041

-0.477

+ 6.49

-133.8

+ 708

vn'

4.J+3J

+0. 00068

-0. 00!^4

+ 1.295

-17.43

+261. 3

-1266

n"

4,5 + 2J

-0. 00030

+0. 0041

-0. 921

+ 11.58

- 95.2

+ 452

i'

4.J+3J-S
+(<>-<?„) an

-0. 00001

+0. 0001

-0. 036

+ 0.58

-5.3

+ 25

^n'

A

0.00000

-0.0004

+0. 052

- 0.44

+ 2.0

n

2d +2 A

+0. 000044

-0. 000,i2

+0. 0266

- 0.212

+ 0.83

v'

2d+ A

-0. 000016

+0. 00038

-0. 0197

+ 0. 170

- 0.-0

f

4J+4J

-0.00031

+0. 0037

-0. 153

+ 1.62

-7.9

')'?'

4,3 + 3J

+0. 00052

-0. 0072

+0. 335

- 3.40

+ 16.9

-)"

id+2A

-0.00015

+0. 0032

-0. 165

+ 1.68

-8.6

m's

m'3

m'^ 7n'2

m'»

m'^, m'

m", m'

m", vi'

134

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Logarithmic.

Table XXXVIII.

Unit= 1 radian.

Cos

UJ-fi

w-s

W-*

u,->

w-»

10-1

!/'»

w

w'

1.5

[3. 909J

4.960

6. 6748„

7. 2764

7. 540,

7.31

v'

2.0

[4. 644„]

[5. 160]

[6. 1.50]

[8.048„]

8.838

8. 655„

8. 100„

., '

1.9

[3. 41„]

[4. 75„

[6. 509]

8. 2077„]

[8. 994]

8. 919„

f

2.8f3„

5.146

[6. 299.,]

7. 994]

8. 740„

[8, 656]

^r,'

J

2.34„

[4. 446]

[4.57]

[6. 728„]

8. 4022]

9.1999„

9. 0854

8.079

,,'=

2!?

1.6

[2. 6„]

5.744

6. 535„

8. 3811

9. 103] „

9,0128

y,

20+ J

0.8„

[3. 068]

[5. 2988]

7. 2212„

[7. 3772]

[8. 0372]

[8. 764„]

8.068

7)^

2.J+ J

2.32„

3.30

5. 886„

6.718

8. 5059.„

9. 2804

9. 201 7„

Y'

2t?+ A

i

5. 301„

6.149

8. 2302„

9. 0]54

8, 938„

f ri'

20+ A

8. 5592

9. 3245„

9. 2428

r,'

2,>+2J

2.48

3.40„

5.422

6. 292„

7.476

8. 664„

8.636

V

2iJ+2J

1.22

[2. 94„]

[5. 1206„]

7. 6416

[7. 9638„]

[7. 083„]

[8. 645]

8. 582„

rir,'^

2,5+2J

1.9

[3.0„]

5.442

6. 328„

8. 0915,„

8. 630„

8.742

fri

2!J+2J

8. 5904„

9. 3489

9. 8024„

9. 6532

vY

20+ZA

2.04„

3.00

4.98„

5.89

8. 0326

8. 1973„

7.69

fr,

20+ A-S

4.51

5.42„

8. 101]

8. 873„

8.792

}' V

20+2A-S

4.04„

5.00

6.89„

8.182

8. 158„

i'

iO+2A

[2. 6<>„]

[2.7]

6. 1031

[8. 4188„]

[8. 5297]

[6. 0]

7.90„

r>Y

AO+-iA

[2. 72]

f4. 369]

6. 2526„

8. 5594

8. 7988„]

7. 94„]

8.287

8.210

■f

i&+4A

[2. 20„]

[4. 624„]

[5. 824]

[8.0924„]

8. 4338]

7.24]

7.74„

8.044„

}'

AO+ZA-I

1.5„

2.45

4.68

7. 1747„

7.301

8.111

8. 127„

r

(,0+iA

5. 301„

6.149

9. 1294.„

9. 7728

9. 0609,,

rtr

6i?+4J

5.92

6.74„

9. 4432

0. 14644„

0. 05077

7,Y

%0+bA

2.0„

3.0

5.93„

6.79

9. 2774„

0. 03298

9. 9494„

f

&0+&A

2.0

3.0„

5.420

6. 292„

8.634

9. 4351„

9. 3608

P Y

&0+iA-2

4.04„

5.00

8. 272„

9. 1028

9. 0334„

fv

QO+hA-I
(0-0^) Bin

4.51

5.42„

8. 0554

8.926„

8.864

vV

A

[2. 60„]

4.71

5.94„

6. 507„

G. 606

r/

20+ A

1.36

[2. 48]

4.49

[.5.255„]

[5. 51]

5. 25„

V

20 +2 A

1.82„

[2. 42]

4.64„

5. 3.50

[5. 5i„]

[5. 16]

n

A0+2A

2. .34

[3. 00]

5.392

[6. 179^]

6. 528„

6.665

vv'

AO+ZA

2.89„

[3.46

5. 702„

6. 467]

6.851

6. 979„

f

AO+AA
(t>-.?o)2co8

2.66

[3.459„]

[5. 357]

6. 127„]

6. 530„

6.653

Y

2.08„

2. OS

5. 546„

5.546

'

2.54

2.54„

5. 396„

5. 396

■on'

A

2.5„

2.5

5.776

5. 776„

m'3

m'3

m'3, ni'2

m'^, m'2

vi'\ m'

71!'^. m'

m'^, m''

to'2, m'

m'^ m'

=IwSriPr)'Qf'Ci cos Arg. +(0-Oo)IwSTjP7j'qftC„ sin Arg. +{0 -Ooyiw^riPq'QJ'td cos Arg.
where C^, (\, C^, represent the respective coefficients.

COMPARISON OF TABLES.

Table XXX. — With the aid of the manuscriiJt the source of all the discrepancies indicated
by brackets has been traced. Coefficients in parentheses are functions of coefficients in paren-
theses in Table XXVII.

Table XXXI. — The function was computed by the fii'st of Z 81, eqs. (137), which is more
rigid than the one following it, which v. Zeipel used. Aside from the addition of omitted terms,
the bracketed coefficients are more accurate by reason of the errors in v. Zeipel's Table XVIII.

Table XXXII. — The computation was performed according to Z 82, eqs. (139) and (140),
in place of eq. (141) which is less rigid. Besides the discrepancies due to the addition of omitted
terms, four bracketed coefficients are of opposite sign. These discrepancies may be due either

No. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 135

to a numerical error or to the number of terms included. The remaining discrepancy is due
to slight inaccuracies of v. Zeipcl's computation.

Table XXXI 11. — The discrepancy in this tabh> foUows from one in Table XVIIL Third
degree terms in Tabic XVIII were not integrated because, in the aggregate, they amount to
very little.

Table XXXIV. — Our table is more extensive. Second degree terms are, however, not
complete, for they do not include second degree terms in

[1J2] cos s + [22] sin e

The discrepancies are of no importance.

The integration of cq. (146) is best performed individually for each planet. The analytical
developments are as follows:

The differential ecpiation can be WTitten

^'^ /ft/oN^Y"' \ , '^/'"^ r >;} n\

By a change of variable

Writing

'^ ^..my '^"^

we have Z 96, eq. (152), in which the last term can be neglected.

For a given planet the factors w, 1/, f and the argument i are known constants. There-
fore 1 +<^ («?) can be expressed as in eq. (153), as a Fourier series of sines and cosines of mul-
tiples of 2i>, in which the nontrigonometrical term is designated by <r.

Expressing eq. (153 ) in terms of exponentials and solving for di-^e — {11 dz'] j by the expansion

of {1 +<P («?)}"', and reintroducing the trigonometric functions, we have the equation
following eq. (153), in which the nontrigonometrical part is taken outside the brackets as a
common factor. The brackets in this equation do not have the special significance which
they have had previously.

The variables e and i? are now separate and the integration can be performed. Trans-
ferring the common factor to the left-hand side of the eauation. nerforming the integration
and adding

n ,
— c — c

n

as the constant of integration, we have the argument 1^ expressed as a function of i? in eq. (154),
where ^ is defined by eq. (155).

The reversion of the series gives ^ as a function of ^. We have by eq. (154)

'^ = C + -^^cis"'^
where ~0 is a small quantity. Given

z = w + (r«l> (s) , where « is small,

we have, by a theorem of Lagrange,

F^^) = F(w)+a0(^v)F'(w) + ^^ ^[{<P{w)yF'iw)]+- ■ ■ ^^„[{(P (w)}''*'F' («;)]+ ■ • •

By means of this theorem eqs. (156), (157) can be derived, where it is to be noticed that

(^— gC + c') is an approximation for i!^ — !^^). In our developments we have used (C — Co)-

136

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

IVol. XIV.

If in Z eq. (155) we add and subtracu -g-e — [n'os'])
1

C =

Substituting this value of f in eq. (156),

(^c-[n'd2']) + '^c-c'+(j.-[n'dz'])

,>-|e + K.V]-^c + c' = — f

(|.-[.'.V])

+ Series

1+2(^^ + 5,^)

Substituting the last equation in eq. (145), we obtain Z 98, eqs. (159), (160), and (161). In
eq. (160) the factor (c — c) is an approximation for - (^ — Zo) '> in our work we have used the latter.

Since [ndz\ is the series in eq. (156) multiplied by the factor

-; >

1 — W

0 = ^[ndzl+{:

Table XXXV. — With the exception of the two coefficients under the heading w', all the
bracketed quantities are functions of other coefficients in parentheses or brackets, or they are
functions of additional terms. The two coefficients excepted seem to be in disagreement through
some numerical error by v. Zeipel.

Table XXXVI. — Since the mass factors have not been kept explicit, it may be well to remark
that only the zero degree term of third order has been included under the heading w~^.

The bracketed quantities are numerous. Aside from the accumulation of discrepancies
already discussed, the disagreements are to be attributed, in general, to the relative extent of
the computations. It is found from computation that as the number of terms included in a
product is increased the resulting coefficient for a given argument is numerically larger. For
the most part our values are larger than v. Zeipel's. Hence the discrepancies are explained by
assuming that our computation is more extensive. On the other hand, the function is com-
puted much more accurately than is necessary, and many of our disagreements are less important
than they appear to be.

Table XXXVII.— The comparison of Tables XXXVII is similar to that for Tables XXX"\T[
with the exception that our values are not, in general, numerically larger. Some are larger
and some are smaller. Below are brief tables showing to what extent we used the necessary
series. The 0, 1, 2 signify the degrees of the terms included.

l^{,nlz-[nSz]]

w-<

w->

w'

w

w'

w'

w'

w

w

0

1

2

0

1

2

0

1
2

0

1

0

1

2

0

1
2

0

m'

m'^

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

137

^{(1 -e cos £) W-[{l-e cos e) U'j}

IffJ

»-•

UJJ

UlO

w

IC«

»»

U)0

w

U,J

w'

w

w'

Id'

0

1
2

0

1
2

0

1
2

0

1
2

0

1

2

0

1
2

0

0

1

0
1

0

1

m'

m'»

m"

Table XXX\TII. — All the l)racketed quantities probably contain only the accumulation
of the discrepancies in Tables XXIX6, XXXVI, and XXXVII. This is a very important
table, and it is from differences in (P (t?) that the perturbations may be expected to differ most.

PERTtJRBATIONS OF THE RADIUS VECTOR.

1«

If TFand ^ E are tabidated and the computation is performed in duplicate, it is not necessary

to make the long developments and the auxiliary tables in Z §(3, 99-114. For this reason the
formulae in §6 have not been checked and the list of errata does not cover this section.
The essential formulae are giA'en in Z 99. By Z 7, eq. (3G),

,= _lTf-is+iw^+

• ■^=m+^p (o,+o, + o,+

) +

In order to parallel the form of ndz, we write

where (di+d^+d^) is given by Z 93, eq. (150)

Hence the computation proceeds as follows: the perturbation is computed by eq. (36),
the argmnent 0 is replaced by t?, and a corrective term which is the product of (di + d^ + d^)
and the derivative of the function with respect to i? is added. The perturbation v is then
expressed as a function of i?. It is tabulated in Table XLIII.

Table XLIII. — If there are no errors of calculation in the construction of the table, all the
discrepancies are due to the accimaulation of other discrepancies previously discussed.

The perturbation v =f{6) includes

tc-i

«•->

IC«

w

at

w"

too

w

U"!

0

0

0

0

0

0

0

1

1

1

1

1

1

1

2

2

2

2

2

3

3

3

3

m'

m'-

where the tabulated numbers signify the degrees of the terms included and where only IF, and E,
are inclusive of third degree.

138

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table XLIII.

Loearltbioic.

[Vol. XIV.

Unit=l"

Cos

iir'

ur'

ur'

w°

w

w'

8.72

[9. 88„]

1. 6349

2. 1070„

2. 2333

')='

9.80

[0. 212„]

2.759

3. 4922„

r,'^

8.9

9,23

2.937

3. 6295„

P

2. 937„

3. 6295

vv'

J

9.66„

9.78

3. 1136„

3. 8440

vv"

29

0. 556„

1. 204

3. 2111„

3. 7970

k

2!>4- i

0.504;,

2. 3472

2. 456„

2. 686„

3. 4735

riW

2<>+ i

0.997

1. 711„

3. 6.559

4. 3103„

^

2i?+ A

0.438

1. 220„

3. 3654

4. 0763„

P V

2S+ i

3. 6975„

4. 3810

ij

2i)+2J

0.438

2. 952„

3. 2.529

[3. 0689„]

3. 3979„

f

2,?+2J

0. 732„

1.497

3, 2410„

4. 0643

U"

2.?+2J

0. 772„

1.589

3.4136

4. 0723

iV

2>?+2J

3. 9048

1. 5649„]

4. 9303

riW

2^+3J

0.505

1.344„

3. 4757„

2, 783]

fn

2,?+ J -I

9. 33„

0.15

2. 938„

3. 58C50]

j' Y

2i(+2J-J

9.20

0.10„

2. 0251

3. 2961„

^

4,)+2J

8.9

1. 2819„

3. 5514

3. 6173„

3. 8147

■J >?'

4^+3i

9.75„

[1. 5024]

3. 7885„

[4. 1394

4. 3110„

ri'

4,? +4 J

9.98

[1. 1342„]

3. 4007

[3. 9091„]

[4. 1480

f

4<J+3J-2

9.64„

2.305

2. 542„

[2. 749„

l"

6^+3J

0.438

1. 220„

4. 2675

4. 7993„

vY'

6^+4J

1. 125„

1.862

4. 6479„

[5. 2324]

■ vW

6^+5J

1.198

1. 947„

4. 5397

5. 1768„]

v'

6^ + 6J

0. 732„

1.508

3. 9457„

4. 6328]

r- r,'

6,?+4J-J

9.20

0.10„

3. 4099

'4. 1710„

pr,

6/)+bJ-j:

9.70„

0.56

3. 2601„

[4. 0542]

r)v'

is+ 0

3. 4878„

4. 1106

4-'+ a+ A

8.3„

2. 2106

2.7I79„

2.919

f

if+ 0-\- A

3. 5709„

4. 2261

ri'

^-+ 0+ J

3. 4.507

4. 1296„

Y'

h+ >?+ ^

3. 5100

4. 1837„

Vr/

i=-+ i9+2J

2, 579„

3. 9270

v

A.'+3i»+2J

0.08

3. 6873

4. 1471„

4. 7839

fl

i£ + 3!?+3J

9.5

3. 5727„

4. 1511

4. 7545„

ri"

i£+5^+3J

4. 5568

5. 1414„

riV

ic-+575+4J

4. 7261„

[5. 4067]

n'

i£+5t?+5J

4. 2862

5. 0418„]

P

i£+5J+4J-J

3. 2570

4. 0005„

r/

-if+ -J

1. 086„

2. 7090

3. 3467„

3. 7098

V

-J^+ ^+ J

0.88

2. 1967„

3. 0952

3. 5836„

ri"

-ij+3^+ J

2.514

4. 1049„

vf

-i£+3i?+2J

4. 0853

[3. 9122

f

-j£+3,3+3J

3. 804I7J

[3. 8118'

P

-is-i-Z9+2A-I

2.416

3. 6926„

1

£

9.62

0.58„

2. 143„

2.682

2. 9151„

r,'

£+ J

9.04„

9.9

2.061

2. 666„

2. 9477

riV

£ + 2.?+ J

0.444

1. 1661„

3. 0588

3. S035„

[4. 2554]

£+2i3+2J

9.487

2. ]744„

2. 7280

2. 972„

2.976

Y

£+2,J+2J

0. 344„

1.1143

2. 692„

[3. 5334]

[4. 0772„]

Y'

£+2^+2J

0. 025„

0.828

2.634

3. 0726

4. 0416„

P

£+2i?+2J

3. 1265

3. 880fi„

4. 3473

rin'

£+2^+3J

9.98

0. 811„

2. 873„

3. 1697

[3. 5856]

vv"

£+4i?+2J

1.105

1.89„

2.864

4. 3477„

r,'

£+4^+3J

8.8„

0.398

2. 8000„

3. 5327

4. 0065„

4. 3207

vW

£+4,?+3J

1. 260„

2.083

3. 0931

4.4160

V

£+4.>+3J

3. 8375

4. 0446„

P n'

£+4^+3J

0.267

1. 15„

3. 9421

4. 6972„

1

£+4>J+4J

9.19

0. 248„

[2. 6356]

3.4317„

3. 9469

4. 2558„

v'

£+4,?+4J

0.774

1.66„

3. 0934„

[3. 7866„]

Vri"

£+4,?+4J

4. 1154„

4. 5547

pr,

£+4,9 +4 J

0. 455„

1.32

3. 8518„

4. 6436

r,W

€+4il+bJ

3. 7579

4. 3244„

Xo. 3.]

Logarithmic.

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

Table XLIII— Continued.

139

Unit-l"

Cos

w->

«r->

w-i

U)0

w

«)»

f^

S+40+ZJ-I

3. 0030

3. 8869„

/■' -i'

s+4t9+4J-^"

2. 4425

1. 85„

r^

£+(ii?4-lJ

9.9S„

0.480

3. 501fi„

4. 3723

4. 9952„

n V

£ + (il? + 5J

0. 29IJ

0. 823„

3. G3C9

4. 5582„]

5. 20!)3

f

e+dd+GJ

9. 95„

0.538

3. 1C85„

4. 1334]

[4. 8131„]

f

e+Gd+5J-l'

8.5„

[9. 15]

2. 114„

3. 0881

[3. 7886^;

,"

e+8.)+5J

4. 2554„

4. 9349

,,'»

e+8iJ+6J

1.320

2. 152„

4. 5657

5. 3010„

riY

£+8^+7J

1. 228„

2.093

4. 3995„

5. 1827

r)'

£+8(5 +8J

0.648

1.54„

3. 7543

4. 5812„

f l'

£+8,?+GJ-J'

3. 28I8„

4. 1442

fr,

e+so+rj-s

3. 0763

3. 9759„

ri"

- £+2,?

0. 305

[1. 1007„1

2.912

3. 4958,,

[3. 8151]

vn'

- i+2d+ A

0. 490„

[1. 3330J

3. 0160„

[3. 7273]

4. 3119

f

- £+2.5 +2 J

0.117

0. 982„

2. 288„

[3. 2375„]

3. 7892

f

- £+2,?+ A-S

9.04

9. 96„

2.636

3. 2817„

3. 6568

,"

- £ + -l,5+ J

3. 2197

3. 9G50„

vv"

- £+4<5+2J

1. 14G„

1.89

3. 0204

4. 2441

riW

- £+4,5+3J

1. 005

1.78„

3. 5247„

4. 0012„

f

- £+4.5+-lJ

0. 290„

1.15

3. 1793

2. 982

f v'

- £+4i)+2J-J

3. 2486

4. 0585„

fl

- S+4.J+3J-J

9.98„

0.8

2. 957„

3. 8580

V

4£+ .5+ J

9.0

2. 3363

[3. 0704„]

[3. 5111]

ri'

l£+ .5 +2 J

9.5

1.500

2. 3585

3. 1842„

nV

|£+3t>+2J

2.779

3. 7820„

f£+3.5+3J

9.28

2. 1614„

3. 0257

3. 6491„

f

^£+3.5+3J

1.32

2. 9G6

r,"

|£+3.5+3J

■ 3.3450

4. 1111„

P

|£+3i5+3J

3. 2309

4. 1965„

nr,'

|£+3.5+4J

3. 2994„

4. 1520

n'

|£+5,?+4J

1.017

3.1fil7„

4. 1907

5. 01G0„

1)

^£+5(5+5J

0.88„

2. 9688

4. 03S0„

4. 8781

rr

l£+7,5+5J

4. 0855„

5. 2422

rir,'

l£+7,5+6J

4. 1991

5. 3823„

r,'

|£+7,5+7J

3. 7114„

4. 9188

J-

f£+7.J+6J-J-

2. 615„

3. 8317

>/ v'

-i£+ 0

3.2411

3. 7872„

v'

-\t+ i5+ J

2. 819„

3. 4476

r-

-l£+ d -^

2. 9181„

3. 4813

"/'

2;

2. 364„

3. 0737

vv'

2£+ J

2.624

3. 3489„

rr

2£+ 2J

2. 207„

2.978

j-

2£+ J+-r

2. 620„

3. 2765

n

2£+2J+2J

9.8„

1.63

2. 362„

2.873

y,'

2-+2t)+3J

9.5

1.79G

2. 303„

2. 1007

vv'

2£+4i>+3J

1.92„

2.700

2£+4<5+4J

8.7

[8.8J

1. 5802„

2.4158

2. 9867„

v'

2£+4;5+4J

2.330

3. 1764„

r,"

2£+4>J+4J

3. 1079

3. 9008„

f

2£+4i>+4J

2.736

3. 6809„

nV

2£+4,5+5J

2. 9881„

3. 8425

r,'

2£+f),?+5J

9.64

[0. 53]

2. 652„

3. 6204

4. 3279„

"J

2£+6t/+6J

9.48„

0. 36„]

2. 4419

3. 4512„

4. 1892

rr

2£+S,5+6J

3. 6135„

4. 6784

r,V

2£+8i5+7J

3. 7124

4. 8075„

r/

2£ + 8,5+8J

3. 2109„

4. 3338

f

2£+8,?+7J-i-

2. 0G8„

3. 2092

|£+5.5+5J

9.3„

1. 140„

2. 0056

2. 5727„

n'

|£+7.5+GJ

0.5„

2. 2749„

3. 2377

3. 9184„

V

4£+7,5+7J

0.3

2. 0542

3. 0565„

3. 7710

5£+7.>+7J

8.1

0.43„

1.346

1.959„

140

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Logarithmic.

Table XLIII — Continued.

V

Unit= 1"

Cos

ur'

«r«

KTl

UJO

w

ufl

(.>-i>o)8in

vn'

A

9.66

0. 810„

2. 7559

3. 3840„

3. 6946

n'

2d}- A

9.79„

0. .54

1)

2<?+2J

9.92

[0. 63„]

r)

e

9.801„

0.425

2. 5970„

3. 1493

3. 41.58„

v'

£

1. 07.5„

2. 045

3. 5201„

4. 3751

U"

e

1. 640„

2.514

4. 106fi„

4. 8961

pil

£

1.063

1. 916„

4. 1006

4. 8961„

5. 4076

V

£+ A

9.36

0. 471„

2. 4824

3. 0830„

3. 3936

nW

£+ A

1.565

2. 456„

3. 9671

4. 7890;j

,'3

£+ A

1.543

2.341„

3. 8942

4. 6760„

P n'

f+ A

0.87„

1.75

4. 0705„

4. 8759

5. 3965„

1)7,"

£+ 2J

1. 441„

2.273

3. 7192„

4. .5456

P v'

£+ ^

0.42

1.36„

3. 7799

4. 5819„

5. 0998

f,

£+ J+i'

0. 695„

1.585

4. 0417„

4. 7827

5. 2543„

^'

E+4.J+4J

9.59„

0.45

V

£+4i>+3J

9.46

0.34„

1

2c-+2<>+2J

9.45

[0. ii„]

V

2£+2.?+3J

9.32„

[0,04]

rN

- £+ J

(.5-.>o)2 cos

1. 255„

2.149

3. 6240„

4.4615

V

£

9.25

0. 117„

n'

£+ A

9.12„

0.02

m'2

m''

m", m'

m'

m'

m'

T^^Iw^TjPrj'gftCi COS Arg. + (a-6(i)IuSTiPrj'QftC2 sin ATg. + (0-d„Y:^tv^vPv'qf'C3 COS Alg.
where C„ Cj, C3 represent the respective coefBcients.

PERTURBATIONS OF THE THIRD COORDINATE.

For the third coordinate the developments are limited to perturbations of the first order
and of the first degree with the exception of some secular terms of second degree. We can
therefore use osculating elements in this section, and use 6 and i? without distinction.

By Z 8 eq. (39), 41, eq. (83) and 8, eq. (41) the equations Z 115, (192) are given, in which
I is defined.

Since

dS^SS dS dd^^
de diie de "

By Z 9, eq. (45) we have, with sufficient accuracy, Z 115, eqs. (193). Within these limits,

dd w,, ,

_=-(l_,cosO.

Substituting this relation in the above equation and in eq. (192) in turn, the differential equation
to be integrated is (194).

Since F, G, H are power series in w, it is evident from eqs. (192) that

g = i-, + 2> + J,w^+

where

-^i ~ ^ i.p.q 'T'U.p.q '' ^^i.p.q

NO. 3.] MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 141

Therefore, eq. (194) becomes

Comparing the coefficients of like powers of w on either side of the equation, it is evident
that the integial must be of the form

S =S-i^^ + S, + S,w + S,u^ +

Substituting this relation in the preceding equation and equating like powers of w, the system
of equations (IGS-,) — (195,) follows.

Within the extent of the following developments one more equation should be written by

analogy.

dW
This system of equations is integrated in a manner similar to that for -j- (see p. 81). Each

equation is broken up into two equations, one a function of s and one independent of e. The
differential equation (194) is then replaced by eight differential equations, the integrals of which
can be obtained in the order,

S_., (5,-[S„]),[5o], (5,-[SJ),[,SJ,

dW
As in the case of -j— , the condition is imposed that

The equivalent equations are (196)-('200).

dW,"
A comparison of the differential equations for (-Sj — [^S",]) with the expressions for — t^

dW "

— Tj— leads to an analogous form of integiation for certain terms. Within the extent of our

developments,
and

D

-i (1 -e cos s) ^J{I,-[I,])d,-[(l -e cos e)^JiI,-[I,])de]

bW" bW"
take the place of -. ' and —^~—, respectively. Without change of notation for the third

coordinate, (iS — [S]) is given by eqs. (201), (202), where F, G, a are computed from F, G, H in

Tables XII-XTV', by means of eqs. (118) and (119). The coefficients V', G, a are tabulated in

Tables L to LII.

The function [S] is obtained from the integration of eq. (203). A constant of integration

is added, which is the same in form as Hansen's constant of integration for the perturbation of

the third coordinate, namely,

Ci(cos i/- — f) +f2 sin 4> Z eq. (204)

where c, and Cj are undetermined.

It
By eqs. (192), the pertubation ■ is derived from

" -. = S

I cos %

The perturbation comprises the computed value of eq. (202), the trigonometric sine series given
by Tables L to LII (which can be written by inspection with the aid of Table XV6), the series
forming Table LIU, and the constant of integration (204), in all of which

142

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table L.

Table 1,1.

Table LII.

[Vol. XIV.
Unit=l".

n

0

1

2

3

-1

.5

j',.„(n+l.-n+l)+>r'
J',.„(n-l.-7i+l)+;r'
J',.o(n+l.-n-l)-7r'
j'i.„(n-l.-n-l)-;r'

+ 52.7
+158. 2
-158. 2
- 52.7

+ 96.0

-191.9
-191. 9

+ 57.0

- 2S5. 0

- 95.0

- 285.0

+ 33.8
-101.4
- 50.7

+ 20.1

- 47.0

- 28.2
+ 140. 9

+ 12.0

- 24.0

- 16.0
+ 48.1

=3

o

"S

[X,

F,.„{n+l.-n+l)+K'
FUn-'i--n+l)+7:'
/..„(n+l.-n-l)-;r'
j',.„(/i-l.-.i-l)-!r'

-201
-812
+812
+201

-352

+897
+513

- 253
+1495
+ 498
+ 355

-176
+594
+297

- 119
+ 305
+ 183
-1473

- 80

+172
+114
-439

Uiiit=l"

Go.oin.-n+l)+,i'

- 26. 37

_

47.98

28.50

- 16. 91

- 10. 06

- 6.02

Go.„(n.-n-l)-;r'

+ 79. 10

+

95.96

+

47.50

+ 25. 36

+ 14. 09

+ 8.02

G,.,(n+l.-n+l)+r/

+ 90.3

+

112.3

+

58.5

+ 29.0

+ 13.6

+ 5.8

G,.„(n-l.-n+l)+;7'

+ 530. 8

+

720.8

+

468.9

+ 311. 7

+ 207. 7

+ 138.0

G,.„(n+l.-n-l)--'

- 124.2

—

120.5

—

53.3

- 21.8

- 7.4

- 1.2

G,.„(n-l.-n-l)-j/

+ 609. 9

-

1549. 1

- 674.1

- 369.6

- 219.0

Go.,(n.-n+2)+7r'.

- 162.4

—

211.6

—

103.8

- 47.7

- 19.7

- 6.4

G„.,{n.-n)+7:'

- 166.5

—

352.6

—

298.2

- 229.0

- 167.2

- 118.3

Gj./ji.- w)— k'

+ 166. 5

+

96.7

+

13.2

- 14.4

- 20.7

- 19.3

G„.,(n.-n-2)-r'

+

1825. 5

+

881.4

+ 516. 6

+ 321. 2

+ 204. 1

G„.„(a.-«+l)+;:'

+ 100.4

+

176.3

+

126.8

+ 87.8

+ 59.6

+ 39.9

G„.„(«.-n-l)--'

- 406. 6

-

448.6

-

249.2

- 148.6

- 91.5

- 57.2

G,.„(n+1. -«+])+;:'

- 432

—

592

—

370

- 218

- 122

- 64

G,.„(n-l.-n+l)+;r'

-2047

-

3188

—

2412

-1811

-1342

- 982

u

G,.„(n+l.-n-l)-j:'

+ 718

+

821

+

440

+ 225

+ 107

+ 44

o

G,.o(n-l.-n-l)-ff'

-2401

+12134

+4939

+2788

+1744

(2

G„.,(n.-n+2)+;:'

+ 693

+

951

+

568

+ 314

+ 158

+ 68

G„.,(?i,-n)+7r'

+ 893

+

1773

+

1607

+1356

+1089

+ 844

G„.,(«. -«)-:/

- 893

—

747

—

254

- 27

+ 68

+ 98

G„.,(n.-n-2)-r'

-13263

—

5889

-3549

-2336

-1586

Uiiit=l".

#„.„(«. -n+l)+;r'

- 79. 10

+ 142.49

+ 50. 72

+ 23. 48

+ 12. 03

fl„.o(n.-n-l)-n:'

+ 26. 37

+ 95. 96

+ 142.49

- 70. 45

- 24. 07

ft,.o(n+l.-n+l)+n'

- 609. 9

- 528.9

- 231.4

- 108.8

- 52.7

- 25. 7

F, .„(«-!. -71+1)+::'

+ 124.2

+ 528. 9

+1121. 6

- 897.4

- 365. 9

/7",.„(n + l.-n-l)-;r'

- 530.8

+ 551. 7

+ 166. 9

+ 64.3

+ 26.5

/r,.„(/i-l.-H-l)-7r'

- 90.3

- 312.4

- 421.4

- 572.6

- 967.8

nUn.-n+2)+7,'

+ 1057. 8

+ 311.5

+ 111.3

+ 39.4

+ 11.6

H,.,{n.~n)+n'

- 166.5

-1057. 8

+1145. 2

+ 501. 5

+ 276. 0

Tla-\{n.-n)-K'

+ 166. 5

+ 290. 1

+ 71.9

+ 62.1

+ 44.9

HUn.-n-2)-^

+ 162.4

+ 608. 5

+ 881.4

+ 1.551. 0

- 1020.6

nUn.-n+l)+r/

+ 406. 6

- 747.8

- 297. 2

- 152.4

- 85.8

n„.„{n.—n-l)-7:'

- 100.4

- 256. 6

- 177.8

+ 739. 1

+ 219. 8

i7,.„(n+l.-n+l)+7r'

+2402

+2483

+1362

+ 740

+ 406

+ 222

g

M

ff,.„(n-l.-n+l)+;r'

- 717

-2483

-4550

+8048

+ 3120

#,.„(n+l.-ji-l)-;r'

+2046

-4336

-1408

- 697

— 298

2?,.o(n-l.-n-l)-7r'

+ 432

+1214

+1481

+1901

+1186

6n

H„.,(n.-n+2)+n'

-3908

-1705

- 753

- 325

- 126

nUn.-n)+7:'

+ 893

+3908

-9.529

-3936

- 2233

H„.,(n.-r,)-K'

- 893

-1855

- 39

- 287

- 270

n,.,{n.-n-2)-r/

- 693

-1987

-2363

- 310

+13643

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

143

Tablu LIII.
[5] — {fii (cos ^— e)+f2Hin y!r)

Umt=l".

Sin

w-i

t/jO

w

..'■+40+3J-n^

- .25. ;!()

+ 123. 2

- 281. 8

>)

-v!'+2fl+ J-n'

v!'+25+3J-l-n'

v!'+2fl+ J-n'
^j+6e+5J-n'

+ 50.7

- 816.8

- 521.8
+ 432. 9
+ 129,9

- 246. 5
+ 3636
+ 2851

- 2034

- 861

+ 563. 6

-8548

-7663

+5237

+2770

n'

4,-26 +n'
(|!'+20+2J4-n'
9''+2e+2J-n'

v''+eo+4J-n'
{O-e^) cos

- 649.4
+ 596.4

- 26. 5

- 214

+ 3096
- 2916
+ 494
+ 1236

-7475
+7216
-2266
-3395

v''+ J+n'

+ 191. 93

- 705. 2

+1302. 6

V

j+n'

- 383.8

+ 1410

-2605

ri'

4>+ J+n'
v.',- J-n'

+6584
-5312

-40060
+29610

vv'

^.'•+ 2J+n'
<P+ n'
v',- n'

-5742
-6024
+6024

+36970
+38180
-38180

V'

,j+ J-n'

+6584
-1656

-40060
[+11860]

P

^+ J+n'

-3002

+ 18970

to'

and by eq. (193),

d-0,=^t

By inspection it is clear that the periodic part of <S' is of the form

I Up.,7,Pjj'^ sin A
and the secular terms are of the form

I TJ^.-q-qPri'i-i^nt cos {U-£) +£} + 2"<- '?^i-o cos A

Expanding cos {{A — e)-\-s}, and collecting coefficients of sin s and cos £, the secular terms can
be written

nt{ Kjicos € — f ) + A'j sin e}
where

/„w

w ,

K, = I C7p.,>)P>?'»^ cos (.4 -e)-~ U,., cos A

K,

'o'J^ .

2=-Jfv7'ZP,'«|sin(.4-£)

Introducing this notation, the perturbation can be written in the form of eq. (205).

The coefTicients Vp.q are given in Table LIV. K, and TT,, which are constants, are tabu-
lated in Tallies LV, and LYu, respectively. For a given planet the factors and arguments are
known. Therefore /f, and K^ reduce each to a single numerical quantity.

Since the Bohlin-v.Zeipcl method is based on the fundamental principles of Hansen, the
constants of integration are dcternuncd by the condition which must be satisfied when the

144

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

(Vol. XIV.

perturbations are developed on the basis of osculating elements, namely, that the perturbations
and their fu-st derivatives shall be zero at the time t = 0. The relations to be satisfied are

« =01

t-f

= 0

ing equations are equivalent relations:

Xl

= 0
= 0

< = 0

I cos 1
d/ u \
d£\i, cos ij

Table LIV

Logarithmic.

I Up.gTjPij'i sin Arg.

Unit-l"

Sin

tc-1

w"

w

- J-n'
-n'

20+ J-W

iO+sJ-n'
iO+2J-n'

1.705

3. 0621„
2. 8235

2. 2831

3. 1591„
3. 2462

3. 7258
3. 5528„

2. 8483„

3. 8608
3. 9166„

iE+ 0 -n'
is+ d+ A-n'

i£+50+3J-n'

3.2112„
2. 5875

2. 2787
3.3155

3. 0779„

3. 8.544
3. 4153„

2. 6304„

3. 5865„
3. 3972

-is- (?-2J-n'
_i£_ 0- J-n'
-is+ 6 -n'

-^£+3^+ A-W

-*£+3fl+2J-n'

3. 1158„
3. 1493

2. 3242

3. 3863
3. 3532„

3. 7378
3. 7544„

3. 0060„

4. 1833„
4. 1452

£+20+ A-n'

£+29+2J-n'
£+40+3J-n''

£+6e+4J-n'

£+69+5J-n'

2.6364
1. 423„

1. 4042„

2. 3306„
2. 1137

3. 3704„
2.706

2. 1720

3. 1922
3. 0138„

3. 8423
3. 4014„

2. 6339„

3. 7582„
3. 6101

1

- £-2fl-3J-n'

- £-29-2J-n'

- £ - J-n'

- £+23 -n'

- £+20+ J -IT

2. 7175
2. 7756„

2. 8125
2. 9121„

3. 4858„
3. 5070
1. 6810
3. 4427„
3. 4958

3. 9484
3. 9456„

2. 2463„

3. 7846
3. 8338„

4£+35+2J-n'
4£+3fl+3J-n'

|£+59+4J-n'

f£+7fl+5J-n'

2. 6058
1.760

1. 7510„

2. 9120„

3. 5312„
1.82„
2. 8113
4. 0813

-|£-3(9-4i-n'
-i£-30-3J-n'

-4£- 5-2J-n'
-|£+ 0- A-n'
-|£+ 0 -n'

2. 8673
2. 9620„
2. 0569„
2. 9275„
2. 9702

3. 8458„
3. 9124

2. 7932

3. 4708
3. 5487„

1
n'

2£+45+3J-n'
2£+40+4J-n'
2£+69+5i-n'

1.640
1.617
1. 206„

2. 731„
2. 340„
2.2110

"J

-2£-4(?-5J-n'
-2£-40-4J-n'
-2£-25-3J-n'

-2£ -2J-n'
-2£ - J-n'

2.4012
2. 5241„

1. 5290„

2. 3174„
2. 3514

3. 3634„
3. 4544
2.3210
3. 0558
3. 0737„

m'

=.If7'p.,ijPij'?ain .-l+n({.K',(cos £-<;)+.?■, sin £}+r,(coa £-«)+f5 8in £

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

145

Logarithmic

Table LV,i.

Unit-l"

Cos

w'

w

W

J-n'
J+n'
J+n'
J+n'
J+n'
2J+n'

2. 9180„

1. 9821

2. 8035

3. 5175
3. 1764„
3. 4580„

3. 7732

2. 5473„

3. 7182„
4.3017„

3. 9772

4, 2GG8

2, 8138

m'

Logarithmic

Ki = Iw'riPri'ij'" COS Arg.
Table LV,n.

Unit- 1"

Sin

u"

W

ufl

'V

vn

J-n'
n'
J+n'
J+n'
J+n'
2 J+n'
J+n'

2. 9180

3. 7799
1. 9821„
3. 7744„
3. 5175„
3. 4580
3. 1764

3. 7732„
4.5819„
2.5473

4. 5420
4. 3017
4. 2668„
3. 9772„

2. 8138„

m' '

K.i = I'W'riP-q'^p' sin Arg.

u

-. = in„.„riPTj"' sin A +nt

I cos ) PVI I

By eq. (205), at the date of osculation,

A'i(cos e — e) + Kj sin £ +c,(cos e — e) +c, sin e

t = 0, 0 = 6^

u

By Hansen,"

e cos %

. = I Up.gijPrj'i sin A + c,(cos £ — e)+Cj sin s

(f/ u V^/ ^^ V^'^-0
d£\^ cos V d^V' cos V (/^

in which v. Zeipel's notation is adopted.

(A)
(B)

From the various parts of S, enumerated above, tj can be computed. Since S contains

the constants of integration

Ci(cos ^ — e) +C2 sin ^

JO

the derivative, jy, contains the constants

— Ci sin £ + C2 cos £

The solution of eqs. (A) and (B) gives c^ and c^. But there is a better way of deter-
mining the derivative of the perturbation. The exposition of this second method is
postponed until a particular example is considered, for the perturbations are not yet in a form
which leads to the development of the equations.

' Auseinandersetzung einer zweckmissigen Methode tut Bercchnung der Absoluten Storungen der kleinen Planeten, Ersta Abhandlung, § 5, p. 3
110379°— 22 10

146 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

COMPARISOX OF TABLES.

Tables L, LI, LII check satisfactorily.

Table LIII. — With one exception, the agreement is satisfactory. The bracketed coefficient
contains a misprint in sign in v. Zeipel's table. That it is a misprint is evident from Table LVi,
in which the correct sign is given to the corresponding coefficient.

The terms included in the last colrnnn are computed from the additional tables, XIIw^,
XIIIio^, 'KlYw' and from first degree terms in Z 116, eq. (200). The latter part, namely,

[e cos sQ^j(I,-[I,])ch-l Aj[Aj(_v„-[2„])4/.)]

is added to both eq. (200) and eq. (203).

Table LIV. — Our table is more extensive. The one bracketed quantity includes an addi-
tional term from Table LIII.

Tables hXi, LVu check satisfactorily.

CONSTANTS OF INTEGRATION IN n3z AND v.
The constants in i were treated in the preceding section by the familiar Hansen method.

COS I

It is the purpose of this section to modify the similar treatment of the constants in the per-
turbations ndz and v so as to incorporate them in the elements Cj, «„, 7:^, (p^,. Through the con-
stants of integration, the constant elements, which have been used from the beginning without
definition, are to be explained.

Since the group method of developing perturbations is built upon the fundamental prin-
ciples of Hansen, his conditions for the determination of the constants of integration must be
fulfilled. These conditions depend upon the choice of initial osculating or mean elements.
Osculating elements are used here. The corresponding conditions are that the perturbations
and their first derivatives, at the date of osculation, (< = 0), shall be zero.

Consider the relation of the constants of integration to the elements. There are two con-
stants in each perturbation since the differential equations are of the second order. The con-
stant added in the first integration is a velocity; the one added in the second integration is a
displacement, or, a perturbation. Now, recalling that the position and velocity of a body for
any time t can be transformed into the constants which are ordinarily called the elements of
the orbit, it is evident, by analogy, that a displacement of the body and the velocity of the
displacement can be transformed similarly into changes in the elements. The four constants
in ndz and v are related to the four elements, a, e, n, c, defining the shape and size of the orbit
and the position in the orbit, and the two constants in the perturbation which is measured perpen-
dicular to the plane of the orbit are related to the elements fi, i, which deternaine the position
of the plane of the orbit. It is possible therefore to modify all six elements, but it is v. Zeipel's
preference to make the transformations only for the first four constants.

It is not necessary to compute

ndz V I
dnSz dv\t = 0
ds irs)

for the following developments perform the transformation analytically, and the changes in
the elements can be computed from auxiliary functions.

Let «„, f„, ::„, c„ be osculating elements; let a, e, tc, c be the osculating elements modified
by the constants of integration in the manner indicated above.

For imdisturbed motion,

■f„sm£ = c„ + V ,, ._ /1+Cp

t9(v-7:,)=^\^Jgie

r cos (i>-;r„)=o. (cos £-f„) r sin (t;-;r„) =a„Vl -«o' sin e

Hansen's choice of ideal coordinates demands that the coordinates and their velocities
shall have the same form of expression for disturbed and undisturbed motion. The ideal polar

No. 3.] MINOR PLANETS— LEUSCIINER, GLANCY, LEVY. 147

coordinates are designated by e or / and r. The relations which are analogous to the above are

■eo sin £ = c„ + n„t+ng8z = c„ + no{t + Sz) /l+c,

f cos/=Oo (cosi — *!o) f sin/=fl„-y/l — ''o^ sin £

f=v—7t„ r — fO +v)

These are the equations for motion in the orbit based on constant osculating elements and
appropriately determined constants of integration.

If, in place of osculating elements and Hansen's ndz and v, v. Zcipel's elements and the
corresponding perturbations are used, the equations are the same in form. In v. Zeipel's
notation s and / take the place of I and /. The omission of the dash over these variables is
permissible, since the physically real values, with which they might be confused, do not occur
in the theory except for the date of osculation, where the subscript zero is added. It is to
be noted that, through v. Zeipel's choice of elements, the coordinates and the perturbations
have values which are numerically different from the Hansen cjuantities of the same designation.

Let the time be the date of osculation and denote the true coordinates by £„, v^, r,. Then
the preceding equations for undisturbed motion become Z 121, equations (206), (207), and
Z 125, equation (230).

Let the disturbed eccentric anomaly and radius vector (s, r) be £, and r„ respectively.
The relations for disturbed motion become Z 121, equation (209), and Z 122, equations (210).

The first derivatives of these expressions are given by equations (208) and (211), respec-
tivelj', and the time rate of e is given by the equation following (209).

The solution of the four equations (210), (211), with the aid of all the others, determines
the four unknown constant elements, a, e, n, c, or, better, a — a^, e — e„, tz — ;r„, and c.

The fact that the adoption of the new elements in connection with the perturbations ndz
and V, as developed in the preceding sections, is equivalent to the use of osculating elements,
follows from the simultaneous solution of the equations for the disturbed coordinates and their
velocities and the corresponding equations for undisturbed motion.

The method of calculating c from the equation

. ,, , ,o (• = £, -e sin £i-W(?2

IS given in the example, page 18.

After many laborious transformations the other three unknowns are expressed in terms of
familiar functions in equations (233)-(23G). In the verification of these equations slight differ-
ences in the numerical coeflicients of certain unimportant terms were found. The magnitudes
of these coefficients depend upon the number of the terms included in making the transforma-
tions. Since it makes little difference whether or not they are included and since v. Zeipel's
values present a more symmetrical form of a later auxihary function, we adopted his coeffi-
cients.

In the functions x, y, z the arguments and factors are functions of ij, r., £„ ^„ J, I, where

but at the beginning of the computation only j;„, r„, f„, tf„, J,,, J^, the corresponding functions of
osculating elements are known.'

' Thorr is a confusion of notation in v. Zeipel's developments. In Z 127, et|uation (238), flo is defined to be the value of 0 at the date of oscu-
lation when osculating elements are used for the planet, and ei signifies the argument if the elements a, e, n, etc., are employed, or bv Z 9
equation (43), ' ' '

and their diTerencc is computed by Z 127, equation (238).
In the collection of formulae by Z 133,

This U an approximation (or the above equation.
Again, mZ, 60,

If the secular terms are counted Irom the date of osculation, the (actor (.o—et) ought to be replaced by (o—M.

Oo-

-jfw-

-fo sin eo)

i-y-

«.-

-e sinti)

-r

.-1

c-c'

1
" 2

(o-f sino)-

mz-

■c'

148 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

By equations Z (43), (235), (236) and the equations preceding (233), the factor tj and the
arguments J, £„ di are given in equations (238) in terms of osculating values and functions of
perturbations, inclusive of first order.

To these should be added

and

3/ 2 \ 1
ri = r + ^l 1 -gTjo cos e^jiy sin s^-z cos £o) + 2')o2+

where i

A=2'i + '^. + ^i

•T =2^0 + ^0 + ^0

The equations (233), (235), (236), and (238) permit the construction of two tables which
determine w, nor a, and e and tc. From here on the developments differ in form from v. Zeipel's
although they are the same in principle. If v, Zeipel's equations (237) and (239) are used, the
term {^2" — iiy^") should read

(V + X3" + x,") - ij (1/2" + 2/3" + y.")

in agreement with Z 91, line 14.

Suppose that w-w^ has been computed by equation (233) and the argument F has been
introduced. The arguments and factors are unknown.

By laylor 3 theorem

w-w,=firi„ r, d„ j„ ^<,)+^/rj,+^jr, + ^Jd,+-l^n,+^^Ji„+ ....

Inclusive of second order in m', the differentiation is for first order terms.

Substituting the values of Jij, JF, Jdo, i^o. ^^0 from equations (238) and the additional

equations above,

/ ?i -f ^ -f ^•f\ 'I / ^ -f ^"f ^^\^
w-w,=f{r)„ F, d„ Jo, ^o) + (^(2^-^^-^j4^2+(^^^+^ + 0-J2'?„3

2(1 -JJo cos ^o'l^ + ^f 1 -fjlo COS ^oj^jCy

sin £„— 2 cos £„) + • •

The order of calculation is: computation of equation (233), in which the arguments and
the factors are given the subscript zero, differentiation of first order terms, computation of the
second order terms in the above equation, and the additon of these second order terms to the
first calculation.

With some foresight the computation can be simplified. The arguments should be arranged
in groups like the following: — n7~' + 2^ + 2J

-(r!,-l)r + 2^ + 2J
-(n-2)r + 2^ + 2J

(7i,-l)r + 2/? + 2J

Then, for whole groups of arguments,

Sf Sf df_
5(2^o) 5J„ 81\

Also for some particular argument in a group, the condition

dJo^SF^S^o
may be satisfied.

No. 8.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

149

Finally, by inspection of the arguments, considerable computation can be avoided if

1

JJo

The function w — Wo is tabulated in Table LVI. Since it is unavoidably a function of w
itself, the determination of w for a given case must be made by successive trials, the first
approximation being y,_^

Logarithmic.

'/o

^o'

Vo'

vor

Cos

r
2r
sr
Ar
hr
ir

-4r+20„+2Jo
-3r+2(?„+2J„
-2r+20„+2J„

- r+2e„+2j„

200+2^0

r+2(?„+2j„

2r+2eo+2J„
3r+29„+2Jo
4r+2eo+2J„
5r+2flo+2Jo
7r+20„+2Jo

-5r4-2»„+ j„

-4r+20„+ J„

-3r+2eo+ j„
-2r+25„+ J,

- r+2ff„+ j„

2^0+ ^0

r+2eo+ ^0
2r+20o+ J„
3r+2fl„+ Jo

4r+2(?„+ Jo
5r+20o+ Jo
7r+2flo+ Jo

r
2r
sr
4r

-4r+46»o+4J„
-3r+4e„+4J„
-2r+40o+4Jo

- r+4eo+4Jo

40„+4Jo
r+4flo+4J„
2r+4^o+4^
3r+4e„+4Jo
4r+4fl„+4Jo
5r+4i9o+4Jo

-4r+4fl„+3Jo
-3r+4(»o+3Jo
-2r+4flo+3Jo

- r+49o+3J„

4ff„+3J„
r+4flo+3Jo
2r+4e„+3Jo
3r+4flo+3Jo
4r+4flo+3J„
5r+4eo+3Jo

Table LVI.

4.360

4.379

4.664

4. 605„

4.71

4.666

4. 516„

[5. 807„]

[6. 2084]

TO-l

[5. 1966„]
4.766
4.446
4.412
4.484

4. 161„
3.19
3.52

5. 1420
7. 6355„
4. 856„

4. 92„

5. 5174„
5. 4248„

4.582
4.674
4.99
5. 4623„
[7. 1987]
5. 0056
4.38
5. 6251
5. 5812

5.83

[8. 0913]

8, 5565„

m",7n'

Unit- I radian.

[5. 7767]

6. 6599

7. 1194
6. 8442
6. 5883
6. 3437
5.875

6. 5090
6.169

6. 8821„

7. 0986„
6.359

8. 2144
8. 0894„
7. 8150„
7. 6056„
7. 4128„
7. 2254„

[6. 8746„]

6. 8776„
6. 8815„
6. 6271„

6. 7985

7. 8314„

8. 2964
8. 0434
7.8458
7.6603
7. 4778
7. 1130

7. 8102
7. 7520„
7. 6172„
7. 7135„

7. 1862

7. 1804
6.817

8. 4680„
8. 8270„
8.7850

[8. 5144]
8. 3274

8. 1627
8.0050

7. 354„

7. 5708„

8.8838

9. 2180
9. 2783„
9. 0241„

8. 8480„
8. 6916„
8. 5401„

m'^, m'

7. 3732„

7. 7572„
7. 5468„
7. 3450„
7. 1490„
6. 7632„

6. 6325„

7. 0658
7. 6078
7. 6970

7. 0722„

8. 4125„
8. 9548
8. 6561
8. 4650

[8. 2958]
8. 1426
[7. 8484]

7. 5604
7. 4536

6. 7816

7. 4732„

8. 1061

9. 1086„
8. 8316„
8. 6564„
8. 5030„
8. 3.544„
8. 0545„

8. 6250„
8. 1242

6. 6043„
8. 2308

7. 9072„
7. 8679„

7. 456„

8. 8822

9. 2073
9. 8236„
9. 4910rt
9. 3006„
9. 1494„
9. 0105„

8. 1083

8. 2084

9.0548„

9. 5174„
0.2833
9. 9635
9. 7850
9.6434
9. 5128

7. 7492

8. 0553
7. 9060
7. 7602
7. 6136
7. 3134

7. 4746„
7. 8698„
7. 9975„
7. 9394„

7. 4480

9. 5668n
9. 2006„
9.0111„

[8. 8561„]

8. 7346„
8. 4936„

7. 8425„
7. 5238„
7. 3174
7. 7966

9. 6833
9. 3296
9. 1558
9. 0248
8. 9050
8. 6668

150

MEMOIRS NATIONAL ACADEMY OF SCIENCES.
Table L VI— Continued.

Logarithmic.

[Vol. XIV.

Urut= 1 radian.

Cos

w-a

u,_s

W-'

w'

w

w'

*)>)'

-ir+ J„
-3r+ j„
-2r+ j„
- r+ j„

7. 7610
7. 420.S

7. 8104^

8. 0479,,

7. 8364„

8. 3915
8. 0268
8. 8018

Jo

4.518„

[5. 886„]

[5. 70„]

r+ J„

7. 1339

7. 8500

2r+ j„

7. 8421

8. 4293„

3r+ Jo

7. 9669

8. 6796„

ir+ 4o

7. 9760

8. 7576„

,'^

-4r+49„+2J„
-3r+40„+2J„
-2r+4fl„+2J„
- r+49„+2J„

6. 9002

7. 1638

8. 1860„

7. 6938„

7. 8502„

8. 4016

400+^^0

3.76

6. 0608„

8. 4157

8. 9760„

9. 1661

+ r+40„+2J„

9. 1714

0. 1382„

+2r+40„+2J„

8. 9358

9. 8333„

+3r+40o+2J<,

8. 7718

9. 6681„

+4r+49(,+2Jo

8. 6236

9. 5372„

,'=

r
2r
sr
4r

3.76

5. 7516

4.7

7. 8677

7. 8610„

8. 1026„
8. 1538„

8. 6727„
8. 2228
8. 7296
8. 8728

p

r
2r
3r
4r

7. 9418„
7. 93r2„
7. 7920„
7. 639„

8. 7337
8.7154
8. 6154
8.5001

f

-4r+45o+3J„-i'o
-3r+45o+3io-^o
-2r+4ff„+3io-^o
- r+40o+3Jo--^o

7.446
7. 1858

7. 6176„

8. 1156„
7. 8677„

7. 9693

490+3^0-^0

4. 804„

7.168

7. 9368„

8. 3724

r+49„+3J„-i'o

7. 7887

8. 8492„

2r+4fl„+3J„-.ro

7.448

8. 453 1„

3/'+4eo+3Jo--^o

7. 1976

8. 2026„

4r+49„+3J„-Jo

6.978

7. 9963„

rio'

2»o+24

5. 4181„

6.292

7. 4754„

8. 6636

6e„+6J„

5. 418„

6.292

8. 6328„

9. 4351

nan''

2«„+ 4a

5.885

6.719„

8. 5059

9. 2804„

2flo+34

4.974

5. 896„

8. 0326„

8. 1975

6fl„+5J„

5.935

6. 780„

9. 2774

0. 0330„

-JoV-

29„

5. 744„

6.535

[8. 3811„]

9. 1030

25„+2J„

5.44„

6. 327

8. 0917

8. 6300

69o+44

5. 919„

6.744

9. 4432„

0. 1464

n"

2flo+ ^0

5.301

6. 149„

8. 2302

9. 0152„

6flo+3Jo

5.301

6. 149„

9.1294

9. 7729„

}'Vo

29„+2io

8. 5904

9. 3492„

9. 8022

20„+ Jo-i'o

4. 502„

5.41

8. 1011„

8. 8726

e^o+sJo-Jo

4. 502„

5.41

8. 0554„

8. 9263

jY

2^0+ 4

8. 5592„

9. 3245

2^0+24-20

4.057

5. 021„

6.887

8. 1804„

6(?„+4Jo-2'o

4.057

5. 021„

8. 2718

9. 1021„

m'»

m"

m'»,m'

m^.m'

m'

m'

w—Wa=2Cw>TjVT)^ft cos Agr., where C representa the respective coefficient.

No. 3.1 MINOR PLANETS— LEUSCHNER, GLANCY, LEVY. 151

Turning now to the determination of e and r, let equations (235), (236) be WTitten in the
form (244), where

^=-^+(4Tr3)-^^+g-+

Multiplying tlie first of these hy sin 9^, the second by cos (j; and adding,

S sin ^ + Ccos </i= —5(2/ cos ^+z sin ^<) +oa;(y cos ij>+z sin ^)

-3«oy(y cos 0+2 sin (j{r) +T— s(y sin ^-z cos 0) + . . .

Here, again, the arguments and factors are functions of the elements a, e, tz, c, and the expansion
in a Taylor's series is necessary.

Let

S sin 0 + Ccos 0 =/()?, r„ e„ i, I)

Then the form of Taylor's series is the same as the expression for w — w^,, (p. 148), with the
following modification. Within first order quantities,

/(j;, r,, (9i, J, 2)= -2(j/cos0 + 2sin9!i)

Hence,

Ssin v'^ + f cos 0=/(,„, r, e,, J„, 2J+(^-^^-^^)^^,

■*'VJ„'^ar'^5s„/2^'^ 5>)4^

4-j(l-,„ cos .0)^ + 2(^1-3 >J„ cos .„j^]-^ +

The order of computation is : calculation of

— 2^'/ cos y''+2 sin 0)

by inspection of the table for W, in which the arguments are to be given the subscript zero,
differentiation of the first order terms, calculation of the necessary products of functions of y, z,
and the partial derivatives, and the addition of these products to the first calculation. The
required function is given in Table LVII.

152

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Logarithmic

Table LVII.
5 sin i^+C cos <!>

Unit-l".

Cos

w-J

W-'

U,-l

UI»

w j

1

ttJ

^_5r+29(,+2J„

8.81

1. 082„

1. 5710

1. 612„ .

v!.-4r+20o+2Jo

9.009

1. 2314„

1. 5492

0. 989„

^!— 3r+2S„+2Jo

9.318

0.931

1. 604„

1.916

4,-2r+20o+2do

9.207

[1. 6478]

2. 1070„

2. 2333

4,- r+29„+2J„

9.711

1.950

2. 3426„

2. 3713

4, +2fl„+2Jo

9.196

2.1712„

2. 5678

2. 565„

0+ r+29o+24

9. 230„

2. 3541„]

3. 1493

3. 7107„

^+2r+29o+2io

9. 220„

1. 9114„

2. 6867

3. 1657„

^+3r4-29„+2J„

9. 724„

1. 5372„

2. 3831

2. 8623„

^;.+4r+20„+2.Jo

9. 494„

1. 2544„

2. 1315

2. 6333„

9!r+5r+29„+2Jo

9. 100„

1. 018„

1.9034

2. 4348„

Vo

V-5r+4fl„+4J„

9. 771„

1. 042„

1. 868

2. 357„

^_4r+40„+4J„

0. 064„

1. 723„

2. 3515

2. 6S14„

v!.-3r+49„+4J„

0. 3185„

2. 1626„

2. 6961

2. 9214„

0-2r+49„+4J„

0. 497„

[2. 7787„]

[3. 0649]

3. 0993„

4>- r+40„+4J„

1. 028fi„

3. 2379„

3. 1223

3. 9385„

,j +49„+4i„

9.199

9.04„

[2. 6172]

[3. 2511„]

[3. 4930]

</.+ r+40o+4J„

0. 7226

3. 1702

4. 1580„

4. 9365

v!'+2r+49„+4J„

0.669

2. 7877

3. 7083„

4. 3605

vi+3r+49o+4J„

0. 9435

2. 5117

3. 4261„

4.0450

^+4r+4eo+4^o

0. 5122

2. 2732

3. 2042„

lo

</.-5r

9. 814„

1.925

2. 634„

2.984

i-4r

0. 0434„

2. 0527

2. 6896„

2. 9432

</i-3r

0.3541„

2.145

2. 675„

2.744

^-2r

9.140

0. 362„

2. 1351

2. 38.50„

2. 4864„

-^- r

0. 4164„

2. 3504„

3. 0929

3. 5397„

}

9. 274„

0. 1436n

<P+ r

0. 3102„

2.497

3. 1875„

3. 5978

4+2r

9. 137„

9.918

1. 9006„

1. 0453

2. 8834

4,+sr

9.4G5

0. 812„

2. 5218„

3.3564

4'+'ir

9.20„

1. 406„

1.729

V

^_5r+49„+3i„

9.476

1.327

1. SS9^

2. 2299

</i-4r+49o+3i„

9.781

1.447

2. 1.^06„

2.5419

Vi-3r+4fl„+3i„

9.811

2. 1070

2. 6:}09„

2. 8608

^5-2r+4eo+3J„

0. 3489

2. 5095

2. 9557„

3. 0952

4,- r+49„+3J„

0. 9511

3. 3599

2. 7758

3. 9726

v!r +49o+3-io

8.76„

[0. 158]

[2. 7932„]

3. 3085

3. 4526„

^!.+ r+49„+3J„

9. 961„

3. 3609„

4.3114

5. 0691„

^+or+49o+3J„

0. 491„

2. 9943„

3. 8728

4. 4922„

^+3r+49o+3Jo

1. 0464„

2. 7293„

3. 6067

4. 1945„

0+4r+4fl„+3J„

0. 678„

2. 4992„

3. 3946

Y

^!.-5r+j„

9.848

2. 0766„

2.712

2. 9697„

4-4r+J„

0. 0792

2. 1609„

2. 6968

2. 7976„

4-zr+Jo

0. 3941

2. 157„

2.491

1.51

4-2r+Ja

9. 013„

0.248

2. 0455

2. 7898„

3. 2380

4'- r+J„

9.901

2.584

3. 2539„

3. 6434

•{> +^0

9. 885„

0. 8518

V>+ Z^+^o

0. 1664

1.836

2.448

3. 3029„

^5+2r+J„

9.009

9.76„

2. 1633

2. 6170„

2. 2433

4,+sr+JO

9.38„

2. 1064

2. 7194„

2. 9212

4,+4r+Jo

1. 9892

2. 6870„

no'

^J-5r+6fl„+6i„

^_4r+6eo+6J„

^i-3r+60„+6J„

4,-2r+6do+eJo
4- r+e^o+eio

2. 3144

2. 9538

3. 3102
[3. 4970]

3. 9455

2. 9730„

3. 3785„
3. 5843„

[3. 8423„]
3. 7269„

^J +6e„+6io

9.95„

1. 1109„

3. 1673„

[3. 9296]

[4. 3377„]

4+ r+6«o+6i„

3. 9144„

5. 0372

^J+2r+69„+6J„

3. 5594„

4. 5942

s/.+3r+69o+6Jo

3. 3121„

4. 3236

No. 3.]

MINOR FLANETS— LEUSCIINER, GLANCY, LEVY.

153

Table

LVIl— Continued.

S ein ^!i-f Ccos 0

Logaiitbmic

Unit- 1".

Cos

«!-l

w->

tc-i

w'

w

«•'

^0^

^-br+2d„+2J„
V'— 4r+20o+2J„
v'.-3r+29„+2Jo
S^-2r+2flo+2Jo
V'— r+20„+2Jo

2. 1657
2. 1255
2.234
2.576
3. 1995

2. 7221„

2. 8004„

3. 1304„
3. 3804„
3. 8325„

^i +29o+2J„

0. 344„

1.017

2. 689„

[3. 4822]

3. 9938„

V>+ r+2flo+2Jo

2. 2480

3. 2S39„

^+2r+20,+2Jo

9.45

3. 1612

3. 8424„

lo'

d-br-20o-2Jo
^-4r-20o-2Ja
v'.-3r-20o-2^o

2. 700„
2. 817„
2. 9247„

3. 5481
3. 6251
3. 6905

^-2r-2flo-2J„

9. 59„

3. 0241„

3. 7470

J,- r-20o-2Jo

3. 1364„

3. 8346

4, -20o-2J„

<p+ r-2eo-2Ao

0,117

0. 95„

2. 297„

[2. 7856„]

3. 6614

2. 8942„

3. 5604

4,+2r-2e^-2Jo

2. 297n

3. 1129

'?o'j'

i-4r+69o+5'4o
<i-3r+60a+bJo

4>-2r+6e„+5Ja

4,- r+69o+5Jo

2. 4885„

2. 976„

3. 6541„
[3. 9514„]

4. 3903„

3. 1691
3. 5560

3. 8829
[4. 1632]

4. 0037„

v!. +69o+5-(„

0.295

1.366

3. 6364

[4. 3301„]

[4. G6C2]

0+ r+6(?o+5J„

4. 4005

5. 4906„

v'.+2r+60o+5J„

4. 0582

5. 0612„

v!.+3r+60o+5Jo

3. 8204

4. 8027„

"Jol'

v!.-5r+29o+ J„
^-4r+2(?o+ io
vJ-3r+29„+ Jo
^J-2r+20o+ ^0

^!.- r+2eo+ io

2. 426„
2. 399„
2. 410„

2. 701„

3. 2842„

3.0684
3. 0310
3. 1305
3. 4602
3. 8558

■

tj +20„+ J„

0.444

1. 188„

3. 0569

[3. 72G6„]

4. 1122

^+ r+20o+ 4

2.8541

3. 5823„

(j+2r+2eo+ io

3. 2191„

3. 7635

VoY

4,-br-2d„- Jo

d,-4r-20o- Jo

4>-zr-20o- Jo

3. 1551
3. 2454

3. 9530„
3. 9948„

3.3100

4. 0023„

d,-2r-20a- Jo

9.93

3. 3277

3. 9401„

4,- r-2fl„- Jo

3. 1976

3. 4598„

4> -2ff„- ^
/,+ r-2fl„- J„

0.490„

1.324

3. 0145„

3. 7326

4. 2787

3. 3632

3. 9402„

v'.+ r-2e„- j„

2. 7792

3. 5224„

VoV'

V'— 5r+29o+3Jo
^_4r+2fl„+3J„

■/.-3r+29o+3^
^_2r+2So+3io
0- r+20o+3Jo

2. 2738„
2. 116n
2. 5858„
2. 809„
2.650„

2.847
3. 0290
3. 3787
3.5429
3. 7297

^ +29o+3J„

9.98

0.60„

2. 873„

[2. 685]

3. 7980

v!.+ r+2flo+3Jo

3. 5126„

4. 2856

v'.+2r+2(?o+3Jo

9.46„

3. 3438„

4. 1208

V

v''-5r+6eo+44
^_4r+6«o+4Jo
v''-3r4-69o+4Jo
s!--2r4-6fl„+4Jo
v!'- r+G0o+4Jo

1.9950

2. 6112

3. 0556

3. 7934

4. 2260

2. 7422„
3. 1949„

3. 5583„
3. 7947„
4.4064

V^. +60o+4J„

9.98„

0.76„

3. 5017„

4. 1098

4. 3552„

^J+ r+60o+4J„

4. 2852„

5. 3521

vJ+2r+6fl„+4Jo

3. 9567„

4. 9249

V"

^J-5r+20o+2Jo
^J-4r+29o+2Jo
v!.-3r+20o+2Jo
%!r-2r+29„+2J„

4,- r+2e„+2j„

2.5018
2.453
2. 4799

2. 9375

3. 2833

3. 0963„
3. 0935„
3. 2779„
3. 6294„
3. 8982„

V- +29„+2J„

0. 025„

0.60

2.634

3. 2781

4. 0439„

^!>+ r4-2flo+2Jo

3. 5607

4. 2381„

^+2r+2eo+2J„

3. 4629

4. 1704„

154

Logarithmic

MEMOmS NATIONAL ACADEMY OF SCIENCES.

Table LVII — Continued.
S sin ^+C cos (j>

[Vol. XIV.

Unit-l".

Cos

U,-3

W-J

w-i

IfO

w

un

v"

^-5r-29„

^_4r-20„

v'— 3r-20„
s!— 2r-20o

0- r-29o

3. 0090„
3. 0676„
3. 0764„

2. 958„

3. 1140

. 3. 7477
3. 7445
3. 6664

3. 3121

4. 0201„

S^- -29o

0.305

1. 127,

2. 912

3. 5491„

3. 9085

4,+ r-2(?„

3. 0396„

3. 6320

^_l-2r-20„

2. 4706„

3. 2330

f

v''-5r+6e„+5.io-^o
^_4r+69„+5Jo-i'o
V'— 3r+6fl„+5J„-Jo
v!— 2r4-6(?„+5J„-i'„

2.006
2.335
2.544
2.718
2.970

2. 7505„

2. 981„

3. 1436„
3. 2445„
2. 9116„

0 +60„+5Jo-i-„

8.6„

9.7

2.114;,

2.923

3. 4067„

V1+ r+6(?o+5Jo--^o

2. 7948„

3. 9420

0+2r+60o+5J„-i'o

2. 3824„

3.4488

}'

,^-.5r+20o+2J„
^_4r+20„+2J„
<j—Zr+20o+2Ao
^_2r+29„+2J„
v!.- r+20„+2J„

9.6

1. 916„

2. 5178„

2. 938„

3. 3406„

2.387
2.911
3. 3047
3. 6294
3. 9330

4, +2<)o+2J„

0. 5910

3. 1266

3. 8021„

4. 1894

^4- r+29„+2J„

3. 4070

4. 3178„

v!.+2r+2(?o+2i(o

3. 0472

3. 9308„

p

4-br-2e„- Jo+Jo

,/.-3r-2flo- J0+-0
^-2r-2flo- Jo+^o
v''- r-2(?„- Jo+J'o

0. 732„
0.35
1.463
2.064
2. 6816

1.085

1. 895„

2. 5146„

3. 0255„
3. 6280„

^ -29o- 4+i'o

9.04

0.11„

2.636

3. 3284„

3. 7399

v^+ r-20„- Jo+i'o

3. 0572„

3. 6430

vJ+2r-2eo- J„+i'o

2. 9121„

3. 5491

n<?

,:. +49o+4J„

0.775

1.65„

3. 1052„

3. 0342„

V', -40„-4Jo

0.29„

1.10

3. 1888

3. 6104„

^?. +89„+8Jo

0.65

1.54„

3. 7520

4. 5812„

rioW

4, +4fl„+5i)o

3. 7577

4. 3244„

^ +400+3^0

1. 260„

2.081

3. 1240

4. 1388

VT- -4e„-3J„

1.005

1. 77„

3. 5356„

3. 3560

^ +8fl„+7Jo

1. 228„

2.093

4. 3980„

5. 1827

Vo-^"

^ +49„+4J„

4. 1155„

4. 5547

4. +4^0+2^0

1.106

1.88„

2.831

4. 1803„

4 -4flo-2J(,

1. 146„

1.88

3.0422

4. 0180

Vf. +8»„+6^„

1.321

2. 152„

4. 5658

5. 3010„

,'3

V'. +4^0+3^0
V^- -43„- J„
^I. +8flo+5J„

3. 8375

3. 2197

4. 2553„

4. 0446„

3. 9650„

4. 9349

f V"

V'' +4flo+3J„-i'o

3.0024

3. 8634„

4 -49„-3J„+i"o

9.98„

0.8

2. 956„

3. 8331

V'. +8fl„+7Jo-i-o

3. 0757

3. 9759„

V'r +49o+4J,

0.46„

1.32

3. 8514„

4.6436

P V

4 +49„+4Jo-2-„
V'' -49„-2Jo+v„
4 +89„+f.J„-i-,

2.442
3. 2486
3. 2818„

1.846„
4. 0585„
4. 1441

^?. +4i9„+3J„

0.27

1.15„

3. 9421

4. 6972„

S sin 4'+C cos </i=2r,w'i)P7)'?j^' cos Arg, where C, represents the coefficient.

No. 3.]

MINOR PLANETS— LEUSCHNER, GLANCY, LEVY.

155

COMPARISON OF TABLES.

Table LVI. — Unless there are errors of caloulation, all the discrepancies are due to the
acciomulation of other discrepancies already discussed. Without going into the details of the
construction, it is suflicient to remark that our table is built from practically all of the available
auxiliary material. Our table includes many more terms than v. Zeipel's table, but it is wanting
in the two arguments (>r and Sf in the first block of terms. These arguments contain 3e and 4£,
respectively, and our series were not inclusive of these higher multiples. It would be more
consistent to include them, since the argument IF is included.

Table LVII. — Unless there are errors of calcxilation, all the discrepancies are due to the
accumulation of discrepancies already discussed. Om- table is built from practically aU the
available auxiliary material. Large disagreements are to be exjjlained by v. Zeipel's use
of the formula following Z 131, equation (244). In this equation the following functions are
omitted :

-J(!// cos ^+2,' sin 0)-H[?/2] cos ip + [z-i] sin <fi).

ERRATA' IN H. v. ZEIPEL, ANGENAHERTE JUPITERSTORUNGEN Ft)R DIE HECDBA-GRUPPE.

With the exception of § 6, Storungen des Radius-vector, all tlie developments have been checked.

Page.

Lino.'

For—

Read—

da

da

1

8a

dx

dJ

da

da

1

9a

dy

¥

3ff»

v'

dv
dc

5

5b

(15)

(16)

dJJ

dV

9

2a

dif,

s<p _

9

lb

.+(1 .)^+i

; '\+w

/, 'l+W

12

2a

C)'

&

12

9a

^^^;l

fl(2i+l)
P n+i

12

10a

^\

0%'

12

6b

(2n+4i+l)7.->-n(4i+4)7;+",

(2n+4t4-l)7j'-»+(4i+4)-y',:;,

13

5a

(2n+4i+3)Ti'-n(4t+4)-)'|+i

(2n+4t+l)7i'-'»+(4i+4)7l;°

14

2b

n'g'

ng'
n=-co

15

8b£t

I

s

16

10a

sin

COS

1 ba

1 pa

16

6b

T'^di

T^'di

dF

dF_

19

6a

doa

dcca

20

5b

T^:?

^::

21

4a

0{">.n

^j3.„

21

4b

Metoden

Methoden

24

9b

IPip.g

-IP*p.q

P„.„rn-f.-n+l]j

P„.o(n-l.-n+l).»

27

21a

Po.o(n-l.-n+lU

34

21a

P„.„(n-l.-n+l)^

42

5b

P„,2(n.-n)

F^Jn.-n)
n,.,{n + \.-n-\)

44

18h

//,.,(n+l.-n+l)

45

20a

,.„(n.-2-n-l)..

,.o(.n— 2.-n-l)^

46

8a

n

G

46

]0a

,.„(n-l--n+2)-j

„.,(n-l--n+2)-»

46

11a

,.o(n-l--n)+a

o.,(n-l--n)-j

' Inclusive of those tabulated bvv. Zcipel. . ^ . .». v ■-

« The number of the lino counting from the top of the paEO is indicated by a, countmg from the bottom of the page by D.

• On page 3 and all following pages ►' is defined by '-'- i-- The error consists in the omission of a statement announcing a change of notation.

See definition of y' given on page 2.

156 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voi.xiv.

Errata in H. v. Zeipel, Angendherte Jupiterstorungen fur die Hecuba-Gruppe — Continued.

Page.

LIno.i

For—

Read—

49

7b

50

6b

a'

r-

a^

50

6b

3 + .j»

3+14,2

61

lb

.So..(n.-n+l)

5o.,(n.-n-l)

5:i

lib

>-

>f

54

5a

H<

•""i

56

4b

H,.,{n-l-n-l)

7/,.,(n-l--n-l)

61

lib

26+20

20+2A

62

17a

(A+60+40

v''+6fl+4i

62

5b

+436

+439

63

9a

[(l-ecosOTF'J

[(1-CC0S£)HV]

65

3a

(106)

(lOGo)

65

5a

(106)

(lOfio)

68

3a

W * W ^ W ' llT' w

n-* w~^ ur^ ic~' w w'

69

6b

sin A

TiPri'lftBln A

69

5b

sin (A — i}i-'r£\
sin (A+iji-t)

r,Pr/9J2<sin (A-ip+s)

69

4b

r/VTj'gj^ Bin (A+(fi—s)

70

lb

W/"

W/'

70

lb

cos A

riPxj'Qj'^ COS A

71

7a

cos A

rjP-n'qjn eos A

75

15a

4o-3

A

75

18a

4oi

■4o-0

75

2b

4io

4o*i

75

lb

^..0

^0*1

79

10b

0

^i

81

8b

1— e cos c)

(1— ecosf)

83

12a

+3744

+3344

86

4a

(I282) iind (130)

(128j), (I283) und (130)

86

Oa

dlF,

dW,

d.>

i>»

91

9a

£ cos £

e cos e

91

11a

W,

'^2'

92

3a

I), cos £

1/1 cos £

92

10b

-l(l-e cos £) (IF-ifl) {W+\3)

-l[(l-e<:os t){W-lS){W+\S)-\

i>W

bW

92

4b

d<>

i>«

93

10a

sin A

-TpTfli'^ sin A

93

10a

S'

X

94

19b

bW

5 If

d&

"55

97

15a

(156)

(154) _

99

4b

— ]}U)8in £)

— ■nw sin £}

100

5a

A'

A

100

6a

-i"i2

-i"2

115

4b

(191)

(192)

116

7a

(195,)
7,(5.-[S.])

(^95-,)
§-,(5,-[5,])

116

10b

119

(')

u,.,

^V1

122

3a

Vo-t:

Va—^

123

4b

(l_f/2_f/a)

(I_f2_f/J)

125

3a

1 —Co cos £0'

1-foCOS £„

128

7a

i°

•^0

129

5a

/'

V

131

7b

W+A-e)

(^-A-z)

131

6b

i'P+A+t)

(<P+A-s)

132

8a

2.9227„

1.9227„

132

26a

5.3376

5.0376

134

9a

(2.>+/4)

(4-J+/4)

w

10

135

10a

2

4

w

w

135

11a

"2

■4

140

2fia

[nSz]

[nSz],

141

Ga

0*8998

0.'8998

■ Ths numter of tbe line counting from the top of the page is indicated by a, counting from the bottom of the page by b.

No. 3.1

MINOR PL.\NETS— LEUSCIINER, GLANCY, LEVY.

157

ERRATA IN KARL BOHLIN, SUR LE D^VELOPPEMENT DES PERTURBATIONS PLAN£TAIRES, § 1-7,

AND TABLES I XX.

rage.

Line.'

For-

Read—

3

5a

j«»=(l+m)a»

/t^={l+m)a'

bW

dW

11

lb

d:

^C

14

11a

1+y"

I—''

20

3a

+ic» cos 2e

-ie^ cos 2«

29

8b

y'n

y'-n

29

2b

eV=I/'

e-^^r

30

11a

a

P

e'

, ^'

30

11a

~k'

+?

30

12a

2n+m— 1

2n+m4-l

30

13a

-($y

+ ©

30

13a

2n-\-m-\

2n+m+l

30

13a

2n+m-2

2n + m+2

30

14a

2n+m-l

2n+m+l

30

5b

eV-i»(T-ir')

gV-lnCT-"-')

33

9a

Kf+'{n-\.n)

rfo+'(n-l.-n)

35

4a

(73)

(74)

36

la

2f j'-igV-l n(«— ir)

2/-^».ngV^'>(»-»')

36

10a

K;.,{n-l.+n+l)

.^,.,(n-l.-n+l)

38

ISa

a

a

38

10b

g-V-iT-o

e--AFi('-T')

38

3b

2(,0/-'

2(^')y-'

38

2b

^n)y'-'

2('j)y'-'

40

3a

KUn.-l-n)

.^,.o(n-l.-n)

41

13a

a

a

45

2a

Ko.o(0.-n)

A'o.o(n.-n)

45

9b

k

46

7a

r(r-l)V-'
1.1.2

r(7-l)V-'
"•" 1.1.2

46

7b

(n-s)(n-s+2) _

1 (n-s)(n-s+2)^^^_.

46

5b

(n-3)

(n-s)

46

4b

V*

r,'*

48

14a

P'j,o(n-2.-n)

P',.o(n-2.-n)

48

7b

P',.o|n+l.-n-2|

P',.j|n+l.-n-2|

48

7b

P\.,Cn+l.-n-2)
P',.,|7i-l.-n-2|

P',.,(n+l.-n-2)

48

6b

P',.2ln-l.-n-21

50

5a

R\.o(n+l.-n-l)-„

P',.o(n+l.-n-l)-,r'

50

9b

R\,o(n-]..-n+l)+e

P',.„(n-l.-n+l)+„'

50

3b

P'„.o(rt.-n+2)+^'

P'„.,(n.-n+2)+,'

61

lb

P'(n+r.-n+s 1

P'|n+r.-n+sl

59

5a

Q'\.o[n+l.-n
S'-'o.An.-n+i .

i?^',.„[n-l.-n]

59

8a

i?3'„.,[n.-n+l

60

6a

</

9'

60

8a

(See footnote ')

«0

9b

24

+(m/')^
24

«0

5b

y(-n+t)

y(-7l+»)(l

61

7b

■Poi n.+n+l]

Po.,[n.-n+l
Po.,[n.-n-l

61

5b

P,,., n.+n-l

«2

la

6

6

«2

7a

P,., (n+2.-n-l)

Pj.,(n+2.-n+l)

«2

7a

(n-2)V
2

(n-l)V
2

62

8a

^.•1

[n-l.-n+l]
*n-l.-n + l)+*

P,.,[n+l.-n+l]

«3

5a

Po-n

Po.o(n-l.-n+l)-*

«3

11a

Px„

n+2.-n-l)+.

P,.o(n+2.-n-l)+4

«3

13a

' 0-0

n — 1. — rj — 1 -*

^u..

n — 1. — n+1 -*

«3

14a

•foo

n — \. — n — l -i

' 0-0

n — 1.— ra+l]_«

63

5b

^ao

n.-n + \ -^

^0 0

n.-n — \ -,r'

«3

2b

Raa

n.—u — \ —z'

^00

7i.-n — 1 -r'

«3

lb

R<ia

'!._n-l]-,'

^00

n.— n — 1 -n'

• The number or the line counting from the
> The argument $ is delmed first by eq. (31)

top of the page is indicated by a, coimtinR from the bottom of the page by b.
, p. 20, secondly by eq. (IC'O, p. 00. The first of these definitions is used in | 8.

158 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi.xiv.

Errata in Karl Bohlin, Sur le Developpement dea Perturbations Planelaires. | 7-7. aTid Tables 7-JfX— Continued.

Page.

Line.i

For—

Read-

64

6a

R2.o[n+l.—n +,r'

Tfj.o n.-n+\]+,-

64

12a

Ro-o n.-u-l -n'

ifoo'J— «—]]-»'

64

14a

(n-n)

(n-2)

64

15a

/?,.,[n + l.-7l]_,'

/?,.,[n+l.-n]w

66

7a

F(n-^T.-n+s

F {n-\-r.-n-\-s)

6G

8a

G{n-\-r.-n-\-s

G{n+r.-n+s)

+3

+3

70

la

-3 Pi4n+l.-n-2)
2

-2 P,.2(n+l.-n-2)
-2

?',.o(n.-n+l)+.

71

4a

f,.„(n. n+l)+.

+3

+3

71

9b

-2 P,.o(n.-n+l)_«
_2

-2 P,.„(n.-n+l)_3
-2

i'',.o(n.-n-l)_„'

73

2a

Fj.„(rt.-n+l)_^'

73

18a, ff.

See foot note. 2

73

4b

flo.o(n. -"+!)+»

i?„.o(n.-n+l)+,:'

73

3b

Ra.o{n.-n+\)+^

.Ro.o(n. -«+!)+»'

74

8a

fii,.oi.-n+l)+>t'

i?o.o(n.— 1+1)+,,'

75

la

Go.o(n+l.-n)

f'0-o("+l.-'0+.-(-ir'

75

lib

Ra.o{n..-n-\-\)-r'

i?o.o(«-— n-l)->r'

78

lb

To'-"

7,i.n

79

lb

-y^m+z.n

7,._,m+2.n

79

*)

n=\

n=0

80

9b

Ti+i™-"

Ti+i""-"

81

12a

*)

81

13a

(120)

(120)*)

135

7a

A'„.3>-<

-^0 3'-'

139

3a

r.-n

2ri'-n

140

2a

A'-n

2ri'-n

154

la

(86)

(93)

161

la

— ar

ar

169

8b

w
//

4^2rA«

170

3a

2^2ri'-»

27i'-'»

170

4b

»

^2V»

171 ff.

See foot note.'

185

2a

3. 27886

3. 27887

185

13b

4

3

188

6b

2. 017 3„

2. 01703„

189

14a

3. 27886

3. 27887

197

16a

0. 146128„

1. 146r28„

197

18b

1. 505151

1. 505150

198

15b

1. 662759„

1. 662758„

198

2b

0. 477121

0. 477121„

' The number ot the line countinR from the top of the page is indicated by a, counting from the bottom of the page by b.
' The space between lines 18 and 19 should read ;'.

'Tables XII, XIII. XIV give the same coefficients in numbers as Tables XVI, XVII, XVIII give in'logarithms,respectiTely. The same factor
should therefore occur in the former.

o

MEMOIRS

or TH2

NATIONAL ACADEMY OF SCIENCES

"Voliame XI^

FOURTH MEMOIR

WASHINGTON

QOVEHNMKNT FEINTING OFFICE

1919

NATIONAL ACADEMY OF SCIENCES.
Volume XIV.

KOrrRTH IVEKMiOIR.

SECOND REPORT ON RESEARCHES ON THE CHEMICAL AND
MINERALOGICAL COMPOSITION OF METEORITES.

BY

GEORGE PERKINS MERRILL,

HEAD CURATOR OF GEOLOGT, UNITED STATES NATIONAL MUSEUM.

CONTENTS,

Page.

Bath, Brown County, S. Dak 1

Bjelokrynitschie, Volhynia, Russia 1

Farmington, Washington County, Kans 2

Forest City, Winnebago County, Iowa 2

Gargantillo, Jalisco, Mexico ' 3

Hartford (Marion), Linn County, Iowa 4

Homestead, Iowa 5

McKinney, Collin County, Tex 6

Ness County, Kans 7

Ochansk, Siberia 7

Ruffs Mountain, S. C 10

Tennasilm, Estland, Russia : 11

Tra\-is County, Tex 11

Waconda, Kans 12

Weston, Conn 15

V

EXPLANATION OF PLATES, MICROSTRUCTURES OF METEORITES.

Page.

Plate I. Figure 1— Bath, Brown County, S. Dak 1

Figiu-e 2 — Bjelokrynitschie, Volhynia, Russia 1

11. Figure 1 — Farmington, Washington County, Kans 2

Figure 2— Forest City, Winnebago County, Iowa 2

III. Figure 1 — Gargantillo, Mexico 3

Figure 2 — Hartford, Linn Countj', Iowa 3

IV. Ness County, Kans. In figure 1, highly magnified, are shown in white, areas (1) of maskelyrdte 7

V. Figure 1 — Tennasilm, Estland, Russia 11

Figure 2 — Waconda, Kans 12

VII

SECOND REPORT ON RESEARCHES ON THE CHEMICAL AND INERALOGICAL

COMPOSITION OF METEORITES/

By GEORGE P. MERRILL,
Head Curator 0/ Geology, United States National Museum,.

The paper here preseated contains the detailed results of studies made during the past
year under a grant from the J. Lawrence Smith fimd. The immediate purpose of the investi-
gation, as noted in my first report, tliese Memoirs, volume 14, 1916, pages 7-29, was the deter-
mination of the presence or absence of sundry reported elements existing in minor quantities,
but naturally it was foimd advisable to extend these boundaries from time to time, as
interesting or important features developed in progress of the work. In several instances results
deemed of special importance have already received publication elsewhere.^

For convenience of reference, the meteorites studied are, in the following pages, considered
alphabetically.

Bath, Brown County, S. Dak. — The fall of tliis stone and the attendant phenomena were
briefly described by Foote.' Later, Brezina,* with even greater breArity, described its litho-
logical features. There is nothing to indicate that he examined the stone in thin sections, and
as it has never been subjected to chemical analysis it seemed a fit subject for further investi-
gation.

Macroscopically the stone is gray, but, owing to oxidation, so filled with rust spots as to give
it a brownish cast. The crust is rough and duU, a characteristic of stones of this class. The
textm-e is firm, but the chondrules, for a large part at least, break free from it when the stone
is fraotm-ed. The most imusual feature, when examined with a pocket lens, is the abimdance
of ghttering crystalline facets of nickel-iron. The slipping faces mentioned by Brezma are not
evident to the imaided eye in the pieces in the museum collection, but in the thin section are
numerous fine black fracture lines, along some of which a differential movement has plainly
taken place.

In thin section the stone is seen to be a spheruliticchondrite with crystalline base. (Fig. 1,
PI. I.) The chondrules are extremely variable in detail, but present no miusual features. The
essential minerals are oUvine and enstatite; more rarely polysynthetically
twinned monoclinic forms appear. Fragmental forms are common, par-
ticularly among the radiating and cryptocrystaUine enstatite types. In
one of the latter was observed a single granule of a distinctly red, trans-
lucent, but not transparent, mineral, of somewhat rounded outline as
though corroded, and complctel}'' isotropic. (Fig. 1.) It is believed to be
a spinel; possibly osbomite; it is impossible to decide from the single occur-
ence of so small an ob j ect. (See further imder Homestead .) The phosphatic
mineral I have of late had so frequent occasion to note occurs but rarely.

Bjehhrynitschie, Volhynia, Russia. — This stone, which fell on January 1, 1SS7, has appar-
ently as yet received but brief notice and been subjected to no chemical analyses. Four ref-
erences to descriptions are cited by Wiilfing, two of which are by B. K. iVgafonov, the others
being even briefer notes in the catalogues of Brezina and Meunier. I have had access to but

I Presented April, 1918; read November, 1918.

• On the Calcium Phosphate of Meteorites, Amer. Journ. Science, voL 43, 1917, pp. 322-324, and Tests lor Fluorine and Tin in Meteorites,
with notes on Uaskclynite and the Eflect of Dry Heat on Meteoric Stones, Proc. Nat. Acad. Sci., vol. 4, no. 6, 1918, p. 176.

• Amer. Journ. Sci., vol. 45, 1893, p. 64.
< Wiener Sammlnne, 1893, p. 269.

111549°— 19 2 1

2 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol. xiv.

one of the papers cited by Agafonov.^ In this the stone is described as composed chiefly of
chondrulcs, entire and fragmental, embedded m a groimd of crystals and crystal fragments.
The mineral composition is given as olivine, bronzite, augite, maskelynite, nickel-iron, and
troilitc. Brozina describes it^ as having suffered from oxidation to a depth of 1-3 cm. below
the original siuiaco, as of a brecciated structure and with strongly developed sUcken-sided
surfaces (Ilarnischflachen) . He classes it a brecciated chondrite (Cib) , though with occasional
black chondniles showing a gradation into the brecciated spheruhtic chondrites (Ccb). One
fragment badly oxidized he seems mclined to class as a crystalline chondrite (Ck). Meunier
in his hst' states that wliile the characteristics are not absolutely identical with those of Tad-
jera, the composition is the same and the differences not sufficiently marked to justify relegat-
ing to a distinct type. Neither says anything of the mineral composition other than is to be
inferred from the classification.

The stone is represented m the national collection only by a small oxidized mass weighing
S grams and a thiix slice of the fresh, unaltered stone weighing 14 grams. A tliin section cut
from this last shows the stone to be of a pronounced chondritic t>^e, the entire mass being
composed of chondndes and fragments of chondiniles closely compressed and with a minimum
amount of fragmental interstitial matter. The mineral composition is nickel-iron, iron sulphide,
olivine, an orthorhombic and a monoclinic pyroxene, the last named polysynthetically twinned.
In two instances intei-stitial areas of the phosphate provisionally called "francoHte" were
noted and there are niunerous areas of the black irresoluble matter wliich Meunier regards as
fayaUte and of secondary origm.

Judging from what has been written and my own oTsservation, the stone is of a somewhat
variable character. From the result of study of this one section I feel disposed to class it as a
veined spheruhtic chondrite (Cca). (See Fig. 2, PI. I.)

Farmington, Washington County, Kans. — This stone belongs to the group of black chondrites
of Brezina, of which but eight representatives are known. The stone was seen to fall and its
history is beyond question. It has been described by several writers among whom only Kunz,
Weinschenk,* and Brezina need here be mentioned. Weinschenk, to whom the microscopic
descriptions are doubtless due, refei-s to the occurrence of " the mineral designated by Tschermak
as 'monticellite-like' formed in the usual way. This contains roimded, colorless mclusions with
bubbles probably of glass." Brezina= says "Auch monticeUitartige Chondren kommen vor."
I am miable in the five thin sections we have of this stone to find the monticeUite-like mineral in
chondrules. It occurs rather hi irregular cavities, sometimes completely filling them and some-
times merely small, colorless crystalhne plates Knmg their waUs. (See Fig. 1, PI. II.) Naturally
there was at once suggested the possibihty that these were a phosphate, a possibihty made a
certainty by treating one of the areas in an uncovered slide with a drop of acid ammonium
molybdate, when the mineral was quite dissolved, giving rise to abimdant crystals of the phospho-
molybdate of ammonium. I have been xmable to detect the "asymmetric feldspar," the
presence of which was thought to be indicated by the chemical analysis. The stnicture is, how-
ever, very obscure, and it is yet possible that a mineral of this nature may exist and be xmrecog-
nizable. Meimier's conclusions relative to the secondary nature of the dark color m the black
chondrites are well supported by a comparison of slides of this stone with those from a roasted
fragment of Homestead.

Forest City, Winnebago County, Iowa. — The only mineralogical description of this stone
that has thus far been given is that of Kunz.' This is incomplete and imsatisfactory, made
evidently without recourse to thin sections and a microscope. He describes it as a "typical
chondrite, apparently of the type of the ParnaUite group of Meimier . . . A broken surface
shows the mterior color to be gray, spotted with brown, black, and white, containing small

> Rev. des Sciences Naturelle, St. Petersburg, no. 1, 1891, p. 41.

! Die Mcteoriten Simmlung, 1895, p. 249 .

' Revision des Pierres Motoorique, 1894, p. 413.

< Min. u. Pet. Mittheil. vol. 12, 1891, pp. 177-182, and Amer. Joum. Sci. ,voI. 43, 1892, pp. 65-67.

' Weiner Sammlung, 1895, p. 253.

« Amer. Joura. Sci., voL 40, 1890, pp. 318-320.

No. 4.1 RESEARCHES ON METEORITES— MERRILL 3

specks of meteoric iron, from 1 to 2 millimeters across. Troilite is also present in small rounded
masses of about the same size. On one broken surface was a very thin scum of black substance,
evidently graphite, soft enough to mark white paper; a feldspar (anorthite) was likewise
observed, and enstatite was also present." Further on, in discussing the analyses by L. G.
Eakins he remarks that " it is of course probable that the Crj O3 represents chromite, and possible
that the alkahcs and alimiina with a Uttle lime represent a soda-hme feldspar." Nothing is
said as to the presence of ohvine, though its presence is to be inferred from the 36.04 percent
soluble in hydrochloric acid.

Under the microscope I find the structui-e very obscure, confused, and, as is so often
the case with meteorites of this class, baffling all efforts at satisfactory descriptions. Few of
the constituent minerals are crystallographically well developed, though occasional smaU forms
in the midst of the chondrules present recognizable crystal faces. The recognized constituents,
aside from the nickel iron and iron sulphide, are olivine and two pyroxenes, one orthorhombic
in crystallization and one monoclinic, the latter polysyntheticaUy twinned. The calcium
phosphate is cormnon in the usual interstitial forms. A black carbonaceous matter in veins
and coating slicken-sided surfaces is not imcommon. Nothing resembling a feldspar is to be
seen in any of the sections examined. (Fig. 2, PI. II.)

GargantiUo (Tomatlan), Jalisco, Mexico. — This stone was described by Shepard,' who
seems to have secured 511 grams out of the total known weight of 780 grams. The mineral
composition as given by him was as follows:

Per cent.

Chrysolite 80. 00

Chladnite (?) 10. 00

Nickeliferous iron 7. 00

Troilite 1

Chromite 3.00

Peroxide of iron J

Total 100. 00

Specific gravity, 3.47 to 3.48.

He noted as a "striking peculiarity . . . the prevalence everywhere of octahedral
crystals of nickeliferous iron," which were "so distinct as to be recognizable with the naked
eye, the brilliant equilateral, triangular faces coming into view by every change of position of
the specimen." No chemical analysis appears to have been made, nor has it apparently been
studied further except by Brezina, who classes it in his catalogue - as a "kugelchenchondrit"
(Cc) and refers to it as having a very loose and friable grotmd mass, thick crust, large chondrules,
many brown flecks, Uke Sai'banovas, and the ii'on abimdant with many crystalline faces.

A httle more may well be added to this description. The stone is so friable and the
abundant chondi-ules so loosely embedded that it is practicaUy impossible to get a satisfactory
section without sacrificing a larger amount of material than is warranted. The microscope
shows an indistinct and confused, fine, granular ground of olivine, enstatite, and occasional
grains of a monoclinic pyroxene, in addition to the metallic constituents and the sulphide.
(See Fig. 1, PI. III.) The fine powder treated on the slide with a drop of ammonium molj'bdate
yields characteristic globules and crystals of phosphomolybdate of ammonium. No feldspars,
even of the maskelynite type, were detected.

A vial of fragments too small for other purposes, found in the Shepard collection, was
sacrificed for the purposes of analysis, with the following results:

Per cent.

Mineral 93. 54

Metal 6. 46

The metal amoxmted to 0.41 grains and consisted of:

Per cent.

Nickel 10. 12

Cobalt 1. 02

Iron (by difference) 88. 86

> Amor. Joum. Sci., vol. 30, 1883, pp. lOi-108. > Die Moteoritensammlung, eto., 1898, p. 256.

4 MEMOIRS NATIONAL ACADEMY OF SCIENCES. rvoi.xiv

The mineral portion amounted to 5.94 giains and consisted of:

Pel cent.

Silica SiOj 41. 16

Alumina AI2O3 3. 97

Ferrous oxide FeO 18. 48

Manganous oxide MnO 0. 39

Chromic oxide Crfii 0. 20

Phosphoric acid P^Oj 0. 30

Sulphuric anhydride SO3 5. 56

Lime CaO 1. 92

Magnesia MgO ' 26. 88

Soda NajO 1. 14

Potash K2O 0. 06

Total 100. 06

A recalculation of these figures gives the following, representing the composition of the

stone as a whole:

Per cent.

SiOa 38.50

AlA 3.71

CrA 0.18

FeO 17. 28

MnO 0.36

MgO 25. 14

CaO 1. 79

NajO 1.06

K2O a 05

P2O, a 28

SO3 5.20

Fe 5.74

Ni 0.66

Co 0.06

Total 100. 01

These figures fall well within the range of chondritic stones. No barium strontium or
other alkaline earths than those mentioned could be detected. No calcium in a water solu-
tion, hence no oldhamite. The mineral composition is olivine, monoclinic and orthorhombic
pyroxene, calcium phosphate (merrUlite of Wherry), chromite, nickel-iron, and troUite.

Hartford (Marion), Linn County, Iowa. — The first descriptions of this stone are by Shepard.'
His determination of its lithological nature is excusable only in consideration of the times and
the means at his command. He wrote: "It appears to contain but a single mineral species of
this (i. e., 'earthy') description, and this one which . . . has imtil now escaped a separate
recognition." For this mineral he proposed the name Tiowardite and gave the complete mineral
composition of the stone as howardite, 83 per cent; nickel-iron, 10.44 per cent; magnetic
pyrite, 5 per cent; olivinoid and anorthite, traces. Some twenty and odd years later Eam-
melsberg ^ reviewed Shepard's work and showed the stone to consist of 10.54 per cent nickel-
iron; 6.37 per cent troilite; 41.58 per cent soluble silicate, and 41.24 per cent insoluble, the
soluble portion being identified as olivine; the insoluble, which was analyzed, being "almost
exactly a bisilicate," but which he does not name.

An examiaation of thin sections from fragments in the Museum collection shows the essen-
tial constituents to be olivine and enstatite, with the usual interstitial calcium phosphate,
nickel-iron, and troilite. The structure is not strongly chondritic. (Fig. 2, PI. III.) No poly-
syntheticaUy twinned pyroxenes were noted. The phosphatic mineral was evident to the naked
eye in two instances as small white spots, perhaps 2 mm. in diameter, on a broken surface of
the stone. These were so soft and friable as to fall down to almost dustlike particles when
touched with a needle point. It is doubtless this brittle property of the mineral, causing it to
break away in the process of grindiag the section, that has prevented its earlier detection. It

• Amer. Joum. Sci., vol. 4, 1847, p. 288, and vol. 6, 1848, p. 403. > Mon.-Ber. Berlin Akad., 1870, pp. 457-459.

No. 4.1 RESEARCHES ON METEORITES— MERRILL. 5

should bo stated that a particle tested by the immersion method showed an index of refraction
of 1.625. Far more abundant than the phosphate is a limpid, colorless mineral, likewise occur-
ring interstitially, but locally so abimdant as to form almost the base in which the other silicates
are embedded. The mode of occurrence and appearance are in every way characteristic of the
so-called maskolynite, but that in many instances the area between crossed nicols breaks up
into granular aggregates which are plainly biaxial and give distinct polarizations in light and
dark, rarely yellowish colors in the thicker sections. The dark cloud, as a rule, sweeps over the
face of the crystal in a manner indicating conditions of strain, and in no case have I been able
to find a satisfactory section showmg the emergence of an optic axis, or other indications of its
optical properties than the indistinct black brushes sweeping across it as the stage is revolved.
It is apparently positive. There are no signs of cleavage, but in a few instances faint lines
were observed traversing the section. In these cases I was able to measure extinction angles
against these lines, of 8° and 10°. But for its very evident doubly refracting properties the
mineral woiild have been set down at once as maskelynite. As it was, additional tests seemed
necessary. Two determinations of its refractive index by the immersion method gave 1.54
and 1.545, which is higher than that of a similar appearing mineral to which I have frequently
referred in other publications. All further doubts as to the nature of the substance are, how-
ever, in this particular case set at rest by the finding of occasional granules still retaining
residual traces of the characteristic twinning bands of a plagioclase feldspar.

Homestead, Iowa. — The Homestead meteoric stone fell on February 12, 1875, and is now
represented by 124,492 grams scattered among 62 collections throughout the world. It has
been the subject of numerous papers, concerning which a reference to Wiilfing's bibliography is
here sufficient. The stone is classed by Brezina as a brecciated gray chondrite (Ccb), and by
Meimier as a limericldte. Wadsworth, who examined the stone in tliin section, states it to
consist of "crystals and grains of ohvine, enstatite, pyrrhotite, iron, and base," and quotes
Lasaulx as stating that it carries plagioclase. Several chemical analyses have been made,
none of which show the presence of any unusual constituents. This is little to be wondered at
when one considers that in the case of Gumbel but 1.5 grams of material were at his disposal.
Much of the interest that is attached to the stone is due to A. W. Wright's work on the gaseous
contents of meteorites.

My own attention was firet drawn to this stone when studying the occurrence of the cal-
cium phosphate concerning which I have of late written several papere, and which, inciden-
tally, I find here in abundant characteristic forms. I do not find the plagioclase feldspar
referred to by Lasaulx, but do fimd in some of the chondrules a polysyn-
thetically t^vinned monoclinic pyroxene which seems to have been wholly
overlooked by previous observer's. The immediate cause of the present
note is, however, the occurrence in each of two shdes examined of a minute,
bright red-brown, scarcely translucent, isotropic mineral embedded in ensta-
tite, as shown in the drawing reproduced here. (Fig. 2.) An attempt at a
definite determination of its mineral nature was a partial failure. Finding it
insoluble in ordinary acids, one of the slides was sacrificed, painting aroimd
the object as closely as possible with vaseline and then covering the exposed
portion with a large drop of fluorhydric acid. The siUcates were all decom-
posed badly, but amidst the gelatinous mass of decomposition products I
was still able to detect the red granule apparently untouched. In an attempt to remove the
granule for further tests and observation, it became hopelessly lost. I can only surmise from
its apparent insolubihty, subtranslucency, color, isotropic nature, and a suggestion of octahe-
dral form, that it may be a spinel. That it is osbomite does not seem probable. The second
slide, from which the accompanying figure was drawn, has been covered and preserved. I may
add that eight other small sections, cut from fragments of the stone in the Barker bequest, gave
no new occurrences of the mineral. This is probably the same mineral noted by Gumbel ' but
thought to be garnet. The decidedly octahedral termination on the form figured in the present
paper seems to warrant its being considered a spinel.

> Sitz. der Uatt-phys. Classe, der K. bayriscben Akad. tu MOncben Dec. 1875, p. 323.

6 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv.

Mc Kinney, Collin County, Tex. — It is remarkable that this interesting stone, which has
been known since 1895, should have been allowed so long to remain unstudied, the bibhography
consisting only of a brief statement by von Hauer ' regarding the acquisition by the Vienna
museimx of upward of 40 kilograms of the material, a description of the stone by Brezina,'
based evidently only on an examination by the naked eye, aided perhaps by a pocket lens, a
brief note by Meimier ^ calling attention to the evidence it afforded of the introduction of the
metal and sulphide after consolidation, and lastly an analysis by Whitfield given in my paper
on the minor constituents of meteorites* the last named made with particular reference to the
possible occurrence of bariima, strontixmi, zirconium, or other of the rare elements.

Macroscopically the stone is fine-grainetl, compact, dull brownish gray, almost black,
looking on a broken surface very much like a piece of hard shale, shoAving here and there a minute
fleok of metal, and with chondrules quite inconspicuous except where it is polished. The
texture is firm and the chondmles break with the stone. On the polished surface they are of
greenish color, suggestive of a sei-pentinous alteration, which, however, microscopic examina-
tion shows not to have taken place.

In the thin section the microscope reveals, in addition to the iron and iron sulphide, three
varieties of pyroxene, one occurring in broad plates ^vith wide (25°-30°) extinction angles, a
polysyntheticaUy twinned variety and normal enstatite, in addition to olivine and the calcimn
phosphate, while the whole mass is here and there so impregnated with a coal black compound
as to give it the dark color referred to.

The chondrules are varied and interestiug. They consist of enstatite ia the common
radiating and cryptocrystaUine forms as well as in good, weU developed phenocrysts in a glassy
or fibrous base. Sometimes the entire chondrule is composed of small, closely compacted
forms -mth httle or no interstitial base. Others are formed wholly of the polysj-nthetically
twinned monocUnic forms. These twinned pyroxenes occur also scattered throughout the
groundmass and under such condition with relation to their associated minerals as to suggest
a dynamic action, a crowding and crushing, and sometimes even raising the question if the twin
structure may not itself be due to this same cause. The occurrence of the twioned forms in
the chondmles where there are no signs of strain forbids, however, the universal apphcation of
any such theory of origin. Still other chondrules are wholly of oUvine. The calcium phosphate
occurs in the usual irregular, interstitial, colorless forms with low relief. The groimdmass
is everj'where so obscured by the black matter that it is impossible to make out a structure for
a certainty. It is apparently fragmental, though if we accept Meimier's views, it may have
been caused by the reheating to which he ascribes this black color. In this connection Brezina
says "Dessen Zugehorigkeit zu den Cs insofeme nicht ganz sichergestellt ist, als die schwarze
Farbe nicht mit Bestimmtheit auf einen Kohlegehalt Zuruckgefuhrt ist". (See further imder
"Effects of dry heat on meteoric stones," Proc. Nat. Acad. Sci., vol. 4, 1918, p. 178.)

This black constituent, which is s\ifficiently abimdant to give the stone a imiform color,
is by no means imiformly distributed, but, as shown in the thin section and figures, is injected
throughout the groimd and along cleavage and fracture lines of the various minerals, being
absent in quantity from the chondrules, forming a dense black, opaque groimd from which
these and the scattered, often fragmental sQicates stand out sharply. An attempt was made
to determine the possible presence of a hydrocarbon, but the facilities at command did not
enable me to arrive at a satisfactory result. One hundred grams of the pulverized stone were
digested for 48 hours, first in ether and next in carbon disulphide. Although care was taken
to use the purest chemicals obtainable, and the filters were fii'st washed in ether, the slight,
colorless extract obtained in the first instance, and the single small drop of a greenish, oil-
like matter in the second, were both felt to be perhaps in part due to impm-ities. Any hydro-
carbon, if present at all, is certainly there in very small quantities. The apparent introduction
(or perhaps better •production) of the coloring matter at a late period in the history of the

' Ann Hof-Mus., vol. 10, 1895, p. 34. ' Revision des Pierres Mcteorique, etc., p. 412.

> Idem, pp. 252, 253. * Mem. Nat. Acad. Sci., vol. 14, 1916, p. 19.

No. 4.)

RESEARCHES ON METEORITES— MERRILL.

Fig. 3.

Fig. 4.

stone is beautifully shown in some of the pyroxene sections where the cleavage and fracture
lines have become so filled as to form a black network between the threads of which the color-
less pyroxenic material standa out sharply and in all its original freshness.

Naturally these observations recalled Meunier's views on the origin of the meteorites of
his tadjerite gi'oup through a preteri'estial heating of aumalites, and the matter seemed of
suflicient interest to warrant a partial repetition of his experiments. The results, I have
given on page 178 of the Proceedings of the Academy as noted above.

Ness Countij, AaTi^.— This is a holocrystalline chondritic stone, of firm texture, the chond-
rules breaking with the matrix. Thin sections show, where not too badly stained by iron
oxides, a granular aggregate of olivine and bronzite with the usual scattering blebs and granules
of nickel-iron and iron sulphide. The chon-
dritic structui'e is very obscure and the chon-
drules themselves present little variation. (Fig.
2, PI. IV.) The stinicture is in places decidedly
cataclastic. Aside from the minerals men-
tioned, I find, rarely, clusters of minute, poly-
synthetically twinned pyroxenes and numerous
limpid, completely colorless interetitial areas,
without crystal outlines or determinable cleav-
age, polarizing onlyin lightand dark colors, often
showing conditions of strain, and giving occa-
sionally biaxial interference figures. It is evi-
dently of the nature of the so-caUed maskelynite.
By careful work with a needle point on an un-
covered section the edge of one of these areas was sufficiently exposed to permit testing by the
immersion method, and found to have an index of refraction of between 1.55 and 1.56, or that
of andesine as given by Iddings. In addition, two of the sections show a completely colorless min-
eral, one of which is isotropic and shows two lines of cleavage cutting at angles of about 56° and
124°, and the other showing extinctions parallel with a single series of cleavage lines and giving
a uniaxial interference figure strongly suggestive of the mineral apatite ' (see Figs. 3 and 4).

Although sought for most carefully, this mineral could not be found in any of the six
other sections examined, and a more exact determination is impossible. It is perhaps the
same mineral referred to by Farrington ' and which he also failed to determine.

OchansJc, Siberia, — ^Through an oversight on my part, this stone in my Handbook and
Catalogue ^ was stated not to have been analyzed as a whole. Since the issue of that publica-
tion, my attention has been called to the paper of Tichomirow and Petrow * in which is given
the analysis quoted below.

My excuse for taking the matter up once more hes in the somewhat unusually high ratio
of nickel to iron^ (1-3.5) which, so far as I now recall, is equalled only by that of the Middles-
borough stone. They also report 0.52 per cent of copper and tin. A quantity of fragments
of not over a gram or so each in weight, the residues from the Ward collection, formed abundant
opportimity for further investigation, which after sundry qualitative tests by myself, was under-
taken in detail by Dr. Whitfield.

As is well known, the stone belongs to the brecciated spherulitic chondrules of Brezina or
canellites of Meunier. The texture seems to be somewhat variable. In a sample received
from De BLroutschoff in 1887, the texture is firm enough to receive a smooth sm-face and a rather
low-grade polish. The samples in the Ward collection, on the other hand, which are fresh
and imoxidized, are quite fiiable. Otherwise, however, both in structure and mineral com-

• A similar mineral described by me in the Mocs meteorite (sec Fig. 5, p. 305, Proc. Nat. Acad. ScL, voL 1, May, 1915) was found to be soluble
In acid and to give solutions reacting for phosphorus and calcium.

> Meteorite Studies I, Field Columbian Museum PubL 64, GeoL Ser., vol. 1, 1902, p. 300.

• Bull. 94, 1916, U. S. National Museum.

• Jour, de russ. phys^hem. Ges. 1S88, Part 1, pp. 513-518.

> See Prior, on the Genetic Relationship and Classification of Meteorites, Uincralogical Magazine, vol. IR, 1916, no. 83, pp. 29 and 33.

8 MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivol.xiv.

position, the stones seem to be identical and there is apparently no reason for doubting the
authenticity of the material now under consideration. A broken siu-face is light ash gray in
color, thickly studded with chondrule,s, some of which are of a dark color and others very light
greenish when broken across. All separate readily from the ground, often in very perfect
spheruUtic forms. No metal is evident to the unaided eye. In thin sections under the micro-
scope the structure Li that of a tufaceous groimd carrying the abundant chondrules, entire and
fragraental, and scattered crystalline particles with the usual spi inkling of metal and metalUc
sulphide. It will be recalled that Siemaschko ' described this last as occmiing in pentago-
dodecahedral forms and, therefore, pyrite. The correctness of this has been questioned (see
Cohen, p. 208). The recognizable silicates are olivine and enstatite, though as often the case
many of the chondrules are densely crypto-crystalline and their mLneralogical natm-e inde-
terminable other than that they are pyroxenic. The powdered stone treated with a drop of
acid ammonium molybdate solution gives rise to abundant leaction for phosphorus, indicative
of a lime phosphate which occurs only in minute interstitial granules quite inconspicuous imless
specially sought under the microscope.

The results of Dr. Whitfield's work are given below. It should be stated that particular
pains were taken, as usual of late, to determine the presence of the rarer elements particularly
tin and copper which the previous investigators had reported, and also the presence or absence
of nickel and cobalt in the silicate portions.

Several grams of the finely pulverized material boiled for half an hour in distilled water
in a platinum vessel yielded no evidences of the presence of oldhamite.

The mineral composition, determined by the usual methods, was foimd to be —

Percent.

Silicate portion (including a small amount of phosphate) 76. 274

Troilite 6. 100

Metallic portion 16. 800

Ohromite (calculated) 0. 766

Total 100. 000

The metallic portion yielded —

Per cent.

Iron 92. 092

Nickel 7. 158

Cobalt 0. 686

Phosphorus 0. 064

Total 100. 000

The silicate portion yielded —

Per cent.

SiO, 44.438

AljO 0.226

CrA => 0. 550

P2O5 0. 503

FeO 13. 675

MnO 0. 376

CaO 1. 505

MgO 27. 204

NiO 0. 678

OoO 0, 066

Na,0 1.186

K,0 0.222

SO, 0. 371

Total 100. 000

' Tschermak's Min. u. petro. Mitt, vol. 11, 1890, p. op. ' Equals 0.766 cUromite.

No. 4.)

RESEARCHES ON METEORITES— MERRILL.

9

By recalculation the composition as a whole is found to be —

Per cent.

SiOj 34.235

AljO 7. 107

Cr,0, 0. 423

PA 0. 387

FeO 10. 535

MnO 0. 289

CaO 1. 159

MgO 20. 958

NiO 0. 563

CoO 0. 058

NajO 0.913

KjO.
SO,.

0.171
0.285

Silicate
portion.

Fe 15. 526]

Ni 1. 196lMetallic

Co 0. llSfportion.

P 0. OllJ

Fe 3.8801

S 2.220

■Troilite.

Total 100. 031

The analysis given by Tichomirow and Petrow is as follows :

Per cent.

SiOj 36. 36

FeO 13. 80

MgO... 18.54

CaO 3.00

Fe 19. 80

Ni 5.55

S 2. 30

P 0.05

C 0.08

CuSn 0. 52

100. 00
The discordance in these results is altogether too large to be accoimted for satisfactorily.
That there must have been some error in the percentage of nickel, as suspected, is evident, as
Dr. Whitfield's analysis of the metallic portion shows but 7.158 per cent of this constituent,
which, when calculated in percentage of the entire stone, amounts to but 1.196 per cent instead
of 5.55. The discrepancy in the calcium oxide (1.159 per cent against 3 per cent) is greater
than should exist in portions from the same mass, but, singularly enough, the amount of troQite
as indicated by the 2.30 per cent of sulphur is about the same and the remaining differences are
perhaps not greater than might be expected with the exception of the alkalies, Whitfield report-
ing 1.084 per cent.

It is to be noted fm-ther that Whitfield reports no traces of tin or copper and that the siH-
cate portion freed from all metal by boiling the finely pulverized mineral in mercuric chloride
still yields 0.744 nickel and cobalt oxide. It may be recalled that in the table of analyses given
in the Memoirs of the Academy ' there are to be found several instances of this character.

To these, at the time, I made no reference in the text, feeling that in some instances at
least they might be due to imperfect separation of the metal from the silicate portion. In
analyses since made especial care has been taken to guard against any such possibility and there
seems no reasonable doubt but that the sUicates-olivines or pjToxenes, or both — in meteorites
carry small quantities of these constituents, as is the case in terrestrial rocks. Such being the
case, it follows that the statement made by Dr. Prior,' together with an explanation by Dr.
Wahl ' to the effect that " the ferromagnesium minerals of chondritic stones contain practically
no oxide of nickel," is founded upon faulty analyses and insufficient data.

' Vol. 14, 1916, pp. 7-27.

> Min. Uag. Nov., 1916, p. 39.

• Uin. u. Petr. Mitthell, vol. 26, 1907.

10 MEMOIRS NATIONAL ACADEMY OF SCIENCES. ivol. xtv.

The Buff's Mountain, South Carolina, meteoric iron, and its included phosphide. — This
beautiful iron was first described by Shepard • and has since been the subject of numerous other
notices, which need reference here only as they bear directly upon the matter in hand. An etched
surface shows it to be a medium octahedrite, after Brezina, or a caillite if we follow Meunier.
It is chiefly distinguished by the broad fields of plessite and the lack of notable quantities of
troilite. The kamacite bands are somewhat swoUen and there are occasional rather incon-
spicuous Reichenbach lamellae. My attention was first drawn to it by the imsatisfactory
nature of Shcpard's analysis and his supposed discovery of potassiimi as one of its constituents.
A slice weighing a little over 150 grams was therefore submitted to Dr. J. E. Whitfield with the
request that he utilize so much as was necessary for an exhaustive analysis. Bulk analysis

yields : rer cent.

Iron 90. 654

Copper 0.018

Nickel 8-550

Phosphorus 0. 233

Cobalt 0. 500

Carbon 0.025

Sulphur 0.020

Silicon None.

Total 100. 000

with no traces of platinimi, palladium, iridium, ruthenium, or the allied elements. Shepard,
it may be recalled, reported 96 per cent iron; 3.121 per cent nickel, with chromium, cobalt,
magnesium, and sulphur in traces. Later Rammelsburg reported a mean of 8.62 per cent
nickel, which is substantially the amount given by Whitfield above. There is nothing of
especial note in this composition unless it be its freedom from the rare elements.

Ninety grams of the iron yielded 1.4843 grams of material insoluble in hydrochloric acid
of one-half ordinary strength. This residue when examined under the microscope was found
to consist largely of schreibersite particles, among which were a few of sufficiently perfect crys-
talline form to permit measurements and determination of crystaUine system. The material
possessed the well-known physical properties of schreibersite (see Cohen, Meteoritenkunde,
pp. 118-131), including the characteristic habit of breaking up readily into cuboidal forms,
and which need not be further discussed.

The particles showing well-developed crystal faces were submitted to Dr. Edgar T. Wherry,
then assistant curator in charge of the Mineral Department, who reported as foUows: ^

The crystals average about one-half millimeter in diameter and are irregularly distorted, some of the faces being
cavernous; the system of crystallization is not evident on superficial examination. The faces yield, however, fairly
good reflections, the positions of which can be located in many cases within 5-10 minutes, unquestionable tetragonal
symmetry being exhibited by the angular relations. The forms observed are: c (001) a (100), m (110), o (111), and
x(362). In addition there are rounded or poorly developed faces of other pyramids and prisms. All of the forms are
incomplete, but there is hardly sufficient regularity in the suppression of faces to justify the assignment of the crystals
to any particular hemihedral class.

Below are given the angles observed, which compare closely with those measured on arti-
ficial crystals by Mallard, Hlawatsch, and Spencer.

Table 1. — Measured and calculated angles of iron phosphide.
Tetragonal, c= 0.346 ±0.001.

No.

Letter.

Symbol.

Crystals.

Measure-
ments.

Angles measured.

Angles calculated.

</>

l>

*

p

0
1
2
3
4

c
a'
m

0
X

001
010
110

in

362

1
2
2
2
1

1
5
5
5
2

0° 00'
90° 00'-
90° 00'-
26° 05'±15'
49° 00'±60'

0° 00'

0°

45°
45°
26°

00'-
00'±15'
00' ±60'
00'±60'

6° 66'

45° 00'
45° 00'
26° 34'

90° 00'
90° 00'
26° 05'
49° 15'

1 Amer. Journ. Sci., vol. 10, 1850 p. 128.

' Amer. Mineralogist, vol. 2, 1917, pp. 80-81; vol. 3, 1198, p. 184.

No. 4.) RESEARCHES ON METEORITES— MERRILL. 11

Several attempts were made at a determination of the chemical composition of this
material, but with results so discordant that the matter must l)o pended awaiting further
investigation.

Tennasilm, Estland, Bussia. — This stone, which fell on the 28th of June, 1872, was described
by G. Baron Schilling some 10 years later.' The acquisition of a fragment weighing nearly a
kilogram, through Krantz, in Bonn, led mo to sacrifice enough for thin sections. An examina-
tion of these leads to conclusions relative to its mineral composition somewhat at variance with
those of Schilling and is the cause of the present note. It should, however, be stated in advance
that Schilling apparently made no use of thin sections, but based his mineralogical determina-
tions wholly upon the results of chemical analysis.

The stone is of a pronounced chondritic type, a veined spherical chondrite (Cca) according
to Brezina, or a limerickite if one follows Meunier. Schilling, as a result of analyses which
need not be repeated here in their entirety, finds the silicate portion of the stone to consist of
54.45 per cent olivine; 32.27 per cent bronzite, and 13.23 per cent labradorite. Cohen ^ seems
to have accepted these results without question and by a further calculation gives the chemical
composition of the labradorite as though it had actual!}' been isolated and analyzed, while as a
matter of fact, as noted later, labradorite, or other feldspar, is wholly lacking, at least so far
as the Museum material is concerned. Meunier ^ apparently accepts this mineralogical deter-
mination, placing the stone in his limerickite group, the mineral composition of which is ensta-
tite associated with bronzite and a feldspathic mineral. My observations are based upon a
study of four thin sections cut from different portions of the mass mentioned. As described,
the stone is of a gray color, plainly chondritic, somewhat soft and friable, the chondrules falling
away readily from the matrix when the stone is broken. The metallic constituents are scarcely
evident to the unaided eye. Under the microscope the chondritic structure is very pronounced
(see Fig. 1, PI. V). The chondrules are in some cases of beautifully limpid, well developed
orthorhombic pyroxenes in a somewhat fibrous base, sometimes of the radiating cryptocrys-
talline forms, sometimes of polj'synthetically twinned monoclinic forms, or again, of olivine.
In no case have I been able to find a feldspar, even in the maskelynite condition.

This occurrence offers an interesting illustration of the danger of calculating the mineral
composition from chemical analyses, and also the weakness of the quantitative classification
when applied to rocks of this type.

Travis County, Tex. — This stone needs a brief reference for the reason that Wiilfing in
his catalogue raises the question if it does not belong to the Bluff, Fayette Coimty, fall.

Such a suggestion is wholly unwarranted, and it is safe to say would never have been
made by one who had seen and compared the two stones. Indeed, if the question of identity
were to be raised it might well be with that of McKanney, in Collin Coimty, which it closely
resembles. Like the McKinney stone it is black in color, very firm and compact, and pre-
sents on a freshly broken surface little to suggest its meteoric nature. It might well be mis-
taken for a fine-grained basalt. The chondritic structure is very obscure and metallic parti-
cles safely identified only with a microscope or pocket lens. Abundant exudations of lawren-
cite, made conspicuous by globules of iron oxide, serve as a fairly safe criterion of its celestial
nature.

Under the microscope the resemblance to the McKinney stone is further augmented. The
ground is everywhere impregnated with a black material, carbonaceous * in part, which per-
meates into the borders of the chondrules and cleavage and fracture lines of the enstatites,
and the olivines have in many cases the same greenish yellow appearance suggestive of a
serpentinous or chloritic alteration. The enstatites of the ground are colorless except where
injected with the black matter which gives the dark hue to the stone. These are interspersed
in a manner difficult of description, with radiating and polysomatic chondrules of both
olivine and pyroxenes often so altered as to break up into scaly and fibrous aggregates when

' Arch. Naturk. Liv. Est. u. Ktirlands, vol. 9, pt. 2, 1882, pp. 95-114.

» Mclooritenkunic, vol. I, p. .110.

» Revision des Picires Mctforiquc, etc., pp. 393-406.

< Roasted in a closed tube tbe powdered stone yields moisture and gives a distinct empyreumatic odor.

12

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

tVoL.XIV.

the stage is revolved between crossed nicols. A monoclinic pyroxene is present in minor quan-
tity, showing indistinct traces of polysynthetic twinning, and there are frequent interstitial,
very irregular areas of calciiun phosphate. It will be noted from Eaktns' analysis that the
stone yields 0.41 per cent PjOj, an unusually large amount. I find nothing that I can with
safety relegate to a feldspar, even of the maskelynite type. The structure is, however, so
obscuie that it will not do to pronounce too definitely on this point. The general resemblance
to the McEonney stone is very close, but in composition, as shown by the two analyses below,
it differs radically in the proportional amounts of alumina and ferrous iron, a difference which
can be explained by the presence of an alimiinous-monoclinic-pyroxene in the stone of McKin-
ney, while magnesian forms prevail in that of Travis County.

Travis
County.

McKin-
ney.

Silica (SiO)

Alumina (8I3O3)

Chromic oxide (CrjOj)

Ferrous oxide (FeO)

Magnesia (MgO)

Lime (CaO)

Manganous oxide (MnO).

Nickel oxide (NiO)

Potash (K^O)

Soda(NajO)

Iron(Fe)

Nickel (Ni)

Cobalt (Co)

Copper (Cu)

Sulphur (S)

Igmtion(HjO)

Phosphoric acid (PjOi)...

Total..
Less O for S.

Percent.

44.75

2.72

.52

16.04

27.93

2.23

Trace.

.52

.13

1.13

1.83

.22

.01

Trace.

1.83

.84

.41

Per cent.

37.900

13.290

1.110

7.400

26.690

1.650

.210

.440

5.070

.920

.050

.004

'6.260

.050

101.11
.92

100.044

Total.

' Chromite.

'FeS.

It is obvious from the above that the Travis County stone is to be classed — following
Brezina — as a black chondrite, rather than a Ckb, as is Bluff.

It is greatly to be regretted that so little is known regarding the fall or fuxding of either of
these interesting stones.

Waconda, Kdns. — ^This stone has been the subject of several papers and briefer references,
of which only those of Shepard, Smith, Wadsworth, and Brezina, are important. Neither
Shepard nor Smith made use of thin sections, a method then practically xinknowa, and their
determinations of mineral composition were surmises based on chemical analyses. Wadsworth
based his brief description evidently on a single section, and there is nothing in Brezina's to
indicate that he made use of other means than perhaps a pocket lens.

As thus far described, the stone is a brecciated crystalline chondrite, or aumaliteof Meunier,
consisting of oHvine, enstatite and a monoclinic pyroxene with the usual sprinkling of metallic
iron and iron sulphide. Smith's analysis, referred to later, showed it to consist of 3.85 per
cent troilite, 5.34 per cent nickel-iron, and 90.81 per cent stony matter. In describing the
appearance of the stone he mentioned as occurring "only on one part" of his specimen a mineral
''in the form of a white, crystalline mass, not exceeding in weight 20 milligrams," which was
soluble in hydrochloric acid, the solution reacting for magnesia and sihca. This mineral he
thought might occupy "the same place among the unisilicates of the meteorites that the ensta-
tite does among the bisilicates."

In looking over a quantity of fragmental material in the Shepard collection my attention
was attracted to a small white area, some 2 mm. in diameter, on one of the fragments, and,
recalling Smith's work, I undertook its determination. The results are given below, and, as
will be apparent, the investigation was much more extended than at first intended.

In the thin section the stone is at once seen to be composed essentially of ohvine and pyrox-
ene with nickel-iron and troilite. The chondritic structure is very evident (Fig. 2, PI. V), the
individual chondrules consisting whoUy of pyroxenes or of olivine in the customary forms,

No. 4.)

RESEARCHES ON METEORITES— AfERRILL.

13

embeckled in a crystalline ground of the same constituents, and the metallic components. Where
not stained by oxidation the silicates are beautifully clear and pellucid . The pyroxene is in part
of the normal enstatite type, though many of the larger forms are monodinic, showing extinction
angles as high as 25°. In almost the first section examined attention was attracted to a minute,
irregular, colorless area traversed by numerous fracture lines, with only moderate rehef, non-
pleochroic, and polarized in faint bluish-gray colors. Its appearance at once suggested the
phosphatic mineral described by me in a previous paper.' Microscopic examination of a con-
siderable number of slides, accompanied in some instances by microchemical tests, showed the
mineral to be a calciiun phosphate, and occurring not infrequently. In no instance was the
mineral found in the crystalline form characteristic of apatite. Nearly altogether it occurs as
an interstitial filling, almost isotropic, and, as in the previous cases which I have described, of
lower refractive indices than normal apatite. Indeed, in many instances the mode of occurrence
and low relief without cleavage or crystal outline causes it to resemble on casual inspection an
interstitial glass, for which doubtless it has heretofore been frequently mistaken. In such cases,
it is onl}' by treating a sUde with a drop of acid and watching the mineral gradually disappear,
then testing the solution, that its true nature can be determined.

Further examination showed the presence of this phosphate in the Waconda stone where
it could not be recognized even microscopically. It was found that when the surface of an
micovered slide was treated with a dilute solution of hydrochloric acid and allowed to stand
for not more than a quarter of an hour, the solution thus obtained would react for phosphorus
and calcium, and the slide when examined be found to contain frequent minute, irregular,
interstitial pits where the material had been dissolved away.

These determinations natujaUy suggested the possible phosphatic nature of the white
spots before noted. An examination with a pocket lens showed these to be composed of aggre-
gates of minute crystals of a faint yellow-green tint. It being obviously impossible to rely
on cutting a thin section including the desired area, recourse was made once more to micro-
chemical tests on minute fragments broken out by a needle point. Reactions for phosphorus
and calcium were easily obtained, the mineral being readily soluble in cold nitric acid and less
so in hydrochloric acid. Dr. E. S. Larsen kindly determined the indices of refraction by solu-
tions, as follows: a =1.627 ±0.003; 7= 1.621 ±0.003. These results are low for normal apatite,
agreeing more closely with those obtained by Dr. Wright on material from the Alfianello and
Rich Mountain stones, as given in the paper before referred to.

As phosphorus was not determined by J. L. Smith in his analysis of either the metallic
or silicate portions of this stone, a second analysis was decided upon. The results as deter-
mined by Dr. J. E. Whitfield are given below, Smith's results being also given for purposes of
comparison.

Preliminary separations yielded:

J. E. WWtflold.

J. I>

Smith.

Fa cejit.

87.80
5.93
6.27

Pe

rcent.
90.81

Kickel-lron . . .

5.34

3.85

100.00

100.00

Nickel-iron yielded:

Iron .

85. .W
13.78
.71
(■)

86.18

Nickel

12.02

Cobalt . .

.91

.04

Total

99.99

99.15

> On the montlceUite-Uke mineral in meteorites, and on oldhamite as a meteoric constituent, Froc. Nat. Acad. Sci., voL 1, 1915, p. 302, and On
the calcium phosphate o( meteoric stones, Amer. Joum. ScL, vol. 43, 1917, p. 322.
< Not determined.

14

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

rvoL-xiv.

Phosphorus not determined in either case.

Smith fm-ther determined the stony portion to consist of 69 per cent soluble in aqua regia
and 41 per cent insoluble, giving analyses of each, from which the bulk analysis given below
was calculated.

J.

E.M'hitDeld.

J

L. Smlth.i

giO,

{

Per cent.

35.05

16.53

.23

4.94

2.25

24.98

.74

.04

.29

.06

.76

.17

l.Bl

6.07

.81

.04

3.99

2.28

}

Per cent.
38.14

FeO '

23.44

p,0

{')

jtljOa

1.02

CaO

(')

MsO

26.69

N^ :::::::::::::::::::::::

CoO

MnO

.47

SO3

(=)

NajO

1.05

KjO

(')

HjO .

h)

Fe

4.64

Ni

.65

Co

.05

F®--\TroiUte .

3.85

S...J

99.84

100.00

1 Analyses recalculEted by Farrington. Smith reported also traces of lithium and copper.

> Not determined.

'Trace.

Five grams of the finely pulverized stone were boiled in distilled water for an hour,
resulting in a solution yielding 0.062 per cent SO3 and 0.012 per cent CaO. A portion of
the SO3 probably came from the decomposed troilite, rendering any calculations imcertain,
while the amoimt of lime (CaO) is too small to make the results moi'e than suggestive of
the presence of a minute quantity of oldhamite. A second 5 grams were boiled for half
an hour in acetic acid of 15 per cent normal strength. The solution yielded 0.08 per cent
P2O5 and 0.122 per cent CaO. Inasmuch as the bulk analysis shows 0.23 per cent P2O5, it is
evident that a complete solution of the phosphate was not accomplished by the acetic acid.
Be this as it may, the relative proportion of acid to base is such as to render it unsafe to draw
definite conclusions.'

It is difficult to account for all the discrepancies between the two analyses. The difference
of some 3 per cent between the amoimt of stony matter and troilite may perhaps be accoimted
for on the supposition that Smith worked, as is so often the custom, on verj^ small amounts
that did not correctly represent the stone as a whole. (Whitfield had 19 grams of selected
material.) The analyses of the metallic portion, it will be noticed, agree fairly well excepting
that Whitfield reports no copper. In the bulk analyses, however, we find a difference of
3 per cent (in round numbers) in the total silica, nearly 7 per cent in the ferrous iron, 3.92 per
cent in the almnina, and 2.25 per cent in the lime, with minor differences, mainly due to
omissions elsewhere. The totals for iron and magnesia do not differ more than might be
anticipated from analyses on separate portions, made by even the same analyst. That Smith
did not determine the nickel and cobalt in the silicate portion is not strange, it being customary
in his day to regard these elements as constituents of the nickel-iron only. The phosphoric
acid, amounting to 0.23 per cent, should in this day certainly not be overlooked.

It does not seem in the least probable that the phosphate to which I have referred above
as evident to the unaided eye can be the white mineral mentioned by Smith. Nevertheless,
a most careful examination of all the material in the Museum and Shepard collections reveals
nothing that is even suggestive of his doubtful unisilicate.

I Since the above was written, Dr. E. T. Wherry ( Amer. Mineralogist, vol. 2, No. 9, 1917) has complimented mo by suggesting that the problem-
atic phosphate be given the name merrillUe. Had I been consulted in the matter I should have suggested a postponement until a more definite
statement of its composition could be given. Incidentally, it may be stated, I had considered the use of Shepard 's name, apetoid (Amer. Journ.
Boi., vol. 2, 1846), but abandoned it because of his definite statement that his mineral contained no phosphorus.

No. 4., RESEARCHES ON METEORITES— MERRILL. 15

Weston, Conn. — Notwithstanding that this is the oldest known of American falls, it is de-
serving of more detailed study than it has yet received either from a mineralogical or chemical
standpoint. The work of Shepard (in 1809, 1846-1848) would natui-ally at this date be consid-
ered faulty. He described the stone as composed principally of howardite and olivinoid, with
scattered grains of magnetic pyrites and nickel-iron. Little advance over this seems to have
been made by subsequent workers, excepting Meunier, who, in classifying the stone as a limer-
ickite, recognized its chondritic character and mineral composition. Brezina classified it as a
spherical chondrite, brecciated, apparently without regard to its composition or microscopic
structure. The breccia-like structme is very evident, and is produced by angular pieces of a
light gray color embedded in the prevailing dark-gray material. The chondritic structure is
equally pronounced in both, and so far as can be determined by the unaided eye or a pocket lens
there are no essential differences between the two kinds of fragments other than that of color.
The mineral composition I find to be chiefly a pyroxene with a low angle of extinction, about
10°, which therefore relegates it to the clino-enstatite of Wahl, a polysynthetically twinned
pyroxene, olivine, "merriUite," nickel-iron, and iron siilphide. No feldspars, even in the form
of maskelynite, were observed.

o

MEMOIRS NATIONAL ACADEMY OF SCIENCES XIV.

FOURTH MEMOIR PL. I.

FIG. 1.

FIG. 2.

MEMOIRS NATIONAL ACADEMY OF SCIENCES XIV.

FOURTH MEMOIR PL. II.

FIG. 1.

FIG. 2.

MEMOIRS NATIONAL ACADEMY OF SCIENCES XIV.

FOURTH MEMOIR PL. III.

FIG. 1.

FIG. 2.

MEMOIRS NATIONAL ACADEMY OF SCIENCES XIV,

FOURTH MEMOIR PL. IV.

FIG. 1.

FIG. 2.

MEMOIRS NATIONAL ACADEMY OF SCIENCES XIV.

FOURTH MEMOIR PL. V.

FIG.

FIG 2.

MEMOIRS

OF THE

NATIONAL ACADEMY OF SCIENCES

"Volume XIV

WASHINGTON

GOVERNMENT PRINTING OFFICE

1921

NATIONAL ACADEMY OF SCIENCES.

Volume XIV.
B^HTTH m;km;oir.

TABLES OF THE EXPONENTIAL FUNCTION AND

OF THE CIRCULAR SINE AND COSINE

TO RADIAN ARGUMENT.

BY

C. E. VAN ORSTRAND.

I

TABLES OF THE EXPONENTIAL FUNCTION AND OF THE CIRCULAE SINE
AND COSINE TO EADIAN ARGUMENT.'

By C. E. Van Ohstrand.

The tables accompanying this paper have been prepared with the expectation of meeting
a twofold requirement. The first was to obtain a few high place values at sufficiently small
intervals of argument for general use in the evaluation of integrals and other functions; the
other object was to obtain a basis for subsequent interpolation to small intervals of argument
for use in the construction of complete 10-place tables which are applicable in the various
fields of pure and applied mathematics. The need of tables meeting these and other require-
ments has been emphasized by various authors.

The most important tables of extended values of the exponential function in which the
exponents are integers or fractions have been constructed by Schulze, Bretschneider, Newman,
Gram, Glaisher, and Burgess. Bretschneider included a few high place values of the circular
sine and cosine to radian argument, but with the exception of these and a few values computed
by Gudermann, there appears to be no extended values of these important functions.

Schulze ' gives values of the ascending exponential at intervals of unity between the limits

1 and 24, inclusive, to 28 or 29 significant figures, and for the special arguments 25, 30, and 60

his values include 32 or 33 figures. In so far as I have been able to ascertain, Schulze gives no

information regarding methods of computation or accuracy of results. Glaisher ^ verified the

first 15 figures of Schulze's value of e'" by direct substitution in the series; the first 13 powers of e

were verified to 22 places of decimals; and the values of e", e", ... e" to 15 places of decimals

by means of the relation

e"— 1

7=e''"' + e""' ... +e + l.

e— 1

Bretschneider * evaluated e, c"', sin 1 and cos 1 each to 105 places of decimals; also values
of the same functions at intervals of unity between the limits 1 and 10, inclusive, to 20 places
of decimals. He corrected the erroneous value of e given in Callet's tables and Vega's Thesaurus
and the slightly erroneous values of sin 1 and cos 1 given to the twenty-fifth decimal by
Gudermann."

Bretschneider obtained his values by direct substitution in the exponential series in con-
nection with the evaluation of the three transcendants,

> Published by permission of the Director of the U. S. Geological Survey.

' I. S. Schulze, Sanunlung logarithmischer Trigonometrischer-Tafeln (Berlin, 1778).

• J. W. L. Qlaisher, Tables of theerponential function. Camb. Phil. Trans., vol. 13 (1883), pp. 243-272. (In Salomon's Tafeln (1827) the Tsluei
of «n, e-n «"•"", ... £«J<»«»n where n has the values 1, 2, ... 9 are given to 12 places.

* C. A. Bretschneider. Berechnung der Grundzahl der natiirlichen Logarithmen, so wie mehrerer anderer mit ihr Eusanunenbangender Zahl-
werthe. Grunert's Archiv der Math, und Phys., Bd. 3 (1S43). pp. 27-34.

> C. Qudennann. Potential oder cyklsch-hyperbolische Functionen. Jour, ftjr die reine imd angewandte Math., Bd. VI (1830), pp. 1-89.

5

MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Voc.xiv.

0^

c- J sin a; J 1 z\ 1 i^ 1 x' ,

Six= / ^-dx=x-3 3-, + ^^-^^ +

t/oO

/t: I COS X ,

Cia;= / rfx

TT fl 2! , 4! 6! ,

= Fr — cos X 5 + -^ 7+ • •

2 Lx J? x^ z'

•

. ri 3!^5! 7!^ 1

•

ri 2! , 4! 6! ,

=sin X 5 + — s — 5+ • • •

\_x x' x" x^

ri 3!^5! 7!^

— cos a; —^ i + —i — 5+ • ■ •

\_x^ X* x" x'

, 1 , . .s , , 1 xV 1 a;V

Eix= / ^ dx=7 + |loge (x')+X+^|| + ^ jj +

ri , 1 , 2! , 3! , 41, 1

known, respectively, as the sine integral, the cosine integral and the exponential integral. The

quantity y is the Eulerian constant 0.5772156

Newman's ' contribution to the subject consists of the following:

18-place values of e""^ from x = 0.0 to x = 37.0 at intervals of 0.1.
12-place values of e-^ from x = 0.000 to x = 15.349 at intervals of 0.001.
14-place values of e~^ from x = 15.350 to x= 17.298 at intervals of 0.002.
14-place values of e"^ from x= 17.300 to x = 27.635 at intervals of 0.005.
16-place values of e^ from x = 0.1 to x = 3.0 at intervals of 0.1.
12-place values of e^ from x = 0.001 to x = 2.000 at intervals of 0.001.
The 18-place table is hardly the equivalent of a 16-place table, as the original computation
included only 18 decimals.

All of Newman's computations are based on formulas of the type

M±N= e-*'i = e-[l ± ^, + J-^ ± |-' + . . .]

wherein h assimies the constant values 1, 0.1, 0.01, . . . dependent upon the interval of inter-
polation. Having given e~^ and e-^+'' the value of e~^~^ is computed from the formula by
putting

M= S^e-^ and iV= S-. e"^,
m! n\ '

m being an even and n an odd integer. The values of the separate terms in these expressions
may be computed by successive divisions. Then the appropriate sumimations give

a known quantity, and

' F. W. Newman. Tables o( the descending exponential function to 12 or 14 places of decimals. Trans. Camb. Phil. Soc, vol. XIII (1883),
pp. 146-241; table of the exponential function e' to 12 places of decimals. Trans. Camb. Phil. Soc., vol. XIV (1889), pp. 237-249.

NO. B.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 7

the quantity to be determined. The equation for M± N provides a check on the values of
M and N, but the sum or difference which is the quantity sought is not verified by this method
until another interpolation is made.

Gram ' gives values of the ascending exponential to 24 places of decimals at intervals of
imity between the limits x = 0 and a; = 20, inclusive; also values of the same function from
z = 5.00 to x = 20.00, inclusive, at intervals of 0.02, the number of tabular decimals ranging
from 4 to 15; and from a; = 0.1 to a; = 15.0, the values are given to one decimal only at inter-
vals of 0.1. Some of the values wore obtained by either repeated multiphcation or logarith-
mic computation, and the remainder were borrowed from Schulze, Bretschneider, and Opper-
mann.

Glaisher ' gives 10-place logarithmic values and 9 significant figures of the natural values
of both the ascending and descending function for the following ranges of argiiment:
From x = 0.001 to a; = 0.100 at intervals of 0.001.
From x = 0.01 to x = 2.00 at intervals of 0.01.
From a; = 0.1 to a; = 10.0 at intervals of 0.1.
From X = 1 to a; = 500 at intervals of unity.

Since the natural values were computed from the logarithmic values, the maximum
tabular error is one unit in the ninth significant figure with the exception of values of e-^ con-
tained in Newman's tables. The remaining values of Glaisher's tables were checked either by
differences or by duplicate computation. Glaisher gives also the reciprocals of the factorials
from 1 to 50, inclusive, to 28 significant figures, and verifies his values by forming the summa-
tions for e and e-K A further verification is obtained by evaluating e"'" to 32 decimal places
by means of the formula

* - 10"
The quantity y X 10~™ is here an approximate value of e~^. The equation gives

loge(y + 7i) = n loge 10 - X
a known quantity. Since log^y is also known, we may evaluate the expressions

Wog^iy + h) -log^y] and y]loge{y + h) -logey].
The expansion of the first by Taylor's series gives

Ji. = yWogtiy + 1)- log,;/] + 2 y ~ 3 p '*' ' " ' '

from which an approximate value of h y-' may be computed by neglecting terms in h beginning
with the square. Finally the substitution of

1 Tv' , 1 , ^'
7; y -; and — -^h-z

2 " y' 3 y'

in the preceding equation gives a corrected value of ^.

Burgess^ gives 30-place values of e~^ for x = 0.5, 1, 2, ... 10; and 14 values of «~*' at
irregular intervals between the limits 1.0 and 3.0, ranging in extent from 23 to 27 decimals.
These values were used in his evaluation of the probabihty integral, but no information seems
to have been given with regard to either method or accuracy of computation.

In his "Rectification of the Circle" (1853), Shanks evaluates the Naperian base by direct
substitution in the series to 137 places of decimals. His second computation * was carried to

' J. p. Gram, Undersogelscr angaaende Maengdcn at Prlmtal under en given Graense. Copenhagen Academy, 6 vol. II (1884), pp. 183-306.
« J. W. L. Glaisher. Tables of the exponential function. Trans. Camb. Phil. Soc., vol. 13 (1883), pp. 244-272.

> James Burgess. On the deflnita Integral _?_ I e"**,!/, with extended tables of values. Trans. Roy. Soc. Edinburgh, vol. 39 (1900), pp.
257-321. -JrJo

' William Shanks. On the extension of the value of the base of Napier's logarithms; of the Naperian logarithms of 2, 3, 5, and 10, and of the
Modulus of Brlgg's, or the Common system of logarithms; all to 205 places of decimals. Proc. Roy. Soc. Vol., VI (1850-1854), p. 397.

8 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv.

the two hundred and fifth decimal by a method given in J. R. Young's Elementary essay on
the computation of logarithms (pp. 13-14). Glaisher ' verified the first resvdt, using the con-
tinued fraction

1 ' 6 10 4n + 2+ ...

but the second result was shown by Boorman ^ to be incorrect after the one hundred and eighty-
seventh decimal.

Boorman's formula is readily deduced. Since we have identically

1 1^1. n+l-m
m mn~n mn
we obtain

m mn n mn
by the substitution

m = n—l

in the numerator of the right-hand member. The series for the Naperian base may thus be
transformed into the series

\ ij l\n mnj I mAn^ m^iij
1 mn m^n\n^ m^n^/

wherein m = 2, n = 3; m, = 4, n, = 5; m^^^, n^^l;

Tich&nek ' and Minks verified Boorman's value of e as far as the two hundred and twenty-
third decimal, making use of Euler's continued fraction

in connection with the relations

2.1^2.3^2.5

1 + F

l-F

e-1 1 1^1 1 ^ 1

2 1 1.7 7.71 71.1001 ' 1001.18089

Gauss* gives values of e"T ranging from 15 to 57 decimals for 13 values of n at irregular
intervals between the limits 1/2 and — 16. He used the formula

giT _ j\^glog O+IO log 6-log c-log 6 . .. -I"

The quantity N is an approximate value of e"" multipUed by aXlO*, and the quantities
c,d,... are the factors of N so selected that their natural logarithms may be taken from
Wolfram's ^ tables.

The present contribution consists of the following tables :

Table I: Values of the reciprocal of n\ to 108 places of decimals at intervals of unity from
1 to 74.

Table II: Values of g^ to 42 significant figures at intervals of unity from 0 to 100.

Table III: Values of e* to 33 significant figures at intervals of 0.1 from 0.0 to 50.0.

> J. W. L. Glalsher. On the calculation of £ from a continued fraction. Brit. Assoc. Eep. (1871), pp. 16-18.

' J. Marcus Boorman. Computation of the Naperian base. Math. Mag. vol. I (1882-1884), pp. 204-205-, see also L'lntermedlalre des mathe-
maticiens, vol. 7 (1900), p. 53; G . Peano, Formelaire de mathematiques. Tome II, No. 3, p. 125.

' F. J. Stndnifika. Ueber^die Berechnung die transcendenten Zahl e. Jahr. iiber die Fort, der Math. Bd. 23 (1891), p. 440: VortrSge fiber mono-
periodische Functionen. Jahr. ilbcr die Fort, der Math. Bd. 25 (1893-1894), p. 736.

' Lemniscatische Fimctionen. Werke 3, pp. 413-432.

'• Logarithmorum Nsturalium. (48 decimals). See Vega's Thesaurus, pp. 641-684.

NO. 0] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 9

Table IV: Values of e* to 62 places of decimals at decimal intervals from 1 X 10-'° to
9 X 10-'.

Table V: Values of er^ ranging from 52 to 62 places of decimals at intervals of unity
from 0 to 100.

Table VI: Values of e~^ ranging from 33 to 48 places of decimals at intervals of 0.1 from
0.0 to 50.0.

Table VII: Values of e~^ to 62 places of decimals at decimal intervals from 1 x 10"'° to
9 X 10-'.

Table VIII: Values of giCnWaeo) ^q 23 places of decimals or significant figures at intervals
of unity from n = 0 to n = 360.

Table IX: Values of «='='" to 25 places of decimals or significant figures for various values
of n.

Table X: Values of sin x and cos x to 23 places of decimals at intervals of unity from 0
to 100.

Table XI: Values of sin x and cos x to 23 places of decimals at intervals of 0.1 from 0.0
to 10.0.

Table XII: Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from
0.000 to 1.600.

Table XIII: Values of sin x and cos x to 25 places of decimals at decimal intervals from
1 X 10-'° to 9 X lO-*.

Table XIV; Miscellaneous values of e^, e~^, sin x and cos x to a great number of decimals,
including Boorman's value of e.

The tabular error of the preceding*tables may in some cases slightly exceed 5 units in the
next succeeding tabular digit as two digits only were dropped from most of the values.

In my preliminary computations, Glaisher's ' table of the reciprocals of the factorials
was used. It contains all of the recurring decimals from n=l to 71=12, inclusive, and from
71=13 to n = 50, inclusive, the values are given to 28 significant figures. This table frequently
failed to give the requisite number of decimals in the vicinity of 7i=13 and upwards, so it was
afterwards extended rouglily to 110 decimals, and the range of the argument extended from
50 to 74, the results being verified by forming the summations for e and e~^, and then computing
the product of these two quantities, in addition to making a direct comparison with the well
known value of e. A further check on the value of e"' consisted in reciprocating e, written in
the form (a + 6)-'.

With the table of reciprocal factorials as a basis, it was easy to compute the value of e*-'
from the series: and afterwards by repeated multiphcations by this factor, in accordance with
the formula,

gZ+Ax=gZ. gAi (2)^

the values of e"', e"*, ... e were obtained and verified by comparing the last computed
value with the well known value of e. Similarly the value of e'" was computed from e and the
value of e'°° was computed from e'°. Values of the descending exponential for the same inter-
vals of argument were determined in the same manner, and the evaluation of both functions
was verified at frequent intervals by means of the product relation, e^e"*. Another check con-
sisted in substituting values of x and Ax in (2). Subsequent interpolations to one- tenth the
previous interval of interpolation provided a further complete check on the entire computation.
The maximum difference between any value and the corresponding value obtained by 10
interpolations did not exceed 15 units in the last decimal or significant figure. Practically all
of the computations were made with a 10-groove computing machine of the millionaire type.

> Z. W. L. Olalsher, Tables of the exponential function. Trans. Camb. Phil. Soc., vol. 13 (1883), pp. 244-272.

• C. F. Degen, Tabularam Enneas (Copenhagen 1824) gives IS-place values of logi« (n!) from n— 1 to n— 1200. De Morgan reprinted the same
to 6 places in his article on " Probabilities " in Encjclopedia MetropoUtana. J. \V. L. Glaisher gives 20-place values of n X nl and lO-place values
ot-log (nXn!) to n-71 in Phil. Trans. Roy. Soc., vol. 160, 1S70, p. 370. Shortrede, Tables (1849, Vol. I) contains 5-place values of log (n!) to
n- 1000 and 8-place values for arguments ending In 0.

10 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [voi..xiv.

The values of e±"W380 contained in Tables VIII and IX were computed in the manner just
described from the values, £>±W36o fi^Q latter function was evaluated for the first 10 decimals
of the exponent by successive multiplication of the appropriate factors taken from Tables
IV and VII. The values for the remaining decimals of the exponents were obtained by sub-
stitution in the exponential series. The product of the two factors thus obtained is the re-
quired result. Checks were applied in the usual manner, also by comparison with the values
given by Gauss.'

Values of the exponential function previously obtained provided an excellent check on the
fundamental values needed in the computation of sin x and cos x. These values were com-
puted at intervals of 0.1 from 0.0 to 1.6, inclusive, by direct substitution in the series and
verified by means of the relation

i~x^ x^ ~i [~x^ x^ n

e* = sina; + cosa; + 2 9i+gj + -- +2 37+7T + - - (3)-

Interpolations were made by means of the formulas

. , . , . Ax (Ax)' .
sin(x + Aa;) =sm x + jy cos x 2^sma; + ,..

Ax . (Ax)'

cos (X + Ax) = cos X — ry Sm X ;j|— COS X + ...

in which Ax assumes the values 0.1, 0.01, ... according to the interval of interpolation. It
will be noted that the two equations together contain terms of the form

(Ax)" sin x/n! and (Ax)" cos x/n\

wherein n assumes successive values of the natural numbers beginning with unity. There are

thus two series of terms,

1 . 1 . 1 .

g-f sm X, ^ sm X, jy sin x, ...

1 1 1

■^ cos X, ^ cos X, jy- COS X, ...

which may be evaluated by dividing the sine or cosine, as the case may be, first by 2, this quo-
tient by 3, the. last by 4, and so on, thus avoiding the use of large factors. The computation
of both functions is made at the same time, and a complete check is obtained on the tenth inter-
polation. The maximum difference between the interpolated values is 10 units, and as there
were two interpolations, the maximum error of interpolation is of the order of magnitude of
20 units in the twenty-fifth decimal. Table X was computed with the assistance of a com-
puting machine by substitution in the trigonometric expansions for sin (x + Ax) and cos (x + Ax),
and verified by assigning various values to x and Ax; also by forming the smn of the squares
of the sine and cosine for several values of the argument. The values of sin x and cos x con-
tained in Tables XIII and XIV were computed by substitution in the respective series and
verified by means of equation (3).

Writers on interpolation emphasize the importance of interpolation by differences while
not much attention is given to interpolation by means of derivatives. This procedure does
not seem justifiable as the time lost in retabulating and differencing the quantities is some-
times much greater than the loss of time due to the possible increased labor and difficvdty of
computation by the derivative formula. Furthermore the check provided by the derivative
formula is much more rehable than that of the difference formula when both the interval of
interpolation and the interpolated values are large. Neither method provides an absolute
check, for experience proves that positive and negative errors of equal or approximately equal
magnitudes very frequently escape detection. The same is true of the various methods of
mechanical quadratures which could be used for the same purpose.

' Loc. cit.

No. 5.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 11

A comparison of the present values with those mentioned in the first part of this paper
shows some interesting results. The values given by Schulzo are generally incorrect in the
last or the next to the last decimal. Newman's 18-place table of the descending exponential
is correct to 16 decimals when the last two decimals are taken into account. His values for
x= —3.5, —26.1, —26.4, and —26.9 contain misprints. The values of f~"-' by Burgess is in
error by approximately one unit in the tliirtieth decimal. Glaisher's value of e"'" computed
from formula (1) is correct. His table of the reciprocals of the factorials contain errors shghtly
in excess of 5 units in the next succeeding decimal for n = 20, 27, 41, and 50. All of Bret-
Schneider's values are correct and the values of e* given by Gram to 24 decimals are correct.
The value of e^P given by Gauss is incorrect in the twenty-third and following decimals.

The present paper was completed before the 1916 report ' of the British Association for the
Advancement of Science was received. The report of the committee on the calculation of
mathematical tables (pp. 59-126) contains the following tables of sin x and cos x to radian
argument :

Table I: Values of sin x and cos x to 11 places of decimals at intervals of 0.001 from 0.000
to 1.600.

Table II: Values of x— sin x and 1 —cos x to 11 places of decimals at intervals of 0.00001
from 0.00001 to 0.00100.

Table III: Values of sin x and cos x to 15 places of decimals at intervals of 0.1 from 0.1
to 10.0.

In one value only, does the tabular error of Tables I and III exceed 10 units in the next
succeeding decimal; the value of sin 9.1 should read 52 instead of 53 in the last two tabular
digits.

Nearly all of the numerical computations were made by A. G. Seiler, piece work computer,
and R. Weinstein and A. T. Harris, aids in the physical laboratory of the Geological Survey. I
am indebted to F. A. Wolff, of the United States Bureau of Standards, Washington, D. C, for
valuable suggestions in regard to the contents of Table IX, and to E, B. Escott, who kindly called
my attention to the omission of several important references which had been overlooked in my
prehminary pubUcations in the Journal of the Washington Academy of Sciences (1912-13). The
values given by Gram and Bretschneider were especially useful as a partial check on certain
values which I had previously carried to a slightly greater number of decimals. No errors were
discovered in my computations.

' The same report, pp. 123-126, contains the following:

10 place values ol the loRarithmic gamma function at i ntervals of 0. 005 from 0. 005 to 1. 000.

10 place values of the integral of the loKarithmic gamma function at intervals of 0. 01 from 0.01 to 1.00,

13 place values of the logarithmic derivate of the gamma function at intervals of unity from 1 to 101, and from 0.5 to 100.5.

12 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [Vol.xiv,

Table I. — Values of the reciprocal ofn\ to 108 places of decimals at intervals of unity from 1 to 74.

n

1

1

1.

2

0.5

3

.16 .

4

.0416

.

5

0. 0083

6

. 00138

7

.00019 84126

8

00002 48015 873

9

.(5) 27557

31922

39858

90652

10

0. 5) 027S5

73192

23985

89065

2

11

.(5) 00250

52108

38544

17187

7

12

.(5) 00020
53476

87675
5645

69878

68098

97921

00903

21201

43231

25434

23654

13

.(5) 00001
34882

60590
8126

43836

82161

45993

92377

17015

49479

32725

71050

14

. (10) 11470

74559

77297

24713

85169

79786

82105

66623

26503

59634

48661

86136

02740

58686

75709

94555

12153

92485

23375

508

15

0. (10) 00764

71637

31819

81647

59011

31985

78807

04441

65100

23976

63244

12409

06849

37245

78380

66303

67476

92832

34891

701

16

. (10) 00047

79477

33238

73852

97438

20749

11175

44027

59693

76498

47702

75775

56678

08577

86148

79143

97967

30802

02180

731

17

. (10) 00002

81145

72543

45520

76319

89455

83010

32001

62334

92735

20453

10339

73922

24033

99185

22302

58703

95929

53069

465

18

. (15) 15619

20696

86862

26462

21636

43600

57333

42351

94040

84469

61685

54106

79112

99954

73461

26483

55329

41837

192

19

. (15) 00822

06352

46624

32971

69559

81236

87228

07492

20738

99182

61141

34426

67321

73681

82813

75025

4.5017

33780

905

20

0. (15) 00041

10317

62331

21648

58477

99061

84361

40374

61036

94959

13057

06721

33366

08684

09140

68751

27260

86689

045

21

. (15) 00001

95729

41063

39126

12308

47674

37360

64303

55287

47379

00621

76510

53969

81365

90911

46131

01297

66032

812

22

. (20) 08896

79139

24505

73286

74889

74425

02468

34331

24880

86391

89841

38816

80971

17768

70278

68240

80274

219

23

. (20) 00386

81701

70630

68403

77169

11931

52281

23231

79342

64625

73471

36470

29607

44250

81316

46445

26229

314

24

. (20) 00016

11737

57109

61183

49048

71330

48011

71801

32472

61026

07227

97352

92900

31010

45054

86268

65217

888

25

0. (25) 64469

50284

38447

33961

94853

21920

46872

05298

90441

04289

11894

11716

01240

41802

19410

74208

716

26

. (25) 02479

59626

32247

97460

07494

35468

47956

61742

26655

42472

65842

08142

92355

40069

31515

79777

258

27

. (25) 00091

83689

86379

55461

48425

71683

64739

13397

86168

71943

43179

33634

92309

45928

49315

39991

760

28

. (25) 00003

27988

92370

69837

91015

20417

27312

11192

78077

45426

55113

54772

67582

48068

87475

64999

705

29

. (30) 11309
98440

96288
43709

64477
74071

16931
34050

55876
88103

46769
438

38316

99244

05014

70865

30

0. (30) 00376
19948

99876
01456

28815
99135

90564
71135

38629
02936

21525

781

64610

56641

46833

82362

31

. (30) 00012
94191

16125
87143

04155
77391

35179
47455

49629
96868

97468
928

66922

92149

72478

51043

32

. (35) 38003
49598

90754
2i293

8o474
48357

35925
99902

93670
154

89278

84129

67889

96345

12318

33

. (35) 01151
86351

63366
46190

20771
71162

9o028
36360

05868
671

81493

29822

11148

18040

76130

34

. (36) 00033
14304

87157
45476

53552
19740

11618
06951

47231

784

43573

33230

06210

24060

02239

35

0. (40) 96775
41299

92958
31992

63189
67.341

09920
480

89816

38092

28748

86401

71492

54694

36

. (40) 02688
17813

22026
86999

62866
79370

36386
597

69161

66613

67465

24622

26985

90408

The numbers in the parentheses represent the number of zeros between the first tabular figure and the decimal
point.

NO.B.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 13

Table I. — Values of the reciprocal of nl to lOS places of decimals at intervals of unity from 1 to 74 — Continued.

n

1
nl

37

0. (40) 00072
27508

05460

48297

17915
29172

30713

178

15382

74503

07228

79043

84513 13254

38

. (40) 00001
58618

91196

64428

32050
87609

40281
794

92510

07223

76506

02080

10118 76664

39

. (45) 04902
29857

46975
15066

65135
918

43397

69415

99397

59027

69490

22478 57913

40

0. (45) 00122
83246

56174
42876

39128
673

38584

94235

39984

93975

69237

25561 96447

41

. (45) 00002
72762

98931
10801

08271
870

42404

51078

91219

14487

21200

90867 36498

42

.(50) 07117
38352

40673

425

12914

39311

40267

12249

69552

40258

74678 54113

43

. (50) 00165
12519

52108
824

67742

19518

86982

95633

71384

93959

50573 91956

44

. (50) 00003
73011

76184
814

28812

32261

79249

61264

40258

74862

71603 95271

45

0. (55) 08359
151

65084

71828

03983

32472

54227

97219

17146

75450 48289

46

. (55) 00181
938

73154

01561

47912

68097

22917

99939

54720

58161 96701

47

. (55) 00003
063

86662

85139

60593

88682

91976

97871

05419

58684 29717

48

. (60) 08055
. (60) 00164

47607

07512

37264

22749

52038

98029

57472

58952 439

49

39747

08316

57903

35158

15347

73429

17499

44060 254

50

0. (60) 00003

28794

94166

33158

06703

16306

95468

58349

98881 205

51

. (65) 06446

95964

04571

72680

45417

78342

52124

50958

455

52

. (65) 00123

97999

30857

14859

23950

34198

89463

93287

663

53

. (65) 00002

33924

51525

60657

72150

00645

26216

30062

031

54

. (70) 04331

93546

77049

21706

48160

09744

74630

778

55

0. (70) 00078

76246

30491

80394

66330

18358

99538

741

56

. (70) 00001

40647

25544

49649

90470

18184

98206

049

57

. (75) 02467

49570

95607

89306

49441

84179

053

58

. (75) 00042

54302

94751

86022

52576

58347

915

59

. (80) 72106

82961

89593

60213

16243

185

60

0. (80) 01201

78049

36493

22670

21937

386

61

. (80) 00019

70131

95680

21683

11835

039

62

. (85) 31776

32188

39059

40513

468

63

. (85) 00504

38606

16493

00643

071

64

. (85) 00007

88103

22132

70322

548

65

0.(90) 12124

66494

34928

039

66

. (90) 00183

70704

45983

758

67

. 90) 00002
. (95) 04032

74189

61880

355

68

20027

652

69

. (95) 00058

43768

517

70

0. (100)83482
. (100)01175

407

71

809

72

. (100)00016

331

73

. (105)224

74

. (105)003

The numbers in the parentheses represent the number of zeros between the first tabular figure and the decimal
point.

u

MEMOIRS NATIONAL ACADEMY OF SCIENCES,

Table II. — Values of e^ to 42 significant figures at intervals of unity Jrom 0 to 100.

[Vol. XIV,

X

ex

0

1. 00000

00000

00000

00000

00000

00000

00000

00000 0

1

2. 71828

18284

59045

23536

02874

71352

66249

77572 5

2

7. 38905

60989

30650

22723

04274

60575

00781

31803 2

3

20. 08553

69231

87667

74092

85296

54581

71789

69879

4

64. 59815

00331

44239

07811

02612

02860

87840

27907

5

148. 41315

91025

76603

42111

55800

40552

27962

3488

6

403. 42879

34927

35122

60838

71805

43388

27960

5900

7

1096. 63315

84284

58599

26372

02382

88121

43244

222

8

2980. 95798

70417

28274

74359

20994

52888

67375

597

9

8103. 08392

75753

84007

70999

66894

32759

96501

148

10

22026. 46579

48067

16516

95790

06452

84244

36635

35

11

59874. 14171

51978

18455

32648

57922

57781

61426

11

12

1

62754. 79141

90039

20808

00520

48984

86783

17020

9

13

4

42413. 39200

89205

03326

10277

59490

88281

78439

1

14

12

02604. 28416

47767

77749

23677

07678

59449

41249

15

32

69017. 37247

21106

39301

85504

60917

21315

50574

16

88

86110. 52050

78726

36763

02374

07814

50350

80272

17

241

54952. 75357

52982

14775

43518

03858

23879

8676

18

656

59969. 13733

05111

38786

50325

90600

33569

2164

19

1784

82300. 96318

72608

44910

03378

87227

03883

620

20

4851

65195. 40979

02779

69106

83054

15405

58684

639

21

13188

15734. 48321

46972

09998

88374

53027

85091

44

22

35849

12846. 13159

15616

81159

94597

84206

89222

69

23

97448

03446. 24890

26000

34632

68482

29752

77649

39

24

2

64891

22129. 84347

22941

39162

15281

18823

40870

2

25

7

20048

99337. 38587

25241

61351

46612

61579

15223

5

26

19

57296

09428, 83876

42697

76397

87609

53427

92036

27

53

20482

40601. 79861

66837

47304

34117

74416

59256

28

144

62570

64291. 47517

36770

47422

99692

88569

0206

29

393

13342

97144. 04207

43886

20580

84352

76857

9694

30

1068

64745

81524. 46214

69904

68650

74140

16500

245

31

2904

88496

65247. 42523

10856

82111

67982

56667

647

32

7896

29601

82680. 69516

09780

22635

10822

42199

562

33

21464

35797

85916. 06462

42977

61531

26088

03692

26

34

58346

17425

27454. 88140

29027

34610

39101

90036

59

35

1

58601

34523

13430. 72812

96446

25774

66012

51762

0

36

4

31123

15471

15195. 22711

34222

92856

92539

07888

6

37

11

71914

23728

02611. 30877

29397

91190

19452

16754

38

31

85593

17571

13756. 22032

86717

01298

64599

95422

39

86

59340

04239

93746. 95360

69327

19264

93424

97019

40

235

38526

68370

19985. 40789

99107

49034

80450

8872

41

639

84349

35300

54949. 22266

34035

15570

81887

9337

42

1739

27494

15205

01047. 39468

13036

11235

22614

798

43

4727

83946

82293

46561. 47445

75627

44280

37081

975

44

12851

60011

43593

08275. 80929

96321

43099

25780

11

45

34934

27105

74850

95348. 03479

72334

06099

53341

17

46

94961

19420

60244

88745. 13364

91171

18323

10181

72

47

2 58131

28861

90067

39623. 28580

02152

73380

43163

7

48

7 01673

59120

97631

73865. 47159

98861

17405

45593

8

49

19 07346

57249

50996

90525. 09984

09538

48447

38819

50

51 84705

52858

70724

64087. 45332

29334

85384

82747

51

140 93490

82426

93879

64492. 14331

23701

68788

6848

52

383 10080

00716

57684

93035. 69548

78619

93898

7056

53

1041 37594

33029

08779

71834. 72933

49379

64398

047

54

2830 75330

32746

93900

44206. 35480

14074

54085

033

55

7694 78526

51420

17138

18274. 55901

29393

99207

077

56

20916 59496

01299

61539

07071. 15721

46737

78152

97

57

56857 19999

33593

22226

40348. 82063

32533

03372

16

58

1 54553 89355

90103

93035

30766. 91117

46200

68363

7

59

4 20121 04037

90514

25495

65934. 30719

16176

84111

1

No. 5] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

Table II. — Values of ex to 42 significant figures at intervals of unity from 0 to 100 — Continued.

15

X

ei

60

11

42007

38981

56842

83662

95718. 31447

65630 19805

61

31

04297

93570

19199

08707

34214. 11071

00372 06295

62

84

38356

66874

14544

89073

32948. 03731

17960 08069

63

229

37831

59469

60987

90993

52840. 26861

36004 6328

64

623

51490

80811

61688

29092

38708. 92846

97448 3139

65

1694

88924

44103

33714

14178

36114, 37197

49489 262

66

4607

18663

43312

91542

67731

84428. 06008

68933 490

67

12523

63170

84221

37805

13521

96074. 43657

67534 89

68

34042

76049

93174

05213

76907

18700. 43505

95373 88

69

92537

81725

58778

76002

42397

91668. 73458

73476 60

70

2

51543

86709

19167

00626

57811

74252. 11296

14074 1

71

6

83767

12297

62743

86675

58928

26677. 71095

59458 4

72

18

58671

74528

41279

80340

37018

12.545. 41194

69464

73

50

52393

63027

61041

94557

03833

21857. 64648

53672

74

137

33829

79540

17618

77841

88529

80853. 89315

7998

75

373

32419

96799

00164

02549

08317

26470. 01434

2778

76

1014

80038

81138

88727

83246

17841

31716. 97577

666

77

2758

51345

45231

70206

28646

98199

02661. 94334

152

78

7498

41699

69901

20434

67563

05912

24060. 45470

466

79

20382

81066

51266

87668

32313

75371

72632. 37469

74

80

55406

22384

39351

00525

71173

39583

16612. 92485

67

81

1

50609

73145

85030

54835

25941

30167

67498. 18994

0

82

4

09399

69621

27454

69666

09142

29327

82904. 32005

4

83

11

12863

75479

17594

12087

07147

81839

40805. 73408

84

30

25077

32220

11423

38266

56639

64434

28742. 46903

85

82

23012

71462

29135

10304

32801

64077

74695. 48629

86

223

52466

03734

71504

74430

65732

33271

47398. 7754

87

607

60302

25056

87214

95223

28938

13027

60752. 6138

88

1651

63625

49940

01855

52832

97962

64858

76706. 963

89

4489

61281

91743

45246

28424

55796

45316

27776. 598

90

12204

03294

31784

08020

02710

03513

63697

53970. 75

91

33174

00098

33574

26257

55516

10785

25919

09603. 01

92

90176

28405

03429

89314

00995

98217

09052

59128. 75

93

2 45124

55429

20085

78555

27729

43110

91534

23487. 6

94

6 66317

62164

10895

83424

48140

50240

87326

26873. 9

95

18 11239

0828S

90232

82193

79875

80988

15925

04790.

96

49 23458

28601

20583

99754

86205

91133

04494

83780.

97

133 83347

19204

26950

04617

36408

70611

50290

7672 .

98

363 79709

47608

80457

92877

43826

76018

57298

9310 .

99

988 90303

19346

94677

05600

30967

13803

71014

0508 .

100

2688 11714

18161

35448

41262

55515

80013

58736

Ill .

16

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table III. — Values of ex to SS significant figures at intervals of 0.1
from 0.0 to 50.0.

[Vol. XIV,

X.

ex

0.0

1.00000 00000 00000 00000 00000 00000 00

.1

1.10517 09180 75647 62481 17078 26490 25

.2

1 22140 27581 60169 83392 10719 94639 67

.3

1.34985 88075 76003 10398 37443 13328 01

.4

1.49182 46976 41270 31782 48529 52837 22

0.5

1.64872 12707 00128 14684 86507 87814 16

.6

1.82211 88003 90508 97487 53676 68162 86

.7

2.01375 27074 70476 52162 45493 88583 07

.8

2.22554 09284 92467 60457 95375 31395 08

.9

2.45960 31111 56949 66380 01265 63602 47

1.0

2.71828 18284 59045 23536 02874 71352 66

.1

3.00416 60239 46433 11205 84079 53588 67

.2

3.32011 69227 36547 48953 07674 29601 64

.3

3 66929 66676 19244 22045 74899 16011 49

.4

4.05519 99668 44674 58722 41088 9.5228 62

1.5

4.48168 90703 38064 82260 20554 60119 28

.6

4.95303 24243 95114 80365 42863 56423 96

.7

5 47394 73917 27199 76079 08626 63009 10

.8

6.04964 74644 12946 08373 10239 53027 72

.9

6.68589 44422 79269 41607 25307 27692 86

2.0

7 38905 60989 30650 22723 04274 60575 01

.1

8.16616 99125 67650 07344 97274 10478 63

.2

9 02501 34994 34120 92647 17771 66888 66

.3

9.97418 24548 14720 73995 76151 56908 86

.4

11.02317 63806 41601 65223 79397 69667 8

2.5

12.18249 39607 03473 43807 01759 51168 0

.6

13 46373 80350 01690 39775 08253 32584 1

.7

14.87973 17248 72834 11186 89930 19468 4

.8

16 44464 67710 97049 87149 80160 10925 0

.9

18.17414 53694 43060 94267 62565 74128 1

3.0

20 08553 69231 87667 74092 85296 54581 7

.1

22.19795 12814 41633 40482 79743 81257 2

.2

24 53253 01971 09348 64356 02637 27964 2

.3

27.11263 89206 57887 42681 83721 10231 2

.4

29.96410 00473 97013 34816 27530 33730 2

3.5

33.11545 19586 92313 75065 32493 50388 6

.6

36.59823 44436 77987 75259 47658 99183 7

.7

40 44730 43600 67390 52889 41892 39039 1

.8

44 70118 44933 00823 03755 78287 29065 3

.9

49.40244 91055 30173 87976 14865 41220 3

4.0

54.59815 00331 44239 07811 02612 02860 9

.1

60.34028 75973 61969 49748 72197 08124 4

.2

66.68633 10409 25141 64502 17346 53992 0

.3

73.69979 36995 95796 91176 19511 70652 5

.4

81.45086 86649 68117 44440 08117 26181 1

4.5

90.01713 13005 21813 55011 54567 45574 4

.6

99.48431 56419 33808 73545 40534 87566 7

.7

109.94717 24521 23498 87972 87004 55366

.8

121.51041 75187 34880 75704 81162 97881

.9

134.28977 96849 35484 84005 86277 74362

5.0

148.41315 91025 76603 42111 55800 40552

.1

164.02190 72999 01743 94514 82613 02021

.2

181.27224 18751 51179 36998 41338 23353

.3

200.33680 99747 91684 83525 66156 38620

.4

221.40641 62041 87087 02509 46801 14279

5.5

244.69193 22642 20387 91518 89495 11839

.6

270.42640 74261 52628 15292 10465 31487

.7

298.86740 09670 60232 67202 80305 55296

.8

330.29955 99096 48654 12024 52287 64816

.9

365.03746 78653 28777 31505 32150 83072

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

17

Table III. — Values of e"' to S3 significanl figures at intervals of 0.1
from 0.0 to 50.6i— JContinued.

r

ei

6.0

403. 42879

34927

35122

60838

71805

43388

.1

445. 85777

00825

16931

79233

21972

16812

.2

492. 74904

10932

56254

57006

20910

66389

.3

544. 57191

01259

29033

05938

86677

33165

.4

601. 84503

78720

82056

60929

82761

16979

6.5

665. 14163

30443

61840

69396

14942

42634

.6

735. 09518

92419

72894

90710

17107

60161

.7

812. 40582

51675

43113

47226

72512

95340

.8

897. 84729

16504

17697

57784

39706

81908

.9

992. 27471

56050

25876

97253

10085

94319

7.0

1096. 63315

84284

58599

26372

02382

8812

.1

1211. 96707

44925

76721

19815

40043

4583

.2

1339. 43076

43944

17829

68735

15152

9872

.3

1480. 29992

75845

45222

83730

58693

3122

.4

1635. 98442

99959

26540

06633

38342

5709

7.5

1808. 04241

44560

63206

90380

14827

7881

.6

1998. 19589

51041

17959

25232

48348

4882

.7

2208. 34799

18872

08523

98030

94345

1393

.8

2440. 60197

76244

99077

24871

55411

2634

.9

2697. 28232

82685

08847

21116

61148

7690

8.0

2980. 95798

70417

28274

74359

20994

5289

.1

3294. 46807

52838

41333

08812

83565

2825

.2

3640. 95030

73323

54721

56857

18339

5742

.3

4023. 87239

38223

09841

54472

32070

1925

.4

4447. 06674

76998

56085

59847

50173

2566

8.5

4914. 76884

02991

34375

43137

36763

4783

.6

5431. 65959

13629

80321

56806

91897

0967

.7

6002. 91221

72610

21980

07565

92099

0448

.8

6634. 24400

62778

85158

52737

29275

5448

.9

7331. 97353

91559

92905

24450

31452

0296

9.0

8103. 08392

75753

84007

70999

66894

3276

.1

8955. 29270

34825

11710

77437

86428

2849

.2

9897. 12905

87439

15886

85434

02479

7437

.3

10938. 01920

81651

83753

33850

61222

010

.4

12088. 38073

02169

84397

55833

57238

533

9.5

13359. 72682

96618

72275

90175

59729

146

.6

14764. 78156

55772

72615

55426

11148

697

.7

16317. 60719

80154

32232

76797

34500

972

.8

18033. 74492

78285

11245

99526

53348

081

.9

19930. 37043

82302

89490

56032

14677

875

10.0

22026. 46579

48067

16516

95790

06452

842

.1

24343.00942

44083

88345

98557

99428

153

.2

26903. 18607

42975

60998

95889

84543

248

.3

29732. 61885

28914

13820

76842

75016

320

.4

32859.62567

44433

12762

26957

08978

804

10.5

36315. 60267

42466

37738

91202

69013

166

.6

40134. 83743

08757

93109

47683

09703

197

.7

44355. 8.5513

02978

66938

62836

34286

021

.8

49020. 80113

63817

18305

10499

68773

316

.9

54176. 36379

66987

33990

00463

83753

492

11.0

59874. 14171

51978

18455

32648

57922

578

.1

66171. 16016

83766

04182

26482

33834

845

.2

73130.44183

34154

97311

60903

28180

212

.3

80821.63754

03135

52465

42612

50238

593

.4

89321. 72336

08055

55699

37363

40540

407

11.5

98715.77101

07604

97428

11026

81147

200

.6

1 09097. 79927

65075

80429

18173

80085

19

.7

1 20571.71498

64506

07884

32987

03867

70

.8

1 33252. 35294

55309

39735

38206

60578

27

.9

1 47266.62524

05526

56665

65566

98194

62

USSM'— 21-

18

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table III. — Values of e^ to S3 significant figures at intervals of 0.1
from 0.0 to 50.0 — Continued.

X

ex

12.0

1

62754. 79141

90039

20808

00520

48984 87

.1

1

79871. 86225

37510

99202

55498

70958 42

.2

1

98789. 1.5114

29545

30399

15171

71329 96

.3

2

19695. 98867

21377

34715

78951

40139 70

.4

2

42801. 61749

83235

41021

99665

43832 72

12.5

2

68337. 28652

08744

56956

47967

37871 50

.6

2

96558. 56029

82029

28131

06698

18068 83

.7

3

27747.90187

38118

24915

27613

20512 30

.8

3

62217.44961

12478

85014

64554

45272 23

.9

4

00312. 19132

98824

57935

63962

48069 30

13.0

4

42413. 39200

89205

03326

10277

59490 88

.1

4

88942.41461

54600

59140

29689

39772 44

^ 2

5

40364.93724

66919

42887

77702

08966 51

'.3

5

97195. 61379

28162

51018

72789

72271 12

.4

6

60003. 22476

61566

27675

08247

66901 27

13.5

7

29416. 36984

77013

31861

08259

40363 04

.6

8

06129. 75912

39902

17000

49212

27173 34

.7

8

90911. 16597

91609

45513

21710

16782 56

.8

9

84609. 11122

90349

84647

14285

05695 62

.9

10

88161.35540

26400

42869

04190

82607 5

14.0

12

02604.28416

47767

77749

23677

07678 6

.1

13

29083. 28081

20933

72415

65547

31032 3

.2

14

68864. 18965

40950

11264

71279

19631 2

.3

16

23345. 98500

84583

73176

94920

55661 4

.4

17

94074. 77260

62144

46062

26766

69215 2

14.5

19

82759. 26353

75687

67141

76278

73256 4

.6

21

91287. 87560

68098

30730

21834

00372 8

.7

24

21747. 63325

24135

50747

88825

38372 5

.8

26

76445. 0.5518

90966

65944

60323

31294 3

.9

29

57929. 23882

23613

37256

83192

42565 6

15.0

32

69017. 37247

21106

39301

85504

60917 2

.1

36

12822. 93074

02438

44330

52318

26886 2

2

39

92786. 83521

09471

82558

38605

78417 8

'.Z

44

12711.89235

04420

61860

72912

49413 9

.4

48

76800. 85327

22664

04847

12229

15576 7

15.5

53

89698. 47628

30123

67815

21079

20761 8

.6

59

56538. 01318

46158

94525

78083

82516 5

65

82992. 58458

37360

04428

51377

35395 1

.8

72

75331.95838

95879

21060

75789

28904 5

.9

80

40485. 29975

85202

66729

31241

77682 7

16.0

88

86110. 52050

78726

36763

02374

07814 5

.1

98

20670. 92207

13565

82889

22079

08745 3

2

108

53519. 89906

44180

45529

12596

65383

'.3

119

94994. 55120

13332

33724

02003

53020

.4

132

56519. 14046

35683

00166

44194

17466

16.5

146

50719. 42895

35169

10097

65773

23551

.6

161

91549. 04176

52861

89444

22585

06037

.7

178

94429. 11955

46139

05552

62473

92240

.8

197

76402. 65849

77754

61390

92622

74676

.9

218

56305. 08232

56648

96058

58443

63455

17.0

241

54952. 75357

52982

14775

43518

03858

.1

266

95351. 31074

27049

13394

12187

57749

.2

295

02925. 91644

54583

71110

68906

11219

.3

326

05775. 72099

58447

95506

06223

13988

.4

360

34955. 08814

16391

55271

54298

50110

17.5

398

24784. 39757

62250

21870

67634

98518

.6

440

13193. 53483

40439

30710

38742

44398

.7

486

42101. 50633

36988

59843

73758

07283

.8

537

57835. 97888

36562

28073

19655

81474

.9

594

11596. 94254

29315

75595

99097

66876

No. 5.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

19

Table III. — Valites of ex to S3 significant figures at intervals of 0.1
from 0.0 to 50.0— Continued.

X

ex

18.0

656

59969. 13733

05111

38786

50325

90600

.1

725

65488. 37232

22497

75110

99891

90593

.2

801

97267.40504

71134

14452

49662

99965

.3

886

31687. 64519

41289

61081

78952

16349

.4

979

53163. 60543

32304

45541

27301

00064

18.5

1082

54987. 75023

07572

48748

04460

1217

.6

1196

402f,4. 19819

05133

97759

51385

3688

.7

1322

22940. 62272

72454

49131

49731

9827

.8

1461

28948. 678G8

13129

20356

77145

8982

.9

1614

97464. 36864

74215

13410

14609

5743

19.0

1784

82300. 96318

72608

44910

03378

8723

.1

1972

53448. 41573

97114

12668

18600

1486

.2

2179

98774. 67921

04573

69563

01720

4717

.3

2409

25905. 95158

92662

02664

76985

1347

.4

2662

64304. 66872

50454

28992

02822

0165

19.5

2942

67566. 04150

88065

66680

80045

3345

.6

3252

15956. 12198

05562

88545

56147

9971

.7

3594

19216. 80017

87860

03058

99120

6792

.8

3972

19665. 80508

38215

53744

05532

4200

.9

4389

95622. 73550

64203

80154

45375

0896

20.0

4851

65195. 40979

02779

69106

83054

1541

.1

5361

90464. 42938

89023

64651

69867

1124

.2

5925

82107. 83683

56144

86124

27127

3255

.3

6549

04512 15323

80392

40495

98782

8846

.4

7237

81420. 94827

82113

22801

67333

7645

20.5

7999

02177. 47550

54067

04598

83728

3990

.6

8840

28623. 85131

39326

49420

01192

2695

.7

9770

02725. 82690

79801

12264

26784

8714

.8

10797

54999. 46453

41371

25566

62697

510

.9

11933

13824. 05498

96018

57459

21390

201

21.0

13188

15734. 48321

46972

09998

88374

530

.1

14575

16796. 05142

39203

84629

61823

210

.2

16108

05175. 60282

86330

43100

62026

135

.3

17802

15034. 76198

29093

45688

56781

729

.4

19674

41884. 33997

16024

55721

30300

926

21.5

21743

59553. 57648

85454

85310

20243

562

.6

24030

38944. 05268

31647

45191

75991

160

.7

26557

68755. 97023

86819

92208

74387

373

.8

29350

78394. 23224

92632

94732

06188

947

.9

32437

63283. 57765

25326

80093

80230

715

22.0

35849

12846. 13159

15616

81159

94597

842

.1

39619

41421. 38043

39369

91055

46827

187

.2

43786

22438. 02895

04595

53310

41691

167

.3

48391

26179. 74308

56773

45193

39107

005

.4

53480

61522. 75056

74038

45957

13828

733

22.5

59105

22063. 02329

06142

72278

94443

044

.6

65321

37094. 69782

08990

20985

63179

874

.7

72191

27949. 94318

43117

28947

94442

179

.8

79783

70264. 14427

69362

61695

05249

058

.9

88174

62789. 57177

77864

45437

53886

624

23.0

97448

03446. 24890

26000

34632

68482

298

.1

1 07696

73371. 15763

45779

38536

25530

80

.2

1 19023

29806. 97713

79397

34848

01024

86

.3

1 31541

08760. 01606

93214

92804

01732

96

.4

1 45375

38454. 77387

81109

95934

22322

54

23.5

1 60664

64720. 62247

86090

61991

59775

50

.6

1 77561

89565. 52034

81110

48593

83852

42

.7

1 96236

24323. 65135

78359

05185

09236

20

.8

2 16874

58909. 74138

08217

35308

44175

78

.9

2 39683

48874. 00676

57400

68251

23095

68

20

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table III. — Values o/e^ to 33 significant figures at intervals of 0.1
from 0.0 to 60.0— Continued.

X

e»

24.0

2

64891

22129. 84347

22941

39162

15281 19

.1

2

92750

07423. 25706

46440

91249

77678 54

.2

3

23538

86830. 63240

94606

10181

71913 49

.3

3

57565

74811. 92562

51762

98574

72157 75

.4

3

95171

26612. 13642

04797

38119

69976 30

24.5

4

36731

79097. 64641

45304

17828

87281 52

.6

4

82663

27438. 62807

18527

03687

39155 11

.7

5

33425

41407. 48840

78591

38870

16973 92

.8

5

89526

25459. 80221

23469

39592

04239 87

.9

6

51527

27202. 37940

92103

46415

77990 24

25.0

7

20048

99337. 38587

25241

61351

46612 62

.1

7

95777

20706. 64333

60674

37733

94535 75

.2

8

79469

82651. 72848

99796

19698

43300 30

.3

9

71964

47559. 19382

99044

89837

30276 11

.4

10

74186

87182. 68578

47334

69615

59214 4

26.5

11

87160

09132. 16965

09652

01023

04023 3

.6

13

12014

80802. 87690

06069

24450

49061 0

.7

14

50000

60991. 79992

16792

65970

81555 6

.8

16

02498

50527. 33242

01261

84906

43941 9

.9

17

71034

74428. 77727

54108

41351

39504 1

26.0

19

57296

09428. 83876

42697

76397

87609 5

.1

21

63146

72147. 05767

28406

29286

74083 0

.2

23

90646

84809. 99645

04520

70140

43285 0

.3

26

42073

37190. 92910

83050

91670

72539 2

.4

29

19942

65405. 62132

14147

79370

61947 9

26.5

32

27035

70371. 15483

07849

19455

52377 3

.6

35

66426

01133. 37854

36755

39770

00888 2

.7

39

41510

30919. 46297

12378

08766

28766 3

.8

43

56042

56701. 72586

52960

40096

29323 2

.9

48

14171

56296. 70645

41109

23997

81144 6

27.0

53

20482

40601. 79861

66837

47304

34117 7

.1

58

80042

42526. 42283

53382

81240

46422 3

.2

64

98451

88545. 30248

85133

02409

44687 4

.3

71

81900

03631. 65428

08266

99454

22658 5

.4

79

37227

05666. 34806

33381

19699

12653 4

27.5

87

71992

51318. 76492

83096

93392

27847 5

.6

96

94551

01915. 23018

62951

00965

77092 2

.7

107

14135

85016. 77547

89905

47059

90553

.8

118

40951

35391. 71069

44133

88465

01118

.9

130

86275

07869. 76518

22787

42513

40126

28.0

144

62570

64291. 47517

36770

47422

99693

.1

159

83612

47516. 40054

90476

75454

54890

.2

176

64623

67334. 23784

68124

17444

60541

.3

195

22428

36252. 86153

64001

24396

78170

.4

215

75620

07648. 18119

79284

16722

73105

28.5

238

44747

84797. 67787

68074

52711

03802

.6

263

52521

87043. 08195

72729

35918

88625

.7

291

24040

78915, 26116

09982

34926

73407

.8

321

87042

89702. 04007

25720

34279

28499

.9

355

72183

74864. 02890

79644

36208

11659

29.0

393

13342

97144. 04207

43886

20580

84353

.1

434

47963

34436. 96185

95024

47682

30942

.2

480

17425

53781. 40567

43880

77859

92347

.3

530

67462

26525. 50089

96447

48663

05030

.4

586

48615

99163. 66652

71853

60117

55890

29.5

648

16744

77934. 32021

79214

42218

51631

.6

716

33581

33446. 16669

80045

98003

56392

.7

791

67350

84845. 35758

16856

01647

30249

.8

874

93453

81880. 23393

20218

01897

03542

.9

966

95220

68253. 50589

75038

08871

22088

No. 5.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

21

Table 111.— Values of ex to S.i significant figures at intervals of 0.1
from 0.0 to 50.0.— Continued

X

cx

30.0

1068

64745

81524. 46214

69904

68650 7414

.1

1181

03809

24255. 46209

01487

22090 7004

.2

1305

24895

28882. 52476

97252

90255 6973

.3

1442

52318

35807. 87724

46546

87919 4322

.4

1594

23467

11433. 85149

18434

29655 0257

30.5

1761

90179

51355. 63141

21609

84760 9319

.6

1947

20262

44891. 01937

17824

87753 5604

.7

2151

99171

21859.31322

47658

79141 9410

.8

2378

31865

62477. 10567

98775

07647 3359

.9

2628

44861

28017. 22881

21183

58524 8783

31.0

2904

88496

65247. 42523

10856

82111 6798

.1

3210

39438

53582. 96612

04636

67150 6906

.2

3548

03451

02513. 33135

61277

07371 6175

.3

3921

18455

70585. 46635

65946

87032 8266

.4

4333

57913

68684. 45663

26741

42381 5711

31.5

4789

34563

32463. 72707

54403

58901 8061

.6

5293

04551

04764. 87666

92400

05063 9297

.7

5849

71996

62294. 84813

15789

57425 7372

.8

6464

94038

55632. 86150

88365

43609 9270

.9

7144

86410

12173. 08287

18592

40090 3405

32.0

7896

29601

82680. 69516

09780

22635 1082

.1

8726

75671

99064. 03195

78574

92376 1833

.2

9644

55773

59617. 86912

13519

98498 2802

.3

10658

88472

74804. 77539

68973

56901 625

.4

11779

88941

99387. 29527

90142

63008 392

32.5

13018

79120

50632. 93871

26745

74701 261

.6

14387

98942

83349. 67368

39501

94593 527

.7

15901

18748

57756. 68324

09867

00264 957

.8

17573

52997

21476. 94368

79223

01141 844

.9

19421

75425

31483. 77619

09187

05800 909

33.0

21464

35797

85916. 06462

42977

61531 261

.1

23721

78421

31044. 37760

36693

34787 219

.2

26216

62603

71890. 35741

51580

61740 413

.3

28973

85266

63661. 34260

27596

09521 264

.4

32021

05935

14764. 11460

39540

04425 132

33.5

35388

74356

12259. 87392

92482

12571 573

.6

39110

61021

11037. 88087

97251

60053 907

.7

43223

90899

35043. 72041

86857

86658 642

.8

47769

80718

51694. 68936

96145

39196 174

.9

52793

80166

31304. 10421

90549

88835 133

34.0

58346

17425

27454. 88140

29027

34610 391

.1

64482

49496

51084. 44647

43293

22486 580

.2

71264

17816

03972. 25326

59900

26513 914

.3

78759

09720

34327. 18570

84270

17314 623

.4

87042

26376

31269. 08969

92798

21172 377

34.5

96196

57855

44776. 41048

71247

85963 930

.6

1 06313

66103

67882. 10250

93915

64891 44

.7

1 17494

76637

20104. 34037

07747

33623 41

.8

1 29851

79882

04385. 00894

18446

81468 97

.9

1 43508

43171

61583. 15354

06356

61404 02

35.0

1 58601

34523

13430. 72812

96446

25774 66

.1

1 75281

59431

73561. 61212

33756

44114 37

.2

1 93716

12051

34757. 28303

31243

64910 33

.3

2 14089

42275

39307. 66424

27337

78435 43

.4

2 36605

40389

52471. 09566

20224

56329 49

35.5

2 61489

41144

45696. 60738

41656

44430 36

.6

2 88990

49291

32558. 10962

41749

67431 50

.7

3 19383

88836

80768. 63352

22189

34573 53

.8

3 52973

78512

63176. 61523

70828

13371 84

.9

3 90096

36216

46888. 64411

37401

49460 41

22

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[VOU SIV,

Table III. — Valius of ex to 33 significant figures at intervals of 0.1
from 0.0 to 50.0— Continued.

X

ex

36.0

4

31123

15471

15195. 22711

34222

92856 93

.1

4

76464

77269

61994. 98745

07283

67350 68

.2

5

26575

01027

13635. 63500

14974

59058 17

.3

5

81955

38753

72964. 47397

44109

37106 06

.4

6

43160

16992

36632. 16821

94211

47485 19

36.5

7

10801

91546

42244. 06486

15833

68420 94

.6

7

85557

60548

35257. 60090

79367

46411 88

.7

8

68175

42005

35355. 65163

79156

01753 73

.8

9

59482

22603

12769. 18144

67304

04477 19

.9

10

60391

85262

02523. 59786

47544

17130 2

37.0

11

71914

23728

02611. 30877

29397

91190 2

.1

12

95165

53352

09485. 45824

65306

72850 . 0

.2

14

31379

28174

12826. 72968

64500

92801 3

.3

15

81918

75491

64744. 53050

66757

27100 8

.4

17

48290

60269

21254. 81056

07962

69122 2

37.5

19

32159

93044

02836. 20844

22759

20919 7

.6

21

35366

96419

36677. 02922

94101

03254 3

.7

23

59945

46824

63243. 03968

93611

44775 8

.8

26

08143

09975

02543. 50501

04825

61149 7

.9

28

82443

90402

36540. 01941

88916

28577 7

38.0

31

85593

17571

13756. 22032

86717

01298 6

.1

35

20624

93461

64588. 56603

83582

49016 4

.2

38

90892

29119

00887. 26618

70087

82885 6

.3

43

00101

00558

80104. 30725

35475

49330 3

.4

47

52346

57616

37170. 45041

48507

32909 5

38.5

52

52155

22859

25158. 15729

58254

60641 7

.6

58

04529

21585

94036. 20184

79777

28230 9

.7

64

14996

88248

82561. 06182

08507

97036 0

.8

70

89667

99407

19634. 02423

10740

15188 6

.9

78

35294

88586

00468. 95973

47814

57630 8

39.0

86

59340

04239

93746. 95360

69327

19264 9

.1

95

70050

78458

77343. 61342

17376

38971 4

.2

105

76541

81163

33982. 46846

15693

64994

.3

116

88886

42402

83560. 86698

34176

84070

.4

129

18217

34052

53920. 49038

85620

11921

39.5

142

76838

11812

91985. 91758

33666

53263

.6

157

78346

29023

02477. 43715

59460

30821

.7

174

37769

45528

92517. 50602

13224

60008

.8

192

71715

67809

35081. 54052

84561

81318

.9

212

98539

70885

14544. 13549

39399

97850

40.0

235

38526

68370

19985. 40789

99107

49035

.1

260

14095

14517

50670. 00948

59692

83355

.2

287

50021

41450

03765. 77508

14124

91637

.3

317

73687

56135

79105. 26803

06064

59178

.4

351

15355

45283

47072. 98994

83315

42259

40.5

388

08469

62436

20324. 02317

21875

72699

.6

428

89992

00386

70710. 64074

96477

64288

.7

474

00771

83917

09565. 01140

65056

82629

.8

523

85954

53099

08701. 59178

65130

82966

.9

578

95433

46328

43124. 57201

85917

65939

41.0

639

84349

35300

54949. 22266

34035

15571

.1

707

13642

11693

40529. 35962

83009

93650

.2

781

50660

77884

47896. 96629

84417

00348

.3

863

69837

52117

44030. 73926

44719

24967

.4

954

53432

62732

08325. 47163

27987

89991

4L5

1054

92357

77020

81418. 45053

91711

6615

.6

1165

87085

88686

56114. 75075

60769

2286

.7

1288

48656

74535

16481. 07600

60987

0147

.8

1423

99788

26807

42680. 14854

81712

4332

.9

1573

76104

73400

54746. 30325

05193

6633

No. S.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

23

Table III. — Values of e^ to S3 significant figures at intervals of 0.1
from 0.0 to 50.0— <!ontinued.

X

ex

42.0

1739

27494

15205

01047. 39468

13036

1124

.1

1922

19608

39061

80476. 58968

93786

0494

.2

2124

35521

07720

08071. 35617

33172

1242

.3

2347

77559

86076

86074. 87538

89370

8665

.4

2594

69331

37488

59588. 85140

73151

4199

42.5

2867

57959

16805

71559. 56393

30187

8547

.6

3169

16556

99926

08018. 31712

57371

5547

.7

3502

46962

25224

63704. 82268

16568

0900

.8

3870

82756

82552

18196. 19954

73844

1926

.9

4277

92605

73211

46063. 77834

40083

1416

43.0

4727

83946

82293

46561. 47445

75627

4428

.1

5225

07068

56173

08600. 14746

7.5513

6950

.2

5774

59616

66338

34529. 15450

27874

7849

.3

6381

91574

69948

30360. 69842

21728

8812

.4

7053

10768

51877

09178. 99594

13708

6927

43.5

7794

88949

57253

06399. 59362

37456

5717

.6

8614

68518

02889

5S825. 49151

03780

4263

.7

9520

69952

96326

24602. 82213

30778

8026

.8

10522

00023

98864

74241. 01297

41703

5134

.9

11628

60866

51075

19279. 13929

36648

7541

44.0

12851

60011

43593

08275. 80929

96321

4310

.1

14203

21469

71275

74732. 70231

52164

8956

.2

15696

97982

64500

13186. 98266

35129

2425

.3

17347

84560

58126

80995. 55797

31144

1650

.4

19172

33445

48105

90107. 58191

10264

4550

44.5

21188

70647

10763

90948. 92010

98100

2595

.6

23417

14218

34749

10750. 51432

14409

6533

.7

25879

94452

56189

427.30. 17190

82453

0870

.8

28601

76205

11251

17788. 91534

78007

8747

.9

31609

83562

46231

64724. 23096

09490

4714

45.0

34934

27105

74850

95348. 03479

72334

061

.1

38608

34041

69043

28227. 13228

24629

719

2

42668

81502

39272

88395. 41377

10669

376

'.Z

47156

33347

31937

07791. 30039

79777

515

.4

52115

80835

76508

83053. 61114

98817

745

45.5

57596

87576

88795

35865. 26851

87821

549

.6

63654

39207

17816

19332. 12347

80923

907

.7

70348

98292

55181

17701. 87293

63572

117

.8

77747

65004

54829

17232. 44579

79926

261

.9

85924

44177

89905

22451. 73279

26071

196

46.0

94961

19420

60244

88745. 13364

91171

183

.1

1 04948

35018

22319

54147. 90908

66367

30

.2

1 15985

86452

14218

49473. 77180

46271

79

.3

1 28184

20437

69374

71210 96429

62509

89

.4

1 41665

45483

40564

33686. 31382

49717

92

46.5

1 56564

54077

85583

41656. 97621

59025

54

.6

1 73030

57727

03314

92805. 23175

74257

30

.7

1 91228

36193

70115

42258. 16763

34724

90

.8

2 11340

02432

40292

75711. 22343

53462

67

.9

2 33566

84870

83171

34964. 40927

95531

96

47.0

2 58131

28861

90067

39623. 28580

02152

73

.1

2 85279

19322

71176

49551. 26703

03268

27

.2

3 15282

26788

66936

88624. 40776

84037

26

.3

3 48440

79345

33095

38552. 71985

37705

33

.4

3 85086

63159

58012

11272. 71553

28309

79

47.5

4 25586

54617

93903

18634. 74249

65238

53

.6

4 70345

87396

17208

02496. 15995

73329

63

.7

5 19812

58133

93678

24359. 56886

42328

44

.8

5 74481

74774

61013

95112. 693.33

79245

43

.9

6 34900

52057

42614

89472. 44726

63512

18

24

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table III. — Values of e^ to S3 significant figures at intervals of 0.1
from 0.0 to 50.0 — Continued.

X

ex

48.0

7

01673

59120

97631

73865. 47159

98861

17

.1

7

75469

24698

67306

37991.61667

18811

74

.2

8

57026

05963

17562

39656. 99046

28407

19

.3

9

47160

27713

79827

71146.82358

95213

41

.4

10

46773

99304

93692

57138. 00626

58991

2

48.5

11

56864

17491

60830

07521. 04037

92992

4

.6

12

78532

64228

08340

57448. 38688

41765

1

.7

14

12997

09405

91929

46708. 68045

97513

8

.8

15

61603

29567

96204

89284 58072

41039

8

.9

17

25838

54795

62031

90250. 54761

43153

1

49.0

19

07346

57249

50996

90525. 09984

09538

5

.1

21

07943

96261

28491

13381. 14667

72236

0

.2

23

29638

36441

28610

87238. 76943

67133

6

.3

25

74648

56998

24118

27728. 21369

31403

7

.4

28

45426

72380

96153

72548. 47735

30218

9

49.5

31

44682

86466

96548

51738. 26948

84493

5

.6

34

75412

04860

37200

08737. 07381

37971

6

.7

38

40924

32444

65405

26989. 78298

37803

3

.8

42

44877

86190

76698

38363. 39647

78017

9

.9

46

91315

56376

34916

34374. 10241

66244

6

50.0

51

84705

52858

70724

64087. 45332

29334

9

No. 5.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 25

Table IV. — Values o/e^ to 62 places of decimals at decimal intervals from 1X10~^° to 9X10~K

ixifr-'".

2

3

4

1X10-'.

2

3

4

IXlOr".

2

3

4

1X10-'.

2

3

4

<p*

1.00000
1. 00000
1. 00000
1.00000

1.00000

1.00000

1. 00000

1. 00000

1. 00000

1. 00000
1.00000
1.00000
1.00000

1. 00000
1. 00000
1. 00000
1. 00000
1.00000

1.00000
1. 00000
1.00000
1.00000

1. 00000
1.00000
1.00000
1. 00000
1. 00000

1.00000
1.00000
1. 00000
1.00000

5 1.00000

6 1.00000

1.00000
1. 00000
1.00000

ixio-«.

2

3

4

1X10-5.

2

3

4

1X10-*.. .

2

3

4

1. 00000
1. 00000
1.00000
1.00000

1. 00000
1.00000
1. 00000
1.00000
1. 00000

1. 00001
1. 00002
1. 00003
1. 00004

1. 0000.5
1. 00006
1. 00007
1. 00008
1.00009

1.00010
1.00020
1.00030
1.00040

00001
00002
00003
00004

00005
00006
00007

oooos

00009

00010
00020
00030
00040

000.50
OOOGO
00070

oooso

00090

00100
00200
00300
00400

00500
00600
00700
00800
00900

01000
02000
03000
04000

05000
06000
07000
08000
09000

10000
20000
30000
40000

50000
60000
70000
80000
90000

00000
00002
00004
00008

00012
00018
00024
00032
00040

00050
00200
00450
00800

00000 00000 50000

00000 00002 00000

00000 00004 50000

00000 oooos 00000

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00000
00000
00000

00001
00001
00002
00003
00004

00005
00020
00045
00080

00125
00180
00245
00320
00405

00500
02000
04500
OSOOO

12500
18000
24500
32000
40500

50000
00001
50004
00010

50020
00036
50057
00085
50121

00166
01333
04500
10667

00012
00018
00024
00032
00040

00050
00200
00450
00800

01250
01800
02450
03200
04050

05000
20000
45000
80000

2.5000
80000
45000
20000
05000

00000
00000
00000
00001

00002
00003
00005
00008
00012

00016
00133
00450
01066

02083
03600
05716
08533
12150

16666
33334
50003
66677

83359
00054
16766
33504
50273

67083
40000
33752
73341

50000
00000
50000
00000
50000

00000
00000
00000
00000

00000
00000
00000
00000
00001

00001
00013
00045
00106

00208
00360
00571
00853
01215

01666
13333
45000
06666

08333
60000
71666
53333
15000

66667
33340
00033
66773

33593

00540
67667
35040
02733

70833
00000
37502
33341

37526
00064
70973
00273
37992

34166
26667
02510
86723

00000
00001
00004
00010

00020
00036
00057
00085
00121

00166
01333
04500
10666

20833
36000
57166
85333
21500

66666
33333
00000
66666

33333
00000
66667
33335
00002

66670
33400
00337
67733

35937
05400
76670
50400
27337

08333
00000
75002
33341

75026
00064
08473
00273
75492

34166
?6666
02501
86672

04188
80064
39330
07030
08238

68055
55558
12543
55880

16606 66666

33333 33334

50000 00003

66666 66677

83333
00000
16666
33333
50000

66666
33333
00000
66667

33335
00005
66676
333.50
00027

66708
34000
03375
77333

59375
54000
66708
04000
73375

83333
00000
50002
33341

50026
00O64
83473
00273
50492

34166
26666
02500
86667

04168
80006
39183
06703
07573

66805
75555
01250
35558

36821
80055
06968
75971
12199

57539
09530
39448
65117

33359
00054
06766
33504
00273

67083
40000
33750
73333

93750
40000
67083
40000
33750

33333
00000
00002
33341

00026
00064
33473
00273
00492

34166
26666
02500
86666

04166
80000
39168
06670
07507

66680
67555
10125
23555

83680
48000
00682
07559
81134

55575
58095
43393
80636

05664
54327
95837
66130
01246

70734
15887
44292
53256

70833
00000
37500
33333

37.500
00000
70833
00000
37.500

33333
00000
00002
33341

00026
00064
33473
00273
00492

34166
26666
02500
86666

04166
80000
39166
06667
07500

66668
66755
01012
72355

88368
64S00
30068
30755
38112

55555
55580
00433
58806

71056
55542
18957
71657
49002

39685
24444
01986
54603

45007
37170
42177
39100
20911

15454
12550
42698
02089

33333 34166 66666 67

00000 26666 66666 76

00002 02500 00001 01

33341 86666 66672 36

00026 04166 66688 37

00064 80000 00064 80

33473 39166 66830 07

00273 06666 67030 76

00492 07500 00738 11

34166 66666 68055 56

26666 66667 55555 56

02.500 00010 12500 00

86666 66723 55555 56

04166 60883 68055 57

80000 00648 00000 06

39166 68300 68065 72

06666 70307 55555 98

07500 07381 12500 95

66666 80555 55557 54

66675 55555 55809 52

00101 25000 04339 29

67235 55555 88063 49

68836
06480
83006
03075
73811

05555

55555
50004
55588

05710
00555
07189
59716
59490

75396
95238
92873
35083

55730
89880
08742
55895
85334

01984
44585
66138
89700

86247
62873
17672
02976
38661

17217
26506
70609
23421

80657
00005
80571
5.5597
25094

55753
80952
33928
06349

56548
42861
56968
57184
01892

82787
73016
41518

17467

71730
00277
07709
91000
23622

40255
53820
40912
46532

45526
80580
36470
67262
76502

83810
33402
44636
81273

10565 48

55428 68

89569 46

16571 47

90017 96

96825 64

38158 73

58770 09

36888 89

58786 97

30857 17

74206 71

46730 53

47699 73

69844 03

88712 52

39955 37

54144 91

41021 76

71446 23

26023 06

27103 65

70826 94

75947 97

10587 14

95086 65

73294 26

60364 36

51744 80

63988 79

81646 47

68646 00

34636 39

81249 62

57909 06

13638 99

26 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv.

Table IV. — Values of e^ to 62 places of decimals at decimal intervals from 1X10'^" to 9X10~' — Continued.

X

ef

5X10-^

6

1.00050
1. 00060
1. 00070
1. 00080
1. 00090

1. 00100
1. 00200
1. 00300
1. 00400

1. 00501
1. 00601
1. 00702
1. 00803
1. 00904

1. 01005
1. 02020
1. 03045
1. 04081

1. 05127
1. 06183
1. 07250
1. 08328
1. 09417

1. 10517
1. 22140
1. 34985
1. 49182

1. 64872

1. 82211

2. 01375
2. 22554
2. 45960

01250
01800
02450
03200
04051

06001
20013
45045
80106

25208
80360

45572
20855
06217

01670
13400
45339
07741

10963
65465
81812
70676
42837

09180
27581
88075
46976

12707
88003
27074
09284
31111

20835
36005
57176
85350
21527

66708
34000
03377

77341

59401
54064
66848
04273
73867

84168
26755
53516
92388

76024
45359
54216
74958
05210

75647
60169
76003
41270

00128
90508
70476
92467
56949

93776
40064
67223
40273
34242

34166
26675
02601
87235

06338
86485
55523
43117
81406

05754
81016
85561
22675

03969
62222
47905
55443
35787

62481
83392
10398
31782

14684
97487
52162
60457
66380

04383

80648
40800
10307
14882

80557
55809
29340
88079

35662
55845
16000
20736
25704

216M
01439
24399
70447

75176
46848
31039
59877
28976

17078
10719
37443
48529

86507
53676
45493
95375
01265

69605
05554
84397
97169
07410

53993
58731
91348
75325

41124
42073
31941
14608
81311

56902
20483
53831
57916

36335

77168
49889
58674
23544

26490
94639
13328
52837

87814
68162
88583
31395
63602

75164
70231
12431
87567
85590

05831
56994
90020

86225

06858
81480
33738
63184

87427

86003
15143
19813

85474

64522
37232
11460
88850
88601

24666
67417
00733
22228

16357
86451
06527
07675
47069

84877
34873
78087
14849
92409

15630

■ 71412

53318

67866

07348
76633
72606
74612
40577

38073
53035
29050
40829

01748
84282
55749
01987
18465

82245
03075
03782
06432

16537
33822
00175
70536
54217

16767
80662
29972
70787
60364

76200
62360

72719
79584

75538
97023
26958
48193
45096

62201
08991
25142
77050

21296
60420
58973
13572
19908

47194
80941
99697
82773

76100
38808
42394
34135
72306

77200
32142
04988
02383

43887

58070
35588
56193

56844

59395
13120

32484
62989
80755

52429
19392
98822
31231

05506
33007
09301
83659
74708

73751
52050
35936
93742

71014
54643
14586
04848
44008

56676
77422
70829
19576
07063

14602
16507
06400
65158

63607
72381
35735
84773
75335

25151
55772
33256
20352

25287
90597
36313
39689
51134

87187
36412
58030
52815

80115
53863
73115
45961
30207

54052 53
18779 52
20697 54
98136 83
86170 96

28514 67
84254 33
58163 87
24520 10

58053 70
28209 31
15364 69
19670 85
36793 79

64404 03
74241 06
69945 48
33957 19

83938 48
72946 22
68582 00
77149 14
95537 27

92863 29
73425 10
49917 99
95633 15

75079 31
20547 48
68989 30
18583 96
48545 74

7

8

9

1X10"'

2

3

4

5

6

7

8

9

1X10"^

2

3

4

5

6

7

8

9

1X10-'

2

3

4

5

6

7

8

9

NO.B.J TABLES OF EXPONENTIAL FUNCTION— VAN ORSTEAND.

Table V.— Values ofe-x ranging from 52 to 62 placet of decimals at intervals of unity from 0 to 100.

27

X

('"■r

0

1.00000

00000

00000

00000

00000 00000

00000

00000

00000

00000 00

1

0. 36787

94411

71442

32159

55237 70161

46086

74458

11131

03176 78

2

. 13533

52832

36612

69189

39994 94972

48440

34076

31545

90957 59

3

. 04978

70683

67863

94297

93424 15650

06177

66316

99592

18842 32

4

.01831

56388

88734

18029

37180 21273

24124

22119

12067

55347 56

5

0. 00673

79469

99085

46709

66360 48423

14842

42488

49585

02735 51

6

. 00247

87521

76666

35842

30451 67430

81666

78915

06479

58553 39

7

. 00091

18819

65554

51620

80031 36084

40928

26264

73724

52743 61

8

. 00033

54626

27902

51183

88213 89125

78086

10193

10900

13372 03

9

. 00012

34098

04086

67954

94976 36690

73003

38260

72152

83228 89

10

0. 00004

53999

29762

48485

15355 91515

56055

06102

37918

08886 66

U

. 00001

67017

00790

24565

93126 35517

36058

08790

77938

04695 93

12

.00000

61442

12353

32820

97586 82308

17880

55323

11223

98931 49

13

. 00000

22603

29406

98105

43257 85277

29053

86894

69353

14242 27

14

. 00000

08315

28719

10356

78840 63985

14256

52622

94607

65836 50

15

0. 00000

03059

02320

50182

57883 71479

49770

22896

39370

82078 08

16

. 00000

01125

35174

71925

91145 13775

17906

01271

91637

94080 07

17

. 00000

00413

99377

18785

16665 96510

27718

95528

06229

36694 37

18

. 00000

00152

29979

74471

26284 36136

62923

35174

31862

17484 33

19

. 00000

00056

02796

43753

72675 40012

98281

62064

63079

78387 37

20

0. 00000

00020

' 61153

62243

85578 27965

94038

01558

20976

37580 73

21

. 00000

00007

58256

04279

11906 72794

17432

41268

12644

29803 62

22

.00000

00002

78946

80928

68924 80771

89130

30644

29320

76931 73

23

. 00000

00001

02618

79631

70189 03039

27527

84061

24977

59833 84

24

. 00000

00000

37751

34544

27909 77516

44969

54752

34067

79168 61

25

0. 00000

00000

13887

94386

49640 20594

66176

37460

86856

91039 98

26

. 00000

00000

05109

08902

80633 24719

87440

01934

79215

76659 41

27

. 00000

00000

01879

52881

65390 83294

75827

04184

22192

62122 87

28

. 00000

00000

00691

44001

06940 20300

94125

84658

74140

92711 82

29

. 00000

00000

00254

36656

47376 92291

03033

85614

85768

16666 03

30

0. 00000

00000

00093

57622

96884 01746

04915

83222

33787

06744 96

31

. 00000

00000

00034

42477

10846 99764

58392

38933

28515

57284 62

32

. 00000

00000

00012

66416

55490 94175

72312

09041

55965

09638 21

33

. 00000

00000

00004

65888

61451 03397

36418

42455

43610

16841 14

34

.00000

00000

00001

71390

84315 42012

96630

27203

42576

04924 12

35

0. 00000

00000

00000

63051

16760 14698

93856

39021

19224

65427 61

36

. 00000

00000

00000

23195

22830 24356

93883

12263

60973

80800 41

37

. 00000

00000

00000

08533

04762 57440

65794

27804

98229

41244 17

38

.00000

00000

00000

03139

13279 20480

29628

70896

46522

31919 65

39

.00000

00000

00000

01154

82241 73015

78598

62624

42063

32386 87

40

0. 00000

00000

00000

00424

83542 55291

58899

53292

34782

85865 80

41

. 00000

00000

00000

00156

28821 89334

98876

80908

82995

10583 41

42

. 00000

00000

00000

00057

49522 26429

35598

06664

38088

05734 23

43

. 00000

00000

00000

00021

15131 03759

10804

86631

40100

70226 51

44

.00000

00000

00000

00007

78113 22411

33796

51571

33167

29279 90

45

.00000

00000

00000

00002

86251 85805

49393

64447

01216

29183 94

46

. 00000

00000

00000

00001

05306 17357

55381

23787

63324

44942 81

47

.00000

00000

00000

00000

38739 97628

68718

71129

31477

49726 91

48

.00000

00000

00000

00000

14251 64082

74093

51062

85321

02803 41

49

. 00000

00000

00000

00000

05242 88566

33634

63937

17180

53028 32

50

0.00000

00000

00000

00000

01928 74984

79639

17783

01734

28165 27

28

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table V. — Values o/e~^ ranging from 52 to 62 places of decimals at intervals of
unity from 0 to 100 — Continued.

I

e-^XlO^"

50

0. 01928

74984

79639

17783

01734

28165

27012

57475 28

51

. 00709

54741

62284

70413

89832

69387

80807

34876 89

52

. 00261

02790

69667

70480

47026

95315

33186

48093 16

53

. 00096

02680

05450

86760

30230

76967

00074

90907 63

54

. 00035

32628

57220

08070

29735

39281

01772

08837 39

55

0. 00012

99581

42500

75030

73600

71340

60714

85530 28

56

. 00004

78089

28838

85469

08127

71770

42317

96289 39

57

. 00001

75879

22024

24311

64895

58751

28803

43631 78

58

. 00000

64702

34925

64546

03261

54039

55292

64893 76

59

. 00000

23802

66408

69440

06058

94324

58880

24963 31

60

0. 00000

08756

51076

26965

20338

48873

28007

39166 04

61

. 00000

03221

34028

59925

16089

00124

77758

48943 75

62

. 00000

01185

06486

42339

81006

28503

07390

97280 99

63

. 00000

00435

96100

00063

08097

36231

24815

88845 96

64

. 00000

00160

38108

90.548

63785

29760

87034

14233 54

65

0. 00000

00059

00090

54159

70613

91401

26029

55584 23

66

. 00000

00021

70522

01130

36394

11986

56925

95727 06

67

. 00000

00007

98490

42456

86978

80839

26942

66474 24

68

. 00000

00002

93748

21117

10802

94660

88806

42392 87

69

. 00000

00001

08063

92777

07278

494.53

66496

16247 34

70

0. 00000

00000

39754

49735

90864

68077

89099

75379 48

71

. 00000

00000

14624

86227

25123

09468

26378

73083 16

72

. 00000

00000

05380

18616

00211

38413

81818

72704 54

73

. 00000

00000

01979

25987

79469

04553

74919

15336 02

74

. 00000

00000

00728

12901

78321

64383

42969

73716 88

75

0. 00000

00000

00267

86369

61808

07794

43444

15201 08

76

. 00000

00000

00098

54154

68611

12580

28938

09797 36

77

. 00000

00000

00036

25140

91914

35592

24240

83319 73

78

. 00000

00000

00013

33614

81550

22613

41453

01407 91

79

. 00000

00000

00004

90609

47306

49280

56613

53873 52

80

0. 00000

00000

00001

80485

13878

45415

17231

21283 57

81

. 00000

00000

00000

66396

77199

58073

44007

02255 27

82

. 00000

00000

00000

24426

00737

74052

76794

40802 61

83

. 00000

00000

00000

08985

82594

40493

80669

66884 82

84

. 00000

00000

00000

03305

70062

67607

34298

45509 64

85

0. 00000

00000

00000

01216

09929

92528

25564

41682 63

86

. 00000

00000

00000

00447

37793

06181

12073

46276 56

87

. 00000

00000

00000

00164

58114

31082

27365

11660 34

88

. 00000

00000

00000

00060

54601

89540

11858

84531 86

89

. 00000

00000

00000

00022

27363

56179

57437

39222 91

90

0. 00000

00000

00000

00008

19401

26239

90515

43036 11

91

. 00000

00000

00000

00003

01440

87850

65374

55326 31

92

. 00000

00000

00000

00001

10893

90193

12136

37945 96

93

. 00000

00000

00000

00000

40795

58667

17756

01577 01

94

. 00000

00000

00000

00000

15007

85762

70739

48875 45

95

0. 00000

00000

00000

00000

05521

08227

70285

32731 72

96

.00000

00000

00000

00000

02031

092G6

27348

10925 69

97

. 00000

00000

00000

00000

00747

19723

37342

99016 06

98

. 00000

00000

00000

00000

00274

87850

07910

21493 00

99

. 00000

00000

00000

00000

00101

12214

92610

44852 99

100

0. 00000

00000

00000

00000

00037

20075

97602

08359 63

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

29

Table VI. — Values ofe~x ranging from SS to AS places of decimalt
at intervals of 0.1 from 0.0 to 50.0.

X

«-*

0.0

1. 00000

00000

00000 00000

00000

00000 000

.1

0. 90483

74180

35959 57316

42490

59446 437

.2

. 81873

07530

77981 85866

99355

08619 039

.3

. 74081

82206

81717 86606

68737

79317 817

.4

. 67032

00460

35639 30074

44329

25147 826

0.5

0. 60653

0G597

12633 42360

37995

34991 180

.6

. 54881

16360

94026 43262

84589

17232 568

.7

. 49658

53037

91409 51470

48000

93397 529

.8

. 44932

89641

17221 59143

01023

85015 563

.9

.40656

96597

40599 11188

34542

39645 626

1.0

0. 36787

94411

71442 32159

55237

70161 461

.1

. 33287

10836

98079 55328

88469

06431 316

.2

. 30119

42119

12202 09664

49776

07083 222

.3

. 27253

17930

34012 60312

23331

67563 350

.4

.24659

69639

41606 47693

98612

39833 768

1.5

0. 22313

01601

48429 82893

32804

70764 013

.6

. 20189

65179

94655 40848

51792

67643 350

.7

. 18268

35240

52734 65022

39008

37758 940

.8

. 16529

88882

21586 53829

68047

20432 214

.9

. 14956

86192

22635 05264

10120

69103 735

2.0

0. 13533

52832

36612 69189

39994

94972 484

.1

. 12245

64282

52981 91021

86473

76072 626

.2

. 11080

31583

62333 88333

41444

25849 939

.3

. 10025

88437

22803 73372

99406

93797 987

.4

. 09071

79532

89412 50337

51722

20079 691

2.5

0. 08208

49986

23898 79516

95286

74467 160

.6

. 07427

35782

14333 88042

82105

70169 975

.7

. 06720

55127

39749 76512

65517

00855 966

.8

. 06081

00626

25217 96499

56213

88183 941

.9

. 05502

32200

56407 22902

99465

30834 175

3.0

0. 04978

70683

67863 94297

93424

15650 062

.1

.04504

92023

93557 80606

83350

92178 335

.2

. 04076

22039

78366 21.516

60792

62144 425

.3

. 03688

31674

01240 00544

56037

04741 515

.4

. 03337

32699

60326 07948

24001

31470 948

3.5

0. 03019

73834

22318 50073

97862

92363 620

.6

. 02732

37224

47292 56080

15630

62435 553

.7

. 02472

35264

70339 39120

27573

82983 403

.8

. 02237

07718

56165 59577

85833

22540 823

.9

. 02024

19114

45804 38847

20275

43743 654

4.0

0. 01831

56388

88734 18029

37180

21273 241

.1

. 01657

26754

01761 24754

19836

98083 451

.2

. 01499

55768

20477 70621

19843

60228 729

.3

. 01356

85590

12200 93175

72305

74525 767

.4

. 01227

73399

03068 44117

89393

86236 542

4.5

0. OHIO

89965

38242 30649

61431

34286 931

.6

.01005

18357

44633 58164

21330

94331 550

.7

.00909

52771

01695 81709

20540

74291 388

.8

.00822

97470

49020 02884

13620

26766 074

.9

.00744

65830

70924 34051

82360

46420 128

5.0

0. 00673

79469

99085 46709

66360

48423 148

.1

.00609

67465

65515 63610

71345

64785 425

.2

. 00551

65644

20760 77241

79937

54667 303

.3

. 00499

15939

06910 21621

22867

25942 076

.4

.00451

65809

42612 66798

16490

18705 780

6.5

0.00408

67714

38464 06699

34647

02684 721

.6

.00369

78637

16482 93082

06926

36441 249

.7

.00334

59654

57471 27276

57324

36020 607

.8

.00302

75-547

45375 81474

81920

44595 488

.9

.00273

94448

18768 36923

27755

20842 145

30

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table VI. — Values ofe~^ ranging from 33 to 4S places of decimals
at intervals of 0.1 from 0.0 to 50.0 — Continued.

X

gx-x

6.0

0. 00247

87521

76666

35842

30451

67430

817

.1

. 00224

28677

19485

80247

32236

16521

454

.2

. 00202

94306

36295

73436

33862

53459

782

.3

. 00183

63047

77028

90682

52279

36299

895

.4

.00166

15572

73173

93449

90832

54173

641

6.5

0. 00150

34391

92977

57244

73829

03332

168

.6

.00136

03680

37547

89341

68557

63685

588

.7

. 00123

09119

02673

48118

46234

76276

674

.8

.00111

37751

47844

80307

87892

19640

468

.9

. 00100

77854

29048

51076

14475

35575

166

7.0

0. 00091

18819

65554

51620

80031

36084

409

.1

.00082

51049

23265

90427

01462

25456

749

.2

. 00074

65858

08376

67936

80906

47515

335

.3

. 00067

55387

75193

84423

78367

24317

781

.4

. 00061

12527

61129

57255

56702

29776

745

7.5

0. 00055

30843

70147

83358

31020

00088

530

.6

. 00050

04514

33440

61069

55020

35820

856

.7

. 00045

28271

82886

79705

79972

07264

598

.8

.00040

97349

78979

78670

84619

67840

934

.9

.00037

07435

40459

08837

44300

21422

977

8.0

0. 00033

54626

27902

51183

88213

89125

781

.1

. 00030

35391

38078

86666

08655

09532

209

.2

. 00027

46535

69972

14232

76277

89393

761

.3

. 00024

85168

27107

95202

08034

74470

637

.4

. 00022

48673

24178

84827

27986

33560

122

8.5

0. 00020

34683

69010

64417

43689

33430

487

.6

. 00018

41057

93667

57912

49547

76189

858

.7

. 00016

65858

10987

63341

14921

30507

125

.8

. 00015

07330

75095

47660

06434

06463

915

.9

. 00013

63889

26482

01144

78477

65082

136

9.0

0. 00012

34098

04086

67954

94976

36690

730

.1

. 00011

16658

08490

11473

56400

85376

178

.2

. 00010

10394

01837

09335

07306

72733

712

.3

. 00009

14242

31478

17333

78629

43248

947

.4

. 00008

27240

65556

63226

27291

71823

338

9.5

0. 00007

48518

29887

70059

14711

89319

355

.6

. 00006

77287

36490

85387

29971

88458

992

.7

. 00006

12834

95053

22209

55132

40931

438

.8

. 00005

54515

99432

17698

18088

77544

465

.9

. 00005

01746

82056

17530

21858

33726

590

10.0

0.00004

53999

29762

48485

15355

91515

561

.1

. 00004

10795

55225

30070

84235

23804

485

.2

. 00003

71703

18684

12670

45551

91140

292

.3

. 00003

36330

95185

71899

37288

28297

935

.4

.00003

04324

83008

40363

65062

22204

232-

ao.5

0.00002

75364

49349

74715

78574

11097

102

.6

.00002

49160

09731

50319

62336

14140

627

.7

.00002

25449

37913

21219

44136

71874

562

.8

. 00002

03995

03411

17193

64237

54489

295

.9

. 00001

84582

33995

78056

47431

58551

471

11.0

0. 00001

67017

00790

24565

93126

35517

361

.1

. 00001

51123

23819

85502

79896

87011

424

.2

.00001

36741

96065

68095

33745

88026

747

.3

. 00001

23729

24261

78823

05171

26757

815

.4

.00001

11954

84842

59094

36391

53266

479

11.5

0.00001

01300

93598

63071

07289

41355

749

.6

. 00000

91G60

87736

24761

44734

00846

298

.7

. 00000

82938

19160

75736

50913

52973

571

.8

.00000

75045

57915

07686

33506

65463

934

.9

.00000

67904

04807

37947

30051

67517

816

No. 5.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

31

Table VI . — Values of e'" ranging from SS to 4S places of decimals
at intervals of 0.1 from 0.0 to 50.0 — Continued.

X

e-a:X10>

12.0

0. 61442

12353

32820

97586

82308

179

.1

•55595

13241

65014

42782

64691

467

.2

.50304

55G07

11144

43312

43226

772

.3

.45517

44463

08323

47633

86288

946

.4

.41185

88707

53570

92504

35497

759

12.5

0. 37266

53172

07867

09929

24851

476

.6

. 33720

15234

13918

32115

08762

687

.7

. 30511

25558

03642

02182

53527

096

.8

. 27607

72572

03720

07929

85038

819

.9

. 24980

50325

86663

59687

66158

817

13.0

0. 22603

29406

98105

43257

85277

291

.1

.20452

30624

52348

88517

46920

038

.2

.18506

01197

58190

67532

46783

110

.3

. 16744

93209

43426

71792

86875

852

.4

. 15151

44112

14324

96163

05769

581

13.5

0.13709

59086

38408

43645

02599

613

.6

.12404

95079

95671

29562

42723

616

. 7

. 11224

46365

23434

33766

81407

425

.8

. 10156

31471

00249

09180

75395

895

.9

. 09189

81357

89795

74280

69218

720

14.0

0. 08315

28719

10356

78840

63985

143

.1

.07523

98299

21642

10566

16958

717

.2

. 06807

98134

39763

37737

59450

189

.3

. 06160

11626

13205

31394

82053

038

.4

.05573

90369

26945

98049

07658

331

14.5

0.05043

47662

56788

80758

92222

233

.6

.04563

52636

79039

92229

75634

380

.7

.04129

24941

58732

68861

37841

627

.8

. 03736

29937

98852

62858

23297

360

.9

. 03380

74348

39047

38111

80934

638

15.0

0. 03059

02320

50182

57883

71479

49770

229

.1

. 02767

91865

85408

06275

11062

90303

601

.2

.02504

51637

23276

19971

12378

74692

052

.3

. 02266

18012

77657

11644

48865

71008

072

.4

. 02050

52457

56119

27503

43940

88106

805

15.5

0. 01855

39136

26159

78240

71710

86473

493

.6

. 01678

82752

99956

62558

32695

60836

212

.7

.01519

06596

75689

62781

87087

19027

273

.8

. 01374

50772

79213

96984

28825

38479

717

.9

. 01243

70602

36028

70065

42338

53941

588

16.0

0. 01125

35174

71925

91145

13775

17906

013

.1

. 01018

26036

93120

00088

98479

40768

981

.2

.00921

36008

34566

12805

18330

14487

091

.3

.00833

68107

89962

77760

99947

21871

855

.4

.00754

34583

49844

24816

62940

02631

941

16.5

0. 00682

56033

76334

86975

53833

89689

872

.6

.00617

60613

35580

37163

68288

57083

203

.7

.00558

83313

92518

26353

28123

48887

751

.8

.00505

65313

48335

52410

44488

14372

545

.9

.00457

53387

69445

80493

73240

17735

568

17.0

0.00413

99377

18785

16665

96510

27718

955

.1

.00374

59705

56295

25069

03124

10250

558

.2

.00338

94943

26196

92178

25752

20721

809

.3

.00306

69412

94563

55723

45060

05384

994

.4

.00277

50832

42240

75246

48379

49380

576

32

MEMOIKS NATIONAL ACADEMY OF SCIENCES

[Vol. XIV,

Table VI. — Values of e ^ ranging from, S3 to 4S places of decimals
at intervals of 0.1 from 0.0 to 50.0 — Continuecf.

X

e

JXIO^

17.5

0. 00251

09991

55743

98180

35473

43740

193

.6

.00227

20459

92773

85882

20171

05083

212

.7

. 00205

58322

29760

44687

85201

09381

462

.8

.00186

01939

26691

55236

15658

41796

203

.9

.00168

31730

69673

75730

08575

95356

349

18.0

0.00152

29979

74471

26284

36136

62923

352

.1

. 00137

80655

54894

57374

37067

27046

561

.2

.00124

09252

78575

09801

76125

51553

128

.3

. 00112

82646

49549

66131

01387

32549

077

.4

.00102

08960

72359

76231

78545

44852

002

18.5

0. 00092

37449

66197

05948

97883

17038

460

.6

. 00083

58390

10137

46206

26295

28090

620

. 7

.00075

62984

11826

51341

18612

28066

415

.8

. 00068

43271

02221

79922

76049

65053

096

.9

.00061

92047

68266

40298

09210

94237

034

19.0

0.00056

02796

43753

72675

40012

98281

621

.1

. 00050

69619

86232

22936

08100

57441

536

.2

.00045

87181

74664

75209

98545

73639

897

.3

.00041

50653

68769

82261

54061

18520

217

.4

.00037

55666

76593

82970

51383

03756

200

19.5

0.00033

98267

81949

50712

25140

73787

681

.6

. 00030

74879

87958

60105

71369

28807

703

.7

.00027

82266

37101

58709

84770

56340

891

.8

. 00025

17498

71943

82798

50011

88884

873

.9

.00022

77927

04120

53677

29238

72891

527

20.0

0. 00020

61153

62243

85578

27965

94038

016

.1

. 00018

65008

92190

27697

33189

22598

761

_ 2

. 00016

87529

85750

85307

37206

62805

645

.3

.00015

26940

15912

66097

18605

61466

414

.4

.00013

81632

59107

95387

89246

35435

639

20.5

0. 00012

50152

86638

67426

28937

55311

923

.6

. 00011

31185

09177

16341

53263

71739

791

.7

. 00010

23538

59775

94154

25949

62818

845

.8

. 00009

26136

02205

67760

50298

19435

554

.9

. 00008

38002

52694

79477

46801

25352

979

21.0

0. 00007

58256

04279

11906

72794

17432

413

.1

. 00006

86098

43996

93450

45164

74732

749

2

. 00006

20807

54094

03619

76867

91816

079

^3

.00005

61729

89244

17303

97323

90528

550

.4

.00005

08274

22551

05926

15332

48072

717

21.5

0. 00004

59905

53786

52316

77907

05925

361

.6

.00004

16139

73942

24154

70246

73925

373

.7

. 00003

76538

80736

11354

32827

62725

700

.8

. 00003

40706

40224

29893

53380

54451

820

.9

.00003

08283

90131

38675

51913

32631

879

22.0

0. 00002

78946

80928

68924

80771

89130

306

.1

. 00002

52401

51068

45210

21742

73861

281

.2

. 00002

28382

33123

61576

64470

64849

105

.3

. 00002

06648

87892

07581

80510

04029

230

.4

. 00001

86983

63804

26844

64135

94383

017

22.5

0. 00001

69189

79226

15130

36130

19439

206

.6

. 00001

53089

25478

79478

29098

85778

291

. 7

.00001

38520

88603

13758

75438

72331

863

.8

.00001

25338

88086

06835

66073

58263

954

.9

. 00001

13411

30933

74976

68297

68776

038

23.0

0. 00001

02618

79631

70189

03039

27527

841

.1

. 00000

92853

32670

14494

21797

43920

867

.2

. 00000

84017

16438

85889

17671

85250

385

.3

.00000

76021

87409

60735

66299

49779

464

.4

.00000

68787

43627

13460

03812

13048

694

NO.B.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

Table VI.— Values of e-x ranging from S3 to 4S places of dedmali
01 intervals of 0.1 from U.O to 50.0— Continued.

33

X

e-^XWo

23.5
.6

.7
.8
.9

0.62241 44622 90778 32321 36689 302
.56318 38950 07427 98158 22576 375
.50958 98614 37966 07815 32879 224
.46109 59744 80822 57885 97649 739
.41721 68910 16002 20720 16549 798

24.0
.1

.2
.3
.4

0.37751 34544 27909 77516 44969 548
.34158 82993 78385 77078 17927 744
.30908 18748 40832 95528 56856 901
.27966 88455 92692 90452 20914 954
.25305 48361 51189 69970 95413 063

24.5
.6
.7
.8
.9

0.22897 34845 64555 28940 85224 694
.20718 37765 72088 85115 75223 970
.18746 76334 52428 00712 00483 363
.16962 77294 18406 63987 44513 620
.15348 55167 14253 44644 47610 507

25.0
.1
.2
.3
.4

0.13887 94386 49640 20594 66176 37460 869
.12566 33126 86023 89591 28573 24497 477
.11370 48673 92667 30575 06855 57781 HI
. 10288 44186 29702 25556 66047 29002 133
.09309 36717 09030 56681 48387 57235 814

25.5
.6

.7
.8
.9

0.08423 46375 44686 47405 87646 52628 816
.07621 86519 45129 01041 86087 42569 880
.06896 54882 32212 00165 51703 34317 985
.06240 25543 05624 06152 61704 36655 883
.05646 41661 16749 62792 72023 10934 402

26.0
.1

.2
.3
.4

0.05109 08902 80633 24719 87440 01934 792
.04622 89492 46686 68940 73403 07967 410
.04182 96830 74887 40238 10483 40957 393
.03784 90624 30743 59536 01635 97079 765
.03424 72479 24915 87477 54037 63906 063

26.5
.6
.7
.8
.9

0.03098 81913 87218 25441 64178 60818 385
.02803 92750 84414 72566 40350 61367 907
.02537 09852 70981 83277 48943 37439 574
.02295 66168 05623 56169 40441 23976 976
.02077 20058 77241 34158 48280 41113 052

27.0
.1
.2
.3

.4

0.01879 52881 65390 83294 75827 04184 222
.01700 66800 14814 06878 50865 79529 609
.01538 82804 33968 11670 38458 16312 667
.01392 38919 35884 98620 87677 77862 518
.01259 88584 28277 88967 69065 98718 199

27.5
.6

.7
.8
.9

0.01139 99185 30443 55345 31786 95696 403
.01031 50728 48906 83550 57811 58485 712
.00933 34638 83457 69077 84065 71852 384
.00844 52673 61639 73721 35688 85617 335
.00764 15939 14129 46027 61067 13778 082

28.0
.1
.2
.3
.4

0.00691 44001 06940 20300 94125 84658 741
.00625 64079 40031 33604 79651 20550 275
.00566 10320 06637 63070 77960 29403 337
.00512 23135 84304 92092 59075 13264 538
.00463 48609 97992 98618 53973 63255 792

28.5
.6

.7
.8
.9

0.00419 37956 58379 54442 52680 72672 186
.00379 47032 35298 56414 35892 66116 403
.00343 35894 77640 25.514 74190 24136 674
.00310 68402 37.543 44761 24890 39628 848
.00281 118.52 98789 04044 93344 23703 026

1153W°— 21-

34

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

(Vol. XIV.

Table VI. — Values of e~^ ranging from 33 to AS places of decimals
at intervals of 0.1 from 0.0 to 50.0 — Continuecl.

X

c-i

X10'°

29.0

0. 00254

36656

47376

92291

03033

85614 868

.1

. 00230

16038

56719

30252

98961

98444 442

.2

. 00208

25772

91055

50034

41929

59577 002

.3

. 00188

43938

58898

98201

66055

82710 392

.4

. 00170

50700

73848

97320

95731

68146 459

29.5

0. 00154

28112

03191

88783

29721

02046 747

.6

. 00139

59933

05613

09997

76718

93894 576

.7

. 00126

31469

78246

44161

41824

90288 582

.8

. 00114

29426

50396

43462

40460

89725 866

.9

. 00103

41772

76747

88631

14871

35865 680

30.0

0. 00093

57622

96884

01746

04915

83222 338

.1

. 00084

67127

40607

93341

71972

67749 165

.2

. 00076

61373

70029

83365

22776

85364 170

.3

. 00069

32297

59758

6.5523

77079

03801 763

.4

. 00062

72602

25925

71015

47054

72075 602

30.5

0. 00056

75685

23263

27224

61872

78872 381

.6

. 00051

35572

37148

02171

53091

68192 004

.7

. 00046

46858

04474

69695

17315

32710 431

.8

. 00042

04951

03518

84753

93216

26484 901

.9

. 00038

04525

58642

21646

74684

33532 907

31.0

0. 00034

42477

10846

99764

58392

38933 285

.1

. 00031

14882

09847

58694

36580

83265 878

.2

. 00028

18461

87547

13372

67019

06804 197

.3

. 00025

50249

76623

42730

31718

36825 785

.4

. 00023

07561

41382

62290

86183

09083 925

31.5

0. 00020

87967

91164

59335

50509

88967 622

.6

. 00018

89271

49411

56410

78926

37742 140

.7

. 00017

09483

54070

45362

63790

25908 984

.8

. 00015

46804

67314

60627

92270

38923 531

.9

. 00013

99606

74665

54398

29638

60755 089

32.0

0. 00012

66416

55490

94175

72312

09041 560

.1

. 00011

45901

08570

22324

18777

38161 942

.2

. 00010

36854

17971

14108

12572

72255 979

.3

. 00009

38184

45884

98657

78521

57413 808

.4

. 00008

48904

40338

71765

13373

59415 985

32.5

0. 00007

68120

46852

02094

90674

25977 989

.6

. 00006

95024

14147

63979

20547

50203 795

.7

. 00000

28883

84964

61G33

73835

11081 070

.8

. 00005

69037

63875

83490

75590

29624 051

9

. 00005

14886

54781

93836

54103

95285 602

33.0

0. 00004

65888

61451

03397

36418

42455 436

.1

. 00004

21553

45104

58862

97123

52302 914

.2

. 00003

81437

33620

85080

38776

35563 724

.3

. 00003

45138

77443

74206

47237

60762 601

.4

. 00003

12294

47752

60511

44037

56739 759

33.5

0. 00002

82575

72871

15611

21020

28754 875

.6

. 00002

55685

09276

69987

34084

22688 697

.7

. 00002

31353

43916

95759

36958

40314 990

.8

. 00002

09337

24855

19385

25874

99128 963

.9

. 00001

89416

17547

84879

72752

75933 317

34.0

0. 00001

71390

84315

42012

96630

27203 426

.1

. 00001

55080

84799

46536

18659

27328 468

.2

. 00001

40322

95408

63094

99211

85099 450

.3

. 00001

26969

45946

66347

89949

81449 683

.4

. 00001

14886

71787

32112

48054

47810 798

34.5

0. 00001

03953

80116

70221

94395

13367 453

.6

. 00000

94061

28904

29918

83443

95415 969

.7

. 00000

85110

17391

47948

70615

64947 459

.8

. 00000

77010

87001

36544

68253

41936 893

.9

. 00000

69682

31678

38580

11813

28543 327

35.0

0. 00000

63051

16760

14698

93856

39021 192

No. 5.]

TABLES OF EXPONENTIAL FUNCTION-VAN ORSTRAND.

Table VI.— Values ofe-x ranging from S3 to 4S places of decimals
at intervals of 0.1 from 0.0 to 50.0— Continued.

35

X

f-^XlO'^

35.0

0.63051 16760 14698 93856 39021 19224 654

.1

.57051 05569 66665 64836 20656 86873 686

2

.51621 92993 27974 97344 74467 39592 088

.3

.46709 45379 44257 03580 57879 67927 598

.4

.42264 46156 92181 08441 86703 53694 809

35.5

0.38242 46628 09713 53519 42886 25672 271

.6

.34603 21444 90013 64814 69714 18553 590

.7

.31310 28321 77790 04439 103C7 50706 083

.8

.28330 71582 47497 90491 45219 11627 172

.9

.25634 69175 79771 01372 82219 04476 870

36.0

0.23195 22830 24356 93883 12263 60973 808

.1

.20987 91048 79305 26873 44597 06680 239

.2

.18990 64673 5S688 94402 30391 82482 057

.3

.17183 44775 93166 33948 23021 57303 245

.4

.15548 22650 34958 57953 08297 21321 002

36.5

0.14068 61712 44614 67672 48913 72822 964

.6

.12729 81119 42342 00516 07241 99744 214

.7

.11518 40949 30761 29059 26337 74188 051

.8

.10422 28790 55958 90584 32069 53886 118

.9

.09430 47607 85267 94412 20739 28211 370

37.0

0.08533 04762 57440 65794 27804 98229 412

.1

.07721 02078 16561 35572 52863 52358 518

.2

.06986 26850 86757 24087 00541 33072 436

.3

.00321 43715 90960 76015 74919 93941 739

.4

.05719 87287 73180 64818 04795 09826 246

37.5

0.05175 55500 58018 68534 85109 07057 388

.6

.04683 03582 83528 48493 54683 16855 962

.7

.04237 38604 74966 82593 34655 87997 980

.8

.03834 14545 04384 98234 92936 93413 804

.9

.03469 27826 97490 91944 53846 38105 277

38.0

0.03139 13279 20480 29628 70896 46522 319

.1

.02840 40481 04287 51936 53646 27988 899

.2

.02570 10455 48452 71119 81485 05469 214

.3

.02325 52676 94886 54372 34296 62932 824

.4

.02104 22363 76776 20152 57806 72554 547

38.5

0.01903 98028 32864 52319 09651 51045 524

.6

.01722 79260 35202 88389 09152 96528 151

.7

.01558 84721 11807 46343 27407 39887 817

.8

.01410 50328 56773 42732 90339 44269 328

.9

.01276 27615 11435 24275 64109 91551 699

39.0

0.01154 82241 73015 78598 62624 42063 324

.1

.01044 92653 43612 05827 71170 02291 972

.2

.00945 48862 73886 56872 67683 46735 659

.3

.00855 51348 83887 15734 63340 96583 931

.4

.00774 10061 59285 82425 54038 35334 841

39.5

0.00700 43520 26168 64522 06111 20117 684

.6

.00633 77998 02373 37888 31289 73849 824

.7

.00573 46784 09208 34299 62399 53206 969

.8

.00518 89516 05054 64108 86055 09785 357

.9

.00469 51575 72631 18967 64182 28650 614

40.0

0.00424 83542 55291 58899 53292 34782 859

.1

.00384 40698 95260 12322 93217 25582 228

.2

.00347 82582 78776 93145 3S283 85687 124

.3

.00314 72582 40230 71953 77048 96788 377

.4

.00284 77570 19982 76205 37354 14757 803

40.5

0.00257 67571 09154 98094 81244 03947 486

.6

.00233 15462 49553 59620 87332 35047 951

.7

.00210 96702 88477 50105 40714 22490 401

.8

.00190 89086 16733 16004 79492 13432 437

.9

.00172 72519 44031 42767 18344 04012 470

36

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

Table VI. — Values of e"^: ranging from S3 to 48 places of decimals
at intervals of 0.1 from 0.0 to 50.0 — Continued.

[Vol. siv.

X

e-x

X10">

41.0

0. 00156

28821

89334

98876

80908

82995 106

.1

. 00141

41542

84892

25895

04128

48638 316

.2

. 00127

95797

11846

40038

13520

71489 987

.3

. 00115

78116

02638

29407

39244

53079 409

.4

. 00104

76312

61103

31040

88099

91446 531

41.5

0. 00094

79359

65350

47559

45429

56113 551

.6

. 00085

77279

31351

14917

47393

06788 324

.7

. 00077

61043

26781

09860

09063

86820 825

.8

.00070

22482

35171

14588

95695

22686 530

.9

. 00063

54204

79932

56898

16115

01639 407

42.0

0. 00057

49522

26429

35598

06664

38088 057

.1

. 00052

02382

88056

36486

16013

12077 002

.2

. 00047

07310

69328

36896

66194

65095 023

.3

. 00042

59350

85360

38765

82354

83665 338

.4

. 00038

54020

02888

41921

20220

99004 186

42.5

0. 00034

87261

53199

44467

34281

84859 880

.6

. 00031

55404

72062

59800

09588

74554 002

.7

. 00028

55128

26026

96901

02510

06254 767

.8

. 00025

83426

88318

39275

69583

84336 693

.9

. 00023

37581

31066

48315

69181

60051 823

43.0

0. 00021

15131

03759

10804

86631

40100 702

.1

. 00019

13849

70686

16334

16651

97089 136

.2

. 00017

31722

82726

55584

82545

15506 796

.3

. 00015

66927

61177

68999

51381

41226 635

.4

. 00014

17814

73448

94625

92151

05608 824

43.5

0. 00012

82891

82360

87848

92767

77284 128

.6

. 00011

60808

52529

36166

06882

27545 051

.7

. 00010

50342

98886

08059

21548

25900 989

.8

. 00009

50389

63809

29842

77008

33343 029

.9

. 00008

59948

10626

01859

42848

36603 861

44.0

0. 00007

78113

22411

33796

51571

33167 293

.1

. 00007

04065

96064

63864

02594

11819 481

.2

. 00006

37065

22595

82837

94977

89144 024

.3

. 00005

76440

45417

65886

78362

28242 312

.4

. 00005

21584

89220

86203

68464

15609 422

44.5

0. 00004

71949

52715

26123

41636

05846 918

.6

. 00004

27037

59159

20617

46712

38215 253

.7

. 00003

86399

59178

04557

49710

54130 889

.8

. 00003

49628

80895

67763

67822

60571 334

.9

. 00003

16357

22876

74373

05000

03529 236

45.0

0. 00002

86251

85805

49393

64447

01216 292

.1

. 00002

59011

39215

04273

31305

92431 185

.2

. 00002

34363

19931

52920

73161

89041 858

.3

. 00002

12060

59215

10958

48008

45775 273

.4

. 00001

91880

35866

91742

41361

20466 612

45.5

0. 00001

73620

52831

00294

72541

72775 788

.6

. 00001

57098

35055

40862

91529

13892 163

.7

. 00001

42148

46589

30674

99007

15384 638

.8

. 00001

28621

25085

64558

58060

81817 263

.9

. 00001

16381

32052

95109

72519

37515 872

46.0

0. 00001

05306

17357

55381

23787

63324 449

.1

. 00000

95284

96620

13365

08941

90311 942

.2

.00000

86217

40279

52610

01613

69548 955

.3

. 00000

78012

73213

50302

88340

07777 956

.4

. 00000

70588

83911

89917

38002

38406 937

46.5

0. 00000

63871

42293

05842

23502

28846 869

.6

.00000

57793

25341

07926

11131

32265 356

.7

.00000

52293

49819

61195

00313

96943 218

.8

.00000

47317

11388

78448

78157

94017 572

.9

. 00000

42814

29515

91910

04355

72624 796

NaS.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

37

Table VI.— Values of e'^ ranging from 33 to ^s' places of
decimals at intervals of 0.1 from o.O to 50.0— Continued.

X

e-^XlO*"

47.0

0. 38739

97628

68718

71129

31477 497

.1

. 35053

38011

81874

44181

14696 776

.2

.31717

60995

95737

66419

90234 132

.3

. 2869!)

28030

20923

62903

85594 584

.4

. 25968

18268

80355

27517

22837 845

47.6

0. 23496

98337

45281

70976

26987 584

.6

. 21260

94976

82419

38687

22606 325

.7

. 19237

70289

32882

68879

79646 073

.8

.17406

99341

49058

66327

58S06 721

.9

.15750

49897

73123

74854

18083 799

48.0

0. 14251

64082

74093

51062

85321 028

.1

. 12895

41788

90489

43765

81148 006

.2

. 11668

25662

72217

70475

74887 795

.3

. 10557

87519

95563

20838

55106 156

.4

. 09553

16053

55124

32778

83159 356

48.5*

0. 08644

05711

30360

94557

72312 023

.6

. 07821

46631

95149

50544

60620 901

.7

. 07077

15538

98051

27453

54748 366

.8

. 06403

67500

99505

46535

26327 711

.9

. 05794

28476

19450

50254

69685 961

49.0

0. 05242

88566

33634

63937

17180 530

.1

. 04743

95912

66955

45835

05016 148

.2

. 04292

51172

74673

23310

79205 757

.3

. 03884

02522

83706

09393

43055 624

.4

. 03514

41135

92253

90440

00963 185

49.5

0. 03179

97090

01977

49498

18163 259

.6

. 02877

35665

87644

17741

59121 756

.7

. 02603

53996

98849

71336

86526 680

.8

. 02355

78038

41041

37407

48630 166

.9

. 02131

59824

02125

48791

98821 454

50.0

0. 01928

74984

79639

17783

01734 282

38 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vol.xiv.

Table VII. — Values o/e~^ to 63 places of decimals at decimal intervals from 1X10'"' to 9X70"'.

X

r^

IXlO-'o. .
2

0. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99998
. 99997
. 99996

0. 99995
. 99994
. 99993
. 99992
. 99991

0. 99990
. 99980
. 99970
. 99960

99999

99998
99997
99996

99995
99994
99993
99992
99991

99990
99980
99970
99960

99950
99940
99930
99920
99910

99900
99800
99700
99600

99500
99400
99300
99200
99100

99000
98000
97000
96000

95000
94000
93000
92000
91000

90000
80000
70000
60000

50000
40000
30000
20000
10000

00000
00001
00004
00007

00012
00017
00024
00031
00040

00049
00199
00449
00799

00000
00000
00000
00000

00000

00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00000
00000
00000

00001
00001
00002
00003
00004

00004
00019
00044
00079

00124
00179
00244
00319
00404

00499
01999
04499
07999

12499
17999
24499
31999
40499

49999
99998
49995
99989

49979
99964
49942
99914
49878

99833
98666
95500
89334

00000
00001
00004
00007

00012
00017
00024

ooo;n

00040

00049
00199
00449
00799

01249
01799
02449
03199
04049

04999
19999
44999
79999

24999
79999
44999
19999
04999

99999
99999
99999
99998

99997
99996
99994
99991
99987

99983
99866
99550
98933

97916
96400
94283
91466
87850

83333
66667
50003
33343

16692
00053
83433
66837
50273

33749
73333
33747
39991

49999
99999
49999
99999

49999
99999
49999
99999
49999

99999
99999
99999
99999

99999
99999
99999
99999
99998

99998
99986
99955
99893

99791
99640
99428
99146
98785

98333
86666
55000
93333

91666
40000
28333
46666
85000

33333
66673
00033
33439

66927
00539
34333
68373
02733

37499
33333
37497
99991

70807
99935
37359
33060
37007

99166
06667
97510
46723

99999
99998
99995
99989

99979
99964
99942
99914
99878

99833
98666
95500
89333

79166
64000
42833
14066
78500

33333
66G66
00000
33333

66666
00000
33334
66668
00002

33337
66733
00337
34399

69270
05399
43337
83733
27337

74999
33333
74997
99991

08307
99935

74859
33060
74507

99166
06666
97501
4667^

29188
20064
94330
27030
93238

68055
55553
12456
55230

83333
66666
50000
33333

16666
00000
83333
66666
50000

33333
66666
00000
33334

66669
00005
33343
66683
00027

33374
67333
03374
43999

92708
53999
33374
37333
73374

49999
33333
49997
99991

83307
99935
49859
33060
49507

99166
06666
97500
46667

29168
20006
94183
26703
92573

66805
75555
01249
35552

36790
79944
06642
75139
10301

53571
01593
60877
49244

33333
66667
00003
33343

66692
00053
33433
66837
00273

33749
73333
33749
39999

27083
39999
33749
73333
33749

99999
33333
99997
99991

33307
99935
99859
33060
99507

99166
06666
97500
46666

29166
20000
94168
26670
92507

66680
67555
10124
23555

83680
47999
00678
07551
81115

55635
53015
56607
30477

05466
45755
15559
45812
00889

45337
65065
00350
37220

37499
33333
37499
99999

70833
99999
37499
33333
37499

99999
33333
99997
99991

33307
99935
«9859
33060
99507

99166
06666
97500
46666

29166
20000
94166
26667
92500

66668
66755
01012
72355

88368
64799
30068
30755
38112

55555
55530
99566
52304

40054

44457
92154
39454
50999

71431
87936
30557
81586

03727
94257
64176
92328
04490

27403
25601
46270
74750

99999
33333
99997
99991

33307
99935
99859
33060
99507

99166
06666
97500
46666

29166
20000
94166
26666
92500

66666
66675
00101
67235

68836
06479
83006
03075
73811

05555
55555
49995
55523

05400

99444
03921
51394
40509

35714
15873
07159
76353

57318
18451
30964
38435
28191

05158
50652
98147
57919

25662
94302
99149
54757
41831

02579
41042
04490
27202

99166
06666
97500
46666

29166
20000
94166
26666
92500

66666
66667
00010
66723

66883
00647
68300
70307
07381

80555
55555
24999
55555

80554
99994
80539
55513
24905

55357
30158
66071
04762

54564

57147
54180
54009
98321

28819
65079
12945
01580

01781
42579
27707
51856
16555

45458
55760
33770
16021

68892
37705
98717
97343
39216

34002
16856
77652
47629

66666 668
66666 756
00001 012
66672 356

66688 368
00064 800
66830 068
67030 755
00737 112

68055 556
55555 556
12500 000
55555 552

68055 540
99999 944
68055 392
55555 139
12499 051

55553 571
55301 587
95660 714
23047 619

00545 636
44571 433
21541 681
94539 724
09982 250

14285 962
73079 365
44484 375
06730 158

46087 544
02285 686
96428 708
86412 329
04840 450

44441 689
35097 002
88616 088
07760 430

49459 812
42873 806
40799 293
87773 848
88237 649

58134 918
14104 217
08484 782
05894 536

76375 056
19692 856
24649 249
88835 463
07633 246

74654 599
39331 863
95014 129
64868 422

3

4

5

6

7

8

9

ixio-»...

2

3

4

5

6

7

8

9

1X10-^..
2

3

4

5

6

7

8

9

1X10-'...
2

3

4

5

6

7

8

9

ixio-»...

2

3

4

5

6

7

8

9

1X10-"...
2

3

4

5

6

7

8

9

1X10-*...
2

3

4

NO. 5.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND. 39

Table VII. — Values oj r^ to 63 places of decimals at decimal intervals from IXKT^" to OXlCr' — Continued.

5X10-*

6

7

8

9

1X10-^

2

3

4

5

6

7

8

9

ixio-"

2

3

4

5

6

7

8

9

IX 10-'

2

3

4

5

6

7

8

9

e"^

0. 99950
. 99940
. 99930
. 99920
. 99910

0. 99900
. 99800
.99700
.99600

0. 99501
. 99401
. 99302
. 99203
. 99104

0. 99004
. 98019
. 97044
. 96078

0. 95122
. 94176
. 93239
. 92311
. 91393

0. 90483
. 81873
. 74081
. 67032

0. 60653
. 54881
.49658
. 44932
. 40656

01249
01799
02449
03199
04048

04998
199S6
44955
79893

24791
79(540
44429
19148
03787

98337
80733
55335
94391

94245
45335
38199
63463
11852

74180
07530
82206
00460

06597
16360
53037
89641
96597

79169
64005
42843
14683
78527

33374
67333
03372
43991

92682
53935
33235
37060
72883

49168
06755
48508
52323

00714

84248
05948
86635
71228

35959

77981
81717
35639

12633
94026
91409
17221
40599

27057
39935
33609
73060
33257

99166
06675
97601
47235

31335

26474
10490
63033
66216

05357
30222
17693
20943

00909
70953
22885
78291
18674

57316

85866
86606
30074

42360

43262
51470
59143
11188

29383
20647
95800
30307
99880

80553
55301
20662
23063

25642
44987
47970
98697
45647

39059
08141
25283
92106

14253
71527
79726
07598
73535

42490
99355
68737
44329

37995
84589
48000
01023
34542

66505
94446
51716
13949
17610

57167
65077
34097
86579

46232
72245
31756
00268
74627

77180
04225
51959
91323

19779
83271
32484
49572
46499

59446
08619
79317

25147

34991

17232
93397
85015
39645

55322
130S2
84631
55747
49663

65597
95442
56091
47756

50418
22520
01454
87164
71266

03655
30886
19433
24588

65216
14970
96785
38881
52061

43662
03942

81687
82607

18045
56787
52896
56279
62598

50247
93730
76235
14313
11297

47023
67564
07417
69165

53859
14254
93S96
93433
41139

77720
62997
34867
60279

06570
60946
43600
00435
02105

11947
43585
21822
19369

34419
53323
17076
59342
78337

26223
94766
05215
50962
83926

55902
61972
74804
28160

08435
35943
51151
59144
11021

79081
12400
36815
72093

87449
88662
68377
83063
85194

05360
91256
51231
80925

18135
11956
67165
14941
03376

02486
25908
89370
82217
91894

36008
92438
89844
35315

97232
49101
39821
20431
68667

25383

46914
52894
71791

34037
54183
74845
14608
82680

98040
26901
99900
21081

48718
69062
71181
27218
17037

90844
96(103
76242
88989
32872

20590
30580
71559
78639

32954
19096
82973
49248
32984

74668
40777
36205
65716

31345
92213
73976
64993
97766

09520

56724
63482
21998

69556
80669
62620
44908
81677

35858 478

96600 982

29612 618

57251 830

95547 938

52028 511

58147 637

07658 983

52494 899

07758 993

12949 836

32899 764

41054 349

92224 925

83878 745

25203 931

32113 500

23439 634

30249 566

74047 235

05493 459

87892 996

63588 265

56257 317

78028 762

95310 067

88910 332

82892 159

80712 117

54711 497

97989 334

46288 648

40

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

mr

Table VIII . — Values ofe ^* to 23 places of decimals or significant figures at intervals of unity

from n=0 to n=360.

TUT

nir

n

eseo

e

"360

0

1.00000

00000

00000

00000

000

1.00000

00000

00000

00000

000

1

. 00870

48344

41532

05452

851

0.99131

13203

96706

33089

086

2

. 01760

64912

05851

57557

922

. 98269

81339

46661

35321

293

3

. 02652

56436

27899

20923

149

.97415

97847

14044

24673

777

4

. 03552

29709

44284

88002

636

. 96569

56224

62250

37866

736

5

1.04459

91583

45014

94512

101

0. 95730

50026

04372

64019

161

6

. 05375

48970

25672

72838

534

. 94898

72861

54113

03484

687

7

. 06299

08842

40056

40824

120

. 94074

18396

77120

78035

553

8

. 07230

78233

53278

26787

366

. 93256

80352

42753

21810

750

9

.08170

64238

95329

35157

955

.92446

52503

76255

85664

364 .

10

1.09118

74016

15113

60646

067

0. 91643

28680

11357

90742

001

11

.10075

14785

34955

62442

319

. 90847

02764

43279

70277

354

12

.11039

93830

05586

13551

147

. 90057

68692

82148

41737

686

13

. 12013

18497

61609

43998

758

. 89275

20454

06818

54556

387

14

. 12994

96199

77457

00326

997

. 88499

52089

19093

61773

205

15

1. 13985

34413

23831

47486

813

0. 87730

57690

98345

66958

381

16

. 14984

40680

24645

42979

844

. 86968

31403

56529

00825

980

17

. 15992

22609

14459

16864

183

. 86212

67421

93584

84944

480

18

.17008

87874

96421

95040

967

.85463

59991

53233

42929

363

19

. 18034

44220

00721

07072

304

. 84721

03407

79150

22453

252

20

1. 19068

99454

43543

23648

548

0. 83984

92015

71522

94334

292

21

.20112

61456

86552

72724

323

. 83255

20209

43985

97863

261

22

.21165

38174

96890

87278

267

. 82531

82431

80929

04404

425

23

. 22227

37626

07701

41621

544

. 81814

73173

95176

74154

744

24

.23298

67897

79186

37185

061

.81103

86974

86035

83770

873

25

1. 24379

37148

60197

02755

267

0. 80399

18420

97707

05373

653

26

.25469

53608

50364

78203

834

.79700

62145

78058

20215

764

27

. 26569

25579

62776

54867

601

. 79008

12829

37755

53050

019

28

.27678

61436

87199

49882

300

. 78321

65198

09750

15963

722

29

. 28797

69628

53859

95957

082

. 77641

14024

09116

53148

702

30

1. 29926

58676

97781

32296

996

0. 76966

54124

93239

80757

377

31

. 31065

37179

23685

86637

747

. 76297

80363

22349

18652

629

32

. 32214

13807

71465

42651

469

. 75634

87646

20394

13493

606

33

. 33372

97310

82225

91314

403

. 74977

70925

36260

55211

Oil

34

.34541

96513

64910

69197

373

. 74326

25196

05323

91514

228

35

1. 35721

20318

63507

91048

389

0. 73680

45497

11336

47638

859

36

. 36910

77706

24846

88483

649

. 73040

26910

48645

61087

260

37

.38110

77735

66988

71089

244

. 72405

64560

84740

43636

504

38

. 39321

29545

48216

30761

162

.71776

53615

23123

85388

162

39

.40542

42354

36629

14676

135

. 71152

89282

66507

18112

526

40

1.41774

25461

80347

96890

877

0. 70534

66813

80324

57596

608

41

. 43016

88248

78334

83212

599

. 69921

81500

56564

47140

580

42

. 44270

40178

51833

88669

716

. 69314

28675

77915

26761

562

43

. 45534

90797

16438

31638

830

. 68712

03712

82222

55056

820

44

.46810

49734

54788

93452

606

. 68115

02025

27255

13050

927

45

1.48097

26704

89909

97123

524

0. 67523

19066

55777

21703

207

46

.49395

31507

59187

63671

005

. 66936

50329

60924

07083

184

47

.50704

74027

88997

09434

472

.66354

91346

51878

49532

865

48

. 52025

64237

69983

52692

842

. 65778

37688

19845

55425

741

49

. 53358

12196

33003

02892

202

. 65206

84964

04322

92403

513

50

1. 54702

28051

25729

10808

292

0. 64640

28821

59664

31222

960

51

. 56058

22038

89930

63039

387

.64078

64946

21933

39577

206

52

. 57426

04485

39427

09338

494

. 63521

89060

76045

75468

076

53

. 58805

85807

38727

16452

016

. 62969

96925

23196

29899

492

54

. 60197

76512

82356

47335

368

. 62422

84336

48569

70835

975

55

1. 61601

87201

74880

69865

091

0. 61880

47127

89331

42525

659

56

. 63018

28567

11630

04461

975

.61342

81169

02896

76423

722

57

.64447

11395

60131

25381

130

. 60809

82365

35475

72070

156

58

. 65888

46568

42253

35813

200

. 60281

46657

90891

08375

381

59

. 67342

45062

17073

42376

336

. 59757

70022

99667

47848

532

NO. 6. J

TABLES OF EXPONENTLVL FUNCTION— VAN ORSTRAND.

41

Table VIII.— Taiues

of e 3

'^toJS

alaces of (ledma

U or sianificant fun

j,res at i

ntervals of unity

from, n

=0 to n=.J60— -Continued.

n

e

ITT

166

e

nw
360

60

1. G8809

17949

C4468

60061 685

0.59238 48471

88388

98366 542

61

. 70288

76400

G9440

84226 286

.58723 78050

49322

99127 427

62

. 71781

31683

07180

7180G 918

.58213 54839

10307

99458 797

63

. 73286

95163

28876

79556 987

. 57707 74952

04903

01162 498

64

. 74805

78307

48277

12786 401

.57206 34537

42796

37068 836

65

1. 76337

92682

29009

43812 068

0.56709 29776

80471

60448 935

66

. 77883

49955

72666

65104 439

.56216 56884

92128

21891 844

67

. 79442

61898

07G64

47944 055

.55728 12109

40855

12193 974

68

. 81015

40382

78877

83281 609

.55243 91730

50054

51732 561

69

. 82601

97387

38062

87426 137

.54763 92060

75114

08702 308

70

1. 84202

44994

35071

61169 047

0. 54288 09444

75325

30485 148

71

.85816

95392

09865

96987 248

.53816 40258

86045

74297 510

72

. 87445

60875

85338

35057 029

.53348 80910

91103

25117 573

73

. 89088

53848

G0945

74952 191

.52885 27839

95439

90737 003

74

. 90745

86822

07164

56095 507

. 52425 77515

97993

65607 591

75

1. 92417

72417

60773

26282 571

0.51970 26439

64815

56963 376

76

. 94104

23367

20970

23901 766

.51518 71142

02420

68493 176

77

. 95805

52514

46334

05834 083

.51071 08184

31370

38617 250

78

. 97521

72815

52633

59432 218

.50627 34157

60084

32185 175

79

. 99252

97340

11495

43450 340

.50187 45682

58879

86159 993

80

2. 00999

39272

49936

09324 583

0. 49751 39409

34237

11586 546

81

. 02761

11912

50766

60790 224

. 49319 12017

03287

55859 665

82

.04538

28676

53877

16465 121

.48890 60213

68524

31010 739

83

. 06331

03098

58409

46730 841

.48405 80735

92732

15419 220

84

. 08139

48831

25824

63003 504

. 48044 70348

74135

38029 032

85

2.09963

79646

83874

44306 209

0.47627 25845

21761

55808 661

86

.11804

09438

31483

92934 532

.47213 44046

31019

36838 163

87

. 13660

52220

44553

17946 512

.46803 21800

59488

63036 447

88

. 15633

22130

82686

52209 299

.46396 55984

02920

68158 185

89

. 17422

33430

96857

15796 707

.45993 43499

71447

28291 620

90

2. 19328

00507

38015

45655 977

0.45593 81277

65996

23676 592

91

.21250

37872

66649

18648 471

. 45197 66274

54911

92236 275

92

.23189

60166

63304

02318 445

. 44804 95473

50778

96776 722

93

. 25145

82157

40072

75057 055

.44415 65883

87447

29355 240

94

. 27119

18742

53061

64705 804

.44029 74540

97256

77852 204

95

2. 29109

84950

15842

62085 369

0. 43647 18505

88459

81301 115

96

.31117

95940

13899

73442 713

.43267 94865

22840

02038 712

97

. 33143

67005

20078

83382 137

.42892 00730

93525

44230 900

98

.35187

13572

11049

07485 054

.42519 33240

02994

49811 129

99

. 37248

51202

84785

21529 362

.42149 89554

41273

04335 996

100

2. 39327

95595

79079

61992 899

0.41783 66860

64320

86718 102

101

.41425

62586

91093

00367 250

.41420 62369

72605

98238 893

102

.43541

68150

97953

01718 646

.41060 73316

89865

07674 338

103

. 45676

28402

78409

85912 539

.40703 96961

42048

50783 996

104

. 47829

59598

35558

17968 202

.40350 30586

36448

23819 430

105

2. 50001

78136

20634

62130 019

0.39999 71498

41007

12101 097

106

. 52193

00558

57900

42433 634

.39652 17027

63807

96093 901

107

. 54403

43552

70618

60808 433

.39307 64.527

32740

78780 699

108

. 56633

23952

08135

32093 526

.38966 11373

75346

79490 198

109

. 58882

58737

74075

03753 213

.38627 54965

98837

40681 099

110

2. 61151

65039

55659

36560 368

0.38291 92725

70286

95517 995

111

. 63440

60137

54159

31073 041

.37959 22096

96997

45396 697

112

. 65749

61463

16490

93361 367

.37629 40546

07033

97887 229

113

. 68078

86600

67964

42149 403

.37302 45561

29929

16861 994

114

. 70428

53288

46196

68320 288

.36978 34652

77555

37864 563

115

2. 72798

79420

36197

66593 929

0.36657 05352

25163

03051 267

116

. 75189

83047

06640

68124 869

.36338 55212

92583

71303 459

117

. 77601

82377

47327

11784 814

.36022 81809

25596

60362 966

118

.80034

95780

07856

00990 098

.35709 82736

77456

79087 012

119

. 82489

41784

37509

02109 953

.35399 55611

90584

09151 846

42

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. SIV,

rlTT

Table VIII. — Valueaofe 300 to23placesofde(Arnals or significant figures at intervals of unity
from n=0 to n=360 — Continued.

TJr

rnr

n

e

JiiO

e

360

120

2. 84965

39082

26361

49747

413

0. 35091

98071

78410

96756 574

121

. 87463

06529

47630

33521

788

. 34787

07774

07388

16090 274

122

. 89982

63147

01269

50400

604

. 34484

82396

79146

67526 631

123

. 92524

28122

58824

16130

382

. 34185

19638

12814

74700 914

124

. 95088

20812

09554

38900

424

. 33888

17216

27488

45804 511

125

2. 97674

60741

07839

68042

567

0. 33593

72869

24854

65602 228

126

3. 00283

67606

21875

40323

374

. 33301

84354

71964

85837 499

127

. 02915

61276

83672

56224

289

. 33012

49449

84158

82840 442

128

. 05570

61796

40372

28530

512

. 32725

65951

08136

52293 545

129

. 08248

89384

06886

55561

611

. 32441

31674

05177

12239 654

130

3. 10950

64436

19876

81476

913

0. 32159

44453

34503

86537 070

131

. 13676

07527

93082

16277

217

. 31880

02142

36793

42076 927

132

. 16425

39414

74008

98402

237

. 31603

02613

17828

54178 719

133

. 19198

81034

01993

93191

080

.31328

43756

32292

75671 061

134

. 21996

53506

67652

30931

867

. 31056

23480

67705

86246 376

135

3. 24818

78138

73723

98777

048

0. 30786

39713

28498

99750 548

136

. 27665

76422

97329

11443

926

. 30518

90399

20228

08131 505

137

. 30537

70040

53645

96356

078

. 30253

73501

33924

41824 435

138

. 33434

80862

61023

39711

728

. 29990

87000

30581

27395 911

139

. 36357

30952

07540

50890

318

. 29730

28894

25775

24304 706

140

3. 39305

42565

19026

13629

581

0. 29471

97198

74421

23663 566

141

. 42279

38153

28551

03522

967

. 29215

89946

55659

92903 789

142

. 45279

40364

47405

62602

350

. 28962

05187

57876

51253 226

143

. 48305

72045

37576

33084

269

. 28710

409S8

63849

61938 250

144

. 51358

56242

85733

63770

467

. 28460

95433

36029

28011 557

145

3. 54438

16205

78745

14106

010

0. 28213

66622

01942

79690 337

146

. 57544

75386

80726

92511

704

. 27968

52671

39727

42063 470

147

. 60678

57444

11646

77322

765

. 27725

51714

63788

72992 081

148

. 63839

86243

27492.

80483

585

. 27484

61901

10583

61985 084

149

. 67028

85859

02021

26069

947

. 27245

81396

24526

81780 282

150

3. 70245

80577

10097

27735

985

0. 27009

08381

44019

85302 330

151

. 73490

94896

12642

61314

616

. 26774

41053

S7601

41601 409

152

. 76764

53529

43204

41037

843

. 26541

77626

40218

05300 877

153

. 80066

81406

96159

20188

329

. 26311

16327

39614

14998 621

154

. 83398

03677

16566

49446

801

. 26082

55400

62840

16975 235

155

3. 86758

45708

91686

38762

187

0. 25855

93105

12878

11462 718

156

. 90148

33093

44175

81243

803

. 25631

27715

05383

19620 141

157

. 93567

91646

26978

10358

430

. 25408

57519

55540

70247 661

158

. 97017

47409

19920

74610

667

. 25187

80822

65037

06147 574

159

4. 00497

26652

28036

26893

504

. 24968

95943

09144

10910 763

160

4. 04007

55875

81621

38818

691

0. 24752

01214

23915

57768 972

161

. 07548

01812

38049

63574

063

. 24536

94983

93494

83007 990

162

. 11120

71428

85352

84208

626

. 24323

75614

37532

87283 989

163

. 14724

11928

47586

97716

905

.24112

41481

98715

69025 103

164

. 18359

10752

91997

98882

778

. 23902

90977

30399

94932 856

165

4. 22025

95584

38003

41550

903

0. 23695

22504

84356

13423 360

166

. 25724

94347

68005

68821

813

. 23489

34482

98618

17666 319

167

. 29456

35212

40053

17615

989

. 23285

25343

85438

65690 883

168

. 33220

46595

02365

17123

644

. 23082

93533

19348

65831 400

169

. 37017

57161

09737

14851

779

. 22882

37510

25321

36583 071

170

4. 40847

95827

41842

78299

274

0. 22683

55747

67038

50727 579

171

. 44711

91764

23449

34735

479

. 22486

46731

35258

74371 974

172

. 48609

74397

46563

26129

116

. 22291

08960

36287

12320 482

173

. 52541

73410

94522

70973

329

. 22097

40946

80544

71968 532

174

. 56508

18748

68054

39580

576

. 21905

41215

71237

58671 293

175

4. 60509

40617

13311

64378

935

0.21715

08304

93124

16295 306

176

. 64545

69487

51911

21830

277

. 21526

40765

01380

27411 572

177

. 68617

36098

12986

37812

013

. 21339

37159

10560

88331 709

178

. 72724

71456

67273

83658

665

. 21153

96062

83657

74925 548

179

. 76868

06842

63252

45548

740

. 20970

16064

21252

15888 956

No. 5.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

43

Table VIII. — Value of e 3oo to 23 places of decimals or significant figures atintervah of unity
from n=o to n=SGO — Continued.

71 IT

TiTT

71

e

3liO

e

360

180

4. 81047

73809

65351

65547

304

0. 20787

95763

50761

90854 696

181

. 85264

04187

94247

68376

546

. 20607

33773

15781

714.56 866

182

. 89517

30086

69266

03886

689

. 20428

28717

65516

24171 015

183

. 93807

83896

52908

51238

967

. 20250

79233

44304

941.57 308

184

. 98135

98291

97523

46992

426

. 20074

83968

81237

92433 351

185

5. 02502

06233

94138

15608

064

0. 19900

41583

79S62

00996 409

186

. 06906

40972

23471

97348

712

. 19727

50750

07976

28001 825

187

. 11349

36048

09149

85162

220

. 19556

10150

87516

24785 622

188

. 15831

25296

73134

98890

260

. 19386

18480

84525

93994 446

189

. 20352

42849

93400

42046

668

. 19217

74445

99217

10355 391

190

5. 24913

23138

G3859

03458

976

0. 19050

76763

56114

78681 754

191

. 29514

00895

56571

83265

967

. 1SS85

24161

94288

54068 539

192

. 34155

11157

S6254

40114

022

. 18721

15380

57668

49883 602

193

. 3S836

89269

77101

73897

017

. 18558

49169

85445

59806 685

194

. 43559

70885

31951

76039

916

. 18397

24291

02555

20809 382

195

5. 48323

91971

03807

97136

379

0. 18237

39516

10243

44604 271

196

. 53129

88808

69741

99716

920

. 18078

93627

76715

45721 131

197

. 57977

9799S

07196

82047

891

. 17921

85419

27864

94992 380

198

. 62868

56459

72711

77144

131

. 17766

13694

38084

27848 651

199

. 67802

01437

83090

49620

933

. 17611

77267

21154

37438 854

200

5. 72778

70502

99033

31615

443

0. 17458

74962

21213

83197 126

201

. 77799

01555

11255

57775

134

. 17307

05614

03806

46081 903

202

. 82863

32826

29113

78243

032

. 17156

68067

47006

62309 877

203

. 87972

02883

71761

47667

295

. 17007

61177

32621

68000 005

204

. 93125

50632

61857

07528

136

. 16859

83808

37470

87729 933

205

5. 98324

15319

21845

98509

259

0. 16713

34835

24740

00589 333

206

6. 03568

36533

72839

59245

528

. 16568

13142

35411

17891 704

207

. 08858

54213

36113

87554

966

. 16424

17623

79767

07278 251

208

. 14195

08645

.37250

60212

885

. 16281

47183

28968

98514 529

209

. 19578

40470

12944

27450

523

.16140

00734

06708

06842 675

210

6.25008

90684

20498

18661

514

0. 15999

77198

80929

10309 347

211

. 30487

00643

50033

16278

404

. 15860

75509

55626

18041 887

212

. 36013

12066

39432

75439

813

. 15722

94607

62709

66992 878

213

. 41587

67036

92048

87908

302

. 15586

33443

53943

85216 098

214

. 47211

08007

97192

09721

086

. 15450

90976

92954

60275 062

215

6. 528^

77804

53430

93262

120

0. 15316

66176

47306

51918 782

216

. 58606

19626

94724

85836

308

. 15183

58019

80648

88688 251

217

. 64378

77054

19415

78406

301

.15051

65493

44929

88641 365

218

. 70201

94047

22103

09921

242

. 14920

87592

72678

44903 714

219

. 76076

14952

28427

54626

483

.14791

23321

69353

17267 840

220

6. 82001

84504

32789

41895

435

0. 14662

71693

05757

71574 243

221

. 87979

47830

39026

80471

041

. 14535

31728

10522

09113 701

222

. 94009

50453

04079

81546

502

. 14409

02456

62649

28792 320

223

7. 00092

38293

84666

97854

660

. 14283

82916

84126

65298 220

224

. 06228

57676

87000

18874

469

. 14159

72155

32601

47001 970

225

7. 12418

55332

19564

85403

134

0. 14036

69226

94120

17811 747

226

. 18662

78399

48992

10085

391

. 13914

73194

75930

67688 869

227

. 24961

74431

59050

14038

971

. 13793

83129

99347

17009 761

228

. 31315

91398

12782

13469

168

. 13673

98111

92677

00436 697

229

. 37725

77689

17818

14127

594

. 13555

17227

84208

96431 763

230

7. 44191

82118

94888

95642

314

0. 13437

39572

95262

49016 520

231

.50714

53929

49569

92130

574

. 13320

64250

33297

28843 804

232

. 57294

42794

47283

00103

068

. 13204

90370

85082

81107 998

233

. 63931

98822

91585

69482

006

. 13090

17053

09927

08276 070

234

. 70627

72563

05775

58586

006

. 12976

43423

32964

36073 612

235

7. 77382

15006

17839

59185

504

0. 12863

68615

38501

11608 165

236

. 84195

77590

48777

23202

201

. 12751

. 91770

63419

82956 263

237

. 91069

12205

04327

48324

230

. 12641

12037

90640

09980 896

238

.98002

71193

70129

05724

452

. 12531

28573

42636

56582 583

239

8. 04997

07359

10344

19219

860

. 12422

40540

75013

15019 892

44

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table VIII. — Valuesofe **to 23 places of (kcimah or ngnijkant figures at intervals of unity

from n=0 to n=360 — Continued.

TJTT

JITT

n

^360

^-360

240

8. 12052

7.3966

69776

31584

700

0. 12314

47110

70133

13364

153

241

. 19170

24748

79512

20337

923

. 12207

47461

30804

57578

317

242

. 26350

13908

66119

52166

269

. 12101

40777

74020

60131

364

243

. 33592

96124

64430

92220

258

. 11996

26252

24754

07477

529

244

. 40899

26554

33946

11833

666

. 11892

03084

09806

19143

782

245

8. 48269

60838

78883

65769

776

0. 11788

70479

51708

51579

606

246

. 55704

55106

71914

37891

883

. 11686

27651

62678

00330

151

247

. 63204

65978

81608

82193

324

. 11584

73820

38624

54497

323

248

. 70770

50572

03631

14405

896

. 11484

08212

53210

57853

367

349

. 78402

66503

95712

37936

932

. 11384

30061

51962

31368

003

250

8. 86101

71897

16436

16666

869

0. 11285

38607

46432

12303

254

251

. 93868

25383

67870

36172

912

. 11187

33097

08411

65419

737

252

9. 01702

86109

42078

24232

692

. 11090

12783

64195

22224

482

253

. 09606

13738

71543

31006

758

. 10993

76926

88893

04573

217

254

. 17578

68458

83541

99102

710

. 10898

24793

00793

89319

680

255

9. 25621

10984

58498

83788

916

0. 10803

55654

55776

71080

766

256

. 33734

02562

92359

13954

448

. 10709

68790

41770

80559

368

257

. 41918

04977

63014

15006

354

. 10616

63485

73264

16236

506

258

. 50173

80554

00814

45758

072

. 10524

39031

85859

47610

938

259

. 58501

92163

63207

32495

911

. 10432

94726

30877

48527

800

260

9. 66903

03229

13534

14816

610

0. 10342

29872

70007

19498

038

261

. 75377

77729

04024

49510

263

. 10252

43780

70002

58267

516

262

. 83926

80202

63023

50721

902

. 10163

35765

97425

38248

614

263

. 92550

75754

86489

76864

133

. 10075

05150

13433

54778

112

264

10. 01250

30061

33801

07274

906

. 09987

51260

68614

99512

953

265

10. 10026

09373

27905

84421

191

0. 09900

73430

97866

23620

359

266

. 18878

80522

59858

30543

655

. 09814

71000

15315

50760

602

267

. 27809

10926

97775

81021

711

. 09729

43313

09290

01199

605

268

. 36817

68595

00257

10415

295

.09644

89720

37326

88724

480

269

. 45905

22131

34300

61111

832

.09561

09578

21227

52368

126

270

10. 55072

40741

97762

18776

707

0.09478

02248

42154

85279

102

271

. 64319

94239

46392

13375

833

. 09395

67098

35773

23400

282

272

. 73648

53048

25491

59412

138

. 09314

03500

87430

56944

153

273

. 83058

88210

06228

84196

705

. 09233

10834

27382

27974

251

274

. 92551

71389

26656

28462

535

. 09152

88482

26056

77720

981

275

11. 02127

74878

37469

39427

194

0. 09073

35833

89362

07576

123

276

. 11787

71603

52549

12522

656

. 08994

52283

54033

18023

577

277

.21532

35130

04329

74439

212

. 08916

37230

83019

90074

479

278

. 31362

39668

04034

36878

174

. 08838

90080

60914

74082

642

279

. 41278

60078

06820

87478

044

. 08762

10242

89420

51121

503

280

11. 51281

71876

81881

21773

686

0.08685

97132

82857

32406

240

281

. 61372

51242

87537

57770

642

. 08610

50170

63708

62544

664

282

. 71551

75022

51379

12769

988

. 08535

68781

58205

92697

753

283

. 81820

20735

55483

60465

906

. 08461

52395

91951

90025

436

284

. 92178

66581

26768

25061

373

. 08388

00448

85581

50085

363

285

12.02627

91444

32515

08210

017

0. 08315

12380

50460

79142

032

286

. 13168

74900

81115

83997

199

. 08242

87635

84423

13630

715

287

. 23801

97224

28082

36923

751

.08171

25664

67542

44305

263

288

. 34528

39391

87368

57954

636

. 08100

25921

57943

12880

954

289

. 45348

83090

48050

54145

018

. 08029

87865

87646

49263

274

290

12. 56264

10722

96411

68161

042

0. 07960

10961

58453

17730

730

291

. 67275

05414

43480

45175

074

. 07890

94677

37861

40714

661

292

. 78382

51018

58068

26138

346

. 07822

38486

55020

69091

440

293

. 89587

32124

05355

88321

175

. 07754

41866

96720

68172

576

294

13. 00890

34060

91076

96265

160

. 07687

04301

03414

88845

924

295

13. 12292

42907

11347

68916

485

0. 07620

25275

65278

93586

651

296

. 23794

45495

08192

11707

660

. 07554

04282

18303

07319

705

297

. 35397

29418

30813

05730

179

. 07488

40816

40418

63376

343

298

. 47101

83038

02658

89895

883

. 07423

34378

47658

15045

833

299

. 58908

95489

94337

16123

604

. 07358

84472

90348

83479

756

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

45

Table VIII. — Valtte o/«*360 (^ 03 places of decimah or significant figwe» at intervals of unity

from n=0 to n=S60 — Continued.

n

nir

e

360

300

13. 70819

56691 02426

02113

374

0. 07294

90608

49339

12960

414

301

. 82834

57346 34235

51186

394

. 07231

52298

32258

04796

701

302

. 94954

88955 98570

53978

601

. 07168

09059

69807

01360

509

303

14. 07181

43822 02548

32482

414

. 07106

40414

12083

92024

232

304

. 19515

15055 54523

33038

599

. 07044

65887

24939

13005

305

305

14. 31956

96583 73173

21391

698

0. 06983

45008

86363

13366

939

306

. 44507

83157 02799

79841

590

. 06922

77312

82905

59665

336

307

. 57168

70356 34899

53854

180

. 06862

62337

06125

51972

697

308

. 69940

54600 36058

43239

420

. 06802

99623

49072

24242

264

309

. 82824

33152 82226

81168

582

. 06743

88718

02797

02216

549

310

14. 95821

04129 99429

92888

587

0. 06685

29170

52894

92312

725

311

15. 08931

66508 10970

75002

865

. 06627

20534

76076

75149

998

312

. 22157

20130 91181

85629

540

. 06569

62368

36770

77612

585

313

. 35498

65717 25783

85622

353

. 06512

54232

83753

97568

760

314

. 48957

04868 78908

21351

505

. 06455

95693

46812

55591

273

315

15. 62533

40077 66842

90294

364

0. 06399

86319

33431

48247

389

316

. 76228

74734 38559

81883

558

. 06344

25683

25512

77747

710

317

.90044

13135 63083

37706

315

. 06289

13361

76122

32962

040

318

16. 03980

60492 23760

27247

885

. 06234

48935

06264

97027

672

319

. 18039

22937 19490

87927

558

. 06180

31987

01687

56990

743

320

16. 32221

07533 72983

31192

026

0. 06126

62105

09709

91134

679

321

. 46527

22283 46001

69911

820

. 06073

38880

36083

09861

317

322

. 60958

76134 62300

76276

270

. 06020

61907

41875

26199

944

323

. 75516

78990 36419

33804

977

. 05968

30784

40384

32227

413

324

. 90202

41717 11546

01993

393

. 05916

45112

94077

57888

523

325

17. 05016

76153 03370

67490

843

0. 05865

04498

11557

88910

155

326

. 19960

95116 51876

11575

491

. 05814

08548

44556

20705

151

327

. 35036

12414 80504

80046

591

. 05763

56875

84950

25362

667

328

. 50243

42852 62855

98504

099

. 05713

49095

61809

09020

719

329

. 65584

02240 96979

33333

821

. 05663

84826

38463

37113

933

330

17. 81059

07405 87331

56566

908

0. 05614

63690

09601

05185

051

331

. 96669

76197 34463

31140

320

. 05565

85311

98388

33142

596

332

18. 12417

27498 32503

91954

067

. 05517

49320

53615

61039

295

333

. 28302

81233 74512

57506

306

. 05469

55347

46868

24636

339

334

. 44327

58379 65764

66793

056

. 05422

03027

69721

89207

412

335

18. 60492

80972 45042

96590

118

0. 05374

91999

30962

20223

646

336

. 76799

72118 14003

75195

206

. 05328

21903

53828

69746

206

337

. 93249

56001 74688

70203

097

. 05281

92384

73282

57537

209

338

19. 09843

57896 75253

89920

315

. 05236

03090

33298

26082

030

339

. 26583

04174 63988

00603

375

. 05190

53670

84178

48896

838

340

19. 43469

22314 51692

24830

545

0. 05145

43779

79892

71674

436

341

. 60503

40912 82495

49996

317

. 05100

73073

75438

65999

091

342

. 77686

89693 13178

40155

162

. 05056

41212

24226

75537

216

343

. 95020

99516 01081

09241

569

. 05012

47857

75487

34785

287

344

20. 12507

02389 00669

79061

689

. 04968

92675

71700

40629

501

345

20. 30146

31476 68838

11393

295

0. 04925

75334

46047

57143

187

346

. 47940

21110 79019

70049

959

. 04882

95505

19886

34218

104

347

. 65890

06800 44189

35867

752

. 04840

52862

00246

20794

325

348

. 83997

25242 48830

65263

134

. 04798

47081

77346

53620

544

349

21. 02263

14331 89948

51294

515

. 04756

77844

22136

02642

314

350

21. 20689

13172 27206

15042

175

0. 04715

44831

83853

54279

980

351

. 39276

62086 42266

24607

222

. 04674

47729

87610

14020

846

352

. 58027

02627 07417

09125

285

.04633

86226

31992

09911

568

353

. 76941

77587 63565

05900

027

. 04593

60011

86684

78696

681

354

. 96022

31013 07675

50090

701

.04553

68779

90117

16507

855

355

22. 15270

08210 89744

88342

326

0. 04514

12226

47126

76165

636

356

. 34686

55762 19387

70332

009

. 04474

90050

26644

93311

316

357

. 54273

21532 82122

45426

202

. 04436

01952

59402

23741

086

358

. 74031

54684 65441

65506

596

.04397

47637

35653

74467

767

359

. 93963

05686 94751

69532

741

.04359

26811

02924

11187

273

360

23. 14069

26327 79269

00572

909

0. 04321

39182

63772

24977

442

46 MEMOIRS NATIONAL ACADEMY OF SCIENCES. [vouxiv.

Table IX. — Values o/e^"'^ to 25 places of decimals or significant figures for various values ofn.

n

e

nir

^-n,r

7/6
13/6
19/6

39. 06361

903. 95906

20918. 23899

33631
99632
06336

89410
95003
20474

86273
11733
74990

103

87

0. 02559
.00110
. 00004

92703
62447
78051

67096

77255
71384

25596
90464
06160

73767
07938
18611

5/4

9/4

13/4

50. 75401

1174. 48316

27178. 35393

95117
53991
28751

34935
39896
52262

60233
15170
56105

883

1

0. 01970
.00085
.00003

28729
14383
67939

86617
42805
86952

11028
15803
62379

26839
58525
65643

4/3

7/3

10/3

65. 94296

1525. 96588

35311. 90760

52000
89887
51944

64414
50315
42270

66050

18599
60088

359
0

0. 01516
. 00065
. 00002

46198
53226
83190

64546
43327
59145

56995
69247
16207

25407
97556
79080

3/2

5/2
7/2

111. 31777

2575. 97049

59609. 74149

84898
65975
28721

56226
70550
55884

02684
92240
50138

10

7

0. 00898
. 00038
. 00001

32910
82032
67757

21129
03926
81524

42788
76624
22578

96650

72325
70825

5/3

8/3

11/3

187. 91462

4348. 47465

1 00626.71551

85023
93769
40705

98509

06427
19800

43960
43192
4780

74
0

0. 00532
. 00022
. 00000

15654
99656
99377

78800
95636

18774

58297
20042
69429

30579
96150
21058

7/4
11/4
15/4

244. 15106

5649. 82470

1 30740.85684

28542
14771
59666

75029
50409

27285

02837
93657
2389

17
8

0. 00409
. 00017
. 00000

58248
69966

76487

89350
41991
18419

83589
13104
96689

25536
12384
60038

11/6
17/6
23/6

317. 21714

7340. 62439

1 69867. 13281

25286
31050
35262

95191
68162
43509

88997
80074
0267

58
8

0. 00315
. 00013
. 00000

24147
62281
58869

52962
93468
54017

28940
02216
74846

00659
28376
24408

2
3
4

535. 49165

12391. 64780

2 86751.31313

55247
79166
66532

64736
97481
99746

50304
50654
6916

93

0. 00186
. 00008
. 00000

74427
06995
34873

31707
17570
42356

98881
30459
20899

44302
92392
54918

5
6

7

66 35623. 99934

1535 52935.39.544

35533 21280.84704

11342
66939
43596

33266
22626
96468

264
2

0. 00000
. 00000
. 00000

01507
00065
00002

01727
12412
81426

53900
13607
84574

64611
99007
85553

8

9

10

8 22263 15585.59499

190 27738 95292. 16129

4403 15058 60632. 02901

52749
16866
14005

6691

54
4

0. 00000
. 00000
. 00000

00000
00000
00000

12161
00525
00022

55670
54851
71101

94093
76006
06832

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

Table X.— Values of sin x and cos x to 23 places of decimals at intervals of unity from 0 to 100.

47

X

sin X

coax

0

0.00000 00000 00000 00000 000

1.00000 00000 00000 00000 000

1

+ .84147 09848 07896 50665 250

+0,54030 23058 68139 71740 094

2

+ .90929 74268 25C81 69539 602

— .41614 68365 47142 38699 757

3

+.14112 00080 59867 22210 074

- .98999 24966 00445 45727 157

4

- .75680 24953 07928 25137 264

- .65364 36208 63611 91463 917

5

-0.95892 42740 C3138 46889 315

+0.28366 21854 63226 26446 664

6

- .27941 54981 98925 87281 156

+ .96017 02866 50366 02054 565

7

+ .65698 65987 18789 09039 700

+ .75390 22543 43304 63814 120

8

+ .98935 82466 23381 77780 812

- . 14550 00338 08613 52586 884

9

+ .41211 84852 41756 56975 627

- .91113 02618 84676 98836 829

10

-0.54402 11108 89369 81340 475

-0.83907 15290 76452 45225 886

11

- .99999 02065 50703 45705 150

+ .00442 56979 88050 78574 836

12

- .53657 29180 00434 97166 537

+ .84385 39587 32492 10465 396

13

+ .42016 70368 2G640 92186 896

+ .90744 67814 5019G 21385 269

14

+ . 99060 73556 94870 30787 535

+ . 13673 72182 07833 59424 893

15

+0.65028 78401 57116 86582 974

-0.75968 79128 58821 27384 815

16

- .28790 33166 65065 29478 446

- .95765 94803 23384 64189 964

17

- .96139 74918 79556 85726 164

- .27516 33380 51596 92222 034

18

- .75098 72467 71676 10375 016

+ .66031 67082 44080 14481 610

19

+ . 14987 72096 62952 32975 424

+ . 98870 46181 86669 25289 835

20

+0.91294 52507 27627 65437 610

+0.40808 20618 13391 98606 227

21

+ .83665 56385 36056 03186 648

- .54772 92602 24268 42138 427

22

- .00885 13092 90403 87592 169

- .99996 08263 94637 12645 417

23

- .84622 04041 75170 03524 133

- .53283 30203 33397 55521 576

24

- .90557 83620 06623 84513 579

+ .42417 90073 36996 97593 705

25

-0.13235 17500 97773 02890 201

+0.99120 28118 63473 59808 329

2G

+ . 76255 84504 79602 73751 582

+ .64691 93223 28640 34272 138

27

+ .95637 59284 04503 01343 234

- .29213 88087 33836 19337 140

28

+ .27090 57883 07869 01998 634

- .96260 58663 13566 60197 545

29

- .66363 38842 12967 50215 117

- .74805 75296 89000 35176 519

30

-0.98803 16240 92861 78998 775

+ 0.15425 14498 87584 05071 866

31

- .40403 76453 23065 00604 877

+ .91474 23578 04531 27896 244

32

+ . 55142 66812 41690 55066 156

+ .83422 33605 06510 27221 553

33

+ . 99991 18601 07267 14572 808

- .01327 67472 23059 47891 522

34

+ .52908 26861 20023 82083 249

- .84857 02747 84605 18659 997

35

-0.42818 26694 96151 00440 675

-0.90369 22050 91506 75984 730

36

- .99177 88534 43115 73683 529

- . 12796 36896 27404 68102 833

37

- .64353 81333 56999 46068 567

+ .76541 40519 45343 35649 108

38

+ . 29636 85787 09385 31739 230

+ .95507 36440 47294 85758 654

39

+ .96379 53862 84087 75326 066

+ .26664 29323 59937 25152 683

40

+ 0.74511 31604 79348 78698 771

-0.66693 80616 52261 84438 409

41

- . 15862 26688 04708 98710 332

- .98733 92775 23826 45822 883

42

- .91652 15479 15633 78589 899

- .39998 53149 88351 29395 471

43

- .83177 47426 28598 28820 958

+ .55511 33015 20625 67704 483

44

+ .01770 19251 05413 57780 795

+ .99984 33086 47691 22006 901

45

+0.85090 35245 34118 42486 238

+0.52532 19888 17729 69604 746

46

+ . 90178 83476 48809 18503 329

- .43217 79448 84778 29495 278

47

+ . 12357 31227 45224 00406 153

- .99233 54691 50928 71827 975

48

- .76825 46613 23666 79904 497

- .64014 43394 69199 73131 294

49

- .95375 26527 59471 81836 042

+ .30059 25437 43637 08368 703

50

-0.26237 48537 03928 78591 439

+0.96496 60284 92113 27406 896

51

+ .67022 91758 43374 73449 435

+ .74215 41968 13782 53946 738

52

+ .98662 75920 40485 29658 757

- .16299 07807 95705 48100 333

53

+ .39592 51501 81834 18150 339

- .91828 27862 12118 89119 973

54

- .55878 90488 51616 24581 787

- .82930 98328 63150 14772 785

■^5

-0.99975 51733 58619 83659 863

+0.02212 67562 61955 73456 356

56

- .52155 10020 86911 88018 741

+ .85322 01077 22584 11396 968

57

+ .43616 47552 47824 95908 053

+ .89986 68269 69193 78650 300

58

+ .99287 26480 84537 11816 509

+ .11918 01354 48819 28543 584

59

+ .63673 80071 39137 88077 123

- .77108 02229 75845 22938 744

48

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vou XIV,

Table X. — Valties of sin x and cos x to 23 places of decimals at intervals of unity from 0 to 100 —

Continued.

X

sin X

cos X

60

-0. 30481

06211

02216

70562

565

-0. 95241

29804

15156

29269 382

61

- .96611

77700

08392

94701

829

- .25810

16359

38267

44570 121

62

- .73918

06966

49222

86727

602

+ .67350

71623

23586

25288 783

63

+ .16735

57003

02806

92152

784

+ .98589

65815

82549

69743 864

64

+ .92002

60381

96790

68335

154

+ .39185

72304

29550

00516 171

65

+0. 82682

86794

90103

46771

021

-0. 56245

38512

38172

03106 212

66

- .02655

11540

23966

79446

384

- .99964

74.559

66349

96483 045

67

- .85551

99789

75322

25899

683

- .51776

97997

89505

06565 339

68

- .89792

76806

89291

26040

073

+ .44014

30224

96040

70593 105

69

- .11478

48137

83187

22054

507

+ .99339

03797

22271

63756 155

70

4-0. 77389

06815

57889

09778

733

+0. 63331

92030

86299

83233 201

71

+ .95105

46532

54374

63665

657

- .30902

27281

66070

70291 749

72

+ .25382

33627

62036

27306

903

- .96725

05882

73882

48729 171

73

- . 67677

19568

87307

62215

498

- .73619

27182

27315

96016 815

74

- .98514

62604

68247

37085

189

+ .17171

73418

30777

55609 845

75

-0. 38778

16354

09430

43773

094

+0. 92175

12697

24749

31639 230

76

+ . 56610

76368

98180

32361

028

+ .82433

13311

07557

75991 501

77

+ .99952

01585

80731

24386

610

- .03097

50317

31216

45752 196

78

+ .51397

84559

87535

21169

609

- .85780

30932

44987

85540 835

79

- .44411

26687

07508

36850

760

- .89597

09467

90963

14833 703

80

-0. 99388

86539

23375

18973

081

-0. 11038

72438

39047

55811 787

81

- . 62988

79942

74453

87856

521

+ .77668

59820

21631

15768 342

82

+ .31322

87824

33085

15263

353

+ .94967

76978

82543

20471 326

83

+ .96836

44611

00185

40435

015

+ .24954

01179

73338

12437 735

84

+ . 73319

03200

73292

16636

321

- .68002

34955

87338

79542 720

85

-0. 17607

56199

48587

07696

212

-0. 98437

66433

94041

89491 821

86

- .92345

84470

04059

80260

163

- .38369

84449

49741

84477 893

87

- .82181

78366

30822

54487

211

+ .56975

03342

65311

92000 851

88

+ .03539

83027

33660

68362

543

+ .99937

32836

95124

65698 442

89

+ .86006

94058

12453

22683

685

+ .51017

70449

41668

89902 379

90

+0. 89399

66636

00557

89051

827

-0. 44807

36161

29170

15236 548

91

+ . 10598

75117

51156

85002

021

- .99436

74609

28201

52610 672

92

- .77946

60696

15804

68855

400

- .62644

44479

10339

06880 027

93

- .94828

21412

69947

23213

104

+ . 31742

87015

19701

64974 551

94

- .24525

19854

67654

32522

044

+ .96945

93666

69987

60380 439

95

+0. 68326

17147

36120

98369

958

+0. 73017

35609

94819

66479 352

96

+ . 98358

77454

34344

85760

773

- . 18043

04492

91083

95011 850

97

+ . 37960

77390

27521

69648

192

- .92514

75365

96413

89170 475

98

- .57338

18719

90422

88494

922

- .81928

82452

91459

25267 566

99

- .99920

68341

86353

69443

272

+ .03982

08803

93138

89816 180

100

-0. 50636

56411

09758

79365

656

+0. 86231

88722

87683

93410 194

NO.B.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

Table XI. — V^aUtes of sin x and cos x to 23 places of decimals at intervals of 0.1 from 0.0 to 10.0.

49

X

sin X

cos r

0.0

+0. 00000

00000

00000

00000

000

+1. 00000

00000

00000

00000 000

.1

. 0!)983

3416G

46828

15230

681

0. 99500

41G52

78025

76009 550

.2

. 1986()

93307

95061

21545

941

. 98006

65778

41241

63112 420

.3

. 29552

020(>fi

61339

57510

532

. 95533

04891

25006

01964 231

.4

. 38941

83423

08650

49166

631

. 92106

09940

02885

08279 853

0.5

+0. 47942

55386

04203

00027

329

+0. 87758

25618

90372

71611 628

.6

. 5G464

24733

95035

35720

095

. 82533

56149

09678

29724 095

.7

.64421

76872

37691

05367

261

. 7C484

21872

84488

42625 580

.8

. 71735

60908

99522

76162

717

. 69670

67093

47165

42092 075

.9

. 78332

69096

27483

38846

138

. 62160

99682

70664

45648 472

1.0

+0. 84147

09848

07896

50665

250

+0. 54030

23058

68139

71740 094

.1

. 89120

73600

61435

33995

180

. 45359

61214

25577

38777 137

.2

. 93203

90859

67226

34907

013

. 36235

77544

76673

57763 837

.3

. 96355

81854

17192

96470

135

. 26749

88286

24587

40699 798

.4

.98544

97299

88460

18065

947

. 16996

71429

00240

93801 675

1.5

+0. 99749

49866

04054

43094

172

+0. 07073

72016

67702

91008 819

.6

. 99957

36030

41505

16434

211

- .02919

95223

01288

72620 577

.7

. 99166

48104

52468

61534

613

. 12884

44942

95524

68408 764

.8

. 97384

76308

78195

18653

237

. 22720

20946

93087

05531 607

.9

. 94630

00876

87414

48848

971

. 32328

95668

63503

42227 883

2.0

+0. 90929

74268

25681

69539

602

-0. 41614

68365

47142

38699 757

.1

. 86320

93666

48873

77068

076

. 50484

61045

99857

45162 094

.2

. 80849

64038

19590

18430

404

. 58850

11172

55345

70852 414

.3

. 74570

52121

76720

17738

541

. 66627

60212

79824

19331 788

.4

. 67546

31805

51150

92656

577

. 73739

37155

41245

49960 882

2.5

+0. 59847

21441

03956

49405

185

-0. 80114

36155

46933

71483 350

.6

. 51550

13718

21464

23525

773

. 85688

87533

68947

23379 770

.7

. 42737

98802

33829

93455

605

. 90407

21420

17061

14798 253

.8

. 33498

81501

55904

91954

385

. 94222

23406

68658

15258 679

.9

. 23924

93292

13982

32818

426

. 97095

81651

49590

52178 111

3.0

+0. 14112

00080

59867

22210

074

-0. 98999

24966

00445

45727 157

.1

+ .04158

06624

33290

57919

470

. 99913

51502

73279

46449 238

.2

- .05837

41434

27579

90913

722

. 99829

47757

94753

08466 166

.3

. 15774

56941

43248

38201

165

. 98747

97699

08864

88393 659

.4

. 25554

11020

26831

31924

990

. 96679

81925

79401

01428 220

3.5

-0. 35078

32276

89619

84812

037

-0. 93645

66872

90790

33769 866

.6

.44252

04432

94852

38426

673

. 89675

84163

34147

00587 029

.7

. 52983

61409

08493

21321

078

. 84810

00317

10408

15883 567

.8

. 61185

78909

42719

07573

369

. 79096

77119

14416

69999 657

.9

. 68776

61591

83973

81809

089

. 72593

23042

00140

12937 233

4.0

-0. 75680

24953

07928

25137

264

-0. 65364

36208

63011

91463 917

.1

. 81827

71110

64410

50426

504

. 57482

39465

33268

91153 503

.2

. 87157

57724

13588

06001

858

. 49026

08213

40699

57765 554

.3

. 91616

59367

49454

98403

171

. 40079

91720

79975

29690 676

.4

. 95160

20738

89515

95403

539

. 30733

28699

78419

68311 914

4.5

-0. 97753

01176

65097

05538

914

-0. 21079

57994

30779

70598 048

.6

. 99369

10036

33464

45613

810

- .11215

25269

35054

51742 991

.7

. 99992

32575

64100

88417

954

- .01238

86634

62890

73715 051

.8

. 99616

46088

35840

67178

160

+ .08749

89834

39446

50932 022

.9

. 98245

26126

24332

51227

638

. 18651

23694

22575

40449 433

5.0

-0. 95892

42746

63138

46889

315

+0. 28366

21854

63226

26446 664

.1

. 92581

46823

27732

29694

615

. 37797

77427

12980

56332 058

.2

. 88345

46557

20153

26467

308

. 46851

66713

00376

95863 909

.3

. 83226

74422

23901

16356

456

. 55437

43361

79160

92944 495

.4

. 77276

44875

55987

36235

847

. 63469

28759

42634

36240 675

5.5

-0. 70554

03255

70391

90623

192

+0. 70866

97742

91260

00002 742

.6

. 63126

66378

72321

31146

367

. 77556

58785

10249

79765 581

.7

. 55068

55425

97637

76122

735

. 83471

27848

39159

68274 923

.8

. 46460

21794

13757

21141

823

. 88551

95169

41319

00416 466

.9

. 37387

66648

30236

35981

485

. 92747

84307

44035

74090 610

11539

4°— 20 1

50

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XI.-

-Values of sin x and cos x to 23 places of decimals at intervals of 0.1 from 0.0 to 10.0 —

Continued.

X

sin X

cos X

6.0

-0. 27941

54981

98925

87281

156

+0. 96017

02866

50366

02054 565

.1

- .18216

25042

72095

54002

413

. 98326

84384

42584

59658 502

.2

- .08308

94028

17496

57800

058

. 99654

20970

23217

47513 940

.3

+ . 01681

39004

84349

89031

097

. 99985

86363

83415

14228 667

.4

.11654

92048

50493

28948

042

. 99318

49187

58192

65859 474

6.5

+0. 21511

99880

87815

52429

695

+0. 97658

76257

28023

49988 631

.6

. 31154

13635

13378

17435

499

. 95023

25919

58529

46621 974

.7

.40484

99206

16598

16163

219

. 91438

31482

35319

44113 790

.8

. 49411

33511

38608

32222

208

. 86939

74903

49825

17244 162

.9

. 57843

97643

88199

87017

378

. 81572

51001

25357

07265 676

7.0

+0. 65698

65987

18789

09039

700

+0. 75390

22543

43304

63814 120

.1

. 72896

90401

25876

15207

599

.68454

66664

42806

34062 180

.2

. 79366

78638

49153

05246

445

. 60835

13145

32254

67100 485

.3

.85043

66206

28564

51751

737

. 52607

75173

81105

18891 541

.4

. 89870

80958

11626

75926

950

.43854

73275

74390

64913 410

7.5

+0. 93799

99767

74738

85794

846

+0. 34663

53178

35025

81097 162

.6

. 96791

96720

31486

42590

346

+ .25125

98425

82255

38005 815

.7

. 98816

82338

77000

36855

239

+ .15337

38620

37864

52597 738

.8

.99854

33453

74604

96343

877

+ .05395

54205

62649

57303 257

.9

.99894

13418

39772

03630

491

- .04600

21256

39536

59449 775

8.0

+0. 98935

82466

23381

77780

812

-0. 14550

00338

08613

52586 884

.1

. 96988

98108

45086

24322

432

.24354

41537

35791

46446 505

.2

. 94073

05566

79772

90115

365

. 33915

48609

83835

20740 049

.3

. 90217

18337

56293

64000

050

. 43137

68449

70620

17370 933

.4

.85459

89080

88280

66283

324

. 51928

86541

16685

29914 480

8.5

+0. 79848

71126

23490

28666

691

-0. 60201

19026

84823

61534 843

.6

. 73439

70978

74113

14371

716

. 67872

00473

20012

70086 447

.7

. 66296

92300

82182

79220

235

. 74864

66455

97399

15731 879

.8

. 58491

71928

91762

25353

093

. 81109

30140

61655

56288 909

.9

. 50102

08564

57884

98201

617

.86543

52092

41112

05963 983

9.0

+0. 41211

84852

41756

56975

627

-0. 91113

02618

84676

98836 829

.1

+ . 31909

83623

49351

77079

400

. 94772

16021

31112

02471 907

.2

+ . 22288

99141

00246

95752

807

.97484

36214

04163

74194 145

.3

+ . 12445

44235

07062

40798

941

. 99222

53254

52603

40775 691

.4

+ . 02477

54254

53358

12107

977

. 99969

30420

35206

47217 795

9.5

-0. 07515

11204

61809

30728

348

-0. 99717

21561

96378

47289 160

.6

. 17432

67812

22979

98512

410

.98468

78557

94126

91002 034

.7

. 27176

06264

10943

12433

774

. 96236

48798

31310

03407 036

.8

. 36647

91292

51927

74816

925

. 93042

62721

04753

51854 938

.9

. 45753

58937

75321

04441

382

. 88919

11526

25361

05463 444

10.0

-0. 54402

11108

89369

81340

475

-0. 83907

15290

76452

45225 886

No. 5.)

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

51

Table XII. — Values of sin x and cos x to SS placet of decimals at intervals of 0.001 from 0.000

to 1.600.

X

sin X

cog X

0.000

0.00000

00000

00000

00000

000

1.00000

00000

00000

00000

000

.001

.00099

99998

33333

34166

667

0. 99999

95000

00041

66666

528

.002

.00199

99986

06666

93333

331

. 99999

80000

00666

66657

778

.003

.00299

99955

00002

02t499

957

. 99999

55000

03374

99898

750

.004

.00399

99893

33341

86666

342

. 99999

20000

10666

66097

778

0.005

0. 00499

99791

66692

70831

783

0. 99998

75000

26041

64496

529

.006

.00599

99640

00064

79994

446

. 99998

20000

53999

93520

004

.007

. 00699

99428

33473

39150

327

. 99997

55001

00041

50326

542

.008

.00799

99146

66939

73291

723

. 99996

80001

70666

30257

819

.009

.00899

98785

00492

07405

100

. 99995

95002

73374

26188

857

0.010

0. 00999

98333

341C6

66468

254

0. 99995

00004

16665

27778

02j6

.011

. 01099

97781

68008

75446

684

. 99993

95006

10039

20617

059

.012

.01199

97120

02073

59289

053

. 99992

80008

63995

85281

066

.013

.01299

96338

36427

42921

659

. 99991

5.5011

90034

96278

551

.014

. 01399

95426

71148

51241

801

. 99990

20016

00656

20901

438

0.015

0. 01499

94375

06328

09109

944

0. 99988

75021

09359

17975

106

.016

. 01599

93173

42071

41340

585

. 99987

20027

30643

36508

430

.017

. 01699

91811

78498

72691

726

. 99985

55034

80008

14243

829

.018

. 01799

90280

15746

27852

832

. 99983

80043

73952

76107

331

.019

. 01899

88568

53967

31431

205

. 99981

95054

29976

32558

650

0.020

0. 01999

86666

93333

07936

649

0. 99980

00066

66577

77841

270

.021

. 02099

84565

34033

81764

335

. 99977

95081

03255

88132

556

.022

. 02199

82253

76279

77175

771

. 99975

80097

60509

19593

878

.023

.02299

79722

20302

18277

769

. 99973

55116

59836

06320

750

.024

. 02399

76960

66354

28999

311

. 99971

20138

23734

58193

002

0.025

0. 02499

73959

14712

33066

217

0. 99968

75162

75702

58624

967

.026

. 02599

70707

65676

53973

517

. 99966

20190

40237

62215

698

.027

. 02699

67196

19572

14955

411

. 99963

5.5221

42836

92299

214

.028

. 02799

63414

76750

38952

746

. 99960

80256

09997

38394

779

.029

. 02899

59353

37589

48577

881

. 99957

95294

69215

53557

207

0.030

0. 02999

55002

02495

66076

853

0. 99955

00337

48987

51627

216

.031

. 03099

50350

71904

13288

752

. 99951

95384

78809

04381

810

.032

. 03199

45389

46280

11602

188

. 99948

80436

89175

38584

710

.033

. 03299

40108

26119

81908

762

. 99945

55494

11581

32936

824

.034

. 03399

34497

11951

44553

435

. 99942

20556

78521

14926

773

0.035

0. 03499

28546

04336

19281

702

0. 99938

75625

23488

57581

460

.036

. 03599

22245

03869

25183

461

. 99935

20699

80976

76116

700

.037

. 03699

15584

11180

80633

489

. 99931

55780

86478

24487

902

.038

. 03799

08553

26937

03228

414

. 99927

80868

76484

91840

819

.039

. 03899

01142

51841

09720

085

. 99923

95963

88487

98862

358

0.040

0. 03998

93341

86634

15945

255

0. 99920

01066

60977

94031

457

.041

.04098

85141

32096

36751

449

. 99915

96177

33444

49770

040

.042

.04198

76530

89047

85918

946

. 99911

81296

46376

58494

043

.043

. 04298

67500

58349

76078

755

. 99907

56424

41262

28564

524

.044

.04398

58040

40905

18626

492

. 99903

21561

60588

80138

853

0.045

0. 04498

48140

37660

23632

066

0. 99898

76708

47842

40921

992

.046

. 04598

37790

49604

99745

054

. 99894

21865

47508

41817

869

.047

. 04698

26980

77774

54095

689

. 99889

57033

05071

12480

849

.048

.04798

15701

23249

92191

340

. 99884

82211

67013

76767

299

.049

.04898

03941

87159

17808

403

. 99879

97401

80818

48087

272

0.050

0. 04997

91692

70678

32879

487

0. 99875

02603

94966

24656

287

.0.51

. 05097

78943

75032

37375

800

. 99869

97818

58936

84647

237

.052

. 05197

65685

01496

29184

649

. 99864

83046

23208

81242

407

.053

. 05297

51906

51396

03981

925

. 99859

58287

39259

37585

623

.054

. 05397

37598

26109

55099

505

.99854

23542

59564

41634

531

0. 055

0. 05497

22750

27067

73387

446

0. 99848

78812

37598

40913

005

.056

. 05597

07352

55755

47070

891

. 99843

24097

27834

37163

704

.057

. 05696

91395

13712

61601

567

. 99837

59397

85743

80900

770

.058

. 05796

74868

02.534

99503

794

. 99831

84714

67796

65862

676

.059

. 05896

57761

23875

40214

896

. 99826

00048

31461

23365

235

52

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Values of sin x and cos x to 2S places of decimals at intervals of 0.001 from 0.000

to i. 600— Continued.

X

sinx

cos

X

0.060

0. 05996

40064

79444

59919

909

0. 99820

05399

35204

165.54 766

.061

. 06096

21768

71012

31380

500

. 99814

00768

38490

34561 437

.062

. 06196

02863

00408

23757

982

. 99807

861.56

01782

86552 769

.063

. 06295

83337

69523

02430

343

. 99801

61562

86542

95687 334

.064

.06395

63182

80309

28803

166

. 99795

26989

55229

92968 628

0.065

0. 06495

42388

34782

60114

361

0. 99788

82436

71301

10999 144

.066

. 06595

20944

35022

49232

601

. 99782

27904

99211

77634 635

.067

. 06694

98840

83173

44449

361

. 99775

63395

04415

09538 592

.068

.06794

76067

81445

89264

458

. 99768

88907

53362

05636 926

.069

.06894

52615

32117

22165

004

. 99762

04443

13501

40472 866

0.070

0. 06994

28473

37532

76397

655

0. 99755

10002

53279

57462 091

.071

. 07094

03632

00106

79734

071

. 99748

05586

42140

62048 084

.072

. 07193

78081

22323

54229

480

. 99740

91195

50526

14757 726

.073

. 07293

51811

06738

15974

250

. 99733

66830

49875

24157 139

.074

. 07393

24811

55977

74838

360

. 99726

32492

12624

39707 777

0.075

0. 07492

97072

72742

34208

684

0. 99718

88181

12207

44522 774

.076

. 07592

68584

59805

90718

980

. 99711

33898

23055

48023 568

.077

. 07692

39337

20017

33972

485

. 99703

69644

20596

78496 785

.078

. 07792

09320

56301

46257

015

. 99695

95419

81256

75551 417

.079

.07891

78524

71660

02252

478

. 99688

11225

82457

82476 279

0.080

0. 07991

46939

69172

68730

688

0. 99680

17063

02619

38497 771

.081

. 08091

14555

51998

04247

389

. 99672

12932

21157

70937 933

.082

. 08190

81362

23374

58826

394

. 99663

98834

18485

87272 823

.083

. 08290

47349

86621

73635

718

. 99655

74769

76013

67091 212

.084

. 08390

12508

45140

80655

638

. 99647

40739

76147

53953 598

0.085

0. 08489

76828

02416

02338

544

0. 99638

96745

02290

47151 570

.086

. 08589

40298

62015

51260

514

. 99630

42786

38841

93367 506

.087

. 08689

02910

27592

29764

492

. 99621

78864

71197

78234 626

.088

. 08788

64653

02885

29594

973

. 99613

04980

85750

17797 412

.089

. 08888

25516

91720

31524

112

. 99604

21135

69887

49872 388

0.090

0. 08987

85491

98011

04969

125

0. 99595

27330

11994

25309 284

.091

. 09087

44568

25760

07600

919

. 99586

23565

01450

99152 586

.092

. 09187

02735

79059

84943

819

. 99577

09841

28634

21703 483

.093

. 09286

59984

62093

69966

323

. 99567

86159

84916

29482 217

.094

. 09386

16304

79136

82662

751

. 99558

52521

62665

36090 844

0.095

0. 09485

71686

34557

29625

724

0. 99549

08927

55245

22976 426

.096

. 09585

26119

32817

03609

347

. 99539

55378

57015

30094 649

.097

. 09684

79593

78472

83083

006

. 99529

91875

63330

46473 881

.098

. 09784

32099

76177

31775

683

. 99520

18419

70541

00679 686

.099

. 09883

83627

30679

98210

683

. 99510

35011

75992

51179 796

0.100

0.09983

34166

46828

15230

681

0. 99500

41652

78025

76609 556

.101

. 10082

83707

29567

99512

975

. 99490

38343

75976

65937 840

.102

. 10182

32239

83945

51074

864

. 99480

25085

70176

08533 469

.103

. 10281

79754

15107

52769

040

. 99470

01879

61949

84132 117

.104

. 10381

26240

28302

69768

897

. 99459

68726

53618

52703 737

0.105

0. 10480

71688

28882

49043

655

0. 99449

25627

48497

44220 501

.106

. 10580

16088

22302

18823

209

. 99438

72583

50896

48325 268

.107

. 10679

59430

14121

88052

588

. 99428

09595

66120

03900 596

.108

. 10779

01704

10007

45835

941

. 99417

36665

00466

88538 307

.109

. 10878

42900

15731

60869

939

. 99406

53792

61230

07909 607

0.110

0. 10977

83008

37174

80866

495

0. 99395

60979

56696

85035 784

.111

. 11077

22018

80326

31964

714

.99384

58226

96148

49459 483

.112

.11176

59921

51285

18131

952

. 99373

45535

89860

26316 578

.113

. 11275

96706

56261

20553

909

. 99362

22907

49101

25308 652

.114

. 11375

32364

01575

97013

636

. 99350

90342

86134

29576 080

0.115

0.11474

66883

93663

81259

372

0. 99339

47843

14215

84471 755

.116

. 11574

00256

39072

82361

097

. 99327

95409

47595

86235 439

.117

.11673

32471

44465

84055

722

. 99316

33043

01517

70568 768

.118

. 11772

63519

16621

44080

790

.99304

60744

92218

OHIO 921

.119

.11871

93389

62434

93496

613

. 99292

78516

36926

57814 950

Na6.J

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

53

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0,000

to 7.6WJ — Oontinued.

X

sinx

cos X

0.120

0.11971

22072

88919

35996

735

0. 99280

86358

53866

25224 810

.121

. 12070

49559

03206

47206

615

. 99268

84272

62252

80653 067

.122

. 12169

75838

12547

73970

447

. 99256

72259

82294

82259 329

.123

. 12269

00900

24315

3.3626

003

. 99244

50321

35193

57029 382

.124

. 12368

24735

46003

13267

407

. 99232

18458

43142

88655 070

0.125

0. 12467

47333

85227

08995

744

0.99219

76672

29329

05314 910

.126

. 12566

68685

49729

25157

389

. 99207

24964

17930

67355 462

.127

. 12665

88780

47372

73569

978

. 99194

03335

34118

54873 474

.128

. 12765

07608

86148

72735

909

.99181

91787

04055

55198 803

.129

. 12864

25160

74174

47043

273

. 99169

10320

54896

50278 123

0.130

0. 12963

41426

19694

85954

121

0. 99156

18937

14788

03959 451

.131

.13062

56395

31083

43179

968

. 99143

17638

12868

49177 481

.132

. 13101

70058

16843

35844

433

. 99130

06424

79267

75039 751

.133

. 13260

82404

85608

43632

907

.99116

85298

45107

13813 659

.134

. 13359

93425

46144

07929

171

.99103

54260

42499

27814 325

0.135

0. 13459

03110

07348

30938

844

0. 99090

13312

04547

96193 339

.136

. 13558

11448

78252

74799

575

. 99076

62454

65348

01628 375

.137

.13657

18431

68023

00677

867

. 99063

01689

59985

16913 714

.138

. 13756

24048

85962

67852

453

. 99049

31018

24535

91451 667

.139

. 13855

28290

41508

32784

107

. 99035

50441

96067

37644 937

0.140

0.13954

31146

44236

48171

799

0. 99021

59962

12637

17189 895

.141

.14053

32607

03861

61995

092

. 99007

59580

13293

27270 829

.142

.14152

32662

30237

76542

691

. 98993

49297

38073

86655 145

.143

. 14251

31302

33359

47427

025

. 98979

29115

28007

21689 546

.144

. 14350

28517

23362

82584

791

. 98964

99035

25111

52197 214

0.145

0. 14449

24297

10526

41263

332

0. 98950

59058

72394

77275 984

.146

. 14548

18632

05272

32992

773

. 98936

09187

13854

60997 551

.147

. 14647

11512

18167

16543

800

. 98921

49421

94478

18007 704

.148

.14746

02927

59922

98870

997

. 98906

79764

60241

99027 617

.149

.14844

92868

41398

34041

627

. 98892

00216

58111

76256 193

0.150

0. 14943

81324

73599

22149

773

0. 98877

10779

36042

28673 498

.151

. 15042

68286

67680

08215

725

. 98862

11454

42977

27245 283

.152

.15141

53744

34944

81070

532

. 98847

02243

28849

20028 611

.153

. 15240

37687

86847

72225

604

. 98831

83147

44579

17178 614

.154

. 15339

20107

34994

54727

267

. 98816

54168

42076

75856 382

0.155

0. 15438

00992

91143

41996

190

0. 98801

15307

74239

85038 006

.156

. 15536

80334

67205

86651

555

. 98785

66566

94954

50224 794

.157

. 15635

58122

75247

79319

902

. 98770

07947

59094

78054 663

.158

. 15734

34347

27490

47428

529

. 98754

39451

22522

60814 736

.159

. 15833

08998

36311

53983

354

. 98738

61079

42087

60855 150

0.160

0. 15931

82066

14245

96331

146

0. 98722

72833

75626

94904 095

.161

. 16030

53540

73987

04906

020

.98706

74715

81965

18284 099

.162

.16129

23412

28387

41960

095

. 98690

66727

20914

09029 574

.163

. 16227

91670

90460

00278

226

. 98674

48869

53272

51905 638

.164

. 16326

58306

73379

01876

705

. 98658

21144

40826

22328 234

0.165

0. 16425

23309

90480

96685

825

0. 98641

83553

46347

70185 554

.166

. 16523

86670

55265

61216

228

. 98625

36098

33596

03560 791

.167

. 16622

48378

81396

97208

916

. 98608

78780

67316

72356 233

.168

.16721

08424

82704

30268

843

.98592

11602

13241

51818 712

.169

. 16819

66798

73183

08481

981

.98575

34564

38088

25966 434

0.170

0. 16918

23490

66996

01015

762

0.98558

47669

09560

70917 193

.171

. 17016

78490

78473

96702

805

. 98541

50917

96348

38117 998

.172

.17115

31789

22117

02607

812

. 98524

44312

68126

37476 -124

.173

.17213

83376

12595

42577

560

. 98507

27854

95555

20391 598

.174

. 17312

33241

64750

55773

865

. 98490

01546

50280

62691 158

0.175

0.17410

81375

93595

95189

433

0. 98472

65389

04933

47463 670

.176

. 17509

27769

14318

26146

505

. 98455

19384

33129

47797 052

.177

. 17607

72411

42278

24778

176

.98437

63534

09469

09416 699

.178

. 17706

15292

93011

76492

317

. 98419

97840

09537

33225 443

.179

. 17804

56403

82230

74417

975

.98402

22304

09903

57745 046

54

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XIL—Valius of sin x and cos x to 2S places of decimals at intervals of 0.001 from 0.000

to 1.600 — Continued.

X

sinx

cos X

o.iso

0. 17902

95734

25824

17834

180

0.98384 36027

88121 41459 272

.181

. 18001

33274

39859

10581

029

.98366 41713

22728 45058 522

.182

. 18099

69014

40581

59452

980

.98348 36661

93246 13586 083

.183

. 18198

02944

44417

72574

233

.98.330 21775

80179 58485 974

.184

. 18296

35054

67974

57756

116

. 98311 97056

65017 39552 448

0.185

0. 18394

65335

28041

20836

370

0.98293 62506

.30231 46781 122

.186

. 18492

93776

41589

64000

231

.98275 18126

59276 82121 799

.187

. 18591

20368

25775

84083

224

. 98256 63919

36591 41132 959

.188

. 18689

45100

97940

70855

554

.98237 99886

47595 94537 971

.189

.18787

67964

75611

05288

013

.98219 26029

78693 69683 022

0.190

0. 18885

88949

76500

57799

285

0.98200 42351

17270 31896 788

.191

. 18984

08046

18510

86484

571

.98181 48852

51693 65751 875

.192

. 19082

25244

19732

35325

424

.98162 45535

71313 56228 034

.193

. 19180

40533

98445

32380

691

. 98143 32402

66461 69777 178

.194

. 19278

53905

73120

87958

485

.98124 09456

28451 35290 214

0.195

0. 19376

65349

62421

92769

058

0. 98104 76695

49577 24965 723

.196

. 19474

74855

85204

16058

510

.98085 34125

23115 35080 479

.197

. 19572

82414

60517

03723

204

.98065 81746

43322 66661 867

.198

. 19670

88016

07604

76404

820

.98046 19561

05437 06062 170

.199

. 19768

91650

45907

27565

917

.98026 47571

05677 05434 796

0.200

0. 19866

93307

95061

21545

941

0.98006 65778

41241 63112 420

.201

. 19964

92978

74900

91597

545

.97986 74185

10310 03887 090

.202

. 20062

90653

05459

37903

151

.97966 72793

12041 59192 306

.203

. 20160

86321

06969

25571

640

.97946 61604

46575 47187 084

.204

. 20258

79972

99863

82615

083

.97926 40621

15030 52742 047

0. 205

0. 20356

71599

04777

97905

397

0.97906 09845

19505 07327 536

.206

. 20454

61189

42549

19110

856

.97885 69278

63076 68803 784

.207

. 20552

48734

34218

50612

330

.97865 18923

49802 01113 156

.208

.20650

34224

01031

51399

175

.97844 .58781

84716 53874 491

.209

. 20748

17648

64439

32944

665

.97823 88855

73834 41879 553

0.210

0. 20845

98998

46099

57060

871

0.97803 09147

24148 24491 614

.211

. 20943

78263

67877

33732

895

.97782 19658

43628 84946 201

.212

. 21041

55434

51846

18932

346

.97761 20391

41225 09554 014

.213

.21139

30501

20289

12409

982

.97740 11348

26863 66806 039

.214

. 21237

03453

95699

55467

398

.97718 92531

11448 86380 882

0.215

0. 21334

74283

00782

28707

677

0.97697 63942

06862 38054 344

.216

. 21432

42978

58454

49764

905

.97676 25583

25963 10511 247

.217

. 21530

09530

91846

71012

439

.97654 77456

82586 90059 555

.218

. 21627

73930

24303

77249

851

.97633 19564

91546 39246 782

.219

. 21725

36166

79385

83368

434

.97611 51909

68630 75378 736

0.220

0. 21822

96230

80869

31995

179

0.97589 74493

30605 48940 602

.221

. 21920

54112

52747

91115

124

.97567 87317

95212 21920 392

.222

. 22018

09802

19233

51671

977

.97545 90385

81168 46034 788

.223

.22115

63290

04757

25146

920

.97523 83699

08167 40857 388

.224

. 22213

14566

33970

41115

484

.97501 67259

96877 71849 392

0.225

0. 22310

63621

31745

44782

417

0.97479 41070

68943 28292 737

.226

. 22408

10445

23176

94494

428

.97457 05133

46983 01125 708

.227

. 22505

55028

33582

59230

720

.97434 59450

54590 60681 052

.228

. 22602

97360

88504

16071

214

.97412 04024

16334 34326 607

.229

. 22700

37433

13708

47642

363

.97389 38856

57756 84008 477

0.230

0. 22797

75235

35188

39540

462

0.97366 63950

05374 83696 773

.231

. 22895

10757

79163

77732

354

.97343 79306

86678 96733 940

.232

. 22992

43990

72082

45933

437

.97320 84929

30133 53085 695

.233

. 23089

74924

40621

22962

869

. 97297 80819

65176 26494 602

.234

. 23187

03549

11686

80075

884

.97274 66980

22218 11536 294

0.235

0. 23284

29855

12416

78273

112

0. 97251 43413

32643 00578 389

.236

. 23381

53832

70180

65586

809

.97228 10121

28807 60642 091

.237

. 23478

75472

12580

74343

904

.97204 67106

44041 10166 529

.238

. 23575

94763

67453

18405

752

. 97181 14371

12644 95675 843

.239

. 23673

11697

62868

90384

520

.97157 51917

69892 68349 034

No. 6.] TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

Table XU.— Values of sin x and cos x to 2-3 places of decimals at intervals of 0.001 from 0.000

to 7.600— Continued.

55

X

sin X

cos X

0.240

0. 23770

262G4

27134

58836

079

0. 97133

79748 52029

60492

618

.241

. 23867

38453

88793

6.5429

334

. 97109

97865 96272

61916

095

.242

.23964

48256

76627

22091

869

. 97086

06272 40809

96210

262

.243

. 24061

55663

196.55

08131

828

. 97062

04970 24800

96928

391

.244

. 24158

60663

47136

67335

933

. 97037

93961 88375

83670

294

0.245

0. 24255

63247

88572

05043

522

0. 97013

73249 72635

38069

313

.246

. 24352

63406

73702

85196

546

. 96989

42836 196.50

79682

233

.247

.24449

61130

32513

27365

389

. 96965

02723 72463

41782

166

.248

. 24546

56408

95231

03750

445

. 96940

52914 75084

47054

425

.249

. 24643

49232

92328

36159

337

. 96915

93411 72494

83195

397

0.250

0. 24740

39592

54522

92959

685

0. 96891

24217 10644

78414

459

.251

. 24837

27478

12778

86007

332

. 96866

45333 36453

76838

955

.252

. 24934

12879

98307

67549

922

. 96841

56762 97810

13822

250

.253

. 25030

95788

42569

27105

742

. 96816

58508 43570

911.54

897

.254

. 25127

76193

77272

88317

722

. 96791

50572 23561

52178

941

0.255

0. 25224

54086

34378

05782

506

0. 96766

32956 88575

56805

375

.256

. 25321

29456

46095

61854

486

. 96741

05664 90374

56434

780

.257

.25418

02294

44888

63424

714

. 96715

68698 81687

68781

ISO

.258

. 25514

72590

63473

38674

587

. 96690

22061 16211

52.599

126

.259

. 25611

40335

34820

33804

209

. 96664

65754 48609

82314

035

0.260

0. 25708

05518

92155

09735

339

0. 96638

99781 34513

22555

822

.261

. 25804

68131

68959

38788

820

. 96613

24144 30519

02595

835

.262

. 25901

28163

98972

01336

401

. 96.587

38845 94190

90687

131

.263

. 25997

85606

16189

82426

844

. 96561

43888 84058

68308

107

.264

. 26094

40448

54868

68386

239

. 96535

39275 59618

04309

520

0.265

0. 26190

92681

49524

43392

399

0. 96509

25008 81330

28964

923

.266

. 26287

42295

34933

86023

278

. 96483

01091 10622

07924

537

.267

. 26383

89280

46135

65779

278

. 96456

67525 09885

16072

584

.268

. 26480

33627

18431

39579

372

. 96430

24313 42476

11288

118

.269

. 26576

75325

87386

48230

942

. 96403

71458 72716

08109

368

0.270

0. 26673

14366

88831

12873

229

0. 96377

08963 65890

51301

623

.271

. 26769

50740

58861

31394

301

. 963.50

36830 88248

89328

696

.272

. 26865

84437

33839

74821

451

. 96323

55063 07004

47727

972

.273

. 26962

15447

50396

83684

915

. 96296

63662 90334

02389

084

.274

. 27058

43761

45431

64354

828

. 96269

62633 07377

52736

246

0.275

0. 27154

69369

56112

85351

302

0. 96242

51976 28237

94814

248

.276

. 27250

92262

19879

73627

557

. 96215

31695 23980

94278

169

.277

. 27347

12429

74443

10825

9S1

. 96188

01792 66634

59286

807

.278

.27443

29862

57786

29507

043

. 96160

62271 29189

13299

879

.279

. 27539

44551

08166

09350

952

. 96133

13133 85596

67778

997

0.280

0. 27635

56485

64113

73331

967

0. 96105

54383 10770

94792

459

.281

. 27731

65656

64435

83865

270

. 96077

86021 80586

99523

878

.282

. 27827

720.54

48215

38926

293

. 96050

08052 71880

92684

682

.283

. 27923

75669

54812

68142

411

. 96022

20478 62449

62830

504

.284

. 28019

76492

23866

28856

909

. 95994

23302 31050

48531

495

0.285

0. 28115

74512

95294

02165

110

0. 95966

16526 57401

10746

590

.286

. 28211

69722

09293

88922

591

. 95938

00154 22179

04351

746

.287

. 28307

62110

06345

05725

374

.95909

74188 07021

50572

193

.288

. 28403

51667

27208

80861

997

. 95881

38630 94525

08568

713

.289

. 28499

38384

12929

50237

384

. 95852

93485 68245

47227

984

0.290

0. 28595

22251

04835

53268

394

0. 95824

38755 12697

16807

013

.291

. 28691

03258

44540

28750

981

. 95795

74442 13353

20481

688

.292

. 28786

81396

73943

10698

841

. 95767

00549 56644

85799

478

.293

. 28882

56656

35230

24153

475

. 95738

17080 29961

36036

308

.294

. 28978

29027

70875

80965

551

. 95709

24037 21649

61457

636

0. 295

0. 29073

98.501

23642

75547

489

0. 95680

21423 21013

90483

768

.296

. 29169

65067

36583

80597

1.55

. 95651

09241 18315

60759

429

.297

. 29265

28716

53042

42792

582

. 95621

87494 04772

90127

632

.298

. 29360

89439

16653

78457

616

. 95592

56184 72560

47507

858

.299

. 29456

47225

71345

69198

389

.95563

15316 14809

23678

590

56

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Valws of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to 1.600 — Continued.

X

sin X

cos X

0.300

0. 29552

02066

61339

57510

532

0. 95533

64891

25606

01964

231

.301

. 29647

53952

31151

42357

025

. 95504

04912

99993

28826

414

.302

. 29743

02873

25592

74716

586

. 95474

35384

33968

84359

763

.303

. 29838

48819

89771

53102

518

.95444

56308

24485

52692

116

.304

. 29933

91782

69093

19051

897

. 95414

67687

69450

92289

242

0.305

0. 30029

31752

09261

52585

026

0. 95384

69525

67727

06164

084

.306

. 30124

68718

56279

67635

045

. 95354

61825

19130

11990

559

.307

. 30220

02672

56451

07447

613

. 95324

44589

24430

12121

945

.308

. 30315

33604

56380

39950

549

. 95294

17820

85350

63513

878

.309

. 30410

61505

02974

53093

365

. 95263

81523

04568

47552

001

0.310

0. 30505

86364

43443

50156

564

0. 95233

35698

85713

39784

281

.311

. 30601

08173

25301

45030

632

. 95202

80351

33367

79558

038

.312

. 30696

26921

96367

57464

615

. 95172

15483

53066

39561

711

.313

. 30791

42601

04707

0S284

189

. 95141

41098

51295

95271

383

.314

. 30886

55200

98932

14579

138

. 95110

57199

35494

94302

111

0.315

0. 30981

64712

27602

84860

120

0. 95079

63789

14053

25664

080

.316

. 31076

71125

39828

14184

658

. 95048

60870

96311

88923

617

.317

. 31171

74430

84966

79252

234

. 95017

48447

92562

63269

094

.318

. 31266

74619

12688

33468

402

. 94986

26523

14047

76481

749

.319

. 31361

71680

72974

01977

833

. 94954

95099

72959

73811

467

0.320

0. 31456

65606

16117

76666

176

0. 94923

54180

82440

86757

531

.321

. 31551

56385

92727

11130

659

. 94892

03769

56583

01754

395

.322

. 31646

44010

53724

15619

332

. 94860

43869

10427

28762

501

.323

. 31741

28470

50346

51938

844

. 94828

74482

59963

69764

173

.324

. 31836

09756

34148

28330

674

. 94796

95613

22130

87164

613

0.325

0. 31930

87858

57000

94315

718

0. 94765

07264

14815

72098

048

.326

. 32025

62767

71094

35507

128

. 94733

09438

56853

12639

034

.327

. 32120

34474

28937

68391

319

. 94701

02139

68025

61918

976

.328

. 32215

02968

83360

35077

048

. 94668

85370

69063

06147

877

.329

. 32309

68241

87512

98012

460

. 94636

59134

81642

32541

351

0.330

0. 32404

30283

94868

34670

020

0. 94604

23435

28386

97152

941

.331

. 32498

89085

59222

32199

224

. 94571

78275

32866

92611

768

.332

. 32593

44637

34694

82047

Oil

. 94539

23658

19598

15765

535

.333

. 32687

96929

75730

74545

756

. 94506

59587

14042

35228

939

.334

. 32782

45953

37100

93468

777

. 94473

86065

42606

58837

502

0.335

0. 32876

91698

73903

10553

241

0. 94441

03096

32643

01006

864

.336

. 32971

34156

41562

79990

386

. 94408

10683

12448

49997

577

.337

. 33065

73316

95834

32882

957

. 94375

08829

11264

35085

413

.338

. 33160

09170

92801

71669

766

. 94341

97537

59275

93637

243

.339

. 33254

41708

88879

64517

288

. 94308

76811

87612

38092

499

0.340

0. 33348

70921

40814

39678

177

0. 94275

46655

28346

22850

264

.341

. 33442

96799

05684

79816

635

. 94242

07071

14493

11062

025

.342

. 33537

19332

40903

16300

519

. 94208

58062

80011

41330

105

.343

. 33631

38512

04216

23460

104

. 94174

99633

59801

94311

834

.344

. 33725

54328

53706

12813

399

. 94141

31786

89707

59229

468

0.345

0. 33819

66772

47791

27257

928

0. 94107

54526

06513

00285

905

.346

. 33913

75834

45227

35228

880

. 94073

67854

47944

22986

218

.347

. 34007

81505

05108

24823

531

. 94039

71775

52668

40365

059

.348

. 34101

83774

86866

97891

850

. 94005

66292

60293

39119

944

.349

. 34195

82634

50276

64093

188

. 93971

51409

11367

45650

473

0.350

0. 34289

78074

55451

34918

963

0. 93937

27128

47378

92003

503

.351

. 34383

70085

62847

17681

237

. 93902

93454

10755

81724

321

.352

. 34477

58658

33263

09467

102

. 93868

50389

44865

55613

841

.353

. 34571

43783

27841

91058

778

. 93833

97937

94014

57391

869

.354

. 34665

25451

08071

20819

319

. 93799

36103

03447

99266

461

0.355

0. 34759

03652

35784

28543

852

0. 93764

64888

19349

27409

412

.356

. 34852

78377

73161

09276

237

. 93729

84296

88839

87337

915

.357

. 34946

49617

82729

17091

064

. 93694

94332

59978

89202

418

.358

. 35040

17363

27364

58840

891

. 93659

94998

81762

72980

716

.359

. 35133

81604

70292

87868

632

. 93624

86299

04124

73578

312

No. B.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

57

Table XU.— Values of sin x and cos x to 2S places of decimals at intervals of 0.001 from 0.000

to 7.600— Continued.

X

sinx

cosx

0.360

0.35227 42332

75089

97684

991

0. 93589

68236

77934

85835 091

.361

.35320 99538

05683

15610

866

. 93654

40815

54999

29438 322

.362

.35414 53211

26351

96384

60S

. 93519

04038

88060

13742 042

.363

.35508 03343

01729

15734

065

. 93483

57910

30795

02492 855

.364

.35601 49923

96801

63913

294

. 93448

02433

37816

78462 165

0.365

0.35694 92944

76911

39203

863

0. 93412

37611

64673

07984 897

.360

.35788 32396

07750

41380

647

. 93370

03448

67846

05404 739

.367

.35881 68268

55391

05142

021

. 93340

79948

04751

97425 922

.308

.35975 00552

86229

93504

354

. 93304

87113

33740

K7371 606

.369

.36008 29239

67042

91160

721

. 93208

84948

14090

19348 871

0.370

0. 36161 54319

64961

97803

729

0. 93232

73456

06034

42320 381

.371

. 36254 75783

47479

21412

373

. 93190

52640

70704

74082 737

.372

.36347 93621

82448

31502

813

. 931G0

22505

70188

65151 560

.373

. 36441 07825

38085

52343

006

. 93123

83054

67499

02553 347

.374

.36534 18384

82970

56131

067

. 93087

34291

26582

73524 125

0.375

0.36627 25290

86047

56137

291

0. 93050

76219

12314

29114 948

.376

.36720 28534

16625

99809

733

. 93014

08841

90501

47704 265

.377

.36813 28105

44381

61843

251

. 92977

32163

27881

98417 211

.378

.36906 23995

39357

37211

920

. 92940

46186

92123

64451 836

.379

.36999 16194

71964

34164

758

. 92903

50916

51824

06312 328

0.380

0.37092 04694

12982

67184

549

0. 92866

46355

76510

24949 253

.381

.37184 89484

33562

49909

881

. 92829

32508

36638

24806 858

.382

. 37277 70550

05224

88020

096

. 92792

09378

03592

76777 471

.383

. 37370 47899

99862

72083

184

. 92754

76968

49686

81063 030

.384

.37463 21506

89741

70366

479

. 92717

35283

48161

29943 792

0.385

0.37555 91367

47501

21610

089

0. 92679

84326

73184

70454 235

.386

.37648 57472

46155

27762

945

. 92642

24101

99852

66966 223

.387

. 37741 19812

59093

46681

397

. 92604

54613

04187

03679 438

.388

.37833 78378

60081

84790

240

. 92566

75863

63138

47019 143

.389

.37926 33161

23263

89706

110

. 92528

87857

54580

07941 297

0.390

0.38018 84151

23161

42823

lis

0. 92490

90598

57313

04145 068

.391

. 38111 31339

34675

51860

671

. 92452

84090

51063

22192 776

.392

.38203 74716

33087

43373

349

. 92414

68337

16481

39537 314

.393

.38290 14272

94059

55222

774

. 92376

43342

35142

86457 070

.394

.38388 49999

93636

29011

366

. 92338

09109

89547

07898 401

0.395

0.38480 81888

08245

02477

888

0. 92299

65643

63117

25225 693

.396

.38.573 09928

14697

01854

707

. 92261

12947

40199

97879 040

.397

.38665 34110

90188

34186

658

. 92222

51025

06064

84939 589

.398

.38757 54427

12300

79611

426

. 92183

79880

46904

06602 584

.399

. 38849 70867

59002

83601

303

. 92144

99517

49S32

05558 150

0.400

0.38941 83423

08650

49166

631

0. 92106

09940

02885

08279 853

.401

.39033 92084

39988

29019

595

.92067

11151

95020

86221 075

.402

.39125 96842

32150

17700

358

. 92028

03157

16118

16919 248

.403

. 39217 97687

64660

43663

363

. 91988

85959

56976

45007 979

.404

.39309 94611

17434

61324

955

. 91949

59563

09315

43137 110

0.405

0.39401 87603

70780

43071

820

0.91910

23971

65774

72800 745

.406

. 39493 76656

05398

71230

202

. 91870

79189

19913

45073 295

.407

.39.585 61759

02384

29995

816

.91831

25219

66209

81253 568

.408

.39677 42903

43226

97324

356

.91791

62067

00060

73416 956

.409

.39769 20080

09812

36782

508

.91751

89735

17781

44875 737

0.410

0.39860 93279

84422

89359

380

0. 91712

08228

16605

10547 564

.411

.39952 62493

49738

65238

251

.91672

17549

94082

37232 150

.412

. 40044 27711

88838

35528

558

. 91632

17704

51081

03796 202

.413

.40135 88925

85200

23958

010

.91.592

08695

85785

61266 649

.414

.40227 46126

22702

98524

766

. 91551

90527

99696

92832 194

0.415

0.40318 99303

85626

63109

550

0.91511

63204

94631

73753 232

.416

.40410 48449

58653

49047

645

.91471

26730

73322

31180 180

.417

. 40501 93554

26869

06660

654

. 91430

81109

39410

03880 251

.418

.40593 34608

75762

96747

939

. 91390

26344

97475

01872 722

.419

.40684 71603

91229

82037

655

.91349

62441

52975

65972 725

58

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV.

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to /. 600— Continued.

X

SI71 X

cos X

0.420

0.40776

04530

59570

18597

279

0. 91308

89403

12308

27243 609

.421

.40867

33379

67491

47203

546

. 91268

07233

82776

66357 915

.422

.40958

58142

02108

84671

703

.91227

15937

72597

72866 996

.423

.41049

78808

50946

15143

980

.91186

15.518

90901

04379 332

.424

.41140

95370

01936

81337

201

. 91145

05981

47728

45647 576

0.425

0.41232

07817

43424

75749

435

0. 91103

87329

54033

67564 373

.426

.41323

16141

64165

31825

593

. 91062

59567

21681

86066 990

.427

.41414

20333

53326

15081

889

.91021

22698

63449

20950 808

.428

. 41505

20384

00488

14189

067

. 90979

76727

93022

54591 701

.429

.41596

16283

95646

32014

301

. 90938

21659

24998

90577 360

0.430

0.41687

08024

29210

76621

692

0. 90896

57496

74885

12247 591

.431

.41777

95595

92007

52231

243

. 90854

84244

59097

41143 638

.432

. 41868

78989

75279

50136

257

. 90813

01906

94960

95366 563

.433

.41959

58196

70687

39579

028

. 90771

10488

00709

47844 729

.434

.42050

33207

70310

58584

774

. 90729

09991

95484

84510 435

0.435

0. 42141

04013

66648

04753

684

0.90687

00422

99336

62385 731

.436

. 42231

70605

52G19

26011

018

. 90644

81785

33221

67577 465

.437

. 42322

32974

21565

11315

146

. 90602

54083

19003

73181 601

.438

.42412

91110

67248

81323

456

. 90560

17320

79452

97096 848

.439

.42503

45005

83856

79016

027

. 90517

71502

38245

59747 647

0.440

0.42593

94650

65999

60276

972

0. 90475

16632

19963

41716 554

.441

.42684

40036

08712

84433

381

. 90432

52714

50093

41286 061

.442

. 42774

81153

07458

04751

750

. 90389

79753

55027

31889 904

.433

.42865

17992

58123

58891

823

. 90346

97753

62061

19473 892

.444

.42955

50545

57025

59317

745

. 90304

06718

99394

99766 305

0.445

0. 43045

78803

00908

83666

443

0. 90261

06653

96132

15457 899

.446

.43136

02755

86947

65073

141

. 90217

97562

82279

13291 573

.447

.43226

22395

12746

82453

917

. 90174

79449

88745

01061 718

.448

. 43316

37711

76342

50745

219

. 90131

52319

47341

04523 319

.449

.43406

48696

76203

11100

244

. 90088

16175

90780

24210 832

0.450

0. 43496

55341

11230

21042

084

0. 90044

71023

52676

92166 884

.451

. 43586

57635

80759

44573

567

. 90001

16866

67546

28580 847

.452

. 43676

55571

84561

42243

681

. 89957

53709

70803

98337 319

.453

. 43766

49140

22842

61170

507

. 89913

815.56

98765

67474 569

.454

.43856

38331

96246

25020

568

. 89870

00412

88646

59552 965

0.455

0. 43946

23138

05853

23944

492

0. 89826

10281

78561

11933 463

.456

.44036

03549

53183

04468

918

. 89782

11168

07522

31966 167

.457

. 44125

79557

40194

59344

542

. 89738

03076

15441

53089 030

.458

.44215

51152

69287

17350

215

. 89693

86010

43127

90836 721

.459

.44305

18326

43301

33053

008

. 89649

59975

32287

98759 714

0.460

0. 44394

81069

65519

76524

151

0. 89605

24975

25525

24253 639

.461

. 44484

39373

39668

23010

752

. 89560

81014

66339

64298 937

.462

.44573

93228

69916

42563

218

. 89516

28097

99127

21110 867

.463

.44663

42626

60878

89618

275

. 89471

66229

69179

57699 918

.464

. 44752

87558

17615

92537

506

. 89426

95414

22683

53342 602

0.465

0. 44842

28014

45634

43101

319

0. 89382

15656

06720

58962 873

.466

.44931

63986

50888

85958

244

. 89337

26959

69266

52423 883

.467

.45020

95465

39782

08029

479

. 89292

29329

59190

93730 459

.468

.45110

22442

19166

27868

603

.89247

22770

26256

80142 134

.469

. 45199

44907

96343

84976

342

. 89202

07286

21120

01196 857

0.470

0.45288

62853

79068

29070

327

0. 89156

82881

95328

93645 402

.471

.45377

76270

75545

09309

736

. 89111

49562

01323

96296 541

.472

.45466

85149

94432

63474

735

. 89066

07330

92437

04773 005

.473

.45555

89482

44843

07100

635

. 89020

56193

22891

26178 292

.474

.45644

89259

36343

22566

671

. 88974

96153

47800

33674 367

0.475

0.45733

84471

78955

48139

307

0. 88929

27216

23168

20970 288

.476

.45822

75110

83158

66969

994

.88883

49386

05888

56721 822

.477

.45911

61167

59888

96047

279

. 88837

62667

53744

38842 074

.478

. 46000

42633

20540

75103

180

. 88791

67065

25407

48723 197

.479

.46089

19498

76967

56473

739

. 88745

62583

80438

05369 212

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTKAND.

59

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals oj 0.001 from 0.000

to 1.600— Continued.

X

sin X

cos X

0.480

0.46177

91755

41482

88913

664

0. 88699

49227

79284

19439 995

.481

.46266

59394

26861

16364

968

. 88653

27001

83281

47206 469

.482

. 46355

22406

46338

56679

522

. 88606

95910

54652

44417 0.51

.483

. 46443

80783

13613

9.5295

430

. 88560

55958

56506

20075 401

.484

.46532

34515

42849

72867

132

. 88514

07150

52837

90129 517

0.485

0. 46620

83594

48672

73849

162

0.88467

49491

08528

31072 223

.486

. 46709

28011

46175

15033

451

. 88420

82984

89343

33453 094

.487

.46797

67757

50915

34040

104

. 88374

07636

61933

55301 874

.488

. 46886

02823

78918

77761

658

. 88327

23450

93833

75463 416

.489

. 46974

33201

46678

90760

024

. 88280

30432

53462

46844 214

0.490

0.47062

58881

71158

03618

136

0. 88233

28586

10121

49570 547

.491

. 47150

79855

69788

21242

715

. 88186

17916

33995

44058 307

.492

. 47238

96114

60472

11121

556

. 88138

98427

96151

23994 541

.493

. 47327

07649

61583

91533

149

. 88091

70125

68537

69230 763

.494

.47415

14451

91970

19709

261

.88044

33014

23984

98588 075

0.495

0. 47503

16512

70950

79950

264

0. 87996

8709S

36204

22574 157

.496

.47591

13823

18319

71693

150

. 87949

32382

79786

96012 154

.497«-

.47679

06374

54345

97532

118

. 87901

68872

30204

70581 529

.498

. 47766

94157

99774

51191

668

. 87853

96571

63808

47270 917

.499

. 47854

77164

75827

05452

099

. 87806

15485

57828

28743 023

0.500

0. 47942

55386

04203

00027

329

0. 87758

25618

90372

71611 628

.501

. 48030

28813

07080

29394

947

. 87710

26976

40428

38630 733

.502

. 48117

97437

07116

30578

414

. 87662

19562

87859

50795 903

..503

. 48205

61249

27448

70881

314

. 87614

03383

13407

39357 847

.504

. 48293

20240

91696

35573

583

. 87565

78441

98689

97748 295

0. 505

0. 48380

74403

23960

15529

617

0. 87517

44744

26201

33418 203

.506

. 48468

23727

48823

94818

170

. 87469

02294

79311

19588 355

.507

. 48555

68204

91355

38243

967

. 87420

51098

42264

46912 391

.508

.48643

07826

77106

78840

928

. 87371

91160

00180

7.5052 318

.509

. 48730

42584

32116

05316

931

. 87323

22484

39053

84166 561

0.510

0. 48817

72468

82907

49450

013

0. 87274

45076

45751

26310 581

.511

.48904

97471

56492

73435

934

. 87225

58941

08013

76750 129

.512

. 48992

17583

80371

57187

006

. 87176

64083

14454

85187 176

.513

. 49079

32796

82532

85582

104

. 87127

60507

54560

26898 565

.514

. 49166

43101

91455

35667

778

. 87078

48219

18687

53787 441

0.515

0. 49253

48490

36108

63810

364

0. 87029

27222

98065

45347 504

.516

. 49340

48953

45953

92799

025

. 86979

97523

84793

59540 132

.517

. 49427

44482

50944

98899

617

. 86930

59126

71841

83584 429

.518

. 49514

35068

81528

98859

309

. 86881

12036

53049

84660 240

.519

. 49601

20703

68647

36861

855

. 86831

56258

23126

60524 189

0.520

0. 49688

01378

43736

71433

446

0. 86781

91796

77649

90038 785

.521

. 49774

77084

38729

62299

043

. 86732

18657

13065

83614 647

.522

. 49861

47812

86055

57189

109

. 86682

36844

26688

33565 898

.523

. 49948

13555

18641

78.596

658

. 86632

46363

16698

64378 779

.524

.50034

74302

69914

10484

518

. 86582

47218

82144

82893 524

0.525

0. 50121

30046

73797

84942

748

0. 86532

39416

22941

28399 561

.526

. 50207

80778

64718

68796

092

. 86482

22960

39868

22644 077

.527

. 50294

26489

77603

50161

411

. 86431

97856

34571

19753 996

.528

. 50380

67171

47881

24954

981

. 86381

64109

09560

56071 436

.529

.50467

02815

11483

83349

596

. 86331

21723

68210

99902 671

0.530

0.50553

33412

04846

96181

366

0.86280

70705

14761

01180 670

.531

. 50639

58953

64911

01306

143

. 86230

11058

54312

41041 248

.532

. 50725

79431

29121

89905

473

. 86179

42788

92829

81312 894

.533

.50811

94836

35431

92741

999

. 86128

65901

37140

13920 311

.534

. 50898

05160

22300

66364

220

. 86077

80400

94932

10201 726

0.535

0. 50984

10394

28695

79260

534

0. 86026

86292

74755

70140 025

.536

. 51070

10529

94093

97962

456

. 85975

83581

86021

71507 760

.537

. 51156

05558

58481

73096

946

.85924

72273

39001

18926 068

.538

. 51241

95471

62356

25387

754

. 85873

52372

44824

92837 581

.539

. 51327

80260

46726

31605

686

. 85822

23884

15482

98393 339

60

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

tVoL. XIV,

Table XU.— Values of sin x cfnd cos x to 23 places of decimals at intervals of 0.001 from 0.000

to i. 600— Continued.

X

sin X

cos X

0.540

0. 51413

59916

53113

10467

728

0. 85770

86813

63824

14253 797

.541

.51499

34431

23551

08484

914

. 85719

41166

03555

41303 947

.542

. 51585

03796

00588

85758

874

. 85667

86946

49241

51282 623

.543

. 51670

68002

27290

01726

969

. 85616

24160

16304

35326 032

.544

. 51756

27041

47234

00855

920

.85564

52812

21022

52425 567

0.545

0. 51841

80905

04516

98283

861

0. 85512

72907

80530

77799 957

.546

. 51927

29584

43752

65410

714

. 85460

84452

12819

51181 787

.547

. 52012

73071

10073

15436

812

.85408

87450

36734

25018 472

.548

. 52098

11356

49129

88849

675

. 85356

81907

71975

12587 703

.549

. 52183

44432

07094

38858

868

. 85304

67829

39096

36027 442

0. 550

0. 52268

72289

30659

16778

838

0. 85252

45220

59505

74280 498

.551

. 52353

94919

67038

57359

653

. 85200

14086

55464

10953 761

.552

. 52439

12314

63969

64065

565

. 85147

74432

50084

82092 114

.553

. 52524

24465

69712

94301

297

. 85095

26263

67333

23867 110

.554

. 52609

31364

33053

44585

976

.85042

69585

32026

20180 431

0.555

0. 52694

33002

03301

35674

635

0. 84990

04402

69831

50182 218

.556

. 52779

29370

30292

97627

180

. 84937

30721

07267

35704 287

.557

. 52864

20460

64391

54824

757

. 84884

48545

71701

886(38 318

.558

. 52949

06264

56488

10933

415

. 84831

57881

91352

58049 047

.559

. 53033

86773

58002

33815

002

. 84778

58734

95285

77652 517

0.560

0. 53118

61979

20883

40385

187

0. 84725

51110

13416

12609 452

.561

. 53203

31872

97610

81418

533

. 84672

35012

76506

06683 799

.562

. 53287

96446

41195

26300

543

. 84619

10448

16165

29136 481

.563

. 53372

55691

05179

47726

585

. 84565

77421

64850

21564 438

.564

. 53457

09598

43639

06347

607

. 84512

35938

55863

44654 991

0. 565

0. 53541

58160

11183

35362

572

0. 84458

86004

23353

24855 579

.566

. 53626

01367

62956

25057

521

. 84405

27624

02313

00958 945

.567

. 53710

39212

54637

07291

168

. 84351

60803

28580

70603 796

.568

. 53794

71686

42441

39926

969

. 84297

85547

38838

36691 Oil

.569

. 53878

98780

83121

91211

553

. 84244

01861

70611

53715 445

0.570

0. 53963

20487

33969

24099

446

0. 84190

09751

62268

74013 376

.571

. 54047

36797

52812

80524

005

. 84136

09222

53020

93925 658

.572

. 54131

47702

98021

65614

465

. 84082

00279

82920

99876 632

.573

. 54215

53195

28505

31859

028

. 84027

82928

92863

14368 839

.574

. 54299

53266

03714

63213

905

. 83973

57175

24582

41893 605

0.575

0. 54383

47906

83642

59158

222

0. 83919

23024

20654

14757 543

.576

. 54467

37109

28825

18694

718

. 83864

80481

24493

38825 019

.577

. 54551

20865

00342

24296

136

. 83810

29551

80354

39176 658

.578

. 54634

99165

59818

25797

231

. 83755

70241

33330

05683 918

.579

. 54718

72002

69423

24232

321

. 83701

02555

29351

38499 807

0.580

0. 54802

39367

91873

55618

270

0. 83646

26499

15186

93465 789

.581

. 54886

01252

90432

74682

851

. 83591

42078

38442

27434 927

.582

. 54969

57649

28912

38538

382

. 83536

49298

47559

43511 337

.583

. 55053

08548

71672

90300

563

. 83481

48164

91816

36205 988

.584

. 55136

53942

83624

42652

424

. 83426

38683

21326

36508 907

0.585

0. 55219

93823

30227

61353

309

0. 83371

20858

87037

56877 861

.586

. 55303

28181

77494

48692

799

. 83315

94697

40732

36143 543

.587

. 55386

57009

91989

26889

504

. 83260

60204

35026

84331 337

.588

. 55469

80299

40S29

21434

637

. 83205

17385

23370

27399 720

.589

. 55552

98041

91685

44380

278

. 83149

66245

60044

51895 332

0.590

0. 55636

10229

12783

77572

254

0. 83094

06791

00163

49524 800

.591

.55719

16852

72905

55827

556

. 83038

39026

99672

61643 346

.592

. 55802

17904

41388

50056

192

. 82982

62959

15348

23660 255

.593

. 55885

13375

88127

50327

409

. 82926

78593

04797

09361 243

.594

. 55968

03258

83575

48880

201

. 82870

85934

26455

75147 786

0.595

0. 56050

87544

98744

23078

004

0. 82814

84988

39590

04193 468

.596

. 56133

66226

05205

18307

516

. 82758

75761

04294

50517 407

.597

. 56216

39293

75090

30821

541

.82702

58257

81491

82974 799

.598

. 56299

06739

81092

90525

792

. 82646

32484

32932

29164 660

.599

. 56381

68555

96468

43709

545

. 82589

98446

21193

19254 799

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

m

Table XII. — Values of tin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to J. 600— Continued.

X

sin X

cosx

0.600

0. 56464

24733

95035

35720

095

0. 82533

56149

09678

29724 095

.601

. 56546

75265

51175

93580

897

. 82477

05598

62617

27022 123

.602

. 5G629

20142

39837

08553

336

. 82420

46800

45065

11146 193

.603

. 56711

59356

36531

18642

028

. 82363

79760

22901

59135 858

.604

. 56793

92899

17336

91043

574

. 82307

04483

62830

68484 934

0.605

0. 56876

20762

58900

04538

687

0. 82250

20976

32380

00471 116

.606

. 56958

42938

38434

31827

607

. 82193

29243

99900

23403 216

.607

. 57040

59418

33722

21808

719

. 82136

29292

34564

55786 102

.608

. 57122

70194

23115

81800

299

. 82079

21127

06368

09403 380

.609

. 57204

75257

85537

59705

300

. 82022

04753

86127

32317 893

0.610

0. 57286

74601

00481

26119

098

0. 81964

80178

45479

51790 075

.611

. 57368

68215

48012

56380

111

. 81907

47406

56882

17114 225

.612

. 57450

56093

08770

12563

221

. 81850

06443

93612

42372 770

.613

. 57532

38225

63966

25415

904

. 81792

57296

29766

49108 549

.614

. 57614

14604

95387

76236

989

. 81734

99969

40259

08915 198

0.615

0. 57695

85222

85396

78697

975

0. 81677

34469

00822

85945 685

.616

. 57777

50071

16931

60606

809

. 81619

60800

88007

79339 051

.617

. 57859

09141

73507

45614

047

. 81561

78970

79180

65565 411

.618

. 57940

62426

39217

34861

330

. 81503

88984

52524

40689 288

.619

.58022

09916

98732

88572

073

. 81445

90847

87037

62551 318

0.620

0. 58103

51605

37305

07584

296

0. 81387

84566

62533

92868 400

.621

. 5S184

87483

40765

14825

522

. 81329

70146

59641

39252 335

.622

. 58266

17542

95525

36729

641

.81271

47593

59801

97147 027

.623

. 58347

41775

88579

84595

681

. 81213

16913

45270

91684 290

.624

. 58428

60174

07505

35888

387

.81154

78111

99116

19458 331

0.625

0. 58509

72729

40462

15480

540

0. 81096

31195

05217

90218 953

.626

. 58590

79433

76194

76836

923

. 81037

76168

48267

68483 556

.627

. 58671

80279

04032

83139

861

. 80979

13038

13768

15067 973

.628

. 58752

75257

13891

88356

252

. 80920

41809

88032

28,536 214

.629

. 58833

64359

96274

18246

006

. 80861

62489

58182

86569 178

0.630

0. 58914

47579

42269

51311

811

0. 80802

75083

12151

87252 371

.631

. 58995

24907

43555

99690

151

. 80743

79596

38679

902S2 722

.632

. 59075

96335

92400

89983

484

.80684

76035

27315

58094 522

.633

. 59156

61856

81661

44033

509

. 80625

64405

68414

96904 569

.634

. 59237

21462

04785

59635

440

. 80566

44713

53140

97676 566

0.635

0. 59317

75143

55812

91193

198

0. 80507

16964

73462

77004 837

.636

. 59398

22893

29375

30315

454

.80447

81165

22155

17917 411

.637

. 59478

64703

20697

86352

425

. 80388

37320

92798

10598 548

.638

. 59559

00565

25599

66873

364

. 80328

85437

79775

93030 752

.639

. 59639

30471

40494

58084

641

. 80269

25521

78276

91556 338

0.640

0. 59719

54413

62392

05188

355

0. 80209

57578

84292

61358 611

.641

. 59799

72383

88897

92681

375

. 80149

81614

94617

26862 715

.642

. 59879

84374

18215

24594

757

. 80089

97636

06847

22056 216

.643

. 59959

90376

49145

04673

426

. 80030

05648

19380

30729 469

.644

. 60039

90382

81087

16496

070

. 79970

05657

31415

26635 842

0.645

0. 60119

84385

14041

03535

151

0. 79909

97669

42951

13571 848

.646

. 60199

72375

48606

49156

949

. 79849

81690

54786

65377 243

.647

. 60279

54345

85984

56561

576

. 79789

57726

68519

65855 159

.648

. 60359

30288

27978

28662

868

. 79729

25783

86546

48612 327

.649

. 60439

00194

76993

47908

070

. 79668

85868

12061

36819 444

0.650

0. 60518

64057

36039

56037

252

0. 79608

37985

49055

82891 760

.651

. 60598

21868

08730

33782

358

.79547

82142

02318

08089 927

.652

. 60677

73618

99284

80505

818

. 79487

18343

77432

42041 183

.653

. 60757

19302

12527

93778

646

. 79426

46596

80778

62180 929

.654

. 60836

58909

53891

48897

929

. 79365

66907

19531

33114 757

0.655

0. 60915

92433

29414

78343

652

0. 79304

79281

01659

45900 987

.656

. 60995

19865

45745

51174

755

.79243

83724

35925

57253 785

.657

. 61074

41198

10140

52364

359

. 79182

80243

31885

28666 909

.658

. 61153

56423

30466

62074

073

. 79121

68843

99886

65458 154

.659

. 61232

65533

15201

34867

307

. 79060

49532

51069

55734 550

62

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to 1.600 — Continued.

X

sin X

cos X

0.660

0. 61311

68519

73433

78861

515

0. 78999

22314

97365

09278 382

.661

. 61390

65375

14865

34819

272

. 78937

87197

51494

96354 080

.662

. 61469

56091

49810

55178

137

. 78876

44186

26970

86436 061

.663

.61548

40660

89197

83019

186

. 78814

93287

38093

86857 558

.664

. 61627

19075

44570

30974

165

. 78753

34506

99953

81380 523

0.665

0. 61705

91327

28086

60071

171

0. 78691

67851

28428

68686 643

.666

.61784

57408

52521

58518

785

. 78629

93326

40184

00789 551

.667

. 61863

17311

31267

20428

576

. 78568

10938

52672

21368 279

.668

. 61941

71027

78333

24475

901

. 78506

20693

84132

04022 017

.669

. 62020

18550

08348

12498

919

.78444

22598

53587

90446 244

0.670

0. 62098

59870

36559

68035

744

0. 78382

16658

80849

28530 294

.671

. 62176

94980

78835

94799

654

. 78320

02880

86510

10376 414

.672

. 62255

23873

51665

95092

281

. 78257

81270

91948

10240 374

.673

. 62333

46540

72160

48154

700

. 78195

51835

19324

22393 698

.674

. 62411

62974

58052

88456

349

. 78133

14579

91581

98907 578

0.675

0. 62489

73167

27699

83921

682

0. 78070

69511

32446

87358 526

.676

. 62567

77111

00082

14094

496

. 78008

16635

66425

68455 830

.677

. 62645

74797

94805

48239

849

. 77945

55959

18805

93590 877

.678

. 62723

66220

32101

23383

477

. 77882

87488

15655

22308 414

.679

. 62801

51370

32827

22288

658

. 77820

11228

83820

59699 786

0.680

0. 62879

30240

18468

51370

418

0. 77757

27187

50927

93718 239

.681

. 62957

02822

11138

18547

018

. 77694

35370

45381

32416 339

.682

. 63034

69108

33578

11028

644

. 77631

35783

96362

41105 566

.683

. 63112

29091

09159

73043

207

. 77568

28434

33829

79438 156

.684

. 63189

82762

61884

83499

207

. 77505

13327

88518

38411 247

0.685

0. 63267

30115

16386

33585

507

0. 77441

90470

91938

77293 390

.686

. 63344

71140

97929

04308

094

. 77378

59869

76376

60473 500

.687

. 63422

05832

32410

43963

552

. 77315

21530

74891

94232 293

.688

. 63499

34181

46361

45549

316

. 77251

75460

21318

63436 286

.689

. 63576

56180

66947

24110

576

. 77188

21664

50263

68154 418

0.690

0. 63653

71822

21967

94023

743

0. 77124

60149

97106

60197 354

.691

. 63730

81098

39859

46216

467

. 77060

90922

97998

79579 541

.692

. 63807

84001

49694

25323

984

. 76997

13989

89862

90904 069

.693

. 63884

80523

81182

06781

899

. 76933

29357

10392

19670 418

.694

. 63961

70657

64670

73855

200

. 76869

37030

98049

88505 132

0.695

0. 64038

54395

31146

94603

464

0. 76805

37017

92068

53315 502

.696

. 64115

31729

12236

98782

185

. 76741

29324

32449

39366 321

.697

.64192

02651

40207

54680

136

. 76677

13956

59961

77279 757

.698

. 64268

67154

47966

45892

698

. 76612

90921

16142

38958 434

.699

. 64345

25230

69063

48031

063

. 76548

60224

43294

73431 759

0.700

0.64421

76872

37691

05367

261

0.76484

21872

84488

42625 586

.701

.64498

22071

88685

07414

902

.76419

75872

83558

57055 252

.702

. 64574

60821

57525

65445

583

. 76355

22230

85105

11442 075

.703

. 64650

93113

80337

88940

870

. 76290

60953

34492

20253 368

.704

.64727

18940

93892

61979

783

. 76225

92046

77847

53166 023

0.705

0. 64803

38295

35607

19561

705

0. 76161

15517

62061

70453 752

.706

.64879

51169

43546

23864

641

. 76096

31372

34787

58298 030

.707

.64955

57555

56422

40438

747

. 76031

39617

44439

64022 815

.708

. 65031

57446

13597

14335

062

. 75966

40259

40193

31253 107

.709

. 65107

50833

55081

46169

354

. 75901

33304

71984

34997 406

0.710

0. 65183

37710

21536

68121

013

0. 75836

18759

90508

16654 146

.711

. 65259

18068

54275

19866

915

. 75770

96631

47219

18942 159

.712

. 65334

91900

95261

24450

173

. 75705

66925

94330

20755 235

.713

.65410

59199

87111

64083

709

. 75640

29649

84811

71940 852

.714

.65486

19957

73096

55888

565

. 75574

84809

72391

28003 128

0.715

0. 65561

74166

97140

27566

883

0. 75509

32412

11552

84730 074

.716

. 65637

21820

03821

93009

463

.75443

72463

57536

12745 203

.717

. 65712

62909

38376

27837

851

. 75378

04970

66335

91983 563

.718

. 65787

97427

46694

44880

853

. 75312

29939

94701

46092 263

.719

. 65863

25366

75324

69585

417

. 75246

47378

00135

76755 558

No. 5.]

TABLES OF EXPONENTIAL FLTNCTION— VAN ORSTRAND.

63

Table XII. — Values n/sin x and cos x to 2S places of dedmah at intervals of 0.001 from 0.000

to .?. 600— Continued.

X

sin X

cos X

0.720

0. 65938

46719

71473

153C1

800

0. 75180

57291

40894

97944 549

.721

. G6013

61478

83004

58862

952

. 75114

59686

75987

70091 576

.722

. 66088

69636

58443

15198

027

.75048

54570

65174

34189 363

.723

. 66163

71185

46973

13079

967

. 74982

41949

68966

45814 983

.724

. 66238

66117

98439

69907

065

. 74916

21830

48626

09078 707

0.725

0. 66313

54426

63349

66778

441

0. 74849

94219

66165

10497 806

.726

. 66388

36103

92872

23443

354

. 74783

59123

84344

52795 369

.727

.66463

11142

38839

73184

280

.74717

16549

66673

88624 209

.728

. 66537

79534

53748

37633

666

. 74650

66503

77410

54215 910

.729

. 66612

41272

90759

01524

309

.74584

08992

81559

02955 103

0.730

0. 66686

96350

03697

87373

259

0. 74517

44023

44870

38879 013

.731

. 66761

44758

47057

30099

195

.74450

71602

33841

50102 364

.732

. 66835

86490

75996

51573

181

. 74383

91736

15714

42167 693

.733

. 66910

21539

46342

35102

739

. 74317

04431

58475

71321 153

.734

. 66984

49897

14589

99849

159

.74250

09695

30855

77713 862

0.735

0. 67058

71556

37903

75177

973

0. 74183

07534

02328

18528 866

.736

. 67132

86509

74117

74942

523

. 74115

97954

43109

01033 791

.737

. 67206

94749

81736

71700

537

. 74048

80963

24156

15559 237

.738

. 67280

96269

19936

70863

650

. 73981

56567

17168

68402 998

.739

.67354

91060

48565

84779

796

. 73914

24772

94586

14660 158

0.740

0. 67428

79116

28145

06748

388

0. 73846

85587

29587

90979 142

.741

. 67502

60429

19868

84968

216

. 73779

39016

96092

48243 787

.742

. 67576

34991

85605

96417

996

. 73711

85068

68756

84181 492

.743

. 67650

02796

87900

20669

485

.73644

23749

22975

75897 532

.744

. 67723

63836

89971

13633

096

. 73576

55065

34881

12335 582

0.745

0. 67797

18104

55714

81235

936

0. 73508

79023

81341

26664 537

.746

. 67870

65592

49704

53032

193

.73440

95631

39960

28591 681

.747

. 67944

06293

37191

55745

803

. 73373

04894

89077

36602 285

.748

. 68017

40199

84105

86745

313

. 73305

06821

07766

10125 695

.749

.68090

67304

57056

87450

880

. 73237

01416

75833

81627 975

0.750

0. 68163

87600

23334

16673

324

0. 73168

88688

73820

88631 184

.751

. 68237

01079

50908

23885

163

.73100

68643

83000

05659 342

.752

. 68310

07735

08431

22423

554

. 73032

41288

85375

76111 160

.753

. 68383

07559

65237

62625

080

.72964

06630

63683

44059 608

.754

.68456

00545

91345

04882

285

. 72895

64676

01388

85978 367

0.755

0. 68528

86686

57454

92691

917

0. 72827

15431

82687

42395 268

.756

. 68601

65974

34953

25484

772

. 72758

58904

92503

49472 750

.757

. 68674

38401

95911

31587

089

. 72689

95102

16489

70515 436

.758

. 68747

03962

13086

40963

419

. 72621

24030

41026

27404 867

.759

. 68819

62647

59922

57950

885

. 72552

45696

53220

31961 494

0.760

0. 68892

14451

10551

33914

776

0. 72483

60107

40905

17233 969

.761

. 68964

59365

39792

39835

383

. 72414

67269

92639

68715 814

.762

. 69036

97383

23154

38826

030

. 72345

67190

97707

55489 548

.763

. 69109

28497

36835

58582

200

. 72276

59877

46116

61298 318

.764

. 69181

52700

57724

63761

700

. 72207

45336

28598

15545 123

0.765

0. 69253

69985

63401

28295

794

0. 72138

23574

36606

24219 693

.766

. 69325

80345

32137

07631

223

. 72068

94598

62317

00753 084

.767

. 69397

83772

42896

10903

039

. 71999

58415

98627

96800 072

.768

. 69469

80259

75335

73038

195

. 71930

15033

39157

32949 410

.769

.69541

69800

09807

26789

802

.71860

64457

78243

29362 010

0.770

0. 69613

52386

27356

74701

988

0. 71791

06696

10943

36337 129

.771

. 696S5

28011

09725

61005

296

.71721

41755

33033

64806 626

.772

. 69756

96667

39351

43442

524

.71651

69642

41008

16757 355

.773

. 69828

58347

99368

65024

972

. 71.581

90364

32078

15581 770

.774

. 69900

13045

73609

25718

983

. 71512

03928

04171

36356 807

0.775

0. 69971

60753

46603

54062

747

0. 71442

10340

55931

36051 117

.776

.70043

01464

03580

78713

256

. 71372

09608

86716

83660 709

.777

. 70114

35170

30469

99923

379

.71302

01739

96600

90273 093

.778

.70185

61865

13900

60948

949

. 71231

86740

86370

39059 972

.779

. 70256

81541

41203

19385

818

.71161

64618

57525

15198 564

64

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to i. 600— Continued.

X

sin X

cosx

0.780

0. 70327

94192

00410

18436

790

0. 71091

3.5380

12277

35721 026

.781

. 70398

99809

80256

58108

374

. 71020

99032

53550

79296 239

.782

. 70409

98387

70180

C6337

280

. 70950

55582

84980

15931 435

.783

.70540

89918

60324

70046

581

. 70880

0503S

10910

36614 737

.784

. 70611

74395

41535

66131

480

. 70809

47405

36395

82877 671

0.785

0. 70682

51811

05365

92374

614

0. 70738

82691

67199

76290 330

.786

. 70753

22158

44073

98290

801

. 70068

10904

09793

47885 059

.787

. 70823

85430

50625

15901

193

. 70597

32049

71355

67509 330

.788

.70894

41620

18692

30436

730

. 70526

40135

59771

73107 880

.789

. 70964

90720

42656

50970

857

.70455

53168

83632

99934 173

0.790

0. 71035

32724

17607

80981

403

0. 70384

53156

52236

09691 278

.791

. 71105

67624

39345

88841

574

. 70313

46105

75582

19602 208

.792

.71175

95414

04380

78239

979

. 70242

32023

04370

31409 812

.793

. 71246

16086

09933

58529

620

. 70171

10917

30026

00300 275

.794

. 71316

29633

53937

15005

776

. 70099

82793

84643

03792 314

0.795

0. 71386

36049

35036

79112

713

0. 70028

47660

41039

70460 123

.796

. 71456

35326

52590

98579

148

. 09957

05524

12728

08742 151

.797

. 71526

27458

06672

07482

391

. 69885

56392

13922

35499 779

.798

.71596

12436

98066

96241

109

. 69814

00271

59535

04061 971

.799

. 71605

90250

28277

81536

630

. 69742

37169

65179

95703 964

0.800

0. 71735

60908

99522

76162

718

0. 69670

67093

47165

42092 075

.801

. 71805

24388

14736

58803

753

. 69598

90050

22499

59652 695

.802

. 71874

80686

77571

43741

255

. 69527

06047

08886

74871 538

.803

.71944

29797

92397

50488

651

. 69455

15091

24727

13123 218

.804

. 72013

71714

64303

73354

263

. 69383

17189

89116

26831 236

0.805

0. 72083

06429

99098

50932

396

0. 69311

12350

21844

23558 425

.806

. 72152

33937

03310

35522

503

. 69239

00579

43394

94027 956

.807

. 72221

54228

84188

62476

322

. 69166

81884

74945

40074 951

.808

. 72290

67298

49704

1947,2

9'35

. 69094

56273

38365

02528 784

.809

. 72359

73139

08550

15721

677

. 69022

23752

56214

89026 151

0.810

0. 72428

71743

70142

51092

818

0. 68949

84329

51747

01754 964

.811

. 72497

63105

44620

85175

959

. 68877

38011

48903

65129 158

.812

. 72566

47217

42849

06266

069

. 68804

84805

72316

53394 472

.813

. 72635

24072

76416

00277

085

. 68732

24719

47306

18165 280

.814

. 72703

93664

57636

19583

027

. 68659

57759

99881

15892 545

0.815

0. 72772

55985

99550

51786

534

0. 68586

83934

56737

35262 969

.816

. 72841

11030

15926

88414

775

. 68514

03250

45257

24529 414

.817

. 72909

58790

21260

93542

651

. 68441

15714

93509

18772 652

.818

. 72977

99259

30776

72343

223

. 68368

21335

30246

67094 544

.819

. 73046

32430

60427

39565

302

. 68295

20118

84907

59742 692

0.820

0. 73114

58297

26895

87938

131

0. 68222

12072

87613

55166 656

.821

. 73182

76852

47595

56503

084

. 68148

97204

69169

07005 802

.822

. 73250

88089

40670

98872

320

. 68075

75521

61060

91008 857

.823

. 73318

92001

24998

51414

329

. 68002

47030

95457

31885 232

.824

. 73386

88581

20187

01366

283

. 67929

11740

05207

30088 213

0.825

0. 73454

77822

46578

54873

150

0. 67855

69656

23839

88530 058

.826

. 73522

59718

25249

04953

477

. 67782

20786

85563

39229 106

.827

. 73590

34261

78008

99391

793

. 67708

65139

25264

69888 949

.828

. 73658

01446

27404

08557

557

. 67635

02720

78508

50409 750

.829

. 73725

61264

96715

93150

579

. 67561

33538

81536

59331 781

0.830

0. 73793

13711

09962

71872

858

0. 67487

57600

71267

10211 246

.831

. 73860

58777

91899

89026

752

. 67413

74913

85293

77928 481

.832

. 73927

96458

68020

82039

434

. 67339

85485

61885

24928 580

.833

. 73995

26746

64557

48913

544

. 67265

89323

39984

27394 537

.834

. 74062

49635

08481

15603

989

. 67191

86434

59207

01352 983

0. 835

0. 74129

65117

27503

03320

808

0. 67117

76826

59842

28712 570

.836

. 74196

73186

50074

95758

049

. 67043

60506

82850

83235 098

.837

. 74263

73836

05390

06248

576

. 66909

37482

69864

56439 445

.838

. 74330

67059

23383

44844

755

. 00895

07761

63185

83438 385

.839

. 74397

52849

34732

85324

932

. 66820

71351

05786

68708 357

Mo. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

65

Table XII. — Values of sin x and cos x to iS places of decimals at intervals of 0.001 from 0.000

to i. 600— Continued.

X

dnx

COST

0.840

0. 74464

31199

70859

32125

657

0. 66746

28258

41308

11792 267

.841

. 74531

02103

63927

87199

.577

66671

78491

14059

32935 396

.842

. 74.597

65554

46848

16798

923

. 66597

22056

69016

98654 482

.843

. 74664

21545

53275

18184

539

. 66522

58962

51824

47240 065

.844

. 74730

70070

17609

86260

385

. 66447

89216

08791

14192 152

0. 845

0. 74797

11121

74999

80133

429

0. 66373

12824

86891

57589 286

.846

. 74863

44693

61339

89598

886

. 66298

29796

33764

83391 100

.847

. 74929

70779

13273

01550

724

. 66223

40137

97713

70674 409

.848

. 74995

89371

68190

66317

368

. 66148

43857

27703

96802 946

.849

. 75062

00464

64233

63922

547

. 66073

40961

73363

62530 783

0.850

0. 75128

04051

40292

70271

207

0. 65998

31458

84982

17039 542

.851

. 75194

00125

36009

23260

432

. 65923

15356

13509

82909 449

.852

. 75259

88679

91775

88815

295

. 65847

92661

10556

81024 321

.853

. 75325

69708

48737

26849

594

. 65772

63381

28392

55410 547

.854

. 75391

43204

48790

57151

380

. 65697

27524

19944

98010 152

0.855

0. 75457

09161

34586

25193

237

0. 65621

85097

38799

73388 013

.856

. 75522

67572

49528

67867

227

. 65546

36108

39199

43373 300

.857

. 75588

18431

37776

79144

450

. 65470

80564

76042

91635 218

.858

. 75653

61731

44244

75659

143

. 65395

18474

04884

48193 134

.859

. 75718

97466

14602

62217

260

. 65319

49843

81933

13861 148

0.860

0. 75784

25628

95276

97229

459

0. 65243

74681

64051

84627 203

.861

. 75849

46213

33451

58068

441

. 65167

92995

08756

75966 794

.862

. 75914

59212

77068

06350

5G6

. 65092

04791

74216

47091 357

.863

. 75979

64620

74826

53141

684

. 6.5016

10079

19251

25131 418

.'864

.76044

62430

76186

24087

122

. 64940

08865

03332

29254 574

0. 865

0. 76109

52636

31366

24465

750

0. 64864

01156

86580

94718 373

.866

. 76174

35230

91346

04168

073

. 64787

86962

29767

96858 196

.867

. 76239

10208

07866

22598

272

. 64711

66288

94312

7.5010 176

.868

. 76303

77561

33429

13500

144

. 64635

39144

42282

56369 276

.869

. 76368

37284

21299

49706

858

. 64559

05536

36391

79782 561

0.870

0. 76432

89370

25505

07814

480

0. 64482

65472

40001

19477 766

.871

. 76497

33813

00837

32779

191

. 64406

18960

17117

08727 234

.872

. 76561

70606

02852

02438

134

. 64329

66007

32390

63447 280

.873

. 76625

99742

87869

91953

834

. 64253

06621

51117

05733 091

.874

. 76690

21217

12977

38182

114

. 64176

40810

39234

87329 202

0.875

0. 76754

35022

36027

03963

458

0. 64099

68581

63325

13035 656

.876

. 7681S

41152

15638

42337

736

. 64022

89942

90610

64049 903

.877

. 76882

39600

1119S

60682

252

. 63946

04901

88955

21244 528

.878

. 76946

30359

82862

84773

027

. 63869

13466

26862

88380 872

.879

. 77010

13424

91555

22769

271

. 63792

15643

73477

15258 639

0.880

0. 77073

88788

98969

29120

965

0. 63715

11441

98580

20801 550

.881

. 77137

56445

67568

68399

506

. 63638

00868

72592

16079 131

.882

. 77201

16388

60587

79051

337

. 63560

83931

66570

27264 710

.883

. 77264

68611

42032

37074

497

. 63483

60638

52208

18529 695

.884

. 77328

13107

76680

19618

049

.63406

30997

01835

14874 218

0. 885

0. 77391

49871

30081

68504

290

0. 63328

9.5014

88415

24894 213

.886

. 77454

78895

68560

53673

706

. 63251

52699

85546

63485 020

.887

. 77518

00174

59214

36552

600

. 63174

04059

67460

74481 571

.888

. 77.581

13701

69915

33343

321

. 63096

49102

09021

53235 256

.889

.77644

19470

69310

78237

045

. 63018

87834

85724

69127 530

0.890

0. 77707

17475

26823

86549

033

0. 62941

20265

73696

88020 355

.891

. 77770

07709

126.54

17776

316

. 62863

46402

49694

94643 540

.892

. 77832

90165

97778

38577

722

. 62785

66252

91105

14919 057

.893

. 77895

64839

53950

85676

211

. 62707

79824

75942

38222 428

.894

. 77958

31723

53704

28683

432

. 62629

87125

82849

39581 242

0.895

0. 78020

90811

70350

32846

443

0. 62551

88163

91096

01810 880

.896

. 78083

42097

77980

21716

.548

62473

82946

80578

37587 545

.897

. 78145

85575

51465

39740

163

. 62395

71482

31818

11458 656

.898

. 78208

21238

66458

14771

667

. 62317

53778

25961

61790 683

.899

. 78270

49080

99392

20508

171

. 62239

29842

44779

22654 524

1153»4''— 21-

66

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

tVOL. XIV,

Table XII. — Valuei of tin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to ;. 600— Continued.

X

nnx

cosx

0.900

0. 78332

69096

27483

38846

138

0. 62160

99682

70664

45648 472

.901

. 78394

81278

28730

22159

796

. 62082

63306

86633

21658 870

.902

. 78456

85620

81914

55501

279

.62004

20722

76323

02558 530

.903

. 78518

82117

66602

18722

439

. 61925

71938

23992

22842 983

.904

. 78580

70762

63143

48518

260

. 61847

16961

14519

21204 658

0.905

0. 78642

51549

52674

00391

817

0. 61768

55799

33401

62045 040

.906

. 78704

24472

17115

10540

713

. 61689

88460

66755

56924 921

.907

. 78765

89524

39174

57664

940

. 61611

14953

01314

85952 792

.908

. 78827

46700

02347

24696

094

. 61532

35284

24430

19111 466

.909

. 78888

95992

90915

60447

888

. 61453

49462

24068

37523 020

0.910

0. 78950

37396

89950

41187

896

0. 61374

57494

88811

54652 118

.911

. 79011

70905

85311

32130

474

. 61295

59390

07856

37447 803

.912

. 79072

96513

63647

48850

789

.61216

55155

71013

27423 839

.913

. 79134

14214

12398

18619

897

.61137

44799

68705

61677 674

.914

. 79195

24001

19793

41660

812

. 61058

28329

91968

93848 110

0.915

0. 79256

25868

74854

52325

499

0. 60979

05754

32450

15011 758

.916

. 79317

19810

67394

80192

738

. 60899

77080

82406

74518 350

.917

. 79378

05820

88020

11086

785

. 60820

42317

34706

00764 999

.918

. 79438

83893

28129

48016

785

. 60741

01471

82824

21909 476

.919

. 79499

54021

79915

72036

860

. 60661

54552

20845

86522 589

0.920

0. 79560

16200

36366

03026

828

0. 60582

01566

43462

84179 741

.921

. 79620

70422

91262

60393

471

. 60502

42522

45973

65991 745

.922

. 79681

16683

39183

23692

319

. 60422

77428

24282

65074 984

.923

. 79741

54975

75501

93169

858

. 60343

06291

74899

16960 980

.924

. 79801

85293

96389

50226

129

. 60263

29120

94936

79945 468

0.925

0. 79862

07631

98814

17797

639

0. 60183

45923

82112

55377 043

.926

. 79922

21983

80542

20660

537

. 60103

56708

34746

07885 466

.927

. 79982

28343

40138

45653

978

. 60023

61482

51758

85549 703

.928

. 80042

26704

76967

01823

638

. 59943

60254

32673

40005 791

.929

. 80102

17061

91191

80485

294

. 59863

53031

77612

46494 584

0.930

0. 80161

99408

83777

15208

432

0. 59783

39822

87298

23849 491

.931

. 80221

73739

56488

41719

806

. 59703

20635

63051

54424 260

.932

. 80281

40048

11892

57726

899

. 59622

95478

06791

03960 905

.933

. 80340

98328

53358

82661

218

. 59542

64358

21032

41397 846

.934

. 80400

48574

85059

17341

371

. 59462

27284

08887

58618 345

0.935

0. 80459

90781

11969

03555

863

0. 59381

84263

74063

90139 324

.936

. 80519

24941

39867

83565

545

. 59301

35305

20863

32740 634

.937

. 80578

51049

75339

59525

671

. 59220

80416

54181

65034 867

.938

. 80637

69100

25773

52827

488

. 59140

19605

79507

66977 785

.939

. 80696

79086

99364

63359

313

. 59059

52881

02922

39319 443

0.940

0. 80755

81004

05114

28687

022

0. 58978

80250

31098

22996 099

.941

. 80814

74845

52830

83153

915

. 58898

01721

71298

18462 976

.942

. 80873

60605

53130

16899

872

. 58817

17303

31375

04967 973

.943

. 80932

38278

17436

34799

758

. 58736

27003

19770

59766 388

.944

. 80991

07857

57982

15321

017

. 58655

30829

45514

77276 748

0.945

0. 81049

69337

87809

69300

383

0. 58574

28790

18224

88177 827

.946

. 81108

22713

20770

98639

669

. 58493

20893

48104

78446 913

.947

. 81166

67977

71528

54920

560

. 58412

07147

45944

08339 436

.948

. 81225

05125

55555

97938

351

. 58330

87560

23117

31310 012

.949

. 81283

34150

89138

54154

591

. 58249

62139

91583

12874 994

0.950

0. 81341

55047

89373

75068

542

0. 58168

30894

63883

49416 618

.951

. 81399

67810

74171

95507

433

. 58086

93832

53142

86928 810

.952

. 81457

72433

62256

91835

411

. 58005

50961

73067

39704 748

.953

. 81515

68910

73166

40081

165

. 57924

02290

37944

08966 253

.954

. 81573

57236

27252

73984

145

.57842

47826

62640

01435 096

0. 955

0. 81631

37404

45683

42959

322

0. 57760

87578

62601

47846 300

.956

. 81689

09409

50441

69980

433

. 57679

21554

53853

21403 511

.957

. 81746

73245

64327

09381

654

. 57597

49762

52997

56176 536

.958

. 81804

28907

10956

04577

644

. 57515

72210

77213

65441 113

.959

. 81861

76388

14762

45701

891

. 57433

88907

44256

59961 007

No. 8.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTKAND.

67

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to 7.600— Continued.

X

sinx

COSX i

0.960

0. 81919

15683

00998

27163

322

0. 57351

99860

72456

66212

505

.961

. 81976

46785

95734

05121

101

. 57270

05078

80718

44551

395

.962

. 82033

69691

25859

54877

569

. 57188

04569

88520

07322

513

.963

. 82090

84393

19084

28189

263

. 57105

98342

15912

36911

940

.964

. 82147

90886

03938

10495

962

. 57023

86403

83518

03741

923

0.965

0. 82204

89164

09771

78067

694

0. 56941

68763

12530

84208

614

.966

. 82261

79221

66757

55069

650

. 56859

45428

24714

78562

699

.967

. 82318

61053

05889

70544

986

. 56777

16407

42403

28733

004

.968

. 82375

34652

58985

15315

328

. 56694

81708

88498

36093

162

.969

. 82432

00014

58683

98799

136

. 56612

41340

86469

79171

417

0.970

0. 82488

57133

38450

05747

662

0, 56529

95311

60354

31303

653

.971

.82545

06003

32571

52898

564

.56447

43629

34754

78229

727

.972

. 82601

46618

76161

45547

087

. 56364

86302

34839

35633

190

.973

. 82657

78974

05158

34034

750

. 56282

23338

86340

66624

480

.974

. 82714

03063

56326

70155

495

. 56199

54747

15554

99167

663

0.975

0. 82770

18881

67257

63479

226

0. 56116

80535

49341

43450

813

.976

. 82826

26422

76369

37592

699

. 56034

00712

15121

09200

110

.977

. 82882

25681

22907

86257

689

. 55951

15285

40876

22937

736

.978

. 82938

16651

46947

29486

397

. 55868

24263

55149

45183

654

.979

. 82993

99327

89390

69534

022

. 55785

276.54

87042

87601

358

0.980

0. 83049

73704

91970

46808

453

0. 55702

25467

66217

30087

666

.981

. 83105

39776

97248

95697

028

. 55619

17710

22891

37806

645

.982

. 83160

97538

48619

00310

290

. 55536

04390

87840

78167

757

.983

. 83216

46983

90304

50142

703

. 55452

85517

92397

37748

295

.984

. 83271

88107

67360

95650

254

. 55369

61099

68448

39160

207

0.985

0. 83327

20904

25676

03744

902

0. 55286

31144

48435

57861

376

.986

. 83382

45368

11970

13205

801

. 55202

95660

65354

38911

453

.987

. 83437

61493

73796

90007

262

. 55119

54656

52753

13672

322

.988

. 83492

69275

59543

82563

379

. 55036

08140

44732

16453

272

.989

. 83547

68708

18432

76889

279

. 54952

56120

75943

01100

969

0.990

0. 83602

59786

00520

51678

926

0. 54868

98605

81587

57534

313

.991

. 83657

42.503

56699

33299

444

. 54785

35603

97417

28224

252

.992

. 83712

16855

38697

50701

883

. 54701

67123

59732

24618

647

.993

. 83766

82835

99079

90248

385

. 54617

93173

05380

43512

268

.994

. 83821

40439

91248

50455

694

. 54534

13760

71756

83362

006

0.995

0. 83875

89661

69442

96654

953

0. 54450

28894

96802

60547

375

.996

. 83930

30495

88741

15567

733

. 54366

38584

19004

25576

412

.997

. 83984

62937

05059

69798

245

. 54282

42836

77392

79237

026

.998

.84038

86979

75154

52241

668

. 54198

41661

11542

88693

907

.999

. 84093

02618

56621

40408

555

.54114

35065

61572

03531

067

1.000

0. 84147

09848

07896

50665

250

0.54030

23058

68139

71740

094

.001

. 84201

08662

88256

92390

268

. 53946

05648

72446

55654

214

.002

.84254

99057

57821

22046

578

. 53861

82844

16233

47828

237

.003

.84308

81026

77549

97169

747

. 53777

54653

41780

86864

465

.004

.84362

54565

09246

30271

873

. 53693

21084

91907

73184

669

1.005

0. 84416

19667

15556

42661

273

0. 53608

82147

09970

84748

188

.006

.84469

76327

59970

18177

851

. 53524

37848

39863

92716

262

.007

. 84523

24.541

06821

56844

116

. 53439

88197

26016

77062

668

.008

.84576

64302

21289

28431

774

. 53355

33202

13394

42130

747

.009

.84629

95605

69397

25943

853

.53270

72871

47496

32136

904

1.010

0.84683

18446

18015

19012

310

0. 53186

07213

74355

46620

673

.011

. 84736

32818

34859

07211

051

. 53101

36237

40537

55841

426

.012

.84789

38716

88491

73284

331

. 53016

59950

93140

16121

808

.013

.84842

36136

48323

36290

466

.52931

78362

79791

85137

984

.014

. 84895

25071

84612

04660

810

.52846

91481

48651

37156

798

1.015

0.84948

05517

68464

29173

940

0. 52761

99315

48406

78219

896

.016

.85000

77468

71835

55845

003

. 52677

01873

28274

61274

932

.017

.85053

40919

67530

78730

164

. 52591

99163

37999

01253

921

.018

. 85105

95865

29204

92646

111

. 52506

91194

27850

90098

832

.019

. 85158

42300

31363

45804

549

. 52421

77974

48627

11734

503

68

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Values of sin x and cos x to 2S places of decimals at intervals of 0.001 from 0.000

to J. 600— Continued.

X

sin X

cos X

1.020

0. 85210

80219

49362

92361

655

0. 52336

59512

51649

56988

961

.021

. 85263

09617

59411

44882

415

. 52251

35816

88764

38461

245

.022

. 85315

30489

38569

26719

808

. 52166

06896

12341

05336

792

.023

. 85367

42829

64749

24308

778

. 52080

72768

76271

58160

502

.024

. 85419

46633

16717

39374

945

. 51995

33413

30969

63497

542

1.025

0. 85471

41894

74093

41057

997

0. 51909

88868

33369

68691

985

.026

. 85523

28609

17351

17949

715

. 51824

39132

36926

16373

373

.027

. 85575

06771

27819

30046

586

. 51738

84213

96612

59061

276

.028

. 85626

76375

87681

60616

931

. 51653

24121

67920

73657

956

.029

. 85678

37417

79977

67982

525

.51567

58864

06859

76899

186

1.030

0. 85729

89891

88603

37214

627

0. 51481

88449

69955

34753

350

.031

. 85781

33792

98311

31744

398

.51396

12887

14248

86768

878

.032

. 85832

69115

94711

44887

626

. 51310

32184

97296

50370

116

.033

. 85883

95855

64271

51283

734

. 51224

46351

77168

40101

715

.034

. 85935

14006

94317

58248

998

. 61138

55396

12447

80821

625

1.035

0. 85986

23564

73034

57043

938

0. 51052

59326

62230

21842

776

.036

. 86037

24523

89466

74054

819

. 50966

58151

86122

61023

635

.037

. 86088

16879

33518

21889

224

. 50880

51880

44242

08807

028

.038

. 86139

00625

95953

50385

634

. 50794

40520

97216

02209

404

.039

. 86189

75758

68397

97536

975

. 50708

24082

06180

18757

138

1.040

0. 86240

42272

43338

40328

079

0. 50622

02572

32778

40373

447

.041

. 86291

00162

14123

4.5486

997

. 50535

76000

39161

57213

919

.042

. 86341

49422

74964

20150

131

.50449

44374

87^86

81451

427

.043

. 86391

90049

20934

62441

124

. 50363

07704

42416

61010

426

.044

. 86442

22036

47972

11963

456

. 50276

65997

66117

93250

711

1.045

0. 86492

45379

52878

00206

699

0. 50190

19263

23261

38600

728

.046

. 86542

60073

33318

00866

385

. 50103

67509

78620

34140

520

.047

. 86592

66112

87822

80077

424

. 50017

10746

97070

07134

396

.048

.86642

63493

15788

46561

037

. 49930

48980

44586

88513

415

.049

. 86692

52209

17477

01685

140

. 49843

82221

87247

26307

766

1.050

0. 86742

32255

94016

89438

141

0. 49757

10478

91726

99029

085

.051

. 86792

03628

47403

46316

092

. 49670

33760

25200

29002

975

.052

. 86841

66321

80499

51123

146

. 49583

52074

65338

95651

499

.053

. 86891

20330

97035

74685

276

. 49496

65430

50311

48726

051

.054

.86940

65651

01611

29477

198

.49409

73836

78782

21490

510

1.055

0. 86990

02276

99694

19162

460

0. 49322

77302

09910

43854

806

.056

. 87039

30203

97621

88046

624

. 49235

75835

13349

55459

008

.057

. 87088

49427

02601

70443

529

.49148

69444

59246

18707

979

.058

. 87137

59941

22711

39954

543

. 49061

58139

18239

31766

732

.059

. 87186

61741

66899

58660

794

. 48974

41927

61459

41446

534

1.060

0.87235

54823

44986

26228

295

0.48887

20818

60527

56191

864

.061

. 87284

39181

67663

28925

947

. 48799

94820

87564

58818

317

.062

.87333

14811

46494

88556

345

.48712

63943

15140

19351

528

.063

. 87381

81707

93918

11299

356

. 48625

28194

16372

07757

202

.064

. 87430

'39866

23243

36468

402

.48537

87582

64825

06632

362

1.065

0. 87478

89281

48654

85179

424

0. 48450

42117

34660

23847

867

.066

. 87527

29948

85211

08932

453

. 48362

91807

00124

05142

311

.067

. 87575

61863

48845

38105

753

.48275

36660

36647

46667

387

.068

. 87623

85020

56366

303G2

492

. 48187

76686

19345

07484

800

.069

. 87671

99415

25458

18969

874

. 48100

11893

24514

22014

811

1.070

0. 87720

05042

74681

61030

706

0.48012

42290

28534

12436

509

.071

. 87768

01898

23473

85627

336

. 47924

67886

08365

01039

904

.072

. 87815

89976

92149

41877

919

. 47836

88689

41447

22529

904

.073

. 87863

69274

01900

46904

963

. 47749

04709

06700

36282

289

.074

. 87911

39784

74797

33716

111

. 47661

15953

79522

38551

762

1.075

0. 87959

01504

33788

98997

101

0.47573

22432

41788

74632

160

.076

. 88006

54428

02703

50816

869

. 47485

24153

71861

50968

911

.077

. 88053

98551

06248

56244

731

. 47397

21126

49538

47223

840

.078

. 88101

33868

70011

88879

619

.47309

13359

55152

28292

396

.079

. 88148

60376

20461

76291

297

.47221

00861

69469

66273

392

No. B.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

69

Table XII. — Valttea of sin x and cos x to S.3 places of decimah at intervals of 0.001 from 0.000

to J.600— Continued.

X

sin X

cos X

1.080

0. 88195

78068

84947

47373

533

0. 47132

83641

73740

02391 353

.081

. 88242

86941

91699

79609

169

.47044

61708

49685

58871 547

.082

. 88289

86990

69831

46247

031

. 46956

35070

79499

50767 810

.083

. 88336

78210

49337

63390

660

.46868

03737

45845

47743 217

.084

. 88383

60596

61096

36998

790

. 46779

67717

31856

75803 727

1.085

0. 88430

34144

36869

09797

534

0. 46691

27019

211.35

28984 862

.086

. 88476

98849

09301

08104

243

. 46602

81651

97750

80991 522

.087

. 88523

54706

11921

88562

972

. 46514

31624

46239

96791 014

.088

. 88570

01710

79145

84791

522

.46425

76945

51605

44159 401

.089

. 88616

39858

46272

53940

000

.46337

17623

99315

05181 235

1.090

0. 88662

69144

49487

23160

860

0. 46248

53668

75300

87702 790

.091

. 88708

89564

25861

35990

371

. 461.59

85088

65958

36738 852

.092

. 88755

01113

13.362

98641

470

. 46071

11892

58145

4.5833 190

.093

. 88801

03786

50807

26207

951

. 45982

34089

39181

68372 764

.094

. 88846

97579

77956

88779

948

. 45893

51687

96847

28855 783

1.095

0. 88892

82488

35422

57470

660

0. 45804

64697

19382

34113 686

.096

. 88938

58.507

64713

50354

274

.45715

73125

95485

84487 142

.097

. 88984

25633

08227

78315

047

. 45626

76983

14314

84956 158

.098

. 89029

83860

09252

90807

488

. 45537

76277

65483

56224 382

.099

. 89075

33184

11966

21527

609

.45448

71018

39062

45757 688

1.100

0. 89120

73600

61435

33995

180

0. 45359

61214

25577

38777 137

.101

.89166

05105

03618

67046

971

. 45270

46874

16008

69206 400

.102

.89211

27692

85365

80240

901

.45181

28007

01790

.30573 730

.103

. 89256

41359

54417

99171

080

. 45092

04621

74808

86868 576

.104

. 89301

46100

59408

60693

678

. 45002

76727

27402

83352 928

1.105

0. 89346

41911

49863

58063

585

0.44913

44332

52361

57327 478

.106

. 89391

28787

76201

85981

812

. 44824

07446

42924

48852 689

.107

. 89436

06724

89735

85553

594

.44734

66077

92780

11424 866

.108

. 89480

75718

42671

89157

146

. 44645

20235

96065

22607 305

.109

. 89525

35763

88110

65223

027

.44555

69929

47363

94616 628

1.110

0. 89569

86856

80047

62924

063

0.44466

15167

41706

84864 374

.111

. 89614

28992

73373

56775

801

.44376

55958

74570

06453 951

.112

. 89658

62167

23874

91147

427

.44286

92312

41874

38633 030

.113

. 89702

86375

88234

24683

120

.44197

24237

39984

37201 474

.114

. 89747

01614

24030

74633

785

.44107

51742

65707

44874 890

1.115

0. 89791

07877

89740

61099

138

0. 44017

74837

16293

01603 891

.116

. 89835

05162

44737

51180

079

. 43927

93529

89431

54849 166

.117

. 89878

93463

49293

03041

321

. 43838

07829

832.53

69812 438

.118

. 89922

72776

64577

09884

230

. 43748

17745

96329

39623 410

.119

. 89966

43097

52658

43829

826

. 43658

23287

27666

95482 777

1.120

0.90010

04421

76504

99711

910

0. 43568

24462

76712

16761 .399

.121

. 90053

56744

99984

38780

263

. 43478

21281

43347

41055 736

.122

. 90097

00062

87864

.32313

880

. 43388

13752

27890

74199 612

.123

. 90140

34371

05813

05144

201

. 43298

01884

31095

00232 420

.124

. 90183

59665

20399

79088

276

. 43207

85686

54146

91323 845

1.125

0. 90226

75940

99095

16291

842

0.43117

65167

98666

17655 197

.126

. 90269

83194

10271

62482

258

. 43027

40337

66704

57257 452

.127

.90312

81420

23203

90131

256

. 42937

11204

60745

05806 078

.128

. 90355

70615

08069

41527

464

.42846

77777

83700

86372 749

.129

.90398

50774

35948

71758

658

.42756

40066

38914

59134 030

1.130

0.90441

21893

78825

91603

708

0. 42665

98079

30157

31037 122

.131

. 90483

83969

09589

10334

160

. 42575

51825

61627

65422 763

.132

. 90526

36996

02030

78425

425

. 42485

01314

37950

91605 376
14410 l40

.133

. 90568

80970

30848

30177

523

. 42394

46554

64178

.134

.90611

15887

71644

26245

348

. 42303

87555

45785

23669 902

1.135

0. 90653

41744

00926

96078

401

0. 42213

24325

88672

03673 585

.136

.90695

58534

96110

80269

960

.42122

56874

99161

42580 219

.137

. 90737

66256

35516

72815

632

. 42031

85211

83998

41784 656

.138

. 90779

64903

98372

63281

260

.41941

09345

50349

25243 478

.139

. 90821

54473

64813

78880

126

.41850

29285

05800

48758 379

70

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol,. XIV,

Table XII. — Valit^s of sin x and cos x to 2S places of decimals at intervals of 0.001 from 0.000

to J. 600— Continued.

X

sin X

cos X

1.140

0. 90863

34961

15883

26459

422

0. 41759

45039

58358

09217 519

.141

. 90905

06362

33532

34395

940

. 41668

56618

16446

53794 933

.142

. 90946

68673

00620

94400

939

. 41577

64029

88907

89108 094

.143

. 90988

21889

00918

03234

153

. 41486

67283

85000

90333 707

.144

. 91029

66006

19102

04326

885

. 41395

66389

14400

10281 852

1.145

0.91071

01020

40761

29314

164

0.41304

61354

87194

88428 529

.146

.91112

26927

52394

39475

912

.41213

52190

13888

59906 732

.147

.91153

43723

41410

67087

073

.41122

38904

05397

64456 120

.148

.91194

51403

96130

56676

684

. 41031

21505

73050

55.331 381

.149

. 91235

49965

05786

06195

821

.40940

00004

28587

08169 395

1.150

0. 91276

39402

60521

08094

403

0.40848

74408

84157

29815 258

.151

.91317

19712

51391

90306

792

. 40757

44728

52320

67107 284

.152

. 91357

90890

70367

57146

165

. 40666

10972

46045

15621 071

.153

. 91398

52933

10330

30107

602

. 40574

73149

78706

28372 706

.154

. 91439

05835

65075

88579

865

.40483

31269

64086

24481 224

1.155

0. 91479

49594

29314

10465

816

0. 40391

85341

16372

97790 397

.156

.91519

84204

98669

12711

431

.40300

35373

50159

25449 945

.157

. 91560

09663

69679

91743

383

. 40208

81375

80441

76456 266

.158

. 91600

25966

39800

63815

143

.40117

23357

22620

20152 779

.159

.91640

33109

07401

05261

556

. 40025

61326

92496

34689 958

1.160

0. 91680

31087

71766

92661

866

0. 39933

95294

06273

15445 164

.161

.91720

19898

33100

42911

136

.39842

25267

80553

83402 355

.162

. 91769

99536

92520

53200

023

.39750

51257

32340

9.3491 775

.163

.91799

69999

52063

40902

883

. 39658

73271

79035

42889 706

.164

. 91839

31282

14682

83374

147

. 39566

91320

38435

79278 377

1.165

0. 91878

83380

84250

57652

941

0. 39475

05412

28737

09066 125

.166

,91918

26291

65556

80075

906

. 39383

15556

68530

05567 898

.167

. 91957

60010

64310

45798

178

. 39291

21762

76800

17146 187

.168

. 91996

84533

87139

68222

492

. 39199

24039

72926

75312 486

.169

. 92035

99857

41592

18336

360

. 39107

22396

76682

02789 366

1.170

0. 92075

05977

36135

63957

301

0. 39015

16843

08230

21533 266

.171

. 92114

02889

80158

08886

071

. 38923

07387

88126

60718 072

.172

. 92152

90590

83968

31967

851

. 38830

94040

37316

64679 599

.173

. 92191

69076

58796

26061

369

. 38738

76809

77135

00821 054

.174

. 92230

38343

16793

36915

902

. 38646

55705

29304

67479 575

1.175

0. 92268

98386

71033

01956

127

0. 38554

30736

15936

01753 942

.176

. 92307

49203

35510

88974

783

. 38462

01911

59525

87293 547

.177

. 92345

90789

25145

34733

097

. 38369

69240

82956

62048 718

.178

. 92384

23140

55777

83468

944

. 38277

32733

09495

25982 487

.179

. 92422

46253

44173

25312

701

.38184

92397

62792

48743 902

1.180

0. 92460

60124

08020

34610

754

0. 38092

48243

66881

77302 960

.181

. 92498

64748

65932

08156

619

.38000

00280

46178

43547 271

.182

. 92536

60123

37446

03329

642

. 37907

48517

25478

71840 534

.183

. 92574

46244

43024

76141

242

. 37814

92963

29958

86542 917

.184

. 92612

23108

04056

19188

645

. 37722

33627

85174

19493 444

1.185

0. 92649

90710

42853

99516

095

0. 37629

70520

17058

17454 471

.186

. 92687

49047

82657

96383

480

. 37537

03649

51921

49518 342

.187

. 92724

98116

47634

38942

352

. 37444

33025

16451

14476 334

.188

. 92762

37912

62876

43819

290

. 37351

58656

37709

48149 962

.189

. 92799

68432

54404

52606

588

.37258

80552

43133

30684 752

1.190

0. 92836

89672

49166

69260

202

0. 37165

98722

60532

93806 568

.191

. 92874

01628

75038

97404

950

. 37073

13176

18091

28040 589

.192
.193

.92911

04297

60825

77546

899

. 36980

23922

44362

89893 026

. 92947

97675

36260

24192

928

. 36887

30970

68273

08995 672

.194

. 92984

81758

32004

62877

403

. 36794

34330

19116

95213 382

1.195

0. 93021

56542

79650

67095

956

0.36701

34010

26558

45714 570

.196

. 93058

22025

11719

95146

303

. 36608

30020

20629

52004 819

.197

. 93094

78201

61664

26876

083

. 36515

22369

31729

06923 698

.198

. 93131

25068

63866

00337

679

. 36422

11066

90622

11604 876

.199

. 93167

62622

53638

48349

974

.36328

96122

28438

82399 631

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

Tl

Table Xll.— Valties of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to /. 600— Continued.

I

sinx

cosx

1.200

0.93203 90859 67226 34967 013

0.36235 77544 76673 57763 837

.201

.93240 09776 41805 91853 542

.36142 55343 67184 05108 539

.202

.93270 19369 15485 54567 367

.36049 29528 32190 27614 189

.203

.93312 19634 27305 9874S 519

.35956 00108 04273 71008 651

.204

.93348 10568 17240 76215 175

.35862 67092 16376 30309 065

1.205

0.93383 92167 26196 5096G 302

0.35769 30490 01799 56527 660

.206

.93419 64427 96013 35090 992

.35675 90310 94203 63341 607

.207

.93455 27346 69465 24584 444

.35582 46.564 27606 33727 018

.208

.93490 80919 90260 35070 567

.35488 99259 36382 26557 166

.209

.93526 25144 03041 37431 162

.35395 48405 55261 83165 039

1.210

0.93561 60015 53385 93341 646

0.35301 94012 19330 33870 301

.211

.93596 85530 87806 90713 291

.35208 36088 64027 04470 775

.212

.93632 01686 53752 79041 926

.35114 74644 25144 22698 521

.213

.93667 08478 99608 04663 095

.35021 09688 38826 24640 616

.214

.93702 05904 74693 45913 598

.34927 41230 41568 61124 730

1.215

0.93736 93960 29266 48199 416

0.34833 69279 70217 04069 578

.216

.93771 72642 14521 58969 959

.34739 93845 61966 52800 358

.217

.93806 41946 82590 62598 617

.34646 14937 54360 40329 260

.218

.93841 01870 86543 15169 574

.34552 32564 85289 39601 140

.219

.93875 52410 80386 79170 848

.34458 46736 92990 69704 455

1.220

0.93909 93563 19067 58093 524

0.34364 57463 16047 02047 5.52

.221

.93944 25324 58470 30937 151

.34270 64752 93385 66500 405

.222

.93978 47691 55418 86621 257

.34176 68615 64277 57501 890

.223

.94012 60660 67676 58302 957

.34082 69060 68336 40132 702

.224

.94046 64228 53946 57600 622

.33988 66097 45517 56153 996

1.225

0.94080 58391 73872 08723 559

0.33894 59735 36117 30011 855

.226

.94114 43146 88036 82507 685

.33800 49983 80771 74807 668

.227

.94148 18490 57965 30357 157

.33706 36852 20455 98234 533

.228

.94181 84419 46123 18091 912

.33612 20349 96483 08479 750

.229

.94215 40930 15917 59701 104

.33518 00486 50503 20093 523

1.230

0.94248 88019 31697 51002 382

0.33423 77271 24502 59823 955

.231

.94282 25683 58754 03206 998

.33329 50713 60802 72418 427

.232

.94315 53919 63320 76390 684

.33235 20823 02059 26391 462

.233

.94348 72724 12574 12870 299

.33140 87608 91261 19759 164

.234

.94381 82093 74633 70486 175

.33046 51080 71729 85740 328

1.235

0.94414 82025 18562 55790 164

0.32952 11247 87117 98424 316

.236

.94447 72515 14367 57139 322

.32857 68119 81408 78405 786

.237

.94480 53560 32999 77695 223

.32763 21705 98914 98386 387

.238

.94513 25157 46354 68328 851

.32668 72015 84277 88743 487

.239

.94545 87303 27272 60431 046

.32574 19058 82466 43066 054

1.240

0.94578 39994 49.538 98628 471

0.32479 62844 38776 23657 769

.241

.94610 83227 87884 73405 063

.32385 03381 98828 67007 475

.242

.94643 17000 17986 53628 942

.32290 40681 08569 89227 042

.243

.94675 41308 16467 18984 738

.32195 74751 14269 91456 764

.244

.94707 56148 60895 92311 309

.32101 05601 62521 65238 364

1.245

0.94739 61518 29788 71844 815

0.32006 33242 00239 97855 712

.246

.94771 57414 02608 63367 118

.31911 57681 74660 77643 341

.247

.94803 43832 59766 12259 472

.31816 78930 33339 99262 871

.248

.94835 20770 82619 35461 479

.31721 96997 24152 68947 423

.249

.94866 88225 53474 53335 262

.31627 11891 95292 09714 116

1.250

0.94898 46193 555S6 21434 849

0.31532 23623 95268 66544 754

.251

.94929 94671 73157 621S0 713

.31437 32202 72909 11534 791

.252

.94961 336.56 91340 96439 444

.31342 37637 77355 49010 665

.253

.94992 63145 96237 75008 528

.31247 39938 58064 20615 601

.254

.95023 83135 74899 10006 196

.31152 39114 64805 10363 979

1.255

0.95054 93623 15326 06166 303

0.31057 35175 47660 49664 355

.256

.95085 94605 06469 92038 225

.30962 28130 57024 22311 242

.257

.95116 86078 38232 51091 729

.30867 17989 43600 69445 729

.258

.95147 68040 01466 52726 783

.30772 04761 58403 94485 052

.259

.95178 40486 87975 83188 287

.30676 88456 52756 68021 196

72f

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Values of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to ;. 600— Continued.

X

sinx

cosx

1.260

0. 95209

03415

90515

76385

682

0. 30581

69083

78289

32688

634

.261

. 95239

56824

02793

44617

416

. 30486

46652

86939

08001

291

.262

. 95270

00708

19468

09200

227

. 30391

21173

30948

95158

833

.263

. 95300

35065

36151

31003

222

. 30295

92654

62866

81822

373

.264

. 95330

59892

49407

40886

709

.30200

61106

35544

46859

693

1.265

0. 95360

75186

56753

70045

767

0. 30105

26538

02136

65060

070

.266

. 95390

80944

56660

80258

512

. 30009

88959

16100

11818

814

.267

. 95420

77163

48552

94039

032

. 29914

48379

31192

67791

595

.268

.95450

63840

32808

24694

963

. 29819

04808

01472

23518

675

.269

. 95480

40972

10759

06289

671

. 29723

58254

81295

84019

121

1.270

0. 95510

08555

84692

23509

018

0. 29628

08729

25318

73355

114

.271

. 95539

66588

57849

41432

673

. 29532

56240

88493

39166

425

.272

. 95569

15067

34427

35209

944

. 29437

00799

26068

57175

182

.273

. 95598

53989

19578

19640

104

. 29341

42413

93588

35661

000

.274

. 95627

83351

19409

78657

170

. 29245

81094

46891

19906

579

1.275

0. 95657

03150

40985

94719

118

0. 29150

16850

42108

96613

869

.276

. 95686

13383

92326

78101

497

.29054

49691

35665

98290

890

.277

.95715

14048

82408

96095

419

. 28958

79626

84278

07609

308

.278

. 95744

05142

21166

02109

886

. 28863

06666

44951

61732

860

.279

. 95772

86661

19488

64678

437

. 28767

30819

74982

56616

726

1.280

0. 95801

58602

89224

96370

075

0. 28671

52096

31955

51277

939

.281

. 95830

20964

43180

82604

453

. 28575

70505

73742

72036

934

.282

. 95858

73742

95120

10371

286

. 28479

86057

58503

16730

332

.283

. 95887

16935

59764

96853

962

. 28383

98761

44681

58895

050

.284

. 95915

50539

52796

17957

320

. 28288

08626

91007

51923

831

1.285

0. 95943

74551

90853

36739

577

0. 28192

15663

56494

33192

303

.286

. 95971

88969

91535

31748

357

. 28096

19881

00438

28157

651

.287

. 95999

93790

73400

25260

814

. 28000

21288

82417

54428

993

.288

. 96027

89011

55966

11427

805

. 27904

19896

62291

25809

577

.289

. 96055

74629

59710

84322

094

. 27808

15714

00198

56310

871

1.290

0. 96083

50642

06072

65890

556

0. 27712

08750

56557

64138

661

.291

.96111

17046

17450

33810

354

. 27615

99015

92064

75651

234

.292

. 96138

73839

17203

49249

056

. 27519

86519

67693

29289

769

.293

. 96166

21018

29652

84528

675

. 27423

71271

44692

79480

997

.294

. 96193

58580

80080

50693

590

. 27327

53280

84588

00512

263

1.295

0. 96220

86523

94730

24982

339

0. 27231

32557

49177

90379

053

.296

. 96248

04845

00807

78203

231

.27135

09111

00534

74605

108

.297

. 96275

13541

26481

02013

782

. 27038

82951

01003

10035

206

.298

. 96302

12610

00880

36103

915

. 26942

54087

13198

88600

711

.299

. 96329

02048

54098

95282

920

. 26846

22529

00008

41057

992

1.300

0. 96355

81854

17192

96470

135

0. 26749

882S6

24587

40699

798

.301

. 96382

52024

22181

85589

331

. 26653

51368

50360

07039

695

.302

. 96409

12556

02048

64366

761

. 26557

11785

41018

09469

650

.303

. 96435

63446

90740

17032

855

.26460

69546

60519

70890

877

.304

.96462

04694

23167

36927

537

.26364

24661

73088

71318

016

1.305

0. 96488

36295

35205

53009

126

0. 26267

77140

43213

51456

761

.306

. 96514

58247

63694

56266

806

.26171

26992

35646

16255

031

.307

.96540

70548

46439

26036

635

. 26074

74227

15401

38427

774

.308

. 96566

73195

22209

56221

061

. 25978

18854

47755

61955

494

.309

. 96592

66185

30740

81411

924

. 25881

60883

98246

05556

626

1.310

0. 96618

49516

12734

02916

926

0. 25785

00325

32669

66133

818

.311

.96644

23185

09856

14689

520

. 25688

37188

17082

22194

242

.312

. 96669

87189

64740

29162

218

. 25591

71482

17797

37244

030

.313

. 96695

41527

20986

02983

276

.25495

03217

01385

63156

911

.314

. 96720

86195

23159

62656

736

. 25398

32402

34673

43517

173

1.315

0. 96746

21191

16794

30085

794

0. 25301

59047

84742

16937

022

.316

. 96771

46512

48390

48019

478

.25204

83163

18927

20348

457

.317

. 96796

62156

65416

05402

607

. 25108

04758

04816

92269

738

.318

. 96821

68121

1G30G

62628

991

. 25011

23842

10251

76046

566

.319

. 96846

64403

50466

76697

879

. 24914

40425

03323

23067

996

NaB.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

73

Table XII. — Valiies of sin x and cos x to 23 places of decimals at intervals of 0.001 from 0.000

to 1.600 — Continued.

X

sinx

cos j:

1.320

0. 96871

51001

18265

26273

590

0. 24817

54516

52372

95957

398

.321

. 96896

27911

71045

36648

340

. 24720

66126

25991

71738

199

.322

. 96920

95132

61115

04608

211

. 24623

75263

93018

44974

865

.323

.96945

52661

41752

23202

252

. 24526

81939

22539

30889

004

.324

. 96970

00495

67204

06414

685

.24429

86161

83886

68450

760

1.325

0. 96994

38632

92687

13740

188

0. 24332

87941

46638

23445

582

.326

. 97018

67070

74387

74662

236

. 24235

87287

80615

91516

463

.327

.97042

85806

69462

13034

465

. 24138

84210

55885

01181

759

.328

. 97066

94838

36036

71365

051

.24041

78719

42753

16828

662

.329

. 97090

94163

33208

35004

060

.23944

70824

11769

41682

448

1.330

0. 97114

83779

21044

56233

768

0. 23847

60534

33723

20751

578

.331

. 97138

63683

60583

78261

900

. 23750

47859

79643

43748

768

.332

. 97162

33874

13835

59117

786

. 236.53

32810

20797

47988

097

.333

. 97185

94348

43780

95451

405

. 23.556

15395

28690

212.58

288

.334

. 97209

45104

14372

46235

282

. 23458

95624

75063

04672

221

1.335

0. 97232

86138

90534

56369

230

0. 23361

73508

31892

95492

805

.336

. 97256

17450

38163

80187

900

. 23264

49055

71391

49935

286

.337

. 97279

39036

24129

04871

12!)

.23167

22276

66003

85946

099

,338

. 97302

50894

16271

73757

046

. 23069

93180

88407

85958

358

.339

. 97325

53021

83406

09557

931

. 22972

61778

11512

99624

085

1.340

0. 97348

45416

95319

37478

787

0. 22875

28078

084.59

46.523

264

.341

. 97371

28077

22772

08238

616

. 22777

92090

52617.

18849

831

.342

. 97394

01000

37498

20994

365

. 22680

53825

17584

84074

691

.343

.97416

64184

12205

46167

522

. 22583

13291

77188

87585

859

.344

. 97439

17626

20575

48173

349

. 22485

70500

05482

55305

819

1.345

0. 97461

61324

37264

08052

713

0. 22388

25459

76744

96286

212

.346

. 97483

95276

37901

46006

501

. 22290

78180

65480

05279

929

.347

. 97506

19479

99092

43832

603

. 22193

28672

46415

65290

729

.348

.97528

33932

98416

67265

423

. 22095

76944

94502

50100

463

.349

. 97550

38633

14428

88217

916

. 21998

23007

84913

26774

007

1.350

0. 97572

33578

266.59

06926

111

0. 21900

66870

93041

58142

002

.351

.97594

18766

15612

73996

110

. 21803

08543

94501

05261

504

.352

. 97615

94194

62771

12353

536

. 21705

48036

65124

298.54

627

.353

. 97637

59861

50591

39095

407

. 21607

85358

80961

96725

291

.354

. 97659

15764

62506

87244

418

. 21510

20520

18281

76154

163

1.355

0. 97680

61901

82927

27405

609

0. 21412

53530

53567

46271

899

.356

. 97701

98270

97238

89325

386

.21314

84399

63517

9.5410

772

.357

. 97723

24869

91804

83352

894

. 21217

13137

25046

24434

790

.358

.97744

41696

53965

21803

706

. 21119

397.53

15278

49048

406

.359

. 97765

48748

72037

40225

805

. 21021

64257

11553

02083

908

1.360

0. 97786

46024

35316

18567

849

0. 20923

86658

91419

35767

598

.361

. 97807

33521

34074

02249

690

. 20826

06968

32637

23964

842

.362

. 97828

11237

59561

23135

125

. 20728

25195

13175

64404

112

.363

. 97848

79171

04006

20406

864

. 20630

41349

11211

80880

089

.364

. 97869

37319

60615

61343

685

. 20532

55440

05130

25435

952

1.365

0. 97889

85681

23574

G1999

774

0. 20434

67477

73521

80524

932

.366

. 97910

24253

88047

07786

196

. 20336

77471

95182

611.51

240

.367

.97930

53035

50175

73954

516

. 20238

85432

49113

16990

457

.368

. 97950

72024

07082

45982

521

. 20140

91369

14517

34489

495

.369

. 97970

81217

56868

39862

027

.20042

95291

70801

38946

217

1.370

0. 97990

80613

98614

22288

769

0. 19944

97209

97572

96568

820

.371

. 98010

70211

32380

30754

328

. 19846

97133

74640

16515

079

.372

. 98030

50007

59206

63540

094

. 19748

95072

82010

52911

545

.373

. 98050

20000

81114

49613

233

. 19650

91036

99890

06852

798

.374

. 98069

80189

01103

68424

652

. 19552

85036

08682

28380

853

1.375

0. 98089

30570

23155

69608

920

0.19454

77079

88987

18444

822

.376

. 98108

71142

52232

42586

155

. 19356

67178

21600

30840

918

.377

. 98128

01903

94276

66065

826

. 19258

55340

87511

74132

912

.378

. 98147

228.52

56212

27452

479

.19160

41577

67905

13553

129

.379

. 98166

33986

45944

42153

343

.19062

25898

44156

72884

094

74

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. XIV,

Table XII. — Values of sin x and cos x to 23 places of decimals itt intervals of 0.001 from 0.000

to J. 600— Continued.

X

sin X

cosx

1.380

0.98185 35303 72359 72787 813

0.18964 08312 97834 36320 915

.381

.98204 26802 45326 48298 791

.18865 88831 10696 50314 508

.382

.98223 08480 75694 82965 850

.18767 67462 64691 25395 757

.383

.98241 80336 75296 95320 221

.18669 44217 41955 37980 715

.384

.98260 42368 56947 26961 571

.18571 19105 24813 32156 930

1.385

0.98278 94574 34442 61276 561

0.18472 92135 95776 21451 016

.386

.98297 36952 22562 42059 162

.18374 63319 37540 90577 542

.387

.98315 69500 37068 92032 708

.18276 32665 32988 97169 360

.388

.98333 92216 94707 31273 673

.18178 00183 65185 73489 451

.389

.98352 05100 13205 95537 148

.18079 65884 17379 28124 404

1.390

0.98370 08148 11276 54484 004

0.17981 29776 72999 47659 616

.391

.98388 01359 08614 29809 722

.17882 91871 15656 98336 311

.392

.98405 84731 25898 13274 870

.17784 52177 29142 27690 484

.393

.98423 58262 84790 84637 207

.17686 10704 97424 66173 860

.394

.98441 21952 07939 29485 405

.17587 67464 04651 28756 976

1.395

0.98458 75797 18974 56974 360

0.17489 22464 35146 16514 467

.396

.98476 19796 42512 17462 083

.17390 75715 73409 18192 681

.397

.98493 53948 04152 20048 145

.17292 27228 04115 11759 690

.398

.98510 78250 30479 50013 670

.17193 77011 12112 65937 830

.399

.98527 92701 49063 86162 846

.17095 25074 82423 41718 833

1.400

0.98544 97299 88460 18065 947

0.16996 71429 00240 93861 675

.401

.98561 92043 78208 63203 840

.16898 16083 50929 72373 233

.402

.98578 76931 48834 84013 966

.16799 59048 20024 23971 842

.403

.98595 51961 31850 04837 776

.16701 00332 93227 93533 854

.404

.98612 17131 59751 28769 609

.16602 39947 56412 25523 303

1.405

0.98628 72440 66021 54406 982

0.16503 77901 95615 65404 770

.406

.98645 17886 85129 92502 294

.16405 14205 97042 61039 544

.407

.98661 53468 52531 82515 912

.16306 48869 47062 64065 184

.408

.98677 79184 04669 09070 631

.16207 81902 32209 31258 571

.409

.98693 95031 78970 18307 486

.16109 13314 39179 25882 568

1.410

0.98710 01010 13850 34142 909

0.16010 43115 54831 19016 356

.411

.98725 97117 48711 74427 198

.15911 71315 66184 90869 577

.412

.98741 83352 23943 67004 304

.15812 97924 60420 32080 359

.413

.98757 59712 80922 65672 895

.15714 22952 24876 44997 336

.414

.98773 26197 62012 66048 706

.15615 46408 47050 44945 751

1.415

0.98788 82805 10565 21328 142

0.15516 68303 14596 61477 752

.416

.98804 29533 70919 57953 120

.15417 88646 15325 39606 967

.417

.98819 66381 88402 91177 144

.15319 07447 37202 41027 471

.418

.98834 93348 09330 40532 586

.15220 24716 68347 45317 231

.419

.98850 10430 81005 45199 170

.15121 40463 97033 51126 135

1.420

0.98865 17628 51719 79273 627

0.15022 54699 11685 77348 698

.421

.98880 14939 70753 66940 521

.14923 67432 00880 64281 559

.422

.98895 02362 88375 97544 222

.14824 78672 53344 74765 840

!423

.98909 79896 55844 40562 021

.14725 88430 57953 95314 499

.424

.98924 47539 25405 60478 351

.14626 96716 03732 37224 747

1.425

0.98939 05289 50295 31560 129

0.14528 03538 79851 37675 648

.426

.98953 53145 84738 52533 174

.14429 08908 75628 60810 986

.427

.98967 91106 83949 61159 714

.14330 12835 80526 98807 514

.428

.98982 19171 04132 48716 941

.14231 15329 84153 72928 666

.429

.98996 37337 02480 74376 619

.14132 16400 76259 34563 848

1.430

0.99010 45603 37177 79485 729

0.14033 16058 46736 66253 390

.431

.99024 43968 67397 01748 121

.13934 14312 85619 82699 275

.432

.99038 32431 53301 89307 176

.13835 11173 83083 31761 733

.433

.99052 10990 56046 14729 460

.13736 06651 29440 95441 799

.434

.99065 79644 37773 88889 346

.13637 00755 15144 90849 940

1.435

0.99079 38391 61619 74754 605

0.13537 93495 30784 71160 849

.436

.99092 87230 91709 01072 941

.13438 84881 67086 26554 495

.437

.99106 26160 93157 75959 459

.13339 74924 14910 85143 546

.438

.99119 55180 32073 00385 060

.13240 63632 65254 13887 244

.439

.99132 74287 75552 81565 735

.13141 51017 09245 19491 852

No. 6.)

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

7S

Table XII. — Values of sin x and cos x to 23 places of denmals at intervals of 0.001 from 0.000

to 1.600 — Continued.

X

sin X

cosx

1.440

0.99145 S3481 91686 46252 760

0.13042 37087 38145 49297 752

.441

.99158 82761 49554 53923 766

.12943 21853 43347 92153 306

.442

.99171 72125 19229 09874 676

.12844 05325 16375 79275 676

.443

.99184 51571 71773 78212 505

. 12744 87512 48881 85098 002

.444

.99197 21099 79243 94748 990

.12645 68425 32647 28105 135

1.445

0.99209 80708 14686 79795 055

0.12546 48073 59580 71654 525

.446

.99222 30395 52141 50856 088

.12447 26467 21717 24785 871

.447

.99234 70160 66639 35228 024

.12348 03616 11217 43017 513

.448

.99247 00002 34203 82494 21G

.12248 79530 20366 29130 391

.449

.99259 19919 31850 76923 086

.12149 54219 41572 33939 548

1.450

0.99271 29910 37588 49766 535

0.12050 27693 67366 57053 287

.451

.99283 29974 30417 91459 118

.11950 99962 90401 47620 080

.452

.99295 20109 90332 63717 946

.11851 71037 03450 05003 327

.453

.99307 00315 98319 11543 325

.11752 40925 99404 79804 068

.454

.99318 70591 36356 75120 114

.11653 09639 71276 73971 735

1.455

0.99330 30934 87418 01619 777

0.11553 77188 12194 42103 061

.456

.99341 81345 35468 56903 143

.11454 43581 15402 91829 237

.457

.99353 21821 65467 37123 830

.11355 08828 74262 84551 407

.458

.99364 52362 63366 80232 355

.11255 72940 82249 36104 618

.459

.99375 72967 16112 77380 893

.11156 35927 32951 17410 313

1.460

0.99386 83634 11644 84228 683

0.11056 97798 20069 55117 465

.461

.99397 84362 38896 32148 075

.10957 58563 37417 32232 463

.462

.99408 75150 87794 39331 194

.10858 18232 78917 88737 835

.463

.99419 55998 49260 21797 223

.10758 76816 38604 22199 915

.464

.99430 26904 15209 04300 286

.10659 34324 10617 88365 556

1.465

0.99440 87866 78550 31137 923

0.10559 90765 89208 01747 983

.466

.99451 38885 33187 76860 141

.10460 46151 68730 36201 884

.467

.99461 79958 74019 56879 043

.10361 00491 43646 25487 846

.468

.99472 11085 96938 37979 012

.10261 53795 08521 63826 230

.469

.99482 32265 98831 48727 437

.10162 06072 58026 06440 584

1.470

0.99492 43497 77580 89785 993

0.10062 57333 86931 70090 698

.471

.99502 44780 32063 44122 430

.09963 07588 90112 33595 391

.472

.99512 36112 62150 87122 898

.09863 56847 62542 38345 147

.473

.99522 17493 68709 96604 762

.09764 05119 99295 88804 678

.474

.99.531 88922 53602 62729 932

.09664 52415 95545 53005 525

1.475

0.99541 50398 19685 97818 664

0.09564 98745 46561 63028 806

.476

.99551 01919 70812 46063 854

.09465 44118 47711 15478 186

.477

.99560 43486 11829 93145 787

.09365 88544 94456 71943 189

.478

.99569 75096 48581 75747 356

09266 32034 82355 59452 948

.479

.99578 96749 87906 90969 720

.09166 74598 07058 70920 484

1.480

0.99588 08445 37640 05648 408

0.09067 16244 64309 65577 623

481

.99.597 10182 06611 65569 851

.08967 56984 49943 69400 641

.482

.99606 01959 04648 04588 337

08867 96827 59886 75526 752

.483

.99614 83775 42571 53643 374

.08768 35783 90154 44661 519

.484

.99623 55630 32200 49677 461

.08668 73863 36851 05477 303

1.485

0.99632 17522 86349 44454 246

0.08569 11075 96168 55002 845

.486

.99640 69452 18829 13277 079

.08469 47431 64385 59004 070

.487

.99649 11417 44446 63607 933

08369 82940 37866 52356 240

488

.99657 43417 79005 43586 693

.08270 17612 13060 39407 518

.489

.99665 65452 39305 50450 815

.08170 51456 86499 94334 076

1.490

0.99673 77520 43143 38855 320

0.08070 84484 54800 61486 832

.491

.99681 79621 09312 29093 143

.07971 16705 14659 55729 907

.492

.99689 71753 57602 15215 811

.07871 48128 62854 62770 926

.493

.99697 53917 08799 73054 448

.07771 78764 96243 39483 234

.494

.99705 26110 84688 68141 099

.07672 08624 11762 14220 152

1.495

0.99712 88334 08049 63530 364

0.07.572 37716 06424 87121 354

.496

.99720 40586 02660 27521 334

.07472 66050 77322 30411 478

.497

.99727 82865 93295 41279 821

.07372 93638 21620 88691 060

.498

.99735 15173 05727 06360 877

.07273 20488 36561 79219 898

.499

.99742 37506 66724 52131 595

.07173 46611 19459 92192 943

76

MEMOIRS NATIONAL ACADEMY OF SCIENCES.

[Vol. SIV,

Table XII. — Vahws of sin x and cos x to 23 places of decimals al intervals of 0.001 from 0.000

to 1.600 — Continued.

X

sin X

cosx

1.500

0. 99749

49866

04054

43094

172

0. 07073

72016

67702

91008 819

.501

. 99756

52250

46480

86109

251

. 06973

96714

78750

12531 065

.502

. 99763

44659

23765

37519

509

. 06874

20715

50131

67342 208

.503

. 9P770

27061

66667

10173

501

. 06774

44028

79447

39990 761

.504

. 99776

99547

06942

80349

750

. 06674

66664

64365

89231 245

1.505

0. 99783

62024

77346

94581

063

0. 06574

88633

02623

48257 343

.506

. 99790

14524

11631

76379

092

. 06475

09943

92023

24928 268

.507

. 99796

57044

44547

32859

104

. 06375

30607

30434

01988 470

.508

. 99802

89585

11841

r.1264

976

. 06275

50633

15789

37280 758

.509

. 99809

12145

50260

55394

397

. 06175

70031

46086

63952 953

1.510

0. 99815

24724

97548

11924

274

0. 06075

88812

19385

90658 160

.511

. 99821

27322

92446

36636

332

. 05976

06985

33809

01748 769

.512

. 99827

19938

74695

50542

912

. 05876

24560

87538

57464 281

.513

. 99833

02571

85033

95912

947

. 05776

41548

78816

94113 053

.514

. 99838

75221

65198

42198

118

. 05676

57959

05945

24248 072

1.515

0. 99844

37887

57923

91859

188

0. 05576

73801

67282

36886 851

.516

. 99849

90569

06943

86092

495

. 05476

89086

61243

97425 545

.517

. 99855

33265

56990

10456

612

. 05377

03823

86301

48297 399

.518

. 99860

65976

53793

00399

163

. 05277

18023

40981

08625 609

.519

. 99865

88701

44081

46683

784

. 05177

31695

23862

74620 716

1.520

0. 99871

01439

75583

00717

231

0. 05077

44849

33579

19672 613

.521

. 99876

04190

97023

79776

634

. 04977

57495

68814

94487 284

.522

. 99880

96954

58128

72136

872

. 04877

69644

28305

27218 360

.523

. 99885

79730

09621

42098

089

.04777

81305

10835

23593 598

.524

. 99890

52517

03224

34913

328

.04677

92488

15238

67036 388

1.525

0. 99895

15314

91658

81616

285

0. 04578

03203

40397

18782 371

.526

. 99899

68123

28645

03749

180

. 04478

13460

85239

17991 291

.527

. 99904

10941

68902

17990

729

. 04378

, 23270

48738

81854 166

.528

. 99908

43769

68148

40684

234

. 04278

32642

29915

05695 871

.529

. 99912

66606

83100

92265

762

. 04178

41586

27830

63073 262

1.530

0. 99916

79452

71476

01592

427

0. 04078

50112

41591

05868 899

.531

. 99920

8230S

91989

10170

755

. 03978

58230

70343

64380 513

.532

. 99924

75169

04354

76285

152

. 03878

65951

13276

47406 277

.533

. 99928

58038

69286

79026

436

. 03778

73283

69617

42326 008

.534

. 99932

30915

48498

22220

463

. 03678

80238

38633

15178 390

1.535

0. 99935

93799

04701

38256

819

0. 03578

86825

19628

10734 312

.536

. 99939

46689

01607

S1817

592

. 03478

93054

11943

52566 435

.537

. 99942

89585

03928

83506

202

. 03378

98935

14956

43115 073

.538

. 99946

22486

77374

53376

306

. 03279

04478

28078

63750 505

.539

. 99949

45393

88654

84360

752

. 03179

09693

50755

74831 796

1.540

0. 99952

58306

05479

05600

596

0. 03079

14590

82466

15762 248

.541

. 99955

61222

96555

95674

180

. 02979

19IS0

22720

05041 568

.542

. 99958

54144

31593

85726

242

. 02879

23471

71058

40314 858

.543

. 99961

37069

81300

62497

095

. 02779

27475

27051

98418 526

.544

. 99964

09999

17383

71251

832

. 02679

31200

90300

35423 217

1.545

0. 99966

72932

12550

18609

586

0. 02579

34658

60430

86673 867

.546

. 99969

25868

40506

75272

821

. 02479

37858

37097

66826 971

.547

. 99971

68807

75959

78656

660

. 02379

40810

19980

69885 184

.548

. 99974

01749

94615

35418

249

. 02279

43524

08784

69229 328

.549

. 99976

24694

73179

23886

150

. 02179

46010

03238

17647 934

1.550

0. 99978

37641

89356

96389

761

0. 02079

48278

03092

47364 391

.551

. 99980

40591

21853

81488

767

. 01979

50338

08120

70061 827

.552

. 99982

33542

50374

86102

606

. 01879

52200

18116

76905 802

.553

. 99984

16495

55624

97539

966

. 01779

53874

32894

3S564 929

.554

. 99985

89450

19308

85428

298

. 01679

55370

52286

05229 507

1.555

0. 99987

52406

24131

03543

342

0. 01579

56698

76142

06628 284

.556

. 99989

05363

53795

91538

676

. 01479

57869

04329

52043 433

.557

. 99990

48321

93007

76575

277

. 01379

58891

36731

30323 849

.558

. 99991

81281

27470

74851

093

. 01279

59775

73245

09896 874

.559

. 99993

04241

43888

93030

623

. 01179

60532

13782

38778 533

No. 5.)

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

77

Table XII. — X^alues nf sin x and cos x to IS places of decimals at intervals n/O.nOl from 0.000

to /. 600— Continued.

I

sinx

cosx

1.5G0

0. 99094

17202

29966

29574

517

0. 01079

61170

58267

44582 392

.561

. 99995

20163

74406

75969

172

. 00979

61701

06636

34527 146

.562

. 99996

13125

66914

17856

344

.00879

62133

58835

95443 014

.563

. 99996

96087

98192

36062

758

.00779

62478

14822

93777 062

.564

. 99997

69050

59945

07529

731

.00679

62744

74562

75597 546

1.565

0. 99998

32013

44876

06142

794

0. 00579

62943

38028

66597 372

.566

. 99998

84976

46689

03461

318

. 00479

63084

05200

72096 784

.567

. 99999

27939

60087

69348

142

.00379

63176

76064

77045 359

.568

. 99999

60902

80775

72499

201

.00279

63231

50611

46023 436

.569

. 99999

83866

05456

80873

162

. 00179

63258

28835

23243 059

1.570

0. 99999

96829

31834

62021

053

+0. 00079

63267

10733

32548 541

.571

. 99999

99792

58612

83315

895

-0. 00020

36732

03695

22583 254

.572

. 99999

92755

85495

12082

337

.00120

30729

14450

59042 804

.573

. 99999

75719

13185

15626

285

.00220

36714

21533

14087 901

.574

. 99999

48682

43386

61164

539

.00320

36677

24944

45343 613

1.575

0. 99999

11645

78803

15654

423

-0. 00420

36608

24688

30802 109

.576

. 99998

64609

23138

45523

419

. 00520

36497

20771

68822 280

.577

. 99998

07572

81096

16298

798

.00620

36334

13205

78129 029

.578

. 99997

40536

58379

92137

261

.00720

36109

02006

97812 142

.579

. 99996

63500

61693

35254

568

.00820

35811

87197

87324 647

1.580

0. 99995

76464

98740

05255

179

-0. 00920

35432

68808

26480 539

.581

. 99994

79429

78223

58361

895

. 01020

34961

46876

15451 796

.582

. 99993

72395

09847

46545

499

. 01120

34388

21448

74764 568

.583

. 99992

55361

04315

16554

408

. 01220

33702

92583

45294 454

.584

. 99991

28327

73330

08844

324

. 01320

32895

60348

88260 743

1.585

0. 99989

91295

29595

56407

893

-0. 01420

31956

24825

85219 553

.586

. 99988

44263

8G814

83504

374

. 01520

30874

86108

38055 737

.587

. 99986

87233

59691

04289

313

. 01620

29641

44304

68973 475

.588

. 99985

20204

63927

21344

232

. 01720

28245

99538

20485 440

.589

. 99983

43177

16226

24106

322

• . 01820

26678

51948

55400 452

1.590

0. 99981

56151

34290

87198

158

-0. 01920

24929

01692

56809 503

.591

. 99979

59127

36823

68657

422

. 02020

22987

48945

28070 065

.592

. 99977

52105

43527

08066

646

. 02120

20843

93900

92788 583

.593

. 99975

35085

75103

24582

972

. 02220

18488

36773

94801 039

.594

. 99973

08068

53254

14867

933

. 02320

15910

77799

98151 502

1.595

0. 99970

71054

00681

50917

259

-0. 02420

13101

17236

87068 552

.596

. 99968

24042

41086

77790

702

. 02520

10049

55365

65939 492

.597

. 99965

67033

99171

11241

891

. 02620

06745

92491

59282 234

.598

. 99963

00029

00635

35248

219

. 02720

03180

28945

11714 764

.599

. 99960

23027

72179

99440

759

. 02819

99342

65082

87922 093

1.600

0. 99957

36030

41505

16434

211

-0. 02919

95223

01288

72620 577

78 MEMOIRS NATIONAL ACADEMY OF SCIENCES. tvouxiv.

Table XIII. — Values of sin x and cos x to 25 places of decimals at decimal intervals from IXIO-'" to 9X^0-*.

1 XlO-'o

2

3

4

5

6

7

8

9

ixio-».

2

3

4

5

6

7

8

9

ixio-«.

2

3

4

5

6

7

8

9

1X10-'.

2

3

4

5

6 ,

7

8

9 ,

ixio-«.

2

3

4

5

6

7

8

9

ixio-*.

2

3

4

5

6

7

8

9

ixio-^.

2

3

4

5

6

7

8

9

0.00000 00001 00000 00000 00000

.00000 00002 00000 00000 00000

.00000 0000,3 00000 00000 00000

.00000 00004 00000 00000 00000

.00000 00005

.00000 00006

. 00000 00007

. 00000 00008

. 00000 00009

0.00000 00010

. 00000 00020

. 00000 000.30

. 00000 00040

. 00000 00050

. 00000 00060

. 00000 00069

.00000 00079

. 00000 00089

0. 00000 00099

.00000 00199

. 00000 00299

. 00000 00399

. 00000 00499

.00000 00599

. 00000 00699

. 00000 00799

. 00000 00899

0.00000 00999

.00000 01999

. 00000 02999

. 00000 03999

. 00000 04999

. 00000 05999

.00000 06999

. 00000 07999

.00000 08999

0. 00000 09999

. 00000 19999

. 00000 29999

. 00000 39999

.00000
. 00000
.00000
. 00000
. 00000

0. 00000
. 00001
. 00002
. 00003

. 00004
. 00005
. 00006
. 00007
.00008

0. 00009
. 00019
. 00029
. 00039

.00049
. 00059
. 00069
. 00079
.00089

49999
59999
69999
79999
89999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99998

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99998
99995
99989

99979
99964
99942
99914
99878

99833
98666
95500
89333

79166
64000
42833
14666
78500

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99998

99997
99996
99994
99991
99987

99983
99866
99550
98933

97916
96400
94283
91466
87850

83333
66666
50000
33333

16666
00000
83333
66666
50000

33333
66666
00002
33341

66692
00064
33473
66939
00492

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
99999
99999
99999

99998
99987
99955
99893

99792
99640
99428
99147
98785

98333
86667
55000
93333

91667
40000
28333
46667
85000

33333
66667
00000
33333

66667
00000
33333
66667
00000

33333
66667
00002
33342

66693
00065
33473
66940
00492

34167
93333
02500
86667

70833
80000
39167
73333
07499

0. 99999
. 99999
. 99999
. 99999

. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999

. 99999
. 99999
. 99999

. 99999
. 99999
. 99999
. 99999
. 99999

0. 99999
. 99999
. 99999
. 99999

.99999
. 99999
. 99999
. 99999
. 99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99998
99995
99992

99987
99982
99975
99968
99959

99950
99800
99550
99200

98750
98200
97550
96800
95950

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99999
99999
99999
99999
99999

99999
99999
99999
99999

99998
99998
99997
99996
99995

99995
99980
99955
99920

99875
99820
99755
99680
99595

99500
98000
95500
92000

87500
82000
75500
68000
59500

50000
00000
50000
00000

50000
00000
50000
00000
50000

00000
00000
00000
00001

00002
00005
00010
00017
00027

99999
99998
99995
99992

99987
99982
99975
99968
99959

99950
99800
99550
99200

98750
98200
97550
96800
95950

95000
80000
55000
20000

75000
20000
55000
80000
95000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00000
00003
00010

00026
00054
00100
00170
00273

00416
06666
33749
06666

60416
39999
00416
06666
33749

50000
00000
50000
00000

50000
00000
50000
00000
50000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00000
00000
00000

00000
00000
00000
00000
00000

00000
00007
000.34
00107

00260
00540
01000
01707
02734

04167
66667
37500
66667

04167
00000
04167
66667
37500

66667
66666
99990
66610

66450
99352
65033
63026
92619

No. 6.]

TABLES OF EXPONENTIAL FUNCTION— VAN ORSTRAND.

79

Table XIV. — Miscellaneous values of ei, c-*, nnx and cos x to a great number of decimals,

including Boorman's value ofe.

f«-'=l. 10517

09180

75647

62481

17078

26490

24666

82245

47194

73751

87187

92863

28944

09679

66747

65430

29891

43318

97074

86536

32917

12048

54012

44536

e-o.'=0. 90483

74180

35959

57316

42490

59446

43662

11947

05360

98040

09520

56257

31705

57799

65344

24836

10125

03446

03609

04572

38478

74531

46483

18498

sin 0.1=0. 09983

34166

46828

15230

68141

98410

62202

69899

15388

01798

22599

92766

86156

16517

44283

29242

76096

62443

80406

30362

67832

50318

09359

89035

C08 0.1=0. 99500

41652

78025

76609

55619

87803

87029

48385

76225

41508

40359

59352

74468

52659

10218

24046

65296

63618

52826

29279

10723

68588

08368

71860

e+'-»=2. 71828

18284

59045

23536

02874

71352

60249

77572

47093

69995

95749

66967

62772

40766

30353

54759

45713

82178

52516

64274

27466

39193

20030

59921

81741

35966

29043

57290

03342

95260

59563

07381

32328

62794

34907

63233

82988

07531

95251

01901

15738

34187

93070

21540

89126

94937

99405

34631

93819

87250

90567

36251

50082

37715

27509

03586

67692

05047

15575

85094

92906

45748

86005

84299

93465

94757

59371

00435

26480

0

e-'-<'=0. 36787

94411

71442

32159

55237

70161

46086

74458

11131

03176

78345

07836

80169

74614

95744

89980

33571

47274

34591

96437

46627

Bin 1.0=0. 84147

09848

07896

50665

25023

21630

29899

96225

63060

79837

10656

72751

70999

19104

04391

23966

89486

39743

54305

26958

54349

COB 1.0=0. 54030

23058

68139

71740

09366

07442

97660

37323

10420

61792

22276

70097

25538

11003

94774

47176

45179

51856

08718

30893

43572

o

MEMOIBS

NATIONAL ACADEMY OF SCIENCES

y~ob'

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