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TRANSACTIONS

CAMBRIDGE

PHILOSOPHICAL SOCIETY.

VOLUME XIII.

ep oo)

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M.pDccc. LXX XIII.

Cambridge:
PRINTED BY C. J. CLAY, M.A. & SONS,
AT THE UNIVERSITY PRESS.

¢ VLEMI ESTA MAS
aR S.A en oy Ald £83
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STINE 1 AVI Ce VeRO A WL Ae) eh ab |
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An avin a

Cayury, Prof. A.

=20, 1—4

EEX TO Vous x TM:

Table of A”O"+T(m) up to m=n

Cavey, Prof. A. On the Schwarzian Derivative, and
the Polyhedral Functions, 5—68
Bibliography 6; Part I. 8—35. Part II. 35—68

Sei’.
§§ 8-15.

§§ 16—20.
§§ 21—45.
§§ 46—62,

§§ 6368.

Part II.

§§ 69—80.
§§ 81—84.
§§ 85—86.

§§ 87—93.

§§ 94—95.

§§ 96—103.
§§ 104—117.

§§ 118—127.
§§ 128—134.

The Derivative {s, x}, 8—9

The Quadric Function of three or
more Inverts, 10—12

The functions P, @, Rk, 12—13

The PQR-Table, and Annex, 14—15

The Differential Equations {z, z} and
{s, x}, 16—25

The Schwarzian Theory, 25—32

Connection with the differential
equation for the hypergeometric
series, 32—35

The Polyhedral Functions, 35

Origin and Properties, 35—39

Covariantive Formule, 39—40

Investigation of the forms f5 and
hs, 41—42

Invariantive property of the Stereo-
graphic Projection, 42—44

Groups of homographic transforma-
tions, 44—45

The Regular Polydedra, 45—50

The groups of homographic trans-
formations, resumed, 51—61

The system of 15 circles, 61—65

The Regular Polyhedra as solid
figures, 65

Cox, Homersham. On the Application of Quaternions
and Grassmann’s Ausdehnungslehre to different
kinds of Uniform Space, 69—143

Addition of Points, 69

Determination of Distance, 77

Multiplication of Points, 83

Determination of Angles, 88

Multiplication of Lines meeting at a Point, 90
General expression for Ratios of Lines and Points,

Formule in Coordinates, 97
Relations between the Sides and Angles of a Tri-

angle, 99

The different kinds of Uniform Space, 101
Imaginary Geometry of Three Dimensions, 104
Spherical Geometry of Three Dimensions, 110
Ordinary Geometry of Three Dimensions, 111
Spaces of Higher Dimensions, 112

Grassmann’s Ausdehnungslehre. The Outer Mul-

tiplication, 115

Application to Systems of Forces and Linear Com-

plexes, 118

The Regressive Multiplication, 122
The Inner Multiplication, 130
Measure of Distance, 136
Imaginary and Flat Geometry, 141

Exponential Function, tables of (Glaisher), 243
Exponential Function, table of descending (Newman),
145

Functions analogous to Tesseral Harmonics (Hill), 273

INDEX.

GuaisHer, J. W. L. Tables of the Exponential Func-
tion, 243—272
Tables of e, e~*, log,e* and log, ye
Table I. From «=0°001 to «=0°100 at intervals
of 0-001, 254—255
Table II. From x=0°01 to x=2°00 at intervals
of 0°01, 256—259
Table III. From x=0:1 to x=10°0 at intervals of
O11, 260—261
Table IV. From x2=1 to #«=500 at intervals of
unity, 262—271
Comparison with Schulze’s table, no errors, 272
< » Vega’s table, errors, 272
Other existing Tables, Schulze, 243
= 5 » Vega, Kohler, Gudermann,
244
Grassmann’s Ausdehnungslehre, applications of (Cox),
69, 115

Hi, M. J. M. On Functions of more than two
variables analogous to Tesseral Harmonics, 273—299

Newman, F. W. Table of the Descending Exponential
Function to Twelve or Fourteen Places of Decimals,
145—241

Part I. From 2=0 to 2=15'349, at intervals of
001 to twelve decimal places, 151—227

Part II. From #=15'350 to «=17'298 at inter-
vals of ‘002, and from w2=17'300 to «=27°635
at intervals of °005 to fourteen decimal places,
228—241

Polyhedral Functions, the (Cayley), 35

Quaternions (Cox), 69

Schwarzian Derivative (Cayley), 5

Table of A"O"+1(m) (Cayley), 1

Tables of e%, e~*, log,e* and log,,e—* (Glaisher), 243
Table of e * (Newman), 145

Tesseral Harmonics, functions analogous to (Hill),
273

CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS.

iVeL,

CONTENTS.

PAGE

PLable of A™O"-—-TI (m) wp to m=n=20. By PROF. CAYLEY ..........00--eceeceeceeceeeecees il

On the Schwarzian Derivative and the Polyhedral Functions. By Prov. Cayury ......... 5
On the Application of Quaternions and Grassmann’s Ausdehnungslehre to different kinds

of Uniform Space. By HomersHam Cox, B.A., Fellow of Trinity College ............... 69
Table of the Descending Exponential Function to Twelve or Fourteen Places of Decimals.

By F. W. Newman, Emeritus Professor of University College, London .................. 145
Tables of the Exponential Function. By J. W. L. GuatsHer, M.A., F.R.S., Fellow of

srittiya® ollewen CAMbTIA Serene onedcacrescenescesesinse o05e-c0e 0s ssc clase sevsceseonctiesonccasessecs 243

On Functions of more than two variables analogous to Tesseral Harmonics. By M. J. M.
15 iti, ELAS Beene see mece EBor ricer COMP OCeOnOO: nde COF CADE E Cee CCDCOCOCRERE ROAD A EcAReACE OBC EMRE AEE 273

ADVERTISEMENT.

Tue Society as a body is not to be considered responsible for any
facts and opinions advanced in the several Papers, which must rest
entirely on the credit of their respective Authors.

Tue Socisry takes this opportunity of expressing its grateful
acknowledgments to the Synpics of the University Press for their
liberality in taking upon themselves the expense of printing this
Volume of the Transactions,

I. Table of A™0"+TI(m) up to m=n=20. By A. Cayzey, Sadlerian Pro-
fessor of Pure Mathematics.

[Read October 27, 1879.]

Tue differences of the powers of zero, A”0", present themselves in the Calculus of Finite
Differences, and especially in the applications of Herschel’s theorem, f(e‘)=f(1+A)é”,
for the expansion of the function of an exponential. A small Table up to A”0” is
given in Herschel’s Examples (Camb. 1820), and is reproduced in the treatise on Finite
Differences (1843) in the Encyclopedia Metropolitana. But, as is known, the successive
differences AO", A?0", A°0",... are divisible by 1, 1.2, 1.2.8,... and generally A”™0” is
divisible by 1.2.3...m,=TII(m); these quotients are much smaller numbers, and it is there-
fore desirable to tabulate them rather than the undivided differences A”"0": it is more-
over easier to calculate them. A Table of the quotients A"0"+II(m), up to m=n=12
is in fact given by Grunert, Crelle, t. xxv. (1843), p. 279, but without any explanation
in the heading of the meaning of the tabulated numbers C,'(=A"0*=II(n)), and without
using for their determination the convenient formula C,'*=nC,'+C,_," given by Bjorling

a

in a paper Crelle, t. XxXvut. (1844), p. 284. The formula in question, say

[Ne gieal A"0" 3 PATOss .

T(m) "IE (m) * I(m—1)’
is given in the second edition (by Moulton) of Boole’s Calculus of Finite Duferences,
(London, 1872), p. 28, under the form

A"0"=m (Arr2 0 ae NEO).
It occurred to me that it would be desirable to extend the table of the quotients
A"0"=II (m), up to m=n=20. The calculation is effected very readily by means of

the foregoing theorem, which is used in the following form; viz. any column of the
table, for instance the fifth, being

A; then following columnis A

B 2B+A
C 3C0+B
D 4D+C
E 5#£+D
+E;
Vou, XIII. Parr I. Li

|

2 Pror. CAYLEY, TABLE OF A"0"+I1(m) UP TO m=n=20.
and then we obtain a good verification by taking the sum of the terms in the new
column, and comparing it with the value as calculated from the formula,

Sum = 244+3B+40+5D+4+62:

observe that in the two calculations we take successive multiples such as 4D and 5D
of each term of the preceding column, and that the verification is thus a safeguard
against any error of multiplication or addition.

Table, No. 1, of A"0"+ II(m).

4 |
=] ol] o}| 0 | ot | oO | os or 08 0° gw gu gle 013 ou
1 Pid 1) ee | ed ae 1 1 1! 1 1 1 1 1
2 1 38/ 7] 15] 31 63 | 127 | 255 511 1 023 2 047 4095 8191
3 1} 6} 25 | 90] 301 966 | 3025 9 330 | 28501 86 526 261 625 788 970
a 1 | 10 | 65 | 350 |1701 |7770 | 34105 | 145 750 611 501 | 2532530 | 10391 745
5 1 15} 140 /1050 |6951 | 42525 | 246730 | 1379400 | 7508 501 40 075 085
+8 1 21 266 | 2646 | 22827 | 179487 | 1323652 | 9321312 | 63 436 373
7 | 1 28 | 462 5 880 | 63 987 627 396 | 5715424 | 49329 280
|; 8] 1 36 750 | 11880 159 027 | 1899612 | 20912320
| 9] 1 45 | 1155 22275 | 359502 | 5135130
| 10 | 1 55 1705 39 325 752 752
1] Bi 66 2431 66 066
12 | 1 78 3 367
13 | 1 91
14 | } 1
15
| 16 | [oe ny
17 |
18 | |
19 |
| 20 |
{
4
5 ou ois ou 018 ye) 020
1 1 ud 1 1 1 1 1
2 16 383 32 767 65 535 131 071 262 143 524287 | 2
3 2 375 101 7 141 686 21 457 825 64 439 010 193 448 101 580 606 446 | 3
4 | 42355 950 171 798 901 694 337 290 2798 806 985 11 259 666 950 45 232115901 | 4
5 | 210766920 | 1096190550 | 5 652 751 651 28 958 095 545 147 589 284 710 749 206 090 500 | 5
6 | 420693273 | 2734926558 | 17 505 749 898 | 110687 251 039 693 081 601779 | 4306078 895 384 | 6
7 | 408 741 333 | 3281 882 604 | 25708 104786 | 197 462483400 | 1492 924 634 839 | 11 143554045 652 | 7
8 | 216627840 | 2141 764053 | 20415995 028 | 189036065010 | 1709 751 003 480 | 15 170 932 662679 | 8
9 | 67128490 820 784 250 | 9528 822303 | 106175395755 | 1144614626 805 | 12011 282644725 | 9
10 | 12662 650 193 754990 | 2758 334 150 37 112 163 803 477 297 033785 | 5917 584964 655 | 10
1] 1 479 478 28 936 908 512 060 978 8 391 004 908 129 413 217 791 1 900 842 429 486 | 11
12 106 470 2757 118 62 022 324 1 256 328 866 23 466 951 300 411 016 633 391 | 12
| 13 4 550 165 620 4910178 125 854 638 2 892 439 160 61 068 660 3880 | 13
| 14 105 6 020 249 900 8 408 778 243 577 530 6 302 524580 | 14
| 15 if 120 7 820 367 200 13 916 778 452 329 200 | 15
16 1 136 9 996 527 136 22 350 954 | 16
17 1 153 12 597 741 285 | 17
18 | 1 171 15 675 | 18
19 | 1 190 | 19
20 | 1 | 20

Pror. CAYLEY, TABLE OF A"0"=II(m) UP TO m=n=20.

Writing down the sloping lines as columns thus:

1 2 3 4 5 6 7 8 etc.
(0) (2) (4) (6) (8) (10) (12) (14) ete.

1

1 1

1 3 1 |

a 6 7 1

i 10 25 15 1

1 15 65 90 31 1

1 21 140 350 - 301 63 il

1 28 266 1 050 1701 966 127

1 36 462 2 646 6951 7770 3 025

1 45 750 | 5880 22 827 | 42 525 34 105

1 55 1155 | 11880 63 987 179 487 246 730

1 66 1705 | 22275 159 027 627 396 1 323 652 |
= 1 2431 | 39325 359 502 1 899 612 5 715 424

1 91 3367 | 66066 752 752 5 135 130 20 912 320

1 105 4550 | 106 470 1 479 478 12 662 650 67 128 490

1 120 6 020 | 165 620 2757118 |. 28 936 908 193 754 990

1 136 7 820 | 249 900 4910178 62 022 324 512 060 978

il 153 9 996 | 367 200 8408 778 | 125 854638 | 1 256 328 866

1 171 | 12597 | 527 136 13 916 77 243 577 530 | 2 892 439 160

1} 190] 15675 | 741285 | 22350954 | 452 329200 | 6 302 524580

17 16 15 14 13 etc,

bo
o
=
©
nH
los}

co

it appears by inspection that in the second column the second differences are constant,
in the third column the fourth differences, in the fourth column the sixth differences,

and so on, are constant; and we thence deduce the law of the numbers in the successive
columns: viz. this can be done up to column 7, in which we have 14 numbers for
taking the 12-th differences: but in column 8 we have only 13 numbers, and therefore

cannot find the 14-th differences. The differences are given in the following

Table, No. 2 (explanation infra).

<4

el Wl Gy se A 5 6 7
|) Teles le an 1 1 1 | 1
1 AN Bi wall a9 62 126
2 1 |12] 611] 240 841 2772
3 10 | 124 | s90| 5060 25 410
4 3] 131 1830] 16990| 127953
5 70 |2226 | 35216 | 401436
6 15 |1600 | 47062 | 836976
7 630 | 40796 | 1196532
8 105 | 21225 | 1182195
9 10930 | 795718

10 945 349020

11 90 090

12 10 395

We have by means of this Table, the general expressions of A’0’, A™'0", A™*0"... up

to A™*0’, viz. the formule are

4 Pror. CAYLEY, TABLE OF A"0"+I1(m) UP TO m=n=20.
A0'+H(r) =1,
va O\e Apes Ww}
A™0' + H(r=1)=1+42(’ ; jail’ : I
ear. f = ~ r—3\ 9 ly — 3\? eee Cat
A 0" + TI (r—2) = 1+ 6 ( ; ) #12 ( : ) +10 el ee
&e., &e.

where the numerical coefficients are the numbers in the successive columns of the table ;

r—m)\*. , 3 ae : [r —m]*
and where for shortness is written to denote the binomial coefficient —-,-— . For
k [k]*

instance, r= 10, we have
A‘0" = II (8) =14+6.74+12.214+ 10.3543. 35, = 750,

agreeing with the principal Table. It will be observed that in the successive columns
of the Table the last terms are 1, 1, 1.3, 1.3.5, 1.3.5.7, 1.38.5.7.9, and 1.3.5.7.9915 This
is itself a good verification: I further verified the last column by calculating from it
the value of A‘“0”=II(14),= 6302524580 as above. The Table shows that we have
A™™0"=II(r—m) given as an algebraical rational and integral function of 7, of the degree
2m. But the terms from the top of a column, AO’=1, A’0'+1.2=2"'-1, &c., are not
algebraical functions of r.

22 October, 1879.

II. On the Schwarzian Derivative, and the Polyhedral Functions. By A. Cayuey,
Sadlerian Professor of Pure Mathematics.

{Read March 8, 1880.]

THE quotient s of any two solutions of a linear partial differential equation of the

2
second order, — pt +qy=0, is determined by a differential equation of the third

order
d's ality =
dz°® dx?
ds (z =-4(p'+2 2-44),
dx dx

where the function on the left hand is what I call the Schwarzian Derivative; or say

this derivative is
le ih 2
{s, a, =5-3(5),

where the accents denote differentiations in regard to the second variable x of the
symbol.

Writing in general (a, b, c ..§X, Y, Z)* to denote a quadric function
(a, b, c, $(a—b-—c), 4 (-—a+b-—oc), 4(—a—b+oe)§X, Y, Z)’,
then, if the equation of the second order be that of the hypergeometric series, generalised

by a homographic transformation upon the variable 2, the resulting differential equation
of the third order is of the form

fs, 2} =(a, b, ¢ |

1 1 1 Y;
z—a’ x£—b’ x—c/}’

and, presenting themselves in connection with the algebraically integrable cases of this
equation, we have rational and integral functions of s, derived from the polygon, the
double pyramid, and the five regular solids, and which are called Polyhedral Functions.

The Schwarzian Derivative occurs implicitly in Jacobi’s differential equation of the
third order for the modulus in the transformation of an elliptic function (Fund. Nova,
1829, p. 79) and in Kummer’s fundamental equation for the transformation of a hyper-
geometric series (Kummer, 1836: see list of Memoirs): but it was first explicitly con-
sidered and brought into notice in the two Memoirs of Schwarz, 1869 and 1873; the

6 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

latter of these (relating to the algebraic integration of the differential equation for the
hypergeometric series) is the fundamental Memoir upon the subject, but the theory is in
some material points completed in the Memoirs by Klein and Brioschi,

The following list of Memoirs relating as well to the Polyhedral Functions as to the

Schwarzian Derivative is arranged nearly in chronological order.

Kummer, Ueber die hypergeometrische Reihe 147 Fes ... Crelle, t. xv. (1836), pp. 39—83
and 127—172.

Schwarz, Ueber einige Abbildungsaufgaben. Crelle-Borchardt, t. Lxx. (1869), pp. 105—120.
Ueber diejenigen Fille in welchen die Gaussische hypergeometrische Reihe eine alge-
braische Function ihres vierten Elementes darstellt. Do. t. uxxv. (1873), pp. 292—336.

Cayley, Notes on Polyhedra. Quart. Math, Jour. t. vir. (1866), pp. 304—316.

On the Regular Solids. Do. t. xv. (1877), pp. 127—131.

Fuchs, Ueber diejenigen Differentialgleichungen zweiter Ordnung welche algebraische Integralen
besitzen, und eine Anwendung der Invariantentheorie. Crelle-Borchardt, t. 81 (1875), pp.
97—142.

Klein, Ueber biniire Formen mit linearen Transformationen in sich selbst. Math. Ann. ¢, 1x.
(1875), pp. 183—209.

Brioschi, Extrait d’une lettre 4 M. Klein. Math. Ann. t. x1. (1877), pp. 111—114.

Klein, Ueber lineare Differential-Gleichungen. Math. Ann. t. x1. (1877), pp. 115—118.

Brioschi, La théorie des formes dans l’intégration des équations différentielles linéaires du second
ordre. Math. Ann. t. x1. (1877), pp. 401—411.

Gordan, Ueber endliche Gruppen linearer Transformationen einer Verinderlichen. Math. Ann.
t. xu. (1877), pp. 23—46.

Biniire Formen mit verschwindenden Covarianten. Math. Ann. t. x1. (1877), pp. 147—166.

Klein, Ueber lineare Differentialgleichungen, Math. Ann. t. x11. (1877), pp. 167—179.

Weitere Untersuchungen iiber das Icosaeder. Math. Ann. t. x11. (1877), pp. 503—560.

Cayley, On the Correspondence of Homographies and Rotations. Math. Ann. t. xv. (1879),
pp. 238—240.

On the finite Groups of linear transformations of a Variable. Math. Ann. t. xvi. (1880),

pp. 260—263, and pp. 439—440.

I propose in the present Memoir to consider the whole theory: and in particular to
give some additional developments in regard to the Polyhedral Functions.

I remark that Schwarz starts with the foregoing differential equation of the third order

1 il aaNe
{s, x} = (a, b, 6] ) (; =R 5 gee A ; =) ;

and he shows (by very refined reasoning founded on the theory of conformable figures,
which will be in part reproduced) that this equation is in fact algebraically inte-
grable for 16 different sets of values of the coefficients a, b, c. It may I think be
taken to be part of his theory, although not very clearly brought out by him, that
these integrals are some of them of the form, z= rational function of s; others of the

AND THE POLYHEDRAL FUNCTIONS. 7

form, rational function of #= rational function of s; the rational functions of s being in
fact the same in these last as in the first set of solutions, and being quotients of
Polyhedral functions.

But as regards the second set of cases, the solution of these (introducing for con-
venience a new variable z in place of s) may be made to depend upon the solution
in the form, x= rational function of z, of an equation of a somewhat similar form, but
involving two quadric functions of 2 and z respectively, viz. the equation

{x, 2}+ (33) (a, b, ¢ ..) ( ! : —, sa) Gabi e; “)(

L%—a

1 1 ise
z—a,’ 2—b,’ =)
and we have the theorem that the solution of this equation depends upon the deter-
mination of P, Q, & rational and integral functions of z (containing each of them multiple
factors) which are such that P+ Q+&=0: (using accents to denote differentiation in regard
to z, this implies P’+ Q + R’=0, and consequently QR’—- QR=RP'—RP=PQ'- PQ):
and are further such that the equal functions QR’— QR, RP’— RP, PQ’—P'Q contain
only factors which are factors of P, Q or R.

In fact, writing f, g, h=b—c, c—a, a—b, the required relation between «, 2 Is
then expressed in the symmetrical form f(#—a) : g(@-—b) : h(@-c)=P: Q: BR.

The last mentioned differential equation is considered by Klein and Brioschi: the
solutions in 13 cases, or such of them as had not been given by Schwarz, were obtained
by Brioschi, and those of the remaining 3 cases (subject to a correction in one of them)
were afterwards obtained by Klein.

The first part of the present Memoir relates, say to the foregoing equation
1 1 ite\e
Cane “)( i

z—a’ @—b’ z—c
although the other form in {z, z} may equally well be regarded as the fundamental

form: and

We consider in the theory :

A. The Derivative {s, x}, meaning as above explained.

: . é 1
B. Quadric functions of any three or more inverts rae

C. Rational and integral functions P, Q, & having a sum =0, and which are

such that QR’-Q’R, = RP'— k'P, = PQ — PQ, contains only the factors of P, Q, R.
D. The differential equation of the third order.

E. The Schwarzian theory in regard to conformable figures and the correspond-
ing values of the imaginary variables s and a.

F. Connection with the differential equation for the hypergeometric series.
The Second part of the Memoir relates to the Polyhedral Functions.

The paragraphs of the whole Memoir are numbered consecutively.

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

is 8]

PART I.
The Derivative {s, 2}, Article Nos. 1 to 7.

he: olf p=" oe (tog $2): then {s, x} = _ 4p

Ss. aa

2. The derivative {s, 2} may be transformed in regard to either or both of the

variables.

Suppose first that s is a function of the new variable S, (hence also S is a function
of x): using subscript numbers to denote differentiations in regard to S, and the accents
as before for differentiations in regard to , we have

SST,
whence, differentiating the logarithms,
iad ”
oat =’ 83 aL S =
7 ;
8 ees}

and again differentiating

But -} (=)= Ss” |- $ ) | - s”: — (=)

and consequently

that is is) cy — (7
the required formula.

In a very similar manner, taking x a function of X, it is shown that

fs o}=(G,) ( X}- fe, Xp.

3. If in this formula we write S for s, and substitute the resulting value of {S, a}
in the former formula, we have

dS\* dX\? dX
fs, 3=(3,) fs, 8}- (7) (a, x}+(]) (8, XI,
which is the formula for the change of both variables, and it in fact includes the
other two: viz. writing X=2, or S=s, and observing that {s, s}={z, «}=0, we have
the other two formule.

4. By putting in the first formula X =s, we obtain

{s, a} = - (2) t@ s},

a formula for the interchange of the variables.

AND THE POLYHEDRAL FUNCTIONS. 9

as +b
cs+d’
in regard to s, we have

5. Writing S=

and using for a moment the accents to denote differentiation
, ad—be 8" —%
(cs+d)?’ 8’ cs+d’

Ge Q” 2 202
and thence gat () “Gra”

-4(%) = — 2c?
2\S'/ ~ (ces+d)*

Consequently {S, s;=0 (whence also {s, S} =0).

Hence in the first formula {S, z}={s, x}, that is

as + b ee laels
es+d’ Te ? 5)

viz. we may in the derivative {s, x} write for s any homographic function (as +0) + (cs + d)
of s.

6. Again if X et then from the second formula

__ (28 — By) :

{8, v} ST GEEaSn » X};
ax+B) (yx+6)* ;
that is {s, ee ree a} 5

and here, changing s into (as+b)+(cs+d), we have finally

as+b ax+B)_ (yx+6)*
{ee seth = Cet a

which is the formula for the homographic transformation of the two variables s, 2.

7. Let s be a given function of x, the equation {S, z}={s, z} is a differential equa-

; : : ; b
tion of the third order in S, and by what precedes, its general integral is S= at q
F F , CG iy
The direct process is as follows: we have a first integral oy ae second
cAs’

integral log S’= log s’ — 2 log (cs + d) + Const., that is S'= 2} and thence a final in-

(cs + d)

tegral S =B-*.,, which is equivalent to the foregoing value of 8.

Vou. XIII. Parr I.

to

10 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

The Quadric Function of three or more Inverts. Art. Nos. 8 to 15.
1 1
a—a’ x£— 6’

them different: it is assumed that the constant term is =0, and also that the sum of

8. We consider a quadrie function of any number of inverts .-» all of

. . val Dg
the coefficients of the linear terms is=0. We have therefore square terms (a —a)’’

product terms : , and linear terms A , where the sum of the coefficients A
x—-a.x—B wa ‘

is =0. Any product term ee is expressible in the form of a difference = aoe

: a of two linear terms, and (the coefficients of these being equal) after it is

a—-B2x-B
thus expressed the sum of the coefficients of the lmear terms is still =0. The function
is thus always in ss aaa in the form
b A B
@—a)t @—Ay tea e—p

where the sum 4+B+... is =0: this may be called the reduced form.

Ss

9. Observe that any particular invert — may disappear altogether from the reduced

form: this will be the case if a=0 (that is if the original form contains no term in =— :

and if also A=0. An invert thus disappearing from the reduced form is said to be
non-essential: and the inverts which do not disappear are said to be essential. The
original form contains in appearance the non-essential inverts, but it is really a quadric
function of the essential inverts only.

10. Imagine the original function expressed as a rational fraction, the denominator
being the product (#—a)*(«—£8)*(x—y)’*... of the squared factors corresponding to all
the inverts (non-essential as well as essential): the numerator will be in general of a
degree less by 2 than that of the denominator, but the coefficients of any one or more
of the higher powers of 2 may vanish, and the numerator will then be of a lower degree.

But this numerator will for any non-essential invert contain the factor (#—v-y)*, or.

dividing the numerator and denominator each by this factor, the difference of the degrees
of the numerator and denominator will remain unaltered; that is the difference will have
the same value whether we do or do not attend to the non-essential inverts; or say it
will have the same value for the original form and for the reduced form.
; . A B C
11. It is to be remarked that the linear terms ——— + + ..., Where
@—-a «-—-B «@-y¥y

A+B+C+...=0, can be (and that in a variety of ways) expressed as a sum of dif-

1 = :
ferences ———~, that is as a sum of product-terms peed ee) . Hence the quadric
z-a «£-B z—-a.2—P

AND THE POLYHEDRAL FUNCTIONS. 11

function can be (and that in a variety of ways) expressed as a homogeneous function

2
(a, fo, reat ms ; we must have in the form all the essential inverts, and we
need have these only. Supposing that this is so, and that the number of the essential
inverts is =n, then the number of constants is =4”(n+1), whereas the number of con-
stants in the reduced form is only =2n—1: hence the coefficients are not determinate;
or, what is the same thing, we may have different quadric functions having each of them
the same reduced function; these quadric functions, as having the same reduced function,
can only differ by multiples of the evanescent expressions

le + a= 8 &e.

a—-B.x2-y “&-y.2-a aw—a.a—f’

In particular if the number of essential inverts is =8, then the quadric function is of
1 1 ee \e ; :
the form (a, oy (Grater ni—, —;, =.) , Which contains one superfluous constant,
z—a’ «—-B’ x-y¥
and equivalent functions differ only by a multiple of
B-¥ fis a—B
PIs cay 29 ee Saree

12. A quadric function such that the degree of the numerator is less by 4 than
that of the denominator is said to be “curtate.”

2
The conditions in order that the function (a by c) £ enh : ; E E ) may
a—a’ «—pP’ w-y
be curtate are easily found to be
at+tb+c+ 2f+ 2¢+2h=0,
a(at+h+g)+A(h+b+f)+y(g+f+o=0;

and by reason of the superfluous constant we are at liberty to assume a third condition:
the three conditions may be taken to be ath+g, h+b+f, g+f+e each=0; and this
being so the values of f, g, h are =}(a—b—c), }(-a+b-—c), }(-a—b+c) respectively.
Hence the form is

(a, fe y(a—b—c), }(—a+b-—c), }(-a-b+of——, a —

which, as already mentioned, we denote by

(a b, ¢ YH: = 2 =).

We have thus the theorem that a curtate function of any number of inverts, but with

only the three essential inverts : ; :
a—a’ «—B’ 2-y¥

b aif 1 1 1 )
(a , Coe Z—a’ z— Bp’ Ly 2

is always expressible in the foregoing

form

12 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

13. It may be remarked that the function (a, b, c ..YY, Y, 7)’ is a function of
the differences of the variables X, Y, Z; and similarly in the case of four variables a
function (a, b, c, d, f, g, h, 1, m, n¥X, Y, Z, W)* for which a+h+g+l, h+b+f+m,
g+f+e+n, l1+m+n+d are each =0 is a function of the differences of the variables
X, Y, Z, W: and so in general. Any such function is said to be “diaphoric:” and it
is easy to see that, taking for the variables any inverts whatever, a diaphoric function
is always curtate.

14. The function

(eae Ga

why 8 | a b c le
tude eae amen
where the coefficients a, b, c... satisfy the relation at+b+e+...=—2, is diaphoric, and
therefore curtate. In fact forming the sum, coeff. = + 4 coeff. aes +..., this is
(x —a) x—a.xz—B
—a—ta’—fab—fac..., =—f$a(2+a+b+c...) which is =0; and similarly the other

conditions are satisfied.

15. The function

b, c c =
+... +4...) ;
a— B, SY Be

1 1

regarded as a function of the inverts Z «where
xZ—-a’ £-a z—B

a a b
, b, ¢ ..() —— +—*+_ + ..., ——
(0 : a—a'a—a,t -a2—-B

at+a,+...=b+b,4+...=c+¢,+...,=k suppose,

1S

is diaphoric, and therefore curtate. In fact the condition in regard to
“—a@

a(a*+aa,+aa,+...) +4 (—a+b—c) (ab+ab,+...)+4(—a—b+c) (ac+ac,+...)=0;

that is
ak {a+}4(—a+b—c)+4(—a—b+c)} =0,

which is satisfied, And similarly the other conditions are satisfied.

The functions P, Q, R. Article, Nos, 16 to 20.

16. We consider P, Q, &, rational and integral fractions of z, such that P+Q4+R=0:
hence, using the accent to denote differentiation in regard to z, we have also P’+ Q’+R’=0:
and therefore QR’—-QR=RkP’—-RP=PQ-—P’Q,=© suppose: and we require to find
P, Q, R, such that the function © contains only the factors of P, Q, R.

AND THE POLYHEDRAL FUNCTIONS. 13

17. It is to be observed that, effecting upon a solution P, Q, R any linear substi-
tution (22+ 8)+(yz+6), and omitting the common denominator, we have a solution ; but
this is regarded as identical with the original solution. The three functions, if not origin-
ally of the same order, can thus be made to be of the same order; or by taking account
of the root z=00, we may in the original case regard them as being of the same order,
and it is convenient so to regard them: say they are taken to be of the same order 8.
And there is clearly no loss of generality in taking the three functions to be prime to
each other; for any common factor of two of them would divide the third, and might
therefore be struck out.

18. We may therefore write
P=FII(z—-J)?, Q = GIL (2 — m)4, R= HII (z—ny’,

where (z—J)? is taken to denote the distinct simple or multiple factors of P, and the
like as regards Q and R; the factors z—/, z—m, z—n are thus all of them different.
And we have = 2p, =Xq, ==r. :

19. It is at once seen that © is of the degree 28—2, and moreover that it con-
tains the factors II (¢—J)?", Il (g—m)*, II (z—n)"*; hence it contains the factor
I (2 —1)?* (g—m)** (g—n)™.
Suppose the number of distinct indices p is =o,, that of distinct indices g is o,, and
that of distinct indices r is =o,; then the degree of the factor is =38—o¢,-—o,-o,;
and if this be =26—2, then © can have no other variable factor: viz. if the numbers

o,, %, 0, of the distinct indices p, q, r respectively are such that o,+o0,+¢,=8+2 (a
relation which is henceforth taken to be satisfied), then we have

O= KT (z— l)P* (2 —m)" (z—n)™7

As already in effect remarked the conclusion extends to the case where P, Q, R are not
of the same degree; the equation P+Q+R=0 here implies that two functions, say P, Q,
are of the same degree, and the third function & of an inferior degree; but, this

being so, we have only to regard & as containing the factor (1 - 2) of the degree ¢ proper
for raising its degree up to that of P or Q.

20. Solutions are given in the following PQR-Table: in which, where required,
the proper factor (1 - =) has been added; the first column headed Ref. No. (Reference

Number) will be explained further on. The Annex to the same Table will also be
explained.

Pror. CAYLEY, ON THE SCHWARIAN DERIVATITE

14

“snoomo.e ore mig «
SON[VA 9} PUB “MOTyRMOo[vo oT} Jo Sutuuiseq oN} 4B AOIIO oTMOS Mo0q OAT 04 TOES PINOA O10} AX OUT spavsor sv qnq ‘utepy Aq poyzemmoreo ae eee

TIX Se] oT} Io¥ oso, :Tyosortg Aq pozw[No[vo oro ‘AX PUG ATX ‘TTX sour oy} ur ydaoxa “wy ‘® ‘g SuOTPOUNZ OY} EGR} OT} JO J[BY puooes oy} uy

(2 = 1) (L+2)g2(""+ 9618 -)

x an a(@896I + 206108 + 220G18 +
a(**— ¢? S61) O0000FE8EI — ae 7 E)a(t+2)s00000F6R8T + e20GL8T + 2GL86 + 9G -—)—| e(€FS- 298 - 4296 - e79S1)(L43-29)-| AX
(2-1) ars
oe ei “* . . 2
(°° + 2960P)a(**+eF9) “1L, SET — (2-1) aT +2) "sh LE —| <(S008T + 246296 + 29188 + 2960F) 2 — a(6h + 2681 +2269) (681 +269) | AIX
a(T — 2) ("368 — 62) ("FL +27) BOL — s(1 — 2) 2801 — a(I +288 — 2288 — 62) — (I+261+22)| IIIX
o te ares ©
Nae 7 UC 6+ 7)(o +2) <2 096 - (2-1) (e-22)19- o(8 — 761 +276 + 62) — (gt+2)e(G+2)e| IIX
(F-1)a-4
“* 70a #7 oe 2
("6 ~ 28) 2(°*2 — 2?) SOT - (2-1) 1-2 e08- a(G + 28 — 226 — 526) - (ite—9)>) Ix
(1-2) (°° 206 2) (8 +2) £9 - (2 -1)0-2)19- a(8 — 208 — ¢?) - (8+2)2} Xx
ao
2(8+2)2(b —2) 16 (-1) PLE a(8 +2) (1-2) - (b-*)} XT
(1-2) (t+2) P- (1-2) (+2) - ($-1)+ IITA ‘TIA
, ‘A ‘TU
Dn _ ee al a(T + o26ES + o17S000T — 0120 +
Ma 7 Us tn) 002) 0F98—| {2 — Tal oA TL +) BLT - oa? S000 ~ s¢ GES — 09?) — a(L + 0° 888 + 1° FOF + o1°8ES oc?) |
oo) * “* bd
Ma 7 Us C0 ~ art) oC" + 02) 946 (2-1)G-#) 801 - a(L + OE — 688 — a1?) - o(I + 2F1 +52) b
D
(F-1) 0-w)eC-ve-p3-¥)
al E- Nt¥)E- NPG (1 +e28- 3-34) — ( = 1) s(t —)%8- NST - o(I +928 — AG +42) €
nD
A z= (t=) + 2) au? Up — a(I - 2) a(T +2) - (2- 1) “ fs
_(?- i) 1-2 Ler — (¢- t)t- we I
= =a =() fof ‘ON JOU

‘WIAVL-YOd AHL

in

AND THE POLYHEDRAL FUNCTIONS,

AVMIOY oY} 0} Surpuodsertoo ‘Ay 0} [ SIoqMINU UBUTOY oY} SN} oaey ySIG OY} Aoxze Sout, ot}
1G

SpIvZar SV Os[v OAIOSqQ “¢ RIM Woy} JO Towa oo1dv AX 0} XT SOU] Surureutor otf} “(ez

pur 6

PF

‘zaeMyog Sq pasn siequinu

:¢ ‘F ‘g ‘GZ SOU] 01} Jo Surmaquinu uvmoyy oy} ‘AT[RIoues XoUUY ony

jo Wortsodsuvdy B SI O19q} TTA Spivsor sv &ao ) G YIM Woy} Jo yove

20188 8 sour, £ (2 puv ® yo womsodsuvy v st oxormy Aquo) F YMA A OUT $e YA soorBe TTT our, on} yvq} ‘uumpoo (0 ‘q ‘v) oy} spxwBor su oAIOSqO ,
IITA Pus ITA souty ST P r 3 "I tT
‘ 2 L+2 fe =? so Uk 057 6 — com FO = —eop (Ome > “ “ “ ry
f ct G ia c = Cg ry oe 9) Ws Be at 0=e4 Oca =e AX
“ “ Cc : set v a0 * . f) 0=¢0 =e 0=24 e=14 o=er oa « “ “ ATX
f_z z 2 Z
z G
“ “ (= ee We og =. OS Od) ered =e =e 0=24 o=er| “ “ “ Ix
oN IL I 1 1é 1G 1G G G G
age OA. 62 ake Go ¢GG a) [om 9 (8G _ 9 = =¢€e 6 _ip “ “ “ a
“ “ (= - : v k : Ea 2 8 g aI I 0=24 0 F De
«“ “ (G (ee, hl a = ts e) we =D 0=24 e=14 O=Ev eS S - XI
2 ¢
“ “ ( “ “ “ w “ a ‘ = ‘ 28) =o 0=C4 ea ale : ie os IIIA
rd
woIpeyesooy, “ “ “ .. OS 6 7) 08 _ 7 0=2 6 1G 6 ne SG 8 <6
puv worpeyroapog i x "16 54 v 7% eh P iv at € v TIA
worpey Buk oh x=. 8 = cee EE) les as SE _te (CE_ie| 6 68 BE
-8}0Q pue aqno ( o G ST ST G o= ST ST v € ST a
1-2 ,O-2 .2 _. Sil <6 2) Gl i Cay C5 || @ <8 <6
2 = IEE 2 2 ——=% = == =e) © =
WOIpay vaya T, af . 1 I v ¢ ; ; gC 0=64 - IT - rs ¢ F III
WoIpat[esooy GG (8 <6 =a at =er| 26 (8 <6
puv uorpeyvoapog TA 61 & id ee v= él € v 2
woapey GE <8 <6 5 = =er| 8 6
-840Q puw eqny AI ST & v eae oe oe cI € v y
worpayeaya,y, Il 2 f 3 : . 0=¢90 0=24 o=ER : : ; : : €
Teo cae = 2 ee Bows ety
proerfg eqnog I 2 5 (c-¢ — 1) $ 0=29 0=24 Oe lie a (c-U- 1)¢ G
uoskyog 0 “G-w-1)¢ “c-w—1)¥¢ 0=19 o=Uq o=ue) o “c_u-1)§ “G-u-1)t|] 1
ON PU (x0) “(bq) “(de) Jo woRemoyen x2 q *) ON JOU

‘ATAVLYOd AHL OL XUINNV

16 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

The Differential Equations {a, 2} and {s, x}. Art. Nos, 21 to 40.

21. In reference to what follows, it is convenient to put P=XP,, P’=X,P,, where
P, is written for I (¢—J?", the G.c.m. of P and P’; and X is consequently =F into the
product II(z—J) of the several factors taken each with the index unity; and so for Q

and R: viz. we write

P, Q,R=XP,, YQ,
PO FX PO alae,
and the foregoing value of © then is
© = KP,Q,F,.

We come now to the investigation of the leading theorem. Take a, b, ¢ arbitrary,
fgoh=b-c, c—a,a —b; P, Q, R functions of z as above ; and write

f(w—a): g(e@—b): h(@-oc) =P: Q: R,

equations consistent with each other, and which determine 2 as a rational function of z.
Using as before the accent to denote differentiation in regard to z, and taking the co-
efficients (a, b, c) arbitrary, it is required to find the value of

en! 1 ieNs
{w, z}+a (a, b, ¢ = a) =):

22, Calculation of the first term {a, 2}.
We have x=a function («% +8) + Ge + 8), and thence {z, j-th i, ={&, 2} for

a moment; then

nese sik [ROR Le,
“\R) yn oe Ree een

Substituting the values
P= (z-)", Q=U (e-m)", B= (e-n)", Z=T (z—n),

we have

and thence

q-1 r+1
es — = -2 (¢- my t +2 (g- ait

2
-3f 2 2S) ye fg ae

zZ—-m™ zZ—n
or say
£ (-255- P-L q-1 @=! ly tye
ra (z—l)? (e—-1)*"° ~(z—my (z—m,)?*" SSeS
-1( p-l Pay ro q=1 4» Gat ees rfl. #41 SE
z-l z—l, zZ—-m_ z—-m™, z—n zZ—n,

a

AND THE POLYHEDRAL FUNCTIONS. 17
where it is to be observed that

S(p-Y+3(¢-Y- (r+ VD, =§-—o0,+6-0,—-(8+0,), =8-0,-¢,-06,, =—-2;

?

consequently the function is diaphoric, and therefore curtate.

It is to be remarked that the function, although presenting itself in a form unsym-
metric in regard to the factors of P and Q, and of R, is really symmetric as regards
the three sets of factors; this is obvious @ priori, and it will be presently verified.

23. Calculation of the second term

ri (a b e+.) : sly =
as z—a’ z—b’ z—c

f@-a), g(e-b), h(e-—c)=OP, 2Q, OR,
where © is a determinate function of z. Hence

x x rif aoa Qa Ea
SG cb 2-0 PO OO” Reo”

We have

and then substituting these values, by reason that the function is diaphoric, the terms in

,

ra) disappear, and we have

which is

We have Sp=Sqg==r, =8: and hence by what precedes, this function considered as a

function of the inverts a &e., is diaphoric, and therefore curtate.

24. We have therefore

2
z}+2” (abe. I Ba —) =
a—b’ z£-¢

(5 ea = +Ec oy)
ey

= g-3 _= =

ra

. P q r i
+ (a, bc ane rer er :
where the whole function on the right hand is curtate.

Vou. XIII, Part I, 3

a
Z—m™m Z2—-N,

18 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

25. We have to bring the function on the right hand into the reduced form

FE = par eee
ae ae

for the purpose of getting rid of the non-essential inverts (if any).

We write

ie) ons La Pee

=e hie =O Gaeta
pal wv (a!
~—9* G1

viz. 2-1 here denotes any particular factor, and z—J, represents any other factor of the
same set; and so in other like cases.

26. The whole coefficient of Sata is
Ca)

—(p—-1)—-4(p—1) +ap*, =4(1—p’) + ap’;

an expression which, regarded as a function of a and p, is represented by (ap): the paren-
theses are used only to avoid ambiguity, and are omitted when p is a number, thus
al=a, a2=—3-+ 4a, and so in other cases.

S ; 1
27. The whole term in Ses comes from

—l

dy Oe ae Gass z= +.)
+P (20x Pe +( a be) a+b ce) =)

2 1 : 1 1 1 Aa :
viz. each term such as San TS is to be replaced by alee giving rise to

The whole

to the coefficient of

1 1 aks ad
the term ;—, ——; or contributing the term ]

t—iz-l , z-—l
coefficient thus is
=-( -1)(2 S57 4325-374")
P Cb 1h, Olea
3” PB: Fes 2a mJ =o ,
+ Aap oy te a b+e)= —— +p( atb—e)s7—

28. Suppose first that z—/ is a multiple factor of P, viz. a factor with an index p

U

; (dis
greater than 1: then for z=1 we have Q+R=0, Q'+ R’=0, and thence o=% that

is > 2 =S——. We have therefore

AND THE POLYHEDRAL FUNCTIONS. 19

p(-a—b+c)% 74 +p (-a+b-0) 3 —

l—n
Sar neg. ei:
82 21a)

and moreover in the top line the terms i= and — = rs destroy each other. The whole

coefficient of = when (z—J) is a multiple factor of P, thus is

=-—(p-1) (3A -3 a2)

1
l—n°

a form which is now symmetrical in regard to the inverts i

29. The value just obtained is

i= ar 1
i eB On \ Gee 32 sy )
ors +20p) (275, me SE

viz. comparing the two forms and reducing, they will be identical if only

l—m l—n

fe PtP, _ yth+p)q—p_ytlt+p)r—p| _
a p+2ap) |S ai > 0,

and it can be shown that the function inside the {} is in fact =0.

30. We have as before ae => i = or writing each of these quantities = ®,

the equation to be verified is

ory Pee
2 ae ere sae eg aly ae
pte i dyekG
We have Peat? se7° =F
si P: [4% __P e
that is TEE E | for z=1,
- (25254
X (z-1) ;

The first derived function of the numerator is X,(z—-/)+X,—pX’, which for
z=l is X,—pX’, which is =0; and for the denominator it is X’(z—1)+4X, which is
also =0; passing to the second derived functions we find

y Ps ee ap xe or

20 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

From the equation =—S

we find in like manner,

wees
t—l Xx’?
and we thence obtain (z being always = /)
wP +Py = xX,
7 yee aes -
so that the equation to be verified becomes
x 1 1
—1 = ee ee
Xx, ae em ae

31. But from the equation ©, = PQ’—P'Q, =KP,Q,R,, we find XY,-X,Y=KR,,
and then, differentiating, X Y,+ X’Y,—X,Y—X,Y’=KR,: writing in these equations
z=I/, they become = = KR.

A’ Y,—-X,/ Y-X,Y'=KR,,
and dividing the second by the first
Be RS ana AOE ay

= an ee pe
Tome Be
: ; Veen g
or recollecting that X,=pX’, and Y=” we have
! TR = Q’
= > => ich
4 r(z Pyhig
: Come r—1 U qd
wk x7? (27-2) te
1
= =>
(p +1) ®-p2—— —p=7_| >

the required relation.

32. The result is that 2-1 being a multiple factor of P the coefficient of the term

ie
ES Roe
n (ed-1 tas it
Pe EP Ae Go are
=2(ap)/3($ +77) aim
33. In the case where z—/ is a simple factor of P we have p=1, and the
coefficient is
= 04S Ae ae re = ree)
= 2a> jae a b+e) =~ +( a+b =,

=a (2x P34 -3,7)-0-9(2 4-2/4).

l—m -m —n

AND THE POLYHEDRAL FUNCTIONS, 21

34. Of course the formule for the coefficients of een and et give at once by

(z—l) z—l
. 1 1 1 1
a mere change of letters those for the coefficients of 3 and ;
(2—m)*? z—m’ (z—n)?? z—n’
and the function in question,
1 1 aN?

12 .
iaeive (2 ee z—a’ x—b’ seat

is now obtained in the required form,

(bg) (cr) A B C

(g—m)?" Carr Weed 2 eres Pp aes

__(ap)
Gi

+

where (ap) denotes 4(1—p*)+ap*, and the like for (bg) and (er); and where z—/
being a multiple factor of P, the coefficient A contains the factor (ap); and_ similarly

for B and C.
35. Suppose that the coefficients a, b, c are no one of them =0; we have al, =a,
which does not vanish; that is, z—J being a simple factor of P, the expression contains

ar or the invert = is essential: and similarly z—m being a simple factor of Q,

‘ ; 1 :
or z—n a simple factor of R, the inverts = aa este essential, But for z—/

Z
= 1 . 5 5
(2—1)? may vanish, viz. this

will be the case if a=}(1 - a and when this is so the coefficient A of the cor-

a multiple factor of P, the coefficient (ap) of the term

; 1 : at lated Bay a ae
responding term —— also vanishes; that is peas non-essential invert. And similarly

z—l
: é 1 1
for any multiple factor z—m of @ or z—n of R, the invert BS gone Dany, be non-
essential.

36. Hf P, Q, # contain each of them only multiple factors of the same index,
say of the indices p, g, r for the three functions respectively, viz. if the functions are
F (il (z—1))?, G(Il (ze —m)), H((z—n))’, the result contains only the six terms written
down: and then if a, b, ¢ are =4(1 a 3 (1 =7): (0 -3) respectively the result

2

is =0: viz. we then have

1 1 1
/2 . —— afi)
{a, at+a (a, lo “ls 5 b° -\= 0,

or we in fact have for the values in question of (a, b, c) a solution

f(z@-a):g(@—-b) :h(@-ec) =P: Q:h

of this differential equation of the third order.

22 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

37. The reasoning applies directly to lines 2, 3, 4 5 of the PQR-Table: and
with a slight variation to line 1; viz. here the factors of R(=—1+2") are all simple
factors, but in virtue of e=0 and a=b, the corresponding inverts disappear, and, the
other inverts also disappearing, the value of the function is =0. Hence lines 1, 2, 3, 4, 5
of the PQR-Table give each of them a result =0, for the values of (a, b, c) appearing
by the table itself, and shown explicitly in the corresponding line of the Annex.

2

Thus line 3 shows that the function « determined by
Fle—a):g(w—b) :h(w—o) = (+2) —32° 41): -12/—-3 (2-2): =(e—2,/=—327+1)

satisfies
sity Bek Aol vad 1 1 ie
{z, a} +a ioe Sobek (—. z—b’ w—-c a

and so for any other of the five lines.

38. The indices of the factors of P, Q, R may be such that for proper values of the
coefficients (a, b, c) there are in all only three essential inverts, say sabe poe j in ;
z—a,’ 2+b,’2-¢,
belonging to the three functions P, Q, R respectively, or it may be two, or three, of them
to the same function. When this is so, the function of these inverts is by what precedes
a curtate function, and it is consequently a function

ou! 1 ane
(a, Jobe Cc, ee i. > ee ? ’
z—a,’ 2-6,’ z-¢,

where a,, b,, ¢, are the values of the three which do not vanish in the series of expres-
sions (ap), (bg), (cr).

The remaining lines (III, V, VII, VIII) and IX to XV of the PQkK-Table give
such values of P, Q, R, the values of (a, b, c), and the calculation of the values of
(a,, b,, ¢,) being shown by the corresponding lines of the Annex. And we have thus
values of « determined by the equations

f(x-a): g(a@—-b): h(e-c)=P: Q: &,
and giving

' a
{z, 2} +2” (a, b, ce. : Z 4) = (a, by, C, oy == i iL

g—a’ 2—b’ 2— z—a,’ z—b,’ z-¢,

39. For instance, from line IX we have

f(e—a) : g(a—b : h(w—c) =(e—-4)" : —(2-1) (2 +8)" : 27e'(1-=),

12

the values of (a, b, c) are 5 Ss 95° and since P, Q, & contain factors with the

AND THE POLYHEDRAL FUNCTIONS. 23

exponents 3; 1, 2; and 1, 2 respectively, the coefficients which present themselves on
the right hand are
ad eplseb2s el, 62,

: 3 PA Al F
which are =())2 5° 0; 55’ 50 respectively.
ya on 12a 21 : 3 1 1 il!
Hence writing a,, b,, = 5) 95° 50 the corresponding inverts are a see

and the result is

2a ee 1 Laiege {meg NMRA yom fey 2
paee(t 2 -( Le

a—a’ £z—b’ x-c QB DH? KB) “we asil? Bary 7 Z

40. It is hardly necessary to remark that an expression

(a ees Af 1 1 1 )
o\Gint ae SNe Ons 20 ee

in fact denotes zh b SSG

C= Can Caner

The particular form of the z inverts is immaterial; we could by a general linear
1 1

—a,’ 2—b,’ z@
arbitrary; or we can give to the a,, 0,, c, any particular values we please: there would
be a propriety in making the inverts to be in every case (as in the foregoing example)
= —= — but the numerical work would be troublesome, and it is not worth while

to effect it.

transformation upon the z make them to be

ae itnthe (Ge bee
oe 4

41. The conclusion is that lines (III, V, VII, VIII) and IX to XV of the PQR-Table,
give for determinate values of (a, b, c) and (a,, b,, ¢,) solutions

f@=a): g@—b):h@—-—c)=P: QO: hk
of the equation

1 1 NG 1 1 1
12 . .
{x, z}+a (a, ld, © a ) = (a, 1Dig ¢ ae Das =I

where a, b, c, a,, b,, c, are or can be made arbitrary, but without any real gain of
generality herein. This is the Differential Equation {z, 2}.

42. Recurring to the results from the Arabic lines of the PQR-Table, but for
convenience writing s instead of z, we have

f(e-a): g(x@—b) :h(w@-c) =P: QO: Rh

(where P, Q, R are now functions of s), a solution of

da\? 1 1 le NE
{ar, 3 +(Z) (a age 2 TR? —) =0.

Tah fie

24 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

But we have

a(t a

and the foregoing is therefore a solution of
1 I 1 Ne
8, a}=(a, b, ec «. ()——_, —{, —
ta (a 4 z—a’ x—b earl
. differential equation of the third order; this is the Differential Equation {s, z}.

43. From the Roman lines if we assume
f(e-a): g(e@-b) :h(z-)=YP: QQ: R

(where 33, @, R are functions of z, not the same functions that P, Q, R are of s,
since they belong to a different line of the Table): we have as before

da\? 1 1 La 1 1 Ls
2} =~ c 52 — oe Bee = “ ,
{x, a} +(F) (a, b, ¢ z—a’ abs —) (a, b, “; = z—b,’ =

44. We may combine any such result with a properly selected result of the pre-
ceding system, the two results being such that (a, b, c) have the same values in each
of them. (See as to this the foot-note referring to the Annex to the PQR-Table.) The

last equation then becomes

nes =e

or since {z, 2} +() {s, z}={s, 2}, this is

-

. 1 1 : :
{s, }=(a, by &% ea z—b,’ zal ’

1

the corresponding relation between s, 2 being of course obtained by the elimination of «

from the two sets of equations

fl(a—a):g(a@—-b):h(e-c)=P:Q:R, and f(e—-a): g(a@-)) :h(e@- =P: OQ: R;

viz. the required relation is

Pog B= eae =a
(where P, Q, R are functions of s; #3 : @ : RM functions of z; and in virtue of
P+Q+h=0, YH+Q+R=0

the relations are equivalent to a single equation between z and s). And writing finally
x in place of z, that is now considering 73, @, WR as functions of z, we have

P:Q:R=-P:Q:k

as a solution of

{s, a} = (a, by & J, aah ee

AND THE POLYHEDRAL FUNCTIONS. 25

a differential equation of the third order of the foregoing form {s, x}=given function of
z, but with different values of the coefficients, (a,, b,, ¢,) instead of (a, b, c).

45. It thus appears that there are in all 16 sets of values of (a, b, c), for which the
equation is solved, viz. the 16 sets of values are shown in the right-hand column of
the Annex. For greater clearness I exhibit the integral equations as follows:

Functions of 2. Functions of s.
1 f(a-a) : g(a—b) : h(a-c) = IP SY gle (ab) Polygon
if 5 = a @ Double Pyramid
II a = * (3) | Tetrahedron
Ill 4¢ : —(a@+1)? : (w-1)? = n (3) x
IV f(a-a@) : g(a@—-b) : h(w-c) = 5 (4) Cube and Octahedron
Vv (v-1)? :-(#+1)?: 4@ = y (4) 3
VI f (v-4a) : g(a—b) : h(w-e) = 6 (5) | Dodecahedron and Icosahedron
Vr 4x :—(a@+1)? : (c«-1)? = ey (5) =
Vill (w-1)? :-(a+1)?: 4a = o (5) =
IX iE : Q : R (IX) — ” (5) ”

The values of the P, Q, R as functions of a, or of s, are taken out of the PQR-
_ Table: only in the lines II, V, VII, VIII, where P, Q, & are given as

=42z, —(2+1), (¢-1)*

and where, as regards V and VIII, there is a transposition of P and R, I have in-
serted the actual values of the «-functions. (See as to this the foot-note referring to
the Annex.)

The Schwarzian Theory. Article, Nos. 46 to 62.

46. Considering the foregoing equation

1 1 irare\ 2
{s, a} = (a, ly © “4 ) )

z—a,’ o—b,

as a particular case of the equation {s, x} = Rational function of 2, =R(«) suppose, then
we have in l, I, IJ, IV, VI solutions of the form x= Rational function of s.

Vou. XIII. Parr I. 4

26 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE
Consider in general a solution of this form, #=F'(s) a rational function of s: s is

then an irrational function of 2, and if s,, s, are any two of its values {s,, 2} = (a),

as, +6
Gina And then

{s,, z}=R (a); that is {s,, 2}={s,, a}, and therefore (ante, No. 7) s,

«= F(s,)= (2%), = F(s,): viz. F'(s) is a rational function of s transformable into
: cs,

se fa : : as +b mee

itself by the transformation s into mae and it is moreover clear that between any two

roots s whatever of the equation «=F (s) there exists a homographic relation of the form
in question. It is moreover clear that these homographic transformations form a group;
and consequently that #'(s) is a rational function of s transformable into itself by the
several homographic transformations of a group of such transformations: viz. taking @ to
be a rational function of a, it is only in the case w=F'(s), a function of the form in
question, that {s, 2} can be equal to a rational function of a.

47, We may in any equation between 2 and s consider these as imaginary variables
p+q and u+vi respectively; considering then (p, g) and (w, v) as rectangular co-
ordinates of points in different planes, we have a first plane the locus of the points a,
and a second plane the locus of the points s: there is between the two planes a corre-
spondence which is in fact the correspondence of conformable figures: to the infinitesimal
element dx drawn from a point @ of the first figure corresponds an infinitesimal element
ds drawn from the corresponding point s of the second figure, and which elements are
in general connected by an equation of the form ds=(a+ bz) dw (a and 6b functions of «
or s); and this signifies that to obtain the pencil of infinitesimal elements or radii ds
proceeding in different directions from the point s, we alter in a determinate ratio the
absolute lengths of the infinitesimal elements or radii proceeding from the corresponding
point «, and rotate the pencil through a determinate angle: this ratio and angle of
rotation, or say, the Auxesis and Streblosis, being of course variable from point to point.
Or, what comes to the same thing, if dz and d,« be consecutive elements of the path of
the point z, and ds, d,s the corresponding consecutive elements of the path of the point
s, then the ratio of the lengths of the elements dw, d,v is equal to that of the lengths
of the elements ds, d,s; and the mutual inclination of the first pair of elements is equal
to that of the second pair of elements. In particular if at any point the path of @ is
a curved line without abrupt change of direction, then at the corresponding point the
path of s is a curved line without abrupt change of direction. In what precedes we
have the relation at ordinary points, but there may be critical corresponding points (a, s),
the relation at a critical point between the corresponding elements dz, ds being of the

form ds=(a+bi)(dx)*, (\ a positive integer or fraction): here the angle between two
elements ds is =X times that between the two elements dw; or, if the path of the point
x through the critical point is without abrupt change of direction (say if the angle
between the two consecutive elements is the flat angle 7) then the angle between the two
consecutive elements ds is =z: viz. there is in the path of the point s an abrupt change
of direction.

AND THE POLYHEDRAL FUNCTIONS. 27

48. I consider the foregomg equation {s, z}= R(x), where R(«) is a rational func-
tion, and is now taken to be a real function of 2: we may assume s’=p'6’e", where
the accents denote differentiation in regard to zw, and where p’, 0, and therefore also 0’,

are real functions of z. We have

” ” vr

ee
Sis

and thence
dine (Se Vay Bty = a G3 7) 6"
a (=) = p (5 cr @’ = +76
IN 2 WA VIN 2 wpe "ar
-3(5)= -3(6) —1 (5) +300 18" iP,
and thence
7 Ul po

- Va tee
fs, a] =[p, a} +{6, a} +40" Fe ie

Putting this = R(x), and assuming that z is real, we have

F "Oy" "Ey
{p, x} + {, a} +40" — Fe =R(a); 0=iPe.

The last equation gives p’@’=0, that is 6 =0, which gives s’=0, and may be disregarded;
or else p’ =0, therefore p’, a real constant, =y suppose, and {p, 2} =0: hence for the solu-
tion of the equation {s, a}=R(z), we have s’=y6'e", @ a real quantity determined by
{6, 2} +40°=R(a): and then integrating the equation for s’ we have s=a+Bi+ye”, a, By
real constants.

49. The conclusion is that if {s, z}= R(x), a real function of 2, and if z be real, that
is if the point x move along a right line (say the g-line) then s=a+Pi+t ye” (6, and
the constants a, 8, y, being real), that is the point s moves in a circle, coordinates of

the centre a, 8, and radius =y.

50. Suppose a, b, ¢ are any real values of x representing points a, b, c on the a-line;
and A, B, C any given imaginary values of s representing points A, B, C in the s-plane:
49

28 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

since {s, 2} = R(«) is a differential equation of the third order, the integral contains three
arbitrary constants, and we may imagine these so determined that to the values «=a, b, ¢
shall correspond the values s=A, B, C respectively.

If there is not on the a-line any critical point, as the point 2 moves continu-
ously along this line the point s will move continuously along a circle, which (inasmuch
as a, b,c and A, B, C are corresponding points) must be the circle through the three points
A. BoG*:

51. If however the points a, }, c are critical points, such that the element ds at the
corresponding points A, B, C are equal to multiples of (dx)*, (da), (dx)’ respectively, then
to the flat angles 7 at a, 6, c correspond in the path of s the angles Az, wz, vr at the
points A, B, C respectively: and (assuming that a, b, ¢ are the only critical points on
the a-line) the path of s is made up of the three circular ares CA, AB, BC meeting
at angles Aw, wa, vw respectively. The arcs are completely determined by these conditions;
for supposing the are BC to make with the chord BC, at the points B and C, the angles
Ff, f, and similarly the ares CA and AB to make with the corresponding chords the angles
g, g and h, h, then the conditions give Aw, pr, pr=cA+gth, -<B+h+f, -C+f+g,
where the angles referred to are those of the rectilinear triangle ABC: we. have thus
the values of f, g, h; and the are BC is the are on the chord BC meeting it at angles ff:
and the like as regards the ares CA and AB respectively.

& ; :
52. The foregoing equation

il 1 Le}
i .
{s, a} =(a,b, ¢ .)- Spe es -) 5

where a, b, c have the values }(1—2*), $(1—y’), 4(1—v’) (A, w, v being real and positive),
has z=a, b, e for critical points of the kind in question: in fact, writing c—a=h, the
equation is of the form

a 6 RL
{s, ia) a tah ea.
which is satisfied by
d ds 14+2Xr
ah 8 ah ht ot ah t bh +...

and we thence obtain an integral of the form
s=kh*(l+kh+k}’+...), =k for shortness,
This is a particular integral, but we have from it the general integral

ga tt Bk
y + ok’

* Since there is no critical point on the z-line there path of s cannot consist of different arcs of circle, the
can be no abrupt change of direction in the path of s, that one continuing the other without any abrupt change of
is the path of s cannot consist of circular arcs meeting at direction.
an angle: but it is in the text further assumed that the

AND THE POLYHEDRAL FUNCTIONS. 29

and if A be the value of s corresponding to h=0, then 8=64A, and we find

ote, =(4+5¢5) aa =A 42a ie.

viz. reducing : to its principal term i”, and then writing ds, dw for s— A, and h(=a—a)

respectively, we have ds=K (dz)*, or «=a is a critical point with the exponent X; and
similarly z=b and «=e are critical points with the exponents mw and vp respectively.

53. Hence in the equation

{s, x} =(a, Dye Gace ! ; i : is

a—a’ «—b’ w£—c

as the point 2, passing successively through a, b, c describes the z-line, the poimt s passing
successively through A, B, C describes the sides AB, BC, CA of the curvilinear triangle
ABC. To points x indefinitely near the z-line correspond points s indefinitely near the
boundary AB, BC, CA of the triangle, viz. to points w indefinitely near to and on one
side, suppose the upper side, of the a-line, correspond the points s indefinitely near to
and within the boundary of the triangle: and in like manner to whole series of the

points 2 on the same upper side of the a-line, correspond the whole series of points s
inside the triangle.

54. We have attended so far only to one of the points s which correspond to a given
point x, but considering the set of points s which correspond to the same point s, we have in
the s-plane entire circles forming by their intersections curvilinear triangles ABC, ABC’, &c.;
we have thus two systems, say ABC, &c, and ABC’, &c., of triangles, such that to a point
az on the upper side of the #-line correspond points s, one of them within each of the
triangles ABC, &., and to a point w on the lower side of the g-line correspond points s,
one of them within each of the triangles ABC’, &c.; and so consequently that to the two
half-planes on opposite sides of the «z-line correspond the two sets of triangles ABC, &c.,
and ABC’, &c., respectively.

55. In order that the relation s and 2 may be an algebraical one it is necessary
that the two sets of triangles should completely cover, once or a finite number of times,
the whole of the s-plane: and this implies that the angles Aw, mwa, vr have certain
determinate values; and, in fact, that dividing the surface of a sphere into triangles, each
with these angles, the curvilinear triangles ABC, ABC’, &c., are the stereographic pro-
jections of these triangles. It was by such considerations as these that Schwarz, in the
Memoir of 1875, p. 328, obtained the series of values I to XV of A, pw, v, giving for
a, b, ce, =2(1—2’), £(1—p’*), 4(1—>*), the series of values mentioned in the Annex of
the PQR-Table: and thus showed d@ priort that the equation

1 1 LENG
{s, a} =(a, by c=: ese —)

is algebraically integrable for these values of a, b, c; and only for these values, or for values
reducible to them.

30 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

56. As an instance take the double pyramid form: the integral equation is
f(w-a) : g(e—b) : h (w—c) = 4s" : —(s"-1)* : (s"+ 1),

C=) C=) ee =):
(a—b)(w—c)— (s"+ 1)?’

or. say

(Ss 3
or if for greater simplicity we assume a, 6, c=1, 0, », this is a= ay)

—(s*—1) =Jz (s" + 1), that is, s*= ; ae , a solution of the differential equation
+/2

fs }=(3, 40-19, 40-09 KE, , ).

“Kz? x—-1’ 2£-—@

st ey
In particular if n=3, we have a= (5-5) or fai ty an integral of

{s a aie Bn sah gg es —.)
: =(5; 9° 9" Xe? g—1° c=") ©

(s"+1)’

or

aay

\

57. We have here the spherical surface divided by the equator and three meridians

into twelve triangles, each with the angles 4a, 47, 1: and then projecting

from the

South pole on the plane of the equator we have the annexed figure of the s-plane,
divided into 12 curvilinear triangles, each with these same angles 90°, 90°, 60°, and which
are by the shading divided into two systems, each of 6 triangles. The figure of the
z-plane is by the z-line divided into two half-planes, one shaded, the other unshaded;
and we bave on the line the point ¢ at «, a at the origin, and b at the distance

unity.

AND THE POLYHEDRAL FUNCTIONS. 3 31

58. Take « real, then if w is positive and less than 1, s° is real and positive, and
we have for s the infinite half-lines at the inclinations 0°, 120°, 240°, while if z is
positive and greater than 1, s® is real and negative, and we have the infinite half-lines
at the inclinations 60°, 180°, 300°. If is real and negative, then s* is of the form
1-ki
1+ ki’

be choc x 1-VJz
circle radius unity. Writing potas, and supposing that the point z moves along the
x

=cos@+7sin@; whence s is of the same form, or the locus of the point s is a

z-line from 6 through a to ¢ at —, and then from ¢ at + to b, the point s describes
the sides BA, AC, CB of the shaded triangle marked K.

59. Suppose that the point 2 is at &, in the shaded half-plane at an indefi-

nitely small distance from a; say we have x=—2«/, (« small), then taking for Jz
: 1—« (1-2) ; ,

_ 5 SENS EE —— _— _—
the value «(1—7) we have s Leela) 1—2«(1—7) nearly, and hence a value of

s is =1—3«+3xi, which belongs to a point K near A, and within the shaded triangle:
we have thus, in respect of this value of s, the shaded half of the z-plane corresponding
to this shaded triangle: to the same value «=—2«% correspond in all six values of s,
giving six points K each lying near a point A within one of the shaded triangles; and
hence the shaded half-plane corresponds to the six shaded triangles, and the unshaded half-
plane corresponds to the six unshaded triangles.

60. Suppose the equation is
Pw ies Y 1 ree ila?
{s, 2} = (a, B, ext =p mothe —):

_—(b-c) (c—a) (a—b) a b ¢c
ee (pee tt cee tac

that is

where a, b, c are real, but a, b, c are imaginary. It is to be shown that if the path of
x is the circle passing through the points a, b, c, then the path of s is a circle passing
through the corresponding three points.

61. We may find a, £, ¥, 4, 9,, 6, such that a, b, ¢ are =a+Pitye%, a+ Bit yer,
a+Pi+ye™ (this is in fact finding a and 8 the coordinates of the centre, and y the radius
of the circle through the three points a, b, c): we then have c=a+/i+ ye", 0 a varia-
ble parameter, the equation which expresses that the point 2 is situate on the circle in
question.

We have c—a=y (e%—e%t), = cyeh(O+6) {eh (0-%)é_ e-3(0-%)i}. the second factor is
" isin} (@—6,), =iP suppose, or the equation is c—a=iPy . ¢2@+%i, say

2—a=iPy.expit(@+8,).
Similarly «—b=iQy expi $(0+9@,), and «—-c=iRy. expi4(9+0,); where P, Q, R denote
sin$(@—9,), sn 4(@—@,), sin} (@—6,) respectively: in like manner b—c, c—a, a—b,
=iFy expi 4(0,+0,), iGy expi $(0,+90,), tHy expi 1(0,+9,), where F, G, H denote

2

sin }(@,—@,), sin }(@,—8@,), sin4(0,—9,) respectively.

wy)
bo

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

We have
b—c.c—a.a-— b= her
~g—-a.a—b.a—c PQR

expi 4 (6,+ 0, + 0, — 38),

b—-c.a—-a_ spp eri - 4(0,+ 0, +9, +4),

with the like values for ote = and ae Hey, Hence the right-hand side of the
c—a.x—b a—b.u—e
equation is
_FGH b
= POR (ert oqat +a) Ss L))

62. Considering now the left-hand side of the equation, we have

1
{s, x} = Tada? ({s, 6}) — {z, 6}),
(i)
or substituting for z its value= a + i+ ye", this becomes

nee “ ei (fs, 6} ~ 4),

that is =—-({s, 6}—4) expi (— 20).

Assume s=Z+Mi+ Ne®%, LZ, M, and N constants; then using the accent to denote

differentiation in regard to 6, we find without difficulty {s, 6}={©0, 6}+ 40%, and the
value of {s, 2} becomes

=-3 (10. 6} +407 4) expi (— 26).

Hence, substituting the values of the two sides of the equation, the imaginary factor
expi (— 26) divides out, and the equation becomes

Lae fers b
he as bd =4=-5on (prt oat +3)

an equation in which everything is real, and which thus determines © as a real function of
6: and we have therefore the theorem in question.

Connection with the differential equation for the hypergeometric series. Art. Nos. 63 to 68.

63. Take p, q given functions of z, and y a function of a determined by the
equation

AND THE POLYHEDRAL FUNCTIONS. 33

again P, Q given functions of z, and v a function of z determined by the equation

dv
ae 2 Qp=0,

and assume
y = we.

Substituting this value of y in the first equation, we obtain for v an equation of
the second order (the coefficients of which contain w), and we may make this identical
with the second equation; viz. comparing the coefficients of the two equations, we thus
have two equations each containing w; and by eliminating w we obtain a differential
equation of the third order between z and 2 This is in fact the basis of Kummer’s
theory for the transformation of a hypergeometric series: the equation between z, x will
be found presently in a different manner.

64. But if with Schwarz, instead of making the equation obtained for v as above
identical with the given equation for v, we merely assume that the two equations are
consistent, then there is nothing to determine the value of z, which may be regarded
as an arbitrary function of 2; y and wv are then functions of a, and w denotes the
quotient y+v of these two functions, and as such satisfies an equation the form of
which will depend on the assumed relation between z and «. In particular if P and
Q denote the same functions of z that p and qg are of w; and if we assume z=a,
P, Q will become =p, g respectively: the given equation in v will be

d’y dv
det | Page Tes

and w will thus denote the quotient of any two solutions of the equation

dy dy
gat Pant w=;

viz. writing X = p’+ oP 49, then by what precedes, the equation for w will be

{w, 2} =—1X.

65. Returning now to Kummer’s problem, and considering y, v as solutions of the
two differential equations respectively, w is a function independent of the particular
solutions denoted by these letters: we have y=wv, and taking any other two solutions
y,=w»,, and therefore ga; calling each of these equal quantities s we have s denot

a 1
ing the quotient of two solutions of the equation in y, and also the quotient of two
solutions of the equation in v; whence writing as before X= p+ 277 — 4g, and similarly

a P427 _ 40, we have

and since in general

Vou, XII). Parr 1, 5

34 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

we obtain

fe aj=-4ax432(%),

as the required equation for the determination of z as a function of z. The process does
not give the value of w, but this can be found without difficulty, viz.
P SPaz—Spadx : dz
w= ve ies
If z, 2 are regarded each of them as a function of the new independent variable
@, then the equation is
a iy
Poa) = =
(=, 0-4 (75) Z={@ 8) -3(G9) x
66. Jacobi’s differential equation of the third order for the transformed modulus 2,
Fund. Nova, p. 78 is

3 (RIN* = NE) — 9 EN” =") +1" | (TA) (FAR) ah 0,

where the accents denote differentiations in regard to an independent variable @: viz.
dividing by 2k'*X” this becomes
ne (LtkN? ‘
fk, +47 (F*%) =p, a +4n7(

which is thus a particular case of Kummer’s equation, k, % corresponding to 2, z
respectively, and the values of X, Z being

1+ A ad i. + a

X=), 2=-(G

a)
Rye ?

67. In the case of the hypergeometric series, the two differential equations of the
second order are

dy y-G@+Rh+ia dy aby _4
dx? a.l—a@ dt “ole

Gu y-@+h+1)2zdv__ abv ip

dz Ail Se dz z2.l=2
_y(z+(1—2))—(a+8+1)c¢_y,y-a-B-1 ._ —28
Hence p= a ae ae — Ose
and hence
49 gee (= 2=8 — Ut ty Seas a =")
do 9 ig (I—a)? oe :
viz. writing
=a, a=1(1—2’),

p= (a =): b=4(1 —-),
=(y—a—f)', c=4(1-),

AND THE POLYHEDRAL FUNCTIONS. 35

and putting in the formula z—1, =—(1— 2), we have
dp L(1—v*) £(1—v’) i (M—-p’?+r*-1)
Sige he =
1(p 2 ae 4) a (a — 1)? a z.a—1 :
a c —a+b—c
Se G1) Det e.a—1 ’

1 2
a, b, Cc A, Fy je
aw’ 2-0’ g—l

with a like formula for 3 (P+ san 4Q). We then have
y = wv,
w= Cx-¥ (1 — x)¥-2-8-1 2 (1 — z)- yates

and the differential equation of the third order for the determination of z is

Bm Septet) -G 2 ety. Ayan

Lo L—8 Ci—

where a,, b,, c, are the same functions of a’, f, y which a, b, c are of a, B, ¥

This is in effect Kummer’s equation for the transformation of the hypergeometric series.

68. And in like manner the Schwarzian equation for the determination of s, the
quotient of two solutions is

al 1
{s, a} =(a, b, ¢ oe. aan ie)

PART II. THE POLYHEDRAL FUNCTIONS.

Origin and Properties. Art. Nos. 69 to 80.

69. The functions in lines 1,...5 of the PQR-Table are connected with the geo-
metrical forms :

1. Polygon or

Double Pyramid*.

3. Tetrahedron.

4, Octahedron and Cube.

5. Dodecahedron and Icosahedron,
(these figures being regarded as situate on a spherical surface,) and with the stereographic
projections of these figures.

Consider a spherical surface and upon it any number of points: take at pleasure

any point as South Pole, this determines the plane of the equator; and the stereo-

* Prof. Klein regards 1 as belonging to the polygon fundamental figure, to which 1 and 2 each of them belong,
and 2 to the double pyramid: it seems to me that the _ is the polygon.

j—

36 Pror, CAYLEY, ON THE SCHWARZIAN DERIVATIVE

graphic projection of any point is the intersection with the plane of the equator of the
line joining the point with the South Pole.

To fix the ideas take the radius of the sphere as unity: let the axes of a and y
be drawn in the plane of the equator in longitudes 0° and 90° respectively, and the
axis of z upwards through the North Pole: the position of a point on the sphere is
determined by means of its N.P.D. @ and longitude f: moreover we take X, Y, Z
for the coordinates of the point on the surface, and a, y for those of its projection;
and we then have

X, Y, Z=sin @cosf, sin@sin f, cos 0;
pple
Wee

= tan $0 cosf, =tan}ésinf,

¥
U1+Z
and conversely,

X, Y, Z=22, 2y, l—a*-y’, +(1+a*+y’).

We represent the point (X, Y, Z) on the spherical surface by means of the magni-
tude «+iy(=tan}@(cos f+isinf)), or say by the linear factor, s— (x+y): and similarly
any system of points on the surface by means of the system of magnitudes «+ ty, or
say by the function II {s—(x+dy)}, denoting in this manner the product of the linear
factors which correspond to the different points respectively.

70. It will presently appear that if (considering a different stereographic projection,
that is, a different position of the South Pole) we take a’, y’ as the coordinates of
the new projection of the point, then a#’+7y is a homographic function

a(a+iy) +b + {e(a@+iy) +d}

of a+zy: and consequently that the functions of s which belong to different projec-
tions are linear transformations one of the other: but at present we consider a single
projection.

It may be proper to remark that the figures in question are spherical figures
having summits which are points on the spherical surface, edges (or sides) which are
ares of great circle joining two summits, and faces, which are portions of the spherical
surface: the mid-points of the sides, and the centres of the faces are of course points
on the spherical surface.

71. (1), (2). Considering a regular polygon formed by n summits on the equator,
the longitude of one of them being 0°, then the stereographic projections correspond with
the points themselves, and the values of #+%y are

(n —1) 2ar

Qar (n—1)27...
- COs hae wage as: sin

DT tees
1, cos — +7sin — ,..
n n
The corresponding function of s is s*—1.
The values of «+7y for the mid-points of the sides are
37r " (2n =i )or
; n

ar et eT Dit ee
cos — +7sINn—, Cos + 7.81In — , «.. CO
n n n n

+7 sin GaSe

?

and the corresponding function is s”+ 1.
5

AND THE POLYHEDRAL FUNCTIONS. 37

The North and South Poles, which form with the 2 points a double pyramid of
n+2 summits, correspond to the values s=0 and s=a. We have thus

s(1-2)(s*-1)

as the function corresponding to the double pyramid.

72. (8) Considering for a moment the tetrahedron as a figure with rectilinear
edges, this is so placed that two opposite edges are horizontal, and that the vertical,
planes passing through the centre and these two edges respectively are inclined at angles
+45° to the meridian: viz. the upper edge has the longitudes 135°, 315° and the lower
edge the longitudes 45°, 225°. We thus explain the position of the spherical figure.

Corresponding to the summits we have the function s*— 27./3s°+1.

In fact the equation s*—2i/3s°+1=0 gives s*=7(/3+2), and hence the values of

s are the four values of 2«+7y shown in the annexed table for the values of X, Y, Z,
and x+y for the summits of the tetrahedron,

lori ge XGuey a Z | w+ty
Tepes rg es lhe Mees
WS Slag

owes —1l47
135 ar =F vB+1
Ses

225° —- — = |———
‘ =i
Siena ae a 14+
V3+1°

Corresponding to the centres of the faces, or summits of the opposite tetrahedron
we have the function s*+ 27/3 s*+1.

Corresponding to the mid-points of the sides we have the function s(1 - =) (st—1);

viz. the points in question are the North Pole s=0, the South Pole s=o, and the four
points s=+1, s=+z on the equator at longitudes 0°, 90°, 180°, 270° respectively.

73, (4) The octahedron is placed with two of its summits as poles, and the other
four summits in the equator at longitudes 0, 90°, 180°, 270° respectively: the values of

s are as in the last case 0, 0, +1, +7, and the function is s(1 -=) (s*— 1).
ee)

The function for the centres of the faces, or summits of the cube is s*+14s'+1.
The function for the mid-points of the sides of the octahedron or of the cube is
s? — 338° — 33s°+ 1.
74. (5) The Icosahedron is placed with 2 of its summits for poles; five summits
lying in a small circle above the plane of the equator at longitudes 0°, 72°, 144°, 288°,

and the remaining 5 summits in the corresponding small circle below the equator at
longitudes 36°, 108°, 180°, 252° and 324°.

38 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

The function for the summits of the Icosahedron is s(1 - =) (s+ 11s°—1).
The function for the centres of the faces of the Icosahedron, or summits of the
Dodecahedron is s* — 228s" + 494s" + 228s° — 1.

The function for the mid-points of the sides of the Icosahedron or the Dodecahedron is
s — 522s + 100058” + 0 s* — 10005s” + 522s° +1.

I give for the present these results without demonstration.

75. Writing ‘A for s so as to obtain homogeneous functions (*Qa, y)"—it will be

recollected that the 2, y of these functions have nothing to do with the a, y of the
foregoing values x+iy—the forms which have thus presented themselves may be denoted
as follows:
(3) f8=C, — 2073, 192%, yl,
h8 = (1, + 27/3, 1a", 9°)’,
t3 = xy (2*—y'),
(4) fe=ay@*—y,
h4=(1, 14, 102%, y')’,
t4=(1, — 33, —33, 1 et y')*,
(5) fo=ay(l, U1, -1h%, x),
h5 = (1, — 228, +494, + 228, —19 2°, 7')%,
t5=(1, —522, 10005, 0, — 10005, 522, 12°, y’)’,

where observe that /* is the same function as 3. In each set of functions f, h, ¢,
we have h and ¢ covariants of f, viz. disregarding numerical factors,

h is the Hessian, or derivative (f, f), and ¢ is the derivative (f, h).

76. Since f+ is the same function as ¢3, we have of course f4, h4 and t4 them-
selves covariants of f3: but it is convenient to separate the two systems.

77. It is to be observed that f3 is a quartic function haying its quadrinvariant
(I) =0; but independently of this, that is qua quartic function, it has only the covariants
h3 and ¢3 the (Hessian and the cubicovariant respectively), viz. every other covariant is
a rational and integral function of f3, h3 and #3. In particular h4 and ¢4 are rational
and integral functions of f3, h3 and #3; but inasmuch as f3 and 43 are not covariants
of f+ this is not a property of h4 and ¢t4 considered as covariants of /4, and the rela-
tion in question need not be attended to.

78. It has just been stated that 73 qua quartic function has (in the sense explained)
only the covariants h3 and t3: f+ quad special sextic function and f5 qua special dode-
cadic function have the like property, viz. f4 has only the covariants h4 and t4; fo
only the covariants 45 and t5. Hence f3, f4, f5 are “Prime-forms” in the sense defined

AND THE POLYHEDRAL FUNCTIONS. 39

in the paper Fuchs, 1875, viz. a Prime-form has no covariant of a lower order than
itself, and also no covariant of a higher order which is a power of a form of a lower
order.

79. The same functions have also the property that they are functions transform-
able into themselves by means of a group of linear transformations, and in this point
of view they were considered in the nearly contemporaneous paper Klein, 1875; it is in
this paper shown that the functions so transformable into themselves must be Polyhedral
functions as above, the linear transformations in fact corresponding to the rotations
whereby the spherical polyhedron can be brought into coincidence with its own original
position, This theory will be presently given.

80. It is to be observed that if U, V are functions (*§a, y)" of the same order n,
then using the accent to denote differentiation in regard to 2, UV'—U’V and (U, V)

differ only by a numerical factor: and further that writing as before ie and in the

expression UV'’—U’V regarding U, V as functions (*%{s, 1)” and the accent as denoting
differentiation in regard to s, we have UV'’—U’V and (U, V) differing by a numerical
factor only. We have in the PQR-table, lines 8, 4, 5, P, Q, R equal to given numerical
multiples of A*, t¥, f%, the indices a, 8, y being such as to make these to be functions
of the same degree: hence neglecting numerical multipliers PQ’—P’Q is equal to a function
(A, t”), which is =h?-'y-1(h, ¢): and the theorem that PQ’-P’Q, =QR’-Q’R, =RP’— RP
contains only factors of P, Q, R is in fact the theorem that (h, t), (h, f), and (¢, f)
are each of them equal to a term or product of f, h, t: which is a result included in
the theorem that f has only the covariants h and ¢ And by this last theorem we
know already how from R, assumed to be known, we can derive P and Q: viz. Risa
power of f; and we thence have h=(f, f) and t=(h, f), equations giving the functions
h and ¢, upon which P and @ depend.

Covariantive Formule. Art. Nos. 81 to 84,
81. The various covariantive formule will be given with their proper numerical

coefficients,

Tetrahedron function. f, h, ¢ stand for the before-mentioned values, /3, h3, ¢3
(2 Q, R=h’, —121 NIB) oe =i'))

iderye. (2, 0, c, d, e)=1, 0, =. eae
4(f, f'=—96i /3.h, 4, WY =9GVB.f, E(t, O° = 25f A,
(fi) = 3% /3.4, Gaprecwl=0, (fF ht = 11527 = 1152.3
Gap 4B,
Ga 4. f?,

he —f? — 1271/3? =0,
fh=(1, 14, 1Yat, y4)? (=f¥).

40 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

It is convenient to remark that ¢, f°, h® being of the same order we have
EOF KF (P, +O, F)=0,
that is C3. PR, AAS .3.2 Wt (h, H+1V.2.3 tft, f) =9,
an equation, which substituting for (f, A), (h, t), (t, f) their values, reduces itself to the

before-mentioned relation h°—f*—12i /3@=0; and we have thus a verification of the
values of (jf, h), (h, t) and (¢, f). The like remark applies to the other two cases, which

fellow.

82. Hexahedron function. ff A, t stand for the before-mentioned “values F4, 1A, t4

(P, Q, R=A', —#, —108f%).
For f4.
4(f, f° =— 25h,
(f, h)=—8t,
(fF }) =—12h?,
(h, t) =—1728/%,
h? —# —108f*=0.

(a, b, ¢, d, e, f, g) =(0, 4, 0, 0, 0, —4, 0).
2G, Fy =8;

£(h, b) = 3.25.77. 7%,
£(@ S28 1L Ph,

4 (f S)= (720). 8

83. Dodecahedron function. f, h, ¢ stand for the before-mentioned values /5, hd, t5

(P, Q, R=h’, —#, —1728f°).
For f5.
4(f, f)? =—121h,
4(£ FP = 0,
(Ff, h) =— 208,
(f, t) =— 80h’,
(h, t) =— 86400f°,
® —f —1728f° =0.
84, We have

Write
then

(a, b,c, d, ef, g, h, t, j, & t, m=(0, +, 0, 0, 0, 0, a, 0, 0,0, 0, —a, OF
4(f, f=0,
(4 Ff)" =0,
1 (h, h)? =173280f°,
L(t, )? =9082800f%,

£(f, f° =4 (024) (720)°. 33 F*
4(F f)? = 4 (924)? (720)". 35%,

t=(a" +") (1, 522, —10006, — 522, 1% 2%, 7°)*
E=(2?+y*).(1, 2, 6, —2, 1a, y)*
t=£(1, —10, 458 f).
E£ _ (a+) (1, 2, 6, —2, 18, »)!

Or putting

that is, E=p/f, then

* The numerical coefficients —% and 2% are Klein’s B
and A: the latter of them the ordinary quadrinvariant of a
dodecadic function; the former is an invariant linear as
regards the coefficients of f, and existing only for the special
form f in question: viz. writing for a moment

f= d(ey + hy’ —zy"),

then (f, f)® contains the factor \7, and (f containing the

Jay (a a 11a*y? =")

s t
fear Tea

,

(Klein.)

factor \) the form is ;
4(f, f)S= (924)? (720). —Fer- Sf,
which is linear as regards \. We have also
4(f, fy? = (924)? (720). SE d*:
say A=25*, B=-,d; or 84B°=A. Of course in the
case of a general dodecadic function f, we have (f, f)®, an
irreducible covariant, not breaking up into factors.

AND THE POLYHEDRAL FUNCTIONS. Al

Investigation of the forms f5 and h5. Art. Nos. 85 and 86.

ise 5—1
85. Writing for shortness* k= tana = ue

, and g=cos 36° +7 sin 36°, then the values

of «+z2y corresponding to the summits of the Icosahedron are
0,
k, kg, kgs” kof, keg’,
KG. kG eG eg, eg"
B25
and the function 75 is thus

=s (1-5) ¢-#)6-F9,

where the product of the last two factors is s”°+(k°—k*)s°—1. We have
k=, (80 /5+176), =4(5 /5+11),

= J, (80 ¥5-176), =4(5 5-11),

and consequently i #°=11; or the function is s(1- =) (e+ 10s°—1).
86. Similarly writing for shortness* / = tan }y, U'=tan ty’, where
cosy = ae 5 sin’y = a and therefore —— : - _ ;

and g=cos36°+7sin36° as before, then the values of #+7y for the summits of the

dodecahedron are

5 9

loan lo, wilgr mtg. \ilge,
Ug, Ug’, Ug’, Ug’, Ug’,
Re 73 9; ia 9, i Oe, Ig e
Ne ig, iq%, l9°, ar
and the function h5 is therefore
=s94+ 8 (F—P)+1 . 8°48 C°—2%) —-1.
(1 + cos y)*° — (1 — cos y)* a2 COS
sin’y sin’y
= 2 cosy 384+ 64/5 _ 128 cos
sin’y * 45 45)- gin” i

(viz, this last identity depends on 32 (3 +95) (6 + /5) = (114450 1/5) sin‘y, that is
160 (3 + /5) (6 +/5) = (114 + 50 1/5) (120 — 40/5),

or 2. (3 +5) (6+ 1/5) = (57 + 25 ¥/5) (3— V5),

or finally (7+3 75) (64175) = 574+25,/5, which is right).

We have -—P=

5 + 10 cos’y + cos*y)
2/ mi

(64/5), =1144+50 5.

* a is the a, y is the y, and 7 the a-f of the Table, No. 99.

Vout. XIII. Part I. 6

42 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

Similarly U*—1®=114—50 5,
and observing that the sum and product of 114+4501/5, 114—50,/5 are = 228 and 496
respectively, the required function of s is
(s® — 1)? — 228 (s* — s°) + 496s”,
= 5” — 228s 4. 4945 + 2285° + 1,

which is the required value of h5.

Invariantive property of the Stereographic Projection. Art. Nos. 87 to 93.
87. The before-mentioned theorem that the functions derived from two different
stereographic projections of the same point are linear transformations one of the other,
may be thus stated:

Considering on the surface of a sphere, two fixed points A and B; and determining
the position of a point C first in regard to A by its distance @ and azimuth f, and

A

(64
next in regard to B by its distance 6’ and azimuth /’, the azimuths from the great
circle ABz which joins the two points A and B, then we have
tan 40 (cosf+isinf), and tan 46’ (cos f'’+isinf’),
homographic functions one of the other: calling them s, s’, and putting the distance
ABz=c, the relation between them in fact is

ga t= tan 4c
~ 14+stan te’
or, what is the same thing, tan ic (1+ 8s’) =s—s';
or observing that ss’=tan $0 tan 46’ {cos(f+/’)+7isin(f+f’)}, we have the two equa-
tions

tan $c {1+tan $0 tan 4 @ cos (f+ f’)} = tan $6 cos f—tan }0' cos f’,
tan}c{ tan}@tan}6@'sin (f+/)} =tan }@ sin f—tan $6’ sin f’.

AND THE POLYHEDRAL FUNCTIONS. 43

88. If we denote the angles of the spherical triangle by C, A, B and the opposite
sides by c (as before), a, 6, then 6, 0'=b, a; f, f’ =A, 7—B, whence
s, s =tan3b(cos Ad +7sin A), —tan $a (cos B—7sin B):

or we have between the sides a, b, c and angles A, B of a spherical triangle the

relations
tan 3c {1 —tan da tan $b cos (A — B)} = tan bcos A + tan da cos B,

tan}c{ —tan}atan}bsin (A —B)}= tan $b) sin A —tan sasin B;
equations which may be verified by means of the ordinary formule of Spherical
Trigonometry.

89. But it is interesting to give the proof with rectangular coordinates.

Taking (X, Y, 7), (X,, Y,, Z,) for the coordinates, referred to two different sets of
axes, of a point on the spherical surface: also 2, y, z,, y, for the coordinates of the
corresponding stereographic projections, we have

iy YZ) =Car ve > yy CCNY, 2)
| g ; B’ ; of
| oe ie yy |
: Zl = 2a) Dye aya ak ey 2
Diana gs. Wier ee pied — ie ay a Tt pi ay

and thence
Dope Le Qax +2By +y (1—a*-7’)
: 2a! 2+ 28 y ty (1—a—y)
> Latte + 22a + 2B’y + y” (1 —2*—-y’).

90. Introducing z, 2, for homogeneity, or writing —, 7 and —, A in place of 2, y
Si
and z,, y,, respectively, we have
z= ne Soe Gee | Cee y,B,4, 0%, y, 2),
w= Sue yey Ae ey 8 Yo Mas a5 Oe 5. Dh

Bae ae ty t+ 24' 24 2B’y ty (e—2#—y’), =d—9y', 1-7’, 149, B', a, 0%, ),
and thence without difficulty

1 ” ” # Les = " mw a Ua @
4 = ppg (tye +@' +B ei dty)2+( a 18") (w+ iy)

ey ttt TY )2+(a' +78") (cx -w)} {1 —y) 2+ 2a" +78") (a+ wy)},

. 1 u" uy “Or . Me u" “OV” .
Uae laa )z—(2" + 1B") (w—ty)} {1 +9") 2+( @”— 78") (w+ w)},

viz. the form is z,: 2,+1y,: 2,—-ty,=MN: NL: LM (L, M, N linear functions of
z, x+y, z—ty): showing that the relation between two stereographic projections of the
same spherical figure is in fact that of a quadric transformation, the fundamental points
in each figure being an arbitrary point and the two circular points at infinity: or, what
is the same thing, to any line in the one figure there corresponds a circle in the other
figure, which is the “circular relation” of Mdébius.

6—2

44 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

91. The actual values are
z,+ty, l+y"” (l—-y’)2-(2"—18") (e+)
zo yteyt” (L+y")2+ (2 —18") w+)’

z,— ty, a: gy” (= 9") 2— (2+ 18") (e— ty)
z, = y—yi . al +7’) PEs (a” + 7B") (2 — zy) >

viz. attending only to the former of these, we have meth a homographice function of
1
r+ wy

4

, which is the before-mentioned theorem.

92. Supposing that the transformation from (X, Y, Z) to (X,, Y,, Z) is made by

a rotation the coordinates of which are X, pw, v (that is, if f, g, h are the inclinations
of the resultant axis to the axes of a, y, 2 respectively, and @ the angle of rotation,
putting A, w, v=tan}@cosf, tan}@cosg, tan}@cosh), then the coefficients of trans-
formation are

(a, By y )=(14 =H 2Aptr) , 2Qv—p) +4 rayter?,

Rae ae) 2(ur—v) , 1-A+y2—v, Q(uy ta) |

Thee Jefe Gy | 2(A+H) , 2(uv—r) , 1—-N—p?+1*|
and substituting these values, the formule become after an easy reduction

z,+ty, —(W+%) (we+ty)+(A+%M) Zz

% (A—t)(w@+ty)+(v—t)z2 ”
z,— iy, _—(v—A (wt) + (X— tn) 2
a FM) (ty) + Ft)2
attending to the former of these, and writing for greater simplicity ah ; tty

1

=s,, s respectively, we have

_—(v+t)s+At+wm)
< (A—t)s+(v—7) ’

or writing this

_As+B
age ae
then A:B: C0: D=—-v-i:rA4+ip:rA—M : v—-2.

93. I call to mind that the condition in order that the homographiec transformation
s,= (As + B)+(Cs+D) may be periodic of the order n is

5)
2 2m

(A + D)?— 4 (AD — BC) cos a
in particular n=2, it is A+D=0: n=3, it is A?+AD+D?+BC=0: n=4, it is
A? + 1?+2BC=0: and n=5, it is (A+ D)?—34(3 +5) (AD-BC)=0.

Groups of homographic transformations. Art. Nos. 94 and 95.

94. The formula just obtained serve to connect the theory of the rotations of a
polyhedron with that of the homographic transformations s into As+B+(Cs+D): and,

AND THE POLYHEDRAL FUNCTIONS. 45

corresponding to the rotations which leave the polyhedron unaltered, we have groups of
homographic transformations. We have thus, corresponding to the cases of the tetrahedron,
the cube and the octahedron, and the dodecahedron and icosahedron respectively, groups
of 12, of 24, and of 60 homographic transformations s mto As+B+(Cs+D). The
group of 60 and the group of 24 include each of them as part of itself the group of
12: it is further to be remarked that the group of 12 may be regarded as that of the
positive substitutions upon four letters abcd, the group of 24 as that of all the substitu-
tions upon the four letters, and the group of 60 as that of the positive substitutions
upon five letters abcde.

95. I call to mind that a group of functional symbols 1, a, §,... can always be
expressed in the equivalent form 1, Sa¥*, S@S“,... where ¥ is any functional symbol what-
ever: clearly, a, §,... being homographic transformations, then, S being any homographic
transformation whatever, the new symbols Sa37*, S837... will also be homographic trans-
formations; and thus the group of homographic transformations can be expressed in
various equivalent forms: these correspond to the different positions of the polyhedron in
regard to the axes of coordinates: and there are in fact three cases which it is proper to
consider, viz. attending for the moment to the dodecahedron, we may have the axis of z
passing through the midpoint of a side, through the centre of a face, or through a sum-
mit; that is, in the language presently explained, the cases are 1°, Pole at a point ©; 2°,
Pole at a point A; 3°, Pole at a point B.

The regular Polyhedra. Art. Nos. 96 to 103.

96. We require a theory of the regular Polyhedra considered as systems of points
on a sphere. I refer to my two papers (1866) and (1877). In the latter paper, I remark
that considering the five regular figures drawn in proper relation to each other on the
same spherical surface, the only points which have to be considered are 12 points A,
20 points B, 30 points ©, and 60 pomts ®. Describing these by reference to the dodeca-
hedron, the points A are the centres of the faces, the points B are the summits, the
points © are the midpoints of the sides, and the points ® are the midpoints of the
diagonals of the faces. Or describing them by reference to the icosahedron, the points A
are the summits, the points B are the centres of the faces, the points © are the midpoints
of the sides (viz. each point © is the common midpoint of a side of the dodecahedron
and a side of the icosahedron, which there intersect at right angles), and the points ®
are points lying by three’s on the faces of the icosahedron, each point ® of the face being
given as the intersection of a perpendicular A® of the face by a line BB joining the
centres of two adjacent faces and which intersects AO at right angles.

97. The points ® are comparatively unimportant, and it is proper in the first in-
stance to attend only to the 12 points A, the 20 pomts B, and the 30 points ©: these
form 6 pairs of opposite points A, 10 pairs of opposite pots B, and 15 pairs of opposite
points ®. Considering the diameters through each pair of opposite points ©, we have thus a
system of 15 axes, which in fact form 5 sets each of 3 rectangular axes: attending to any
one of such sets. the diametral plane at right angles to one of the three axes contains of

46 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

course the other two axes: it contains also two axes each through a pair of opposite points
A. and two axes each through a pair of opposite points B. If imstead of the plane we
consider its intersection with the sphere, we have thus on the sphere 15 circles each
containing 4 points @, 4 points A and 4 points B. The fifteen circles intersect by fives
in the pairs of opposite points A, by three’s in the pairs of opposite points B, and by
two's in the pairs of opposite points ©; the mutual inclinations of successive circles at
the points A, B, @ being =36°, 60° and 90° respectively. The whole number 15.14,=210
of the intersections of the circles two and two together is thus made up of the 12 points
A each counting 10 times, the 20 points B each counting 3 times, and the 30 points ©
each counting once; 210= 120+ 60+ 30.

98. The angular magnitudes which present themselves are all obtained from the
dodecahedral pentagon, shown in the annexed figure, and in which the angle subtended
by a side at the centre is =72°, and the angle between two adjacent sides is = 120°.

We write AO=a, BO=8, AB=y, B.B,=a, 4 B,BB=0, OB,=g, -OBB=¢.
From the triangle A@B, the angles of which are 36°, 90°, 60° and the opposite sides
B, y, a, we find the values of a, f,-y, and these are such that 2+ @+y=1.

From the triangle B,BB, where the sides B,B, BB,, and the included angle are 2£,
28, 120°, we have the opposite side z, and the other two angles each =0.

From the triangle BBO, where the sides B,B, BO, and the included angle are 28,
8, 120°, we find the opposite side g, the angle BB,O,=¢, and the angle B,OB, = 45°.

Hence each of the angles BOB, B,OB, being =45°, the angle BOB is =90°: in this
triangle the hypothenuse B,B, is =a, and each of the other two sides is =g: whence
we have cosz=cos*g as is in fact the case, and moreover the values give «+2qg=180°.
Also each of the other angles is found to be =60°; that is we have 4 B,BO=60°,
or the whole angle at B, being =120°, the sum of the remaining angles B,B,B, and
BBO is =60°: that is 0+ 6 =60°.

AND THE POLYHEDRAL FUNCTIONS. AT

From the triangle ©8,©’ where the two sides and the included angle are £, 8, 120°,
we find 00’ = 36°.

And from the triangle ©6,0’, where the two sides and the included angle are g, g
and (120° — 2¢=) 20, we find ©0’= 60".

99. We thus arrive at the following Table:

sin cos
5-—/5 5+,/5
0 Y SEs
AO a 31° 43 Ny. 10 10
[5-1 5+1
20° 55’ a &
Be B 0° 55 a5 ORE
a 10-2,/5 A oo
099!
a Y Sie N/- 15 15
2/2 1
- 0 9’ N —
(BB) x 70° 32 3 3
J2 i
(Be) g 54 44 NE A
/3 we
40 Ae! N
BBB 6 37° 46 22 2,2
/3 (V5 -1) J5+3
2 eens psec Es
BoB $ 22 14 4/3 1/3
2 1
2 J5
28 41 50 5 =
% 2 (/5 +1) A-/5
Ze ek 3.5 = rah/ 51
Seen 10+2,/5
a-p 15 aT
5-1 5 A/D
199 5 ee
60 36 5—/5 /5+1
8 4

Where as above

a+B+y=90°,
2+2¢g =180°,
O+¢ = 60.

100. We now construct three figures of the points A, B, @; viz. these are stereo-
graphic projections, each showing the Northern hemisphere projected on the plane of the
equator by lines drawn to the South Pole: hence for any pair of opposite points not on
the equator only the point in the Northern hemisphere is shown: but for a pair ot
opposite points on the equator the two points are each of them shown. In fig. 1 the
North Pole is taken to be a point @; in fig. 2 it is a point A; and in fig. 3 it is
a point B. The Position of any point on the sphere is determined by its N.P.D. and
its longitude (measured from an arbitrary origin, say from the point # of the centre
left-handedly): and in the three figures the positions are as follows,

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

101. Fig. 1. Pole at ©.

N.P.D.’s Longitudes.

2A a= 31° 43’ 0°, 180°

24 90°-a= 58 17 90, 270

4A 90 (0, 180)+a=31° 43’
2A 90°+a=121 43 90, 270

2A 180°-—a=148 17 0, 180

4B g= 54 44 45, 135, 225°, 315°
2B 90°-B= 69 5 0, 180

4B 90 (90, 270)+B=20° 55’
2B 90°+8=110 55 0, 180

4B 180 -g=125 16 45, 135, 225, 315
2B 180°—- B=159 5 90, 270

16 —

40 (90°, 270°) +a =319 43’
40 (0, 180)+#8=20 55
40 (90, 270)+a=31 43
40 0, 90, 180°, 270°
40 (90, 270)+a=81 43
40 (0, 180)+8=20 55
40 (90, 270)+a=31 43

2B B= 20° 55’ 90°, 270°

AND THE POLYHEDRAL FUNCTIONS. 49

102. Fig. 2, Pole at A.

2

a ss)

N.P.D.’s Longitudes.
0 =
2a=639 26’ 0° 72° 1449 9169 288°
180°-2a=116 34 36 108 180 252 324
180 -
y= 37 22 36 108 180 252 324

90°-a+fB= 79 12
90 +a—B=100 48

180 -a=144 17

36 108 180 252 324
0 72 144 216 288

180 —y=142 38 0 72 144 216 288
a= 31 43 0 72 144 216 288
909-a= 58 17 36 108 180 252 324

90 (36 108 180 252 324) +18?
90 +a=121 43 0 72 144 216 288

36 108 180 252 324

Vou. XIII. Parr I.

50 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

103. Fig. 3, Pole at B.

a

<

D>

Dn ele

esa BS ae
ea i Ziv Sf
AQ / \ ae / Wa
x oe H jin re
. : a
} LO
e Pee \ i ; Pak ()
ete Syne
N.P.D.’s Longitudes,
| 34 y= 37°22 | 30015092700
34 90°-a+fB= 79 12 90 210 330
34 90 +a—B=100 48 30 150 270
34 180 -y=142 38 90 210 330
B 0 be
3B 28= 4150 | 90 210 330
6B z= 70 32 (30 150 270)+3=37° 46
6B 180°- «=109 28 (90 210 330)£3=37 46
3B 180 -28=138 10 | 30 150 270
B 180 —
30 B= 2055 | 90 210 330
| 60 g= 5444 | (90 210 330)+gp—29014'
| 30 90°-B= 69 5 | 30 150 270
| 60 90 0 60 120 180°240° 300°
| 8e 90 +B=110 55 90 210 330
_ 60 180 -g=125 16 | (30 150 270)+@=22°14'
30 180 -B=159 5 | 30150 270

ee)

AND THE POLYHEDRAL FUNCTIONS. 51

The groups of homographic transformations, resumed. Art. Nos. 104 to 117.

104. The axes of rotation for the dodecahedron and the icosahedron are 15 axes
each through a pair of opposite points @, 6 axes each through a pair of opposite points A,
and 10 axes each through a pair of opposite points B; or say 15 @-axes, 10 B-axes and
6 A-axes: the corresponding angles of rotation are 180°, 72° and 120°; so that (excluding
in each case the original position or that of a rotation 0) we have in respect of each
@-axis 1 position, in respect of each A-axis 4 positions, and in respect of each B-axis 2 posi-
tions; in all, including the original position, 14+ 15+ (6 x 4) +(10 x 2), =60 positions, that
is a group of 60 rotations.

To find in any one of the three forms the group of homographic transformations,
we can in each case obtain from the foregoing tables the values. cosf, cosg, cosh of the
cosine-inclination of an axis of rotation to the axes of coordinates, and thence calcu-
late the values of

A, #, v=taniScosf, tan $9 cosg, tan $3 cos h,
and thence the values of
A, B,C, D=—v-%i, A+%M, A-t, v—-i,
viz.: in the case of a ©-axis 3 is=180°, (so that here tan}S=o0, or the values of
A, B, C, D are =—v, X+ip, X—t, v, that is —cosh, cosf+icosg, cosf—icosg, cosh);
in the case of a B-axis the values are $=120°, 240°, and therefore tan} S=+,/3; and in
the case of an A-axis, they are 3=72°, 144°, 216°, 288°, and therefore

J/10 + 2/5 10-25

19 =
Re expen 5/5 I

105. The ©-form was first given in my paper of 1879, but in obtaining it I used
results given in the paper of 1877. As regards the identification with the substitution-
symbols, since there is nothing to distinguish inter se the letters a, b, c,d, e, any trans-
formation A, B, C, D of the fifth order might have been taken for abcde, but No. 37
of the group having been taken for this substitution abcde, I do not recall in what manner
I found that consistently herewith the transformation No. 2 (—1, 0, 0, 1, that is s into —s)
of the second order could be taken for ab.cd. But there is no sub-group of an order
divisible by 5; and hence, these two transformations being identified with the two sub-
stitutions, the other transformations correspond each of them to a determinate substitution.

106. Homographic Transformations. The group of 60: Pole at ©.

(Ax +B) + (Cx +D)
1 1 0 0 1 i
2}/ -1 0 0 1 ab.cd
3 0 1 1 0 ac.bd
4 0 1 -1 0 ad.be
5 2 —34+/54+7( 1-./5) | —-8+/5+7(-14./5) | -2 be.de
6 2 2 ae.be

-—34+/5+7(-14+/5) | -84+/5+7( 1-./5) | -

~~ eR ob
Go =I or

r=
o

or
Oo

or

aang a

DoF WN

oro
=>)

|

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

to po to to bo bo to bo bo to

-1

z

—1=J/5+2( 34/5)
14+/5+7( 3+4/5)
1+r/5 +7(—3—n/5)
—1-,/5+7(-3-,/5)
=—3+.J5+7( 1=,/5)
—3+,/5+7(-14+4/5)
3—/5+7(-14+,/5)
3=/5+7( 1=n/5)

bo bo bo bo

-1-,/5+7(-3-,/5)
| —1-J/5+7(

1—,/5)
—1-,/5+7(-1+,/5)
—1-,/5+7( 3+4,/5)
1+/5+i( 34/5)
1+,/5+7(-1+.,/5)
14+/5+7( 1-4/5)
1+./5+7(-3-,/5)
—1-./5+7(-1+,/5)
—34+,/5+7(-1+,/5)
8—J/54+7(-1+,/5)
1+./5+7(-1+,/5)

14+/5+47( 1-4/5)
3-J5+i( 1-,/5)
—~34+/5+¢( 1—,/5)
=i Jeet les)
2
2
2
2
2
2
2
2

3-./54+7(-14,/5)
3-/5+4+7( 1-,/5)
—1-/5+7( 1-,/5)
—1—,/5+7(-1+,/5)
1+/5+2 (—1+./5)
14+/5+7( 1-./5)
-1-,/5+7(-3-,/5)
-1-/5+7( 3+4/5)
14 /54+7( 3+4/5)
1+,/5+7(-3-/5)

—1-/5+1(-1-+44/5)
1+/5+¢( 1/5)
-1-,/5+i( 1-,/5)

1+./5+7(-14+4/5)

WNNWONHNNNNNNNNYNND NN WW

| —34/54+0(—140/5)
| -1-,/5+7( 34,/5)

1+/5+7(-3-,/5)
3-J/5+7( 1-2/5)
-~34+/5+7( 1-,/5)
~1-/5+i(-3-,/5)
14+/54+7( 3+,/5)
3-/5+7(-1+4/5)

3-/5+4+7( 1-4/5)
3-/54+7(-1+,/5)
-1-V5+7(-1+4/5)
~1-J5t¢( 1-n/5)
1+/5+7( 1-./5)
14+/5+7 (—1+4/5)
-1-/5+7( 3+4/5)
-1-/5+7(-3-,/5)
1+./5+7(-3-,/5)
14+ /5+7( 3+,/5)

14+/5+7(-1+,/5)
-1-,/54+7( 1-,/5)
1+,/5+7( 1-,/5)

—1-,/5+7(-1+4/5)

3-./5+7(-1+,/5)
14+./5+7( 3+,/5)
-1-,/5+7(-3-,/5)
-34+,/5+7( 1-,/5)
3-J5+7( 1-4/5)
1+./5+7(-3-,/5)
—1-J5+2( 3+,/5)
-8+4./5+7(-1+4,/5)

-2
-2
-2
-2
-2
-2
-2
-2
-2
-2

—1-,/5+7(-3-,/5)
1+,/5+7(-3-,/5)
1+/5+7( 3+4/5)

—1=/5+7( 3-4/5)
3-/5+7( 1-./5)
3-/54+7(-14+,/5)

—34+/5+7(-14/5)

=3+/54+4( L=a/d)

2
2
2
2

1/544 (=3=0/5)

1+4+:/54+7¢( 1-,/5)
14+/5+7(-1+,/5)
1+/5+7( 3+,/5)
—1-/54+7( 3+,/5)
—1-,/5+7(-1+,/5)
-1-/5+7( 1-,/5)
=a) 122 (= Sin a)
-1-,/5+7( 1-5)
—34+/5+7( 1-,/5)
3-/5+7( 1-5)
14+/5+7( 1—,/5)
14+/5+7(-1+4,/5)
3-—/5+7(-1+,/5)
~34+,/5+7(-1+,/5)
-1-,/5+7(-1+4,/5)
2

2
2
2
2

5

2
2
2 ;

aeb
abe

abcde
acebd
adbec
aedcb
adceb
achde
aedbe
abecd
acbed
abdce
aecdb
adebe
aechd
acdeb
abede
adbce
aebde
abced
adecb
acdbe
abdec
adcbe
ached
acedb

AND THE POLYHEDRAL FUNCTIONS. 53

107. Taking out of the foregoing group of 60 a group of 12 contained in it, viz.
that corresponding to the positive substitutions of the four letters abcd, it is easy to see
that there is a transformation (7,0, 0,1), that is, s into 7s, which can be taken for the
substitution adbe, and to complete thence the group of 24. And we have thus the follow-
ing Table.

Groups of 12 and 24. Pole at ©.

(Ax +B) + (Cx +D)

1 1 0 10) 1 1

2] —1 0 0 1 ab.ed

3 0 1 1 (0) ac.bd

4 (0) 1 -1 0 ad.be

Bee? t 1 1 abe

6| -1 z 1 Zz ach

if il —2 1 Z ade

8 | -—z -2 1 al acd

9 z z 1 =) adb
10 v 1 —i abd
ll| - —7 1 —1 bed
12 —t 1 1 bde
13 t 0 0 1 adbe
14} -7¢ 0 0 1 achd
15 0 t 1 0 cd
16} 0 i —1 | © ab
17 1 = 1 1 acdb
18 | —7 =i 1 z bd
19 zt 1 1 t abed
a0) 1 1 1 i be
21; -1 =I 1 —l abde
22 z =H 1 =f ae
93 | -—72 1 1 —7 adcb
24; -1 1 1 1 ad

108. The group of 60 was obtained in the A-form by Gordan, in his paper. The
passage from the ©-form to the A-form is made as follows: let X, Y, Z be the coordi-
nates of a point when the axes are as in the ©-form, X,, Y,, Z, the coordinates of the
same point when the axes are as in the A-form: we may write

BAZ EM —-aZ + VsiaX te bau wheres, | oe pee
then if the equations of an axis of rotation referred to the first set of coordinates are
X:Y:Z=L:M:N, those of the same axis referred to the second set of coordinates
are bX,+aZ,: Y,:—-aX,+cZ,=L:M:N; or taking these to be X,: ¥,:4=L,:M,:N,
we may write L,, M,, N,=bL+aN, M, —aL+bN: these values are such that L?+M?+N?
=I*+ M*+N*, and hence A, pw, v and A, w,, », being the rotations, we may write

a4 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

L, M, N=, Su, 8; L,, MU, V,=9A,, Y,, 9,; where S has the same value in each set
of equations. From the equations
A:B:0:D=—v—-t:rA4+ Ww: A— i v—-4,
we have
B+C:B-C:D=-A:D+A=A: iv 1-4
=f:iM: N:-,
and similarly
B,+ C,: B,- C,: D,—A,: D,+ A, = D,:1M,: N,:-%.
Hence we may write

B,+C,= b(B+C)+a(D- 4),

B,-C,= B-C,
D,—A,=—a(B+ C)+b(D-A),
D,+A,= D+A,

Or say,
A,= a(B+ C)—b(D-—A)+(D0+A4),
B= b(B+C)+a(D-—A)+ (B- 0),
C,= b(B+C)+a(D-A)-(B- 0),
D,=—a(B+C)+b(D—A)+(D+A4),
which are the values for a transformation (4,, B,, C,,.D,) in the A-form: of course as
only the ratios are material, the values may be multiplied by any common factor.

109. The results are exhibited in terms of € an imaginary fifth root of unity: taking
e=cos 72° +7sin 72°, we have ;

where the upper signs belong to ¢, e and the lower to e’, «. It may be remarked that

sy pall Tal Saas tp ye ae ea
a 2 ee 2 2a ae 2 ae D) .

For instance, we have in the @-group (A, B, C, D)=(—1,0, 0,1); ab.cd: and thence in
the A-group A,, B,, C,, D,=(— 2b, 2a, 2a, 2b); ab.cd: or say this is (- i + = 1),
=(-—1, e+e, e+, 1); which is in the Table given as (-e, & +e, &+e', &); ab.cd.

By effecting the passage to the A-group in this manner we of course obtain the
proper substitution corresponding to each transformation: but I found it easier starting

from two transformations and the corresponding substitutions, to obtain thence by successive
compositions the entire group.

AND THE POLYHEDRAL FUNCTIONS.

110. Homographic Transformations. The group of 60. Pole at A.

@No. (ds +B) +(Cs +D)
1 Ded 1 1
2 4/0 —] 1 0 ad. be
3 13 | 0 —et 1 0 ac. be
4 980 -é 1 0 ae.cd
5 10/0 -é 1 0 ab. de
6 14/0 —€ 1 0 bd . ce
7 6 | e+e et 1 — (e+e?) ae. be
8 5 | e+eé 1 et —(e+e?) be . de
9 16 | e+e € 3 —(e+e) ac. de
10 3 | e+e 2 2 —(e+e*) ac. bd
il 15 | e+e 8 € —(e+e) ae. bd
12 12 | —1 e+e e+e 1 ab. ce
13 ll | -e +1 ete € be .cd
14 7|-é 1l+e e+e e ad . ce
15 2)|-é ete ete & ab.cd
16 8 | -e ete e+ ef et ad . be
We 21 | B+1 € 1 —(e+e) adb
18 35 | 241 2 et — (e+e) aeb
19 30 | &+1 é e —(e+e) bee
20 34 | 341 é! 2 —(e+e3) ced
21 19} &+1 1 € — (e + e°) ade
22 33 | e+e 2 1 — (e+e) ede
23 20 | e+e é et —(e+e) acd
24 22 | e+e! & & = (e+e) abd
25 36 | ete! 1 é Bee) abe
26 29 | e+e € € —(e+e*) bee
27 3l | -e 2 +64 e+et 1 aed
28 17 | -& ette e+e e abe
29 27 | —é e+e ete 2 bed
30 25 | —e4 +l etet 3 aec
31 23 | -1 1+é e+e é bed
32 24 | —é¢4 1+e eet 1 bde
33 34 || =a ete e+e? € ade
34 Ws! || 6 ete e+e € acb
35 28) — e+e e+e 3 bde
36 26 | - 3 e+] tet et ace
37 44 le 0 0 a abecd
38 43 | 2 0 0 1 aedbe
39 42 | 3 0 0 1 achde
40 41 | & 0 0 1 adceb
41 38 | 2+e 1 1 —(e+e°) acebd
42 46 | 2+ € et —(e+e*) abdce
43 58 | e+e e e —(e+e°) adebe
44 55 | +e & € —(e+e) adecb
45 50 | e?+e4 et € —(e+e°) acdeb
46 51 | 1+e? & 1 —(e+e°) abede
47 39 |} 1+2 es as —(e+e*) adbec
48 47 |} 1+é 1 3 — (e+e) aecdb
49 59 | l+é? € e — (e+e?) aebed
50 54 | 1+? 2 € —(e+e°) abeed

51 | 56 | -é
52 49 | -é
53 37 | -e
54 45 | -1
55 57 | -e
56 48 | -&
57 60 | -é
58 53 | -l
59 52 | -e
60 40} -é

é&+1
l+é
ete
ete
e+
efte
ete
é&+1
l+é
ete

e+ef
e+e
e+el
e+e
e+e

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

1 acdbe
€ aecbd
é abcde
é& ached
et abdec
1 adebe
€ acedb
é aebde
é adbce
é! aedch

111. Selecting the transformations which correspond to the positive substitutions abed,
and completing the group of 24 we have

Homographic Transactions. The groups of 12 and 24. Pole at A.

(As +B) +(Cs +D)

1 0 0 1 ul

2 -1 1 0 ad . be

3 a ee —(e+e) ac. bd

4 e+e e+e S ab. cd

5 e+et e+e € abe

6 e+e e+et Se ach

7 é é —(e+e) acd

8 1 € —(e+e°) ade

9 é é —(e+e) abd

10 € 1 —(e+e&) adb

ll 1+é2 eet S bed

12 l+é e+e 1 bde

13 142 1+2e -1 ab

14 l+e+3det -1-3.«- 4 &- cad

15 8+e+e2 -1-3e-& —+e ac

16 —1—e?+2e4 1+-28 1-2 bd

17 —-2-22- QeteP+2Qet Qe+2e? + 3 ad

18 2+ + 2et —2e—-22— 3 Qe +8 + Det be

19 -eté -e+e e+e5—2et abed

-1 abde

acdb
achd
adbe
adceb

As an example of the calculation we have (A, B, C, D) =(0, 7, —1, 0); ab. Hence

A,B, 0, D,= (a2), bG=1) +i+1, bG-1)— (+), ~a(¢+)), = (1, OS

pee ay)

The second and third coefficients are

2

2

? SS

a

5+1 .4../5ae eee
> a +i ae which in

b+z ).

virtue of the values of e and e* are =1+42e and 1+2¢ respectively: or the result is as

above (1, 1+ 2e*, 1+2e, —1).

AND THE POLYHEDRAL FUNCTIONS. 57

112. In lke manner for the passage from the ©-form to the B-form, if X, Y, 7
be the coordinates of a point on the spherical surface in regard to the @-axes, X,, Y,, Z,
those of the same point in regard to the B-axes, we may write

X: Y:Z=X,:bY,+aZ,: —aY,+bZ, where a, b= ay ae :
Hence X¥: Y:Z7=L: M:N, being the equations of an axis of rotation in the first set
of coordinates, those of the same axis in the second set of coordinates will be
X,: bY,:aZ,:—aY,+bZ,=L:M;N, or calling these X,: Y,: Z,=L,: M,:N,, we have
L,, M,, N,=L:bM—aN:aM+bN: these values are such that 1+ M?+NZ= [4+ M*4. N*,
or A, 4, V, A,, 4, ¥, being the rotations, we have L, M, V =), Sp, SSE Dy, MW, N= SA, 32,505,
where S has the same value in the two sets of equations, We have thus

Ba C ¢ B= C 21) s=ALs NEA jh oe Wil 2 Ai & GS;
BL+C,: B,—C,: D,—A,: D,+A,=L,: 2M,: Ny: -A,

and hence
B, + C,= BeaG,
B,-C,= b(B-C)—a(D-A),
D,—A,=—ai(B—C)+b(D—-A),
D,+ A,= D+A;

and thence

A ai(B—C)—b (D—A) + (D+A4),
B,= b(B-—C)—-ai(D—A)+(B+0),
C,=—b(B-C)+a(D— A) + (64+ 0),
D, =—ai (B—C)+ b(D—A) + (D+A4).

113. As an example of the transformation take

(A. B: C, D)=(2,-3+/5+é(1—¥/5), Eien 2/5), - 2) [be. de]:

on BC REC DA eA Cis, 82 ot,5), 29, Ostend thence
A,= 57g ( 6-2V5) +575 ( 2+2¥/),
B= ae C4), aia ie (21 (1 + v5) +(-3+4/5)),
C=573( 4) ty 7g(241-W5)+(-3+¥9)),
D.= 575 (- 6 + 2/5) + xg c 2-23),

viz. multiplying by 2/3, these are

8, 71-6425) +273 (-34+/5), ¢(6—-2/5)4+2,/3(-34+ 75), —8,
that is, 8, (~64+2Vd)(¢+ 1/3), (—6+21/5)(—74+ 73), —8,
or since 24+/3=—2iw and —2+/3=2iw", dividing by 4 these are

2, ¢(8-—¥V5) 0, 1(—-34+75) 0", — 2,
as in the table.

Wor, GUD Rei Ie 8

tw © ww bo tS tO
BEF SSas

| 33 |

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

114. Homographic Transformations.

DNWDNDNNNNNNNNWNNOOS

eee

| 3-7/5

—/3- iJ/d

—J/3-7,/5
V3 -t/5

Seeks
J3=ind
Qu”

2

oh (= Lk ta/8).

The group of 60. Pole at B.

+B) +(Cs +D)
0 0 1
1 1 0
@ 1 0
o 1 0
i( 3 -,/5) z( -3+,/5) -2
t(-8-./5) t( 3+4,/5) -2
t( 3-./5)o t( -3+4,/5) wo? =n
i(-8-VJ5)o t( 3+./5) -2
Z( 3-,/5) o t( -—3+,/5)@ —2
Z(-3-./5) wo? Z( 3+,/5) -2
(-/3-7,/5) o (-V/3+7,/5) o? -2
SEONG: ~J3+745 -2
(—/3 -7,/5) wo (-V3+7,/5) o -2
MB=15 J34i/5 -2
(/3-7,/5) (/3+7,/5) wo? -2
(/3 —7,/5) w? (/3+7,/5) @ -2
0 0 1
0 0 1
2 a) J3+i 5
2 -2 —J/3+t/J5
Qe? =Be SS hiAD
Qu? —2@ V3+7¢/5
Qa — 2? —/34+7/5
2a — 20 V3-7/5
-/3-7%,/5 —J/3 47/5 — 2
—J/3-15 —N3847¢/5 — 20"
-J/3-7/5 (-V3+7,/5) o? =o
-—/3-7/5 (-/3+7,/5) wo? — 2?
N38-2t/5 3847/5 —20*
N38 -t/5 V3+2,/5 — 2
J3-iJ5 ( J3+i4/5) a? -2
M3-t/5 ( /3+%,/5) o — 20°
—/3-7,/5 (-/347,/5) @ — 20
—/3-7t,/5 (-/3+7,/5) @ -2
/3-7,/5 ( /3+7,/5) 0 -2
3-7/5 ( /3+7,/5)@ — 20
Z( 3-A/5) w? Z(-3+,/5) — 2
+2w* —2 (-V3+2¢,/5) w?
5 Ae ( /3+i,/5)0
i( 8-./5) t(-3+/5)@ —20
t( 3-,/5)@ Z(-3+4/5) —2
2o -2 (-/3+7,/5)
2 — 20 ( J3+7,/5) wo?
i( 3-,/5) Z(-3+4/5) wo? — 20?
i( 3-,/5) ow i(-—3+4/5) wo” —2o
2a" — 200? ( V8+7%/5) @
2@ —2o (-/347,/5)
z( 3-/5)o i(-3+,/5) @ — 20°
i(-3-,/5)o t( 3+/5)o — 20*

ade
ade
aed
bce
bee
cde
ced

adceb
achde
aedbe
abeed
aedch
adbec
acebd
abede
adebe
aecdb
abdce
ached
acdeb

aNtToauk wd

AND THE POLYHEDRAL FUNCTIONS.

59

50 3-7/5 20 —20 (/3+74/5) wo” adbce
51 aide Qe0? — 22 (-V3-iN5) @ aechd
52 2 i(-—3-A5) wo? Z( 3+n/5) w? — 20 abede
53 2 i(-38-./5) @ z( 3+,/5) —2@ aebed
54 —J/3-7/5 Qo =2 (-V3+7,/5) abdec
55 3-7/5 z — 2w? ( 4347/5) wo? acedb
56 2 i (-3—A/5) i( 34,/5) 0? — Qo? adebe
57 2 i(-3-,/5) z( 3+,/5) — 20 adech
58 —J/3-17,/5 2 -—2@ (-/3+4+7,/5) @ aebde
59 V3 -iJ5 20” -2 ( V8+7%/5) wo acdbe
60 2 i (<8 —A/b)a8 é( 3+A5) — Qe?
115. We hence derive
Homographic Transformations. The groups of 12 and 24, Pole at B.
(As +B) +(0s +D)
1 0 0 1
2 i( 3—,/5) ( —3+A/5) -2
0 1 1 10)
2 i(3— a5) i( 34/5) -2
Qu? —/3-7,/5 —/347,/5 —20
Qe Boones VB+i/5 —20?
Bea ie/5 20 — Deo? SAS eiAs
3-7/5 2a — 20? J34+7/5
Qo B= 6/5 N8+0/5 — Deo?
Dw” 3-7/5 J38+t/5 —2a
V38—72,/5 20” — 2 J3+7/5
—/3-7,/5 2e? —2 —/34+7/5
2 V3 ( 145)+ (—3-A/5)/ J3( 14+75)+7( 3+42/5)/—2
2 NBG eVB\5 (E88 ( SiN) (CREE) 2
J5 -i D —/5
1 %/5 —t7J5 -—1
2 J8 (-14+45)+7( 8—/5)|,/3 (—1 4/5) +7 (—38+./5)| —2
g V3 (1-5) 4+7( 8-W/5))/3 (1-5) +7 (—3+,/5)|-2
1 (3 zt 1
1 —7i —t 1
WS ( 1=—r/5)+7( 34+4/5)} 2 —2 V3 ( 1—r/5) +7 (—3+4/5)
W/3( 1+/5)+7(—34+,/5)} 2 -2 J3( 1+/5)+7( 3+4/5)
3 (—1405)+7( 3—-./5)| 2 ~2 NBN (eleva) e 0 (SBE hy)
eat /5) 4-7 (—3—./5)| 2 -~2 NE (iiais) ete BEaN)

116. I give also the group of 12, (abce), slightly modifying the form: viz. I write

first /3 + ¢/5 =2,/2k, and therefore V3 i= 22.7: then for « I write Xz, and divide

the A and B by 2: the A and B then contain = and the C and D contain x, and

ie. r ‘ : ; : E
~ =7, we have ;=—z. For instance in the transformation corresponding to

r k
abe, the Avw+B and Ox+D, =2w°x—(V3+%/5) and (-V/3+21/5)a—20 become first
8—2

assuming

60 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

Qw°x —2/2k, and — 2/2 : x—2@, and then (omitting also the factor 2) ov — yae and
N 1 A on ae :
I aie they are wv —it/2 and «.¢/2—@; that is the values of

A, B, C, D are o*, —i2, 12, —ow. The group is

— /2 —a-o, viz. when -

Group of 12. Pole at B.

1 0 0 1 1
@ 0 0 1 ace
| o* 0 0 1 aec
| 2 —tw /2 to »/2 —o abe |
1 —tw*/2 to? /2 —@ ach
i) a -ti-w,/2 trJ/2 -—@ abe
| . 9 9
l —t/2 tw” /2 —o aeb
1 —tw* /2 7/2 —o bee
| 1 —t,/2 tw /2 -—o bce
eal —to /2 to? »/2 —l ab. ce
| 4 —tw* /2 to /2 —1 ae. be
1 —t/2 tJ/2 =I ac. be

117. From the Table of the Groups of 12 and 24, ©-form, it appears that the
group of 12 is

1 1 t(@—-1) -te—-1) t(@4+1) -d(etl) w+t a—-1 —(xe+%) —(w—-1)
«, mig ee ee’g Tea aa > } > > Ct J ane Se (5 5 = >
za+1 a—l1 z—-1 “2—-t’ £41 z—2 x2+e

£ £ z+i1

and if we proceed to form the product of the twelve factors s—a, Ss sta, &., we

have first the three products

1 a—1\? a2+1\? a + i\? a2—t\?
\ 2 Cees 2 2 = ee ales
S-—-aZ.Ss 3P? + (S55) e+ (E45) 5 s (4 )--s (=)

=si'+as°+1; si+ Bs +1; sys? +1;
if for shortness,
1 a'+6a°+1 a —627+1
eR ee 9 Bs ema eS
BEY (# +3) . (@?—1)? ’ : (a = IVINS

The product of the three quartic functions is
= (st +1)? + (s*+1)?8" (a+ B+y) + (s'+ 1) 5* (Byt ya+a8) 458°. a®B'q’;
and we have

3227 (x* + 1) — (a — 33s° — 33s* + 1)

Pinar ay 7 a+B+y= @ (a*—1)? ,
_ —4(a* — 340*+1) _ — 82a? (a + 1)° _—36e(at-1)? _
By = n Wai) > a(B+y)= a(e—1yf ’ By + ya+ a8, = a (a*—1)? Wea 36,
apy = 4 t= 338 — 338 +1)

a (a — 1)

AND THE POLYHEDRAL FUNCTIONS. 61

Hence the product is found to be
12 8 Q,4
=k ao aes _ Bfeat_ pe 2 — 880 — 88xh+ 1
= (s* — 338° — 338° + 1) —s°(s*- 1)’. eee

which is

=o? (ot — 12 0338" = 83841 _ a = B32" — BBat +1
s’(s*—1)? a” (a — 1)?

We thus verify that the twelve transformations x into a, into 2 &e., give each of them

a transformation of the function
a” — 332° — 3327+ + 1
a (a — 1)?

into itself.

The system of 15 circles. Art. Nos. 118 to 127.

118. It has been already remarked that we can from the coefficients (A, B, C, D)
of the homographic transformation pass back to the position of the axis of rotation :
viz., we have

A:B:C: D=-v-t: r+ Ww: r-—W: v-7,
and thence
Ree pe le B+C:—-i(B-C) :D-A_ :1(D+ A),
that is
A, BM, v=—27(B+C), — (B-C), -—i(D—A); +(D+A4).

The equations of the axis thus are
Gis BU wh 2
B+C B-C D-A’

and the equations of the central plane at right angles to the axis are

—~(B+0)2+i(B—C)y+(A—D)2=0.

119. In particular we may find the equations of the 15 planes at right angles to
the @-axes: these are in fact the before-mentioned 15 planes, intersecting the sphere in
great circles the projections of which are the circles in the three figures respectively.
Taking the equation of the plane to be La+ My+Nz=0, it is at once seen that the
equation of the projecting cone (vertex at the South pole) is

N (a? +y? + 2 —1) —2 (2 +1) La + My + Nz) =0,
and hence, writing z=0, we find
N (a + y? —1) —2 (Lx + My) =0

for the equation of the circle in the plane figure. We have thus the equations of a
system of 15 circles related to each other in the manner before referred to.

62 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

120. Taking the @-form, the equations of the 15 planes are at once found: and
we thence obtain the equations of the 15 circles: viz. writing for shortness

QO =2°+ 7? -1,
the equations are

[( 83-5) a+( 1-¥5)y]=9,
(—1—/5)x#+(-—14+¥5)y+2z2=0, (ab. ce) [(—-1—/5) #+(—14+¥5) y]=9,
(1+5)e+( 34/5) ¥+2z2=0, (ac. be) Q-[( 14+75)e+( 34¥75)y]=0;

2z=0, (ab.cd) O=0,

z=0, (ac. bd) « =0,

y=0, (ad. be) y = 0,’
(3—V5)e+( 1-5)y+2z2=0, (ae.be) OQ -
O-

(—34+/5)e@+(-14+5)y+2z=0, (ad. be) and similarly for the other circles.
(1+/5)e@+( 1-1/5) y+2z=0, (ab. de)
(-1—/5)2+(—3-—/5)y+22=0, (ae.bd)

(-34+/5)e+( 1-15)y+2z=0, (ad.ce)
(1+ ¥5)a+ (-14-5) y+2z=0, (ae.cd)
(-1—/5)27+( 34/5) y+2z2=0, (ac.de)

(8-5) e+ (—14+15)y+2z2=0, (be.de)
(-1-/5)2+( 1-15)y+2z=0, (be.cd)
(1+/5) @+(-—3-—5) y+2z2=0, (bd.ce).

121. Observe that the arrangement is in sets of 3 planes, or circles, intersecting at
right angles. One of the circles is the circle 0, =a”*°+y*—1, =0 corresponding to the
equator, and two of them are the right lines «=0 and y=0. The equations of the
remaining 12 circles may be written in the somewhat different form

0+ (W5—Ily—4(y5—1) «]=0,
2-5 -Dly- 4 (/5 +3) 2] =
— (v5 +3)[y+2(v5—1) 2] =0,
— (v5 —T)[y—4(v5 —1) @] = 0,
2+ (V5 -1)[y—3 (v5 + 8) 2] =
0 + (V5 + 3) [y+ 3(V5 — 1) 2] =0,
0 + (V5 =Dly +4 (V3 -1) a] =0,
O = (V5 —V [y+ (V5 +3) 2] =
— (v5 + 8)[y — 3(v5 — 1) #] =0,
—W5—V)[y+3(v¥5—1) 2] =
O + (V5 —1)[y +4 (v5 + 8) 2] =0,
0+ (v5 + 3)[y— 375 — 1) @] =
and it hence appears that 4 and 4 circles have with Q=0 the common chords

y¥+3(¥5—1)c2=0, y—3(¥5—1)c=0 respectively: and that 2 and 2 circles have with
2=0 the common chords y+4(¥i5+3)c=0, y—4(V¥5 +3) @=0 respectively.

AND THE POLYHEDRAL FUNCTIONS. 63

122. The equations of the 12 circles are in fact
0+ (J5-Ylyt}(V5—1)a]=0, 4 (V5+3)[Ly +t HV5—1) 2] =0,
0 + (V5 —1)[y £4(V5 +3) 2] =0:
hence the radii are =/5—1, 2 and /5+1 respectively.
The construction of the 12 circles is as follows: starting with a circle radius 1,

Lay down the diameters y+4(¥5—1)e¢=0 (AA in the figure), and through the
extremities of each describe 2 pairs of circles with the radi /5—1, /5+1 respectively.

Lay down the diameters y+3(/5+3)c=0 (BB in the figure), and through the
extremities of each describe a pair of circles with the radius 2.

123. For the A-form, the equations of the fifteen planes are at once found to be

y =0, ad. be

—2 + (e+e) 2=0, ac. bd

(e +) a + A= (0, ab .cd
(e-—@)a -—i(’+e)y =(); ac. be
—(+e) a2 +i(?—e&)y+2 (e+e) 2=0, ae. be
—a+i(e+eé—e—e&) y+ 27 = (0). ab. ce

(e—¢') & —i(ete)y =0, ab.de
—(e+e)a +i(e —e)yt+2 (e+) z=0, ae. bd
+(f@+e+2)e¢ —i(@-—e&)yt+ 773 =). ad . be
(e—eé)ax +i(ete)y =), ae .cd
—(e+e') x —ti(e —“)y+2(e+e) z=0, ac.de
(@+é+2) a +¢(F—&) y+ 27= 0; ad. ce
(f—e) a +i(e+e)y =0, bd. ce
—(f+e)a —i(’—e&)y+2(e+e) z=0, be . de

—a2-i(+e—e—e) y+ 2), Bo Goh

where as before the three planes of each set intersect at right angles.

124. Passing to the circles, the first plane of each set gives a right line, and we
have thus five of the circles reducing themselves to right lines inclined to the axis of
x at angles 0°, 36°, 72°, 108° and 114° respectively.

The remaining 10 circles form 5 pairs, the circles of a pair having different radii,
but the two radii being the same for each pair, and so that for the several pairs the
‘common chords with the circle 2=0, are the diameters inclined to the axis of «a at
the angles 18°, 54°, 90°, 126° and 162° respectively. Considering the two circles for
which the inclination is 90°, these arise from the planes —a# + (e+e) z=0, (e+e) 7+2=0
respectively. The equations of the circles thus are (e€+¢*)Q+2c%=0, O-2 (e+e) x=0,

or, recollecting that 2(e+e')=/5—1 and therefore rare V5 +1, the equations are
c

a +y?— (V5 -1)e#-1=0, a +y?+(V¥5+1)2=0; hence for the first circle the a-co-

64 Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE

ordinate of the centre is $(V¥5—1) and radius is =4./(10—24/5); for the second circle
the «-coordinate of the centre is =$(/5+41), and radius =}./(10+2./5). We have thus
the construction of these two circles, and consequently the construction of all the 12

circles.

125. For the B-form after some easy reductions, and attending to the relation
o —@ =tV3, the equations of the 15 planes become

2 = (0), ac. bd
(-3+/5)y+ 23 = (0), ab.cd

(B+/5) y+ 22 = 0, ad . be

(30+ Vo y+ 2z = 0, ac . be

— (14+ 75) /382+( 3-1/5) y+ 42=0, ab. ce
(-14 5) /3e+ (-3—/5) y+ 42 = 0, ae. be
a+ V3y = 0), ae. bd

—V3a+ yt (38+1/5) 2=0, ad . be

V32— y + (8—1/5) 2=0, ab. de

—V3a+ V5 yt 22— 0; ac. de
(1-1/5) /8a+(-—3—/5)y+ 42=0, ad . ce
(l+75)V8e+ (8—W5)y+ 42=0, ae.cd
x V/3y =(), bd . ce

30+ yt (8+ 175) 2=0, be . de

—/3e— y+ (8—/5)2=0, be . ed.

126. Of the 15 circles 3 are the lines e—y/3=0, x=0, r+yV3=0, viz. these
are lines at inclinations 30°, 90°, 150° to the axis of #. The equations of the remaiming
12 circles are

O+ (3-V5)y=0,
Q- (84+/5)y=0,
(3415) A-2(y—a3)=0,
(3/5) 2-42 (y—wW/3) =0,
(3-+/3) 2-2 (y +23) =0,
(3-9/5) 242 (y+e9/3)=0,

viz. these are pairs of circles having for their common chords with =0 the diameters
at inclinations 0, 60°, 120° respectively. And lastly we have the circles

20 —[(-14 V5) V/3a—(38+/5)y]=0, | 20+[(-14+ 75) /3a+ (345) y] =0,
2-[ — 3a + V5yJ=0, | O-[ 30+ v5y] = 0,
204[( 14/5) /3e—- (8-1/5) yJ=0, | 20—-[( 1 + /5)/320 + (3 — 5) y] = 0.
127. The first three of these have for common chords with Q=0, the diameters
whose equations are

(-14/5)/3a—(34+75)y=0, —V3e+V5y=0, (14+75)V/3e—(3—V5)y =0:

AND THE POLYHEDRAL FUNCTIONS. 65

viz. these equations are y= (—2+/5) 2/3, nel y=(24+75)x2/3. If, as in a fore-

VS)
L =, aes V/3 V5. V3 be My, ey
going table = 37° 46’, sin = 2/2’ cos = 5 NDE and therefore tan 0 = JB’ then the inclinations of

these diameters to the axis of w are respectively 60°—6, 6 and 120°—8, or say 30°— (8 — 30°),
30° + (@—30°) and 90°—(@—30°), where 6—30°=7° 46, i.e. the inclinations are 30°+7° 46’
and 90°—7°46’. And for the other three circles the common chords are the diameters at
the same inclinations taken negatively. The geometrical construction of the fifteen circles
for the B-case in question is thus not so simple as in the @- and A-cases.

The Regular Polyhedra as Solid figures. Art. Nos. 128 to 134.

128. I annex some results relating to the polyhedra considered as solid figures
bounded by plane faces; or say results relating to the regular solids: s is in each case
taken for the length of the edge of the solid.

Tetrahedron. Cube. Octahedron. Dodecahedron. _Icosahedron.
Edge 8 8 8 8 8 |
: : J/3 glee V3 (/5+1) B+/5
Rad. of circum. sphere, Fi) s 2,9 8.4/3 8 2 8 ri s a
; 1 1 345 [esas
— sos } ————
Rad. of inters. sphere, p | s 2/2 8 2 8.4 er | 2
i .. 1 1 O55 | Seex/d
Rad. of inscribed sph.,7 | s MERE 8.4% DINE: 8 a 10 Is
Rad. of circle circum. to | , 1 Ags ae ; e, 5 +/5 aoe
face, It’ V3 /2 V3 10 AB
Rad. of circle inscribed | , niles Hest A ules 8 5+2/5 ls aoe
to face, 7” "2/3 2 2/3 20 2,/3
Incl. of adjacent faces | cos"! 4=70°28'| 90° cos-1—3=1099 32! |
Incl. of edge to adjacent -1 1 _,4046'| 900 -1_ | _j950 44’
Rie s cos v3 54° 46"| 90 cos 3 125° 44

But we require further data in the cases of the dodecahedron and the icosahedron
respectively.

129. For the dodecahedron, taking as before the edge to be =s, then in the penta-

gonal face
diagonal, g is =s.4(/5+1),
altitude, &k , =s.4/(5+2/5),
segments of do. e , =s.$,/(10—24/5),

Ff » =8.4V7010+2¥5),
where k=e+f=R +r’.

130. The section through a pair of opposite edges is a hexagon, as shown in the
figure, viz. this is constructed by taking the four equal distances OO, =p, =s.}4(3+//5),
meeting at right angles in O; then drawing the double ordinates BB, each =s, through ©,

Vou. XIII. Parr I. 9

66 Pror, CAYLEY, ON THE SCHWARZIAN DERIVATIVE

and @, respectively, and joining their extremities with ©, and ©,: the sides O,B and 0,B
are then each =k, =s.4,/(5+2,/5); and inserting upon them the points A, ® from the
figure of the pentagon, we have several geometrical relations; viz. the line AA cuts the

2
F
2

’
/
en had
‘
,

f= er erasers el BEN oe

Z

parallel sides BO,, BO, at right angles, and when produced passes through the inter-
section of BO, and BO,: we have OA, OB, OO=r, R, p respectively: the four points ®
form a square, the side of which is g, =s.}(/5+1).

131. We find also

= 5+ 5
AM=s 10?
AQ=s ——s

95 5
ou=s,/P2>*, =r.2/2,
09=8,/—4F™, PROS:

5/5
oM=s,/ a.

uB=s,/-O*E).

AND THE POLYHEDRAL FUNCTIONS. 67

It may be remarked that in the figure BO,, B®, are the projections of pentagonal
faces, at right angles to the plane of the paper, having their centres at the points A, 4,
and the perpendicular distance between them = AA: the points Q, Q (only one of them
shown in the figure) determine the directions of the 5+5 sides which abut on these
pentagonal faces respectively; and the 5+5 pomts B which are the other extremities
of these sides respectively form two pentagons, centres M, M in the planes MB and MB
respectively: the remaining 10 sides of the dodecahedron are the skew decagon obtained
by joining in order these 10 points B. We have thus the means of making the per-
spective delineation of the dodecahedron.

132. The dodecahedron is built up from the cube, by placing on each face a
figure of two triangular and two quadrangular faces, the orthogonal projection of which

on the face of the cube is as in the figure: the side of the square is g, =s.4(V5+1):
the slope-breadths of the triangular faces are e, =s.}/(10—2,/5), and those of the
quadrangular faces are f, =s.4/(10+2,/5); the lines represented by the other lines of
the figure are in actual length each=s. We have thus a section which is an isosceles
triangle, base = g, other sides each =f; and the square of the altitude is thus = f? — 449°

B B

eas |

B

= 4s’, or altitude =4s; viz. the altitude of the ridge-line BB, above the face of the
cube is =4s, the half-side of the dodecahedron: we have in this result the most simple
means of forming the perspective delineation of the dodecahedron.

o
wD

Pror. CAYLEY, ON THE SCHWARZIAN DERIVATIVE, &c.

133. For the icosahedron the section through two opposite edges is a hexagon, as
shown in the figure: to construct it we take the four distances O® each =p
=s.}(1+5) meeting at right angles; and then the distances AO,, AO, each =}s8;
and complete the hexagon. This gives the sides A@,, AO, each =s.}1/3, the altitude

of the triangular face, side=s; and then taking 0,B one-third of this, =s or we

have OB at right angles to 40,, and OA, OB, OO= Ki, 7, p respectively.

Moreover, joining A,@,, and OA, we have these lines cutting at right angles in a
point WM: we find

A,O, =s8.$V(5 +25),
Me, = s [2

Pha

ey ere

AM = 8 y/ 10?
om as /PtX?, ep

134. It may be remarked that A,O,, A,O, are the projections of two pentagons in
planes perpendicular to that of the paper, their centres being M, M: producing OM,
OM to the points A,, A, respectively, we have a pentagonal pyramid, summit A,, standing
on the first pentagon, and an opposite pyramid, summit A,, standing on the other pen-
tagon: the 545 triangular faces of the two pyramids are 10 of the faces of the
icosahedron, and the remaining ten faces are the triangles each having for its base a
side of the one pentagon, and for its vertex a summit of the other pentagon, viz. the
sides are the sides of the skew decagon obtained by joining in order the angular points
of the two pentagons. We have thus a convenient method of forming the perspective

delineation of the icosahedron,

III. On the Application of Quaternions and Grassmann’s Ausdehnungslehre to
different kinds of Uniform Space. By Homersuam Cox, B.A., Fellow of
Trinity College, Cambridge.

Read February 20, 1882.
¥Y

INTRODUCTION.

THE object of this paper is, following Grassmann, to establish a pure algebraical calculus,
the laws of which will coincide with those of actual geometry. Ordinary algebra may
be considered a calculus of one dimensional space, either of lengths measured along a
line or of time, or of any other quantity capable of only one kind of variation. It
might even arise as a calculus of discrete objects for starting with the series of natural
numbers, fractional, negative and imaginary quantities would arise as the results of inverse
operations, although no interpretation could be found for them. A geometrical calculus
of two or more dimensions must have more than one independent unit. It must include
the calculus of one dimension, and therefore all algebraical quantities; even imaginary
quantities, for they will arise as an indication that lines do not intersect. Whatever
symbols then a proper geometrical calculus may contain, the algebraic J =1 will always
exist apart and in addition to them. Starting with different independent units, the straight
‘line, the plane, &c. may all have purely analytical definitions given of them, and this has
been done by Grassmann. I have endeavoured to combine with the ideas of Grassmann Prof.
Cayley’s theory of distance and the applications of it made by Dr Klein. It is shewn
that there are three different ways in which distance may be introduced, and consequently
three different kinds of uniform geometry. These are, ordinary geometry, spherical geometry,
and the non-Euclidean geometry of Lobatschewsky and Bolyai. Besides the works referred
to, I have constantly used Prof. Tait’s “Introduction to Quaternions” and Hamilton’s
“Elements.” In this first part chiefly Quaternion methods are employed; in the second
part those of the Ausdehnungslehre.

ADDITION OF PoINTs.

Suppose A and B to be the distinct independent quantities, so that neither can be
derived from the other by multiplication with an algebraical quantity.
We will call A and B points.

Vou. XIII. Parr: If. 10

70 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S
The expressions 24, 3A, 44, &c., which have no distinct meaning except in combination
with other symbols, will be called multiples of the point A.
Every expression of the form pA+qB, where p and q are numbers, is defined to be
some multiple of a point.

pA+qB and rd+sB will be considered different points if “is not equal to a but if

q

then they are multiples of the same point.
Hence all the points included in the series pA +qB (neglecting the number of times
they may happen to be multiplied), form a singly infinite series, each point depending on the

value of = This series of points will be defined to be a line.

If a point be considered to have no dimensions, a line will be a manifold of one

dimension.

It follows from the original assumption that any equation of the form pA+qB=rA+sB

: . : »— 7 }
involves the equations p=r, g=s, for otherwise we should have B=! Ee and this
was supposed not to be possible.

We take as the definition of the addition of points and their multiples the equation
(pA + qB) + (rd + 8B) = (p+r)A+(qt+s) B.
It follows at once that addition is commutative and associative.

Similar definitions can be given for subtraction and multiplication by an algebraical
quantity.

Any point zA4+yB can be derived from any two points p4+qB, r4+sB on the
same line by means of the equation

tA+yB=a2(pA+qB)+y'(rA+sB),
which gives e=pat+ry, y=qu+sy’.

There is then nothing peculiar about the points A, B; nothing to distinguish them
from any other points on the same line. It follows that without violating analogy we may
call the series of points pA+qB a straight line. In ordinary Euclidean geometry the
series of points might lie either on a straight line or a circle. Both would be included
in the definition; for, taken in themselves without reference to outside points there is
nothing to distinguish them, except that the one is infinite and the other finite; and
this the definition says nothing about.

Let C be a new point not lying on the line joining A and B, or, in other words,
not expressible in the form p4+qB.

Then all the points included in the expression pA+qB+rC, leaving out of con-
sideration the number of times each point is taken, form a doubly infinite number. They
will be said to constitute a plane.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 71

If X=1A+mB+4+n0, Y=lA+mB+'C,

be any point in the plane, then
Z=pX+qVY = (lpt+lq) A+ (mp+m'q) B+ (np+ng)C

will also be a point in the plane; or the plane containing any two points contains also
the straight line joining them. This theorem justifies the use of the word plane.

The condition for three points lying in a straight line may be written symmetrically
AX+yuV+vZ=0,

where 2X, pw, v are any numbers.

Suppose P=xA+yB+2zC, then 2, y, 2 or their ratios may be called the co-ordinates
of the point P If Q=a,A+y,B+z2,0, R=27,A+y,B+z2,C be two other points; then
applying the condition that the three points P, @, & should lie on a straight line, and
equating the coefficients of A, B, C to zero, we have

Ae + px, + vx, = 0

Ay + wy, + vy, = 0

Az + wz, + v2, = 0.
Therefore

| i Op) |

| Ty Yy 2
By Ya 2

If x, y, z be considered variable this is the equation to a straight line and may be
written in the form lx+my+nz=0 where J, m, n are constant.
Now suppose P=),A+y,B, Q=r,A+p,B, then the ratio = : = or HAs will: be
1
called the anharmonic ratio of the points ABPQ taken in that order. Thus we have a
definition of anharmonic ratio independent of the idea of distance*.

2 By

Also the ratio is not altered by taking in place of the points A, P, B, @ any
multiples of them; for putting pA for A and oB for B, we have

FN, gf) ll — Ms
spy sammie Ce pe ee

and this gives for the new anharmonic ratio

PR, , P By _Hyry
Xr

; as before.
GN, oe : :

2

So that in determining the anharmonic ratio of four points we may use any multiples
of the points instead of the points themselves.

* Felix Klein, ‘‘ Ueber die Nicht-Euclidische Geo- als Strecken-Verhiiltnisse definirt werden, da diess die
metrie,” Mathematische Annalen, Vol. tv. p. 624, says Kenntniss einer Massbestimmung vorayssetzen wiirde.”
“Die Doppel-Verhiiltnisse diirfen dabei natiirlich nicht

10—2

72 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Now let O be any new point, join OA, OP, OB, OQ, and let a new line cut these
lines in 4’, P’, BY, Q'; put

v >

A’=p0+A, B=c0+B;
then P’ (the intersection of OP and 4'B’)=2,A'+ B= (A,p + w,0) 0+ P,
Q=r,A'+4,B = (Apt pc) 0O4+Q;
Mz
NE.

we
This ratio may then be called the ratio of the pencil of lines OA, OP, OB, OQ, since
it is the same for all lines meeting that pencil.

therefore the anharmonic ratio of A’, P’, B’, Q’ is , the same as that of A, P, B, Q.

If P=27,dA+y,B, Q=27,Aty,B, R=2,A+y,B, S=x,A+y,B be four points on the
line AB, and we wish to find the anharmonic ratio of the points P, Q, R, S, we must
solve the equations ‘
©, = 2,2, + B,v,, Yo= M1 + LyYos
1, =A,L, + MLs, Ys = MY + HY»
By, r, = (2,4, = LeY,) (ay, = LYs)

A He (Ls — Lao) (ey, 77 ©,Y,) .

whence

The quantities 2,y,—,y, which occur in this expression are invariants, for if
P=2,A+y,B, Q=2,A+ y, B,
where A’=pA+qB, B’=rA+sB;
then XY. — LY, = (ps 7 q") (2{Yo ap yy, )-

To find the anharmonic ratio of four straight lines lw+my+nz=0, &c., inter-
secting in a point, we may take the points n,B—m,C, n,B—m,C where they cut the line
BC. This gives for the ratio

(m,n, —m,n,) (m,n, — m,n,)
(m,n, — m,n,) (mn, — myn.) "

We should have also the expressions

(n,l, —n,l,) (nl, — n,1,) (lm, — 7,m,) (l,m, — l,m,
(nl, —n,!,) (nl, — 2,1) (l,m, — l,m.) (lym, + lm,) °

These are all equal, since, if 2, y, z be the point through which the lines pass,

x y Zz
— — - =
mn,—mn, n,/,—n,l, lm, —l,m,

x y Zz

Mg, — My = nl, in Nel, e Lm, — l,m,
We may find now the anharmonic ratio of the four lines joining the point 2, y, z
to (2, Y,> %) (Ger Yor Zs) (La2 Yor 2) (Mer Yur 23 4, m, 0, 1, m,, n, are now the minors of
(x, oY, 2
| Z,, Yp z,
! Ly Yo zy

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 73

Therefore MN, — MN, =| LZ, Y, 2 | 2,
Ly Yp 4,
Zo, Yo 2 |

and the required ratio is

S

2?

z Zz
a 7: Ty Yo %
Ty Ys 2s Gy Ya 4

If we put this equal to a constant k, the locus of the point a, y, z becomes

1] ,
x, y, 2 || a, y, 2 |=k| 2, y, 2 iy Oh A Me
i]
| TM Yyp % || Ts, Yx = | Loy Yo %, Lp Yo %
' ty Yy 2, vy Yas 25 Ly) Yy oN yy Yp z,

an equation of the second degree.

‘It is equivalent to the most general equation of the second degree, for it involves
besides the four points through which the curve passes another constant k, making five
in all.

By means of this theorem or directly from the equation can be proved the other
descriptive theorems about curves of the second degree, such as Pascal’s Theorem, harmonic
properties of poles and polars, self-conjugate triangles, &e.

It may be shewn in the usual way (by taking a pomt X+aAz, Y+Ay, 7+ Xz on the
line joining (X, Y, 7) and (a, y, z)) that

a ee
a ea

is the tangent at any point 2, y, z of the curve F(z, y, z)=0.
Similarly, as usual, are proved all descriptive or projective theorems concerning higher

plane curves, such as Pliicker’s equations, the Hessian, the nine points of inflexion of
a cubie lying three by three on straight lines, &c.,

Sir W. Hamilton’s theory of nets can be easily derived from the addition of points.
If 0=aA+bB+cC be a fourth point in the plane of ABC, then the intersections of
OA, BC; OB, CA; OC, AB give three new points

D,=bB+cC, D,=cC+aA, D,=aA +bB,
which are called points of the first construction.

D,D,, D,D,, DD, intersect DC, CA, AB, OA, OB, OC in six points
E,,=bB—cC, E,,=cC—aA, £,,=aA —bB,
E,,=O+aA £,,=0+6B E,,= O0+cC

=2a4+bB+cC, =2bB+cC+aA, =2cC+aA + bB.

74. Mx COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

E.

93

They lie by threes on four new straight lines (Z,,2,,2,,), (Z,,H,,/,,), &c. The ratios
E,,BD,C, E,,D,E,,D,, &e. are harmonic.

The intersections of the line Z,,B with CA, D,D,, D,D, give three new points,

E,,=2aA+c0, £,,=2aA—bB+cC, and £,,=2aA4+3bB+cC;

BE,,E,,,E,,, form an harmonic range.

In this way an indefinite number of points can be found, including every point of
the form zad+ybB+2zcC, where x, y, 2 are whole numbers, and an indefinite number
of lines including every line the coefficients of whose equation are multiples of a, b, c.

The quantities a, y, z are called by Hamilton the anharmonic co-ordinates of a point.
If P be the point aA +ybB+2cC, and Q,, Q, Q, be the points where AP, BC; BP, CA;
CP, AB intersect; then

is the anharmonic ratio of the points BD,CQ,,
z
pa CD,AQ,,
¥y
ce Tettaseeeaseseaecaseesseeceeeeseeeeseseee eens AD,BQ,.

Let us now take a fourth point D not connected with the other points by a linear
relation. Then we have a space or manifold of three dimensions every point of which
can be represented by 24+ yB+2C+wD.

A single equation will represent a plane, two equations a straight line.

A plane can be determined by four homogeneous co-ordinates, a straight line by six
homogeneous co-ordinates connected by a homogeneous relation.

In this way we may proceed till we have n points A,, A,,....4, unconnected. Every
point #,A,+a,A,...+2,A, will belong to a space of (n—1) dimensions, which may be
called an n pomt space. (See H. D’Ovidio, Mathematische Annalen, Vol. Xt.)

A point can be determined by the ratios of its n co-ordinates, that is by (n—1)
quantities. A straight line will be determined by two points on it, but as each of these
two points may be anywhere on the line we shall have determined two more quantities than
is necessary to fix the straight line. Therefore a straight line requires 2 (n—1)—2=2(n—2)
quantities to fix it.

A plane can be determined by three points, but in determining each of these the
two quantities necessary to fix its position in its plane will have been determined in
excess of what is needed; therefore a plane requires 3(n—1)—-3.2=3(n—8) quantities
to fix it.

In general an r point space requires r(n—1)—r(r—1)=r(n—r) quantities to
determine it.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 79

Or we may begin with the (n—1) point space which is represented by the general
equation
a,0,+0,0,+...+a,0, =0.

This is determined by (n—1) quantities the ratios of a@,, a... d,.

n

An (n—2) point space
is determined by two equations,

Q,0,+4,0,+...+ 4,0, = 0,

b,x, + b,x, +... + 6,2, = 0,
but as 2, may be eliminated from the first equation and «, from the second, these may
be put in the form

CRS 000 am Ohta, (0);

bz, + 6,0, +... + 6,0, =0,

so that the space required 2(n—2) points to determine it.
And in this way the previous result may be confirmed.

The number of quantities requisite to determine an r point space is identical with
that requisite to determine an (n—7) point space.

If a p point space and a qg point space have an r point space in common, together
they contain p+g-—vr independent points. Now this number cannot be greater than n.
Therefore if p+gqg be greater than n the spaces must have at least p+q—n points in
common, and in general will have just that number.

If, however, p+qg—r be less than n, the number of points required to determine
the p point space in the p+q-—r is p(p+q—r-—p),
the q point space in the p+q—r is q(p+q-r-q),
and p+q—r point space in the nm point space (p+q—r) (n+r—p—q).

The difference between the sum of these numbers and the number of quantities
required in general to determine a p point space and a q point space in an n point
space will give the number of conditions that these two spaces have an r point space for

intersection.
It is

p(n—p)+q(n—q)—p(p+q—r—p)—-a(pt+q—-r—-g—(ptq—-r)(nt+r-p-49q)
=r(n+r—p—q).

The n points A,, A,... A, will constitute what may be called an n-hedron.

n

It will contain n points, and n, (n—1) point spaces ;
=e =
i ie lines, and ———- (n—2) point spaces ;
—1)(n-—2 n— -2 ;
oe planes, and eal (n—35) point spaces.

76 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

If we take another point O=a,A,+a,4,+...+a,A, we shall be able to construct a

geometrical net.
The intersections of OA, and (A,A,... A,), &c. give m points B,=a,A,+...4+ 4,4).
B,B, intersects A,A, in a,A,—a,4,=C,,,
and OA, intersects B,B,...B, in B,+B,+... B,=(n-2)0+a,4,=C,,.

n(n+1)

5 new points which lie on straight lines in threes.

We then get

can be constructed,

“nn a

In this way every point of the form a,a,4,+2,a,4,+...+2,0,A,
where 2,7,...2, are whole numbers.

The equation to any (n—J) space will be
F (a,2,...2,) = 90,

and the equation to the tangent space derived from the (n—1) points near ‘z,a,... 7
will be

n

, dF oR = GAY 0:

X,-7-+X,=—+...+7,5—=
vada: ia da. a SOE,
In all that precedes, though the words points and lines are used, it is not necessary
to suppose actual points and lines to be meant. For instance, we might suppose A and
B to be two liquids capable of mixing in any proportions. Then pA+qB will be a
mixture of the two, and r4+sB will be the same or a different mixture according as

* is equal or different from 1° since the quantity is not considered. —pA+B would in
r p

this case have no meaning if the liquids were altogether different; but we might suppose
each of them to be mixture, and then some of one could be supposed contained in some
of the other up to certain limits. This is an instance in which quantities of the form
pA+qB+rC would not always have a real meaning; so that the theorems which have
been mentioned are to be regarded as purely analytical, and we cannot say in each case
without knowing more about the special circumstances of each manifold whether they will
have a real interpretation or not.

Again, suppose A, B denote two conics in the same plane, and therefore intersecting
in four (ordinary) points, pA+qB may represent any conic passing through those four
points, and all these conics are equivalent to what has been called a straight line.

All the conics in a plane form a manifold of 5 dimensions, and if s,s,...8, denote

6
six of them, any other may be represented by 2,8, +2,8, +... + 2,5.

We have so far substantially followed Grassmann. We now proceed to see how and
under what restrictions the idea of distance can be determined.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 77

DETERMINATION OF DISTANCE.

If A and B be two points each taken once only, it is natural to define A+B to
be some multiple of a point midway between them or bisecting them. Also all points
of the form pA+qB, where p and q are positive, will be said to lie between A and B.
This is in accordance with ordinary language, for “something between the two” is said
where there is no reference to position in space.

Now, if q be greater than p, pd+qB can be written p(4+B)+(q—p)B. The
point pA+qB then lies between the middle point of AB and Bb. It may be said to
be nearer to B than to A, or the distance AC may be said to be greater than the
distance CB, if C=pA+qB. AC is therefore less, equal or greater than CB, according

as . is less, equal or greater than 1. Supposing the coefficient of A to be the negative,

then, if we put —pd+qB=C, we shall have Bay C +74, so that B les between A

and C, and C may be said to lie in AB produced. Similarly pA —qB will lie in BA
produced.

We make these assumptions, Ist, that if C lie between A and B,
distance AC + distance CB = distance AB,
2nd. If A, B, C be single points (not multiples of point) and 4’, BY, C’ other

single points, also 4+B=)C, A’+ B=2C’, then the distance between A and B is equal
to the distance between A’ and B’.

The distance AB is thus some function of A, or we may say inversely X is some
function of the (unknown) distance. Put AC=CB=z, and X=¢(c). Take a point 4,
between C and A and a point A, in CA produced, and let A,AA=AA,=y.

Similarly take points B,, B, so that B.B=BB,=y; then
| 7 ; AC = 0B, =2-y,
A,C=CB,=a+y.
Therefore A,+ B, = ¢(«-y) GC, A,+B,= d(x+y)C,
A,+ B,+A,+B, = {o(e@—-y) + d(x+y)} C.

But A,+4,=$%)A, B+ B,=G(y)B, 0+ B=$(2)C;
A, +A,+ B,+B,= $(2) $(y)C.
Hence b(o+y) + $(z—y) = $(x) $(y).

This is Poisson’s functional equation.

Vou, XIII. Parr II. 11

78 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S
Its solutions are

(x) =2 sinh F,

p(x) = 2,

(x) = 2sin7,

where & is some constant.

We may next determine in an equation pP+qQ=rR the connection between the
quantities p; g, 7 and the distances PR, RQ, PQ.

Take a point R, between Q and P, such that PR,=RQ; also points S and 7 in
QP and PQ produced, such that SP=PR,, R,Q= QT.

Also suppose PR=mé@, RQ=n0, where m and n are whole numbers. Divide SP,
PR,, RR, RQ, QT into equal parts @. Consider for the moment the first case only, and

P Pees
which must be taken 2 sinh = times.

take each point of division sinh 5, times, excepting Meg yA

k:

The points of division from S to R,, each taken frie! times, are equivalent to a point

Qk
ee ty) 6 20 nO a nee Oe
at P taken sinh yr (1 +2 cosh Et 2 cosh aa +...+2 cosh *) = sinh ep times.

The points of division from R, to 7, taken sinh = times, are equivalent to a point
s

me 2 meee |
= sinh =———— — jimes.

at Q taken sinh é @ +2 ee ... +2 cosh ai

k

Again, the points of division from S to 7, taken sinh 57, times, are equivalent to a

ae Lea.
point at R taken sinh (14+200sh 7 +... + 2 cosh men 6) = sinh oem salie ti

Ihe k k ie fas

We have then the equation

S 9 :
sinh 22+} 2 P+sinh2™*1 = Meat = Bstgigh = oe
Let PR=a, RQ=8, PQ=y% atB=y.

Suppose m, n to increase indefinitely and 6 to diminish indefinitely. Then in the
limit

aati e. P + sinh Q = sinh YR,

and

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE.

Or, if pP+qQ =rR,
p q r

eM Gy Mig :
sinh = sinh ~

k k k

sinh
These equations may be replaced by
G=p+¢ + 2pq cosh ~,
enc! :
p sinh age! sinh i

In the other two cases the equations are respectively —

r=pt+q,
pz — GB;
r= p+q t+ 2pq cos 7,
5 te 4/83
psin;,=qsin;..

79

We may also determine the connection between the distances of points and the
quantities p, q, r by a method almost identical with that used by Klein im his article
“Ueber die sogenannte nicht Euclidische Geometric.”

Suppose, as before, A, 6, and C are unit points, and A+B=2XC; also suppose that
by a transformation A and C become altered into C and B respectively, and any point
P.=2,A+y,C into P,=a2,C+y,B. Such a transformation will be called a translation, and
it will be assumed that the distance between P, and P, is the same as that between
A and @ or C and B.

0

We have P=2,0+yB=20+y,(aC—A)=2,A+y,C,
where 0, —=—4n,
41 = % = AYo-

Repeating the transformation we get a point P,=«,A+y,B,

where L,=—-Y,;

Y.= 2, + AY; 5

so that x,—drax, +”, =0,

Ya — AY, + Yo = V.
In this way we get a point P,=2,4+y,C with the equations

“2 —re

n m—1

st Th» = 0,

Yo MY ris + Yn=2 = 0.

11—2

80 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

If x, = be the roots of the equation 27—Az+1=0,
we shall have x, = Az" + Bz",
and Yn te Cras mos (Az™* ot Ba).

Now if we put p(x AtyC)+¢(«,4+y,C) =r (x,A + y,C),

y Vie
then — See Oe :
ZY, — XY, TY, — 2,Yo ZY n— YoUn
or ie = q — ao r
a a = TU ay Til, 4, = vx, Tolar = Ly

Now 2,2,,,—%@4, = (Az + Be) {A2*"+ Bae} — (Az? + Bz) {Az + Bey

= AB(z —2z") {nr — Pegi

< ¢ p q = B
so that at =r) = Fr at

—zZ 2-2 a

We will suppose as before the distance between P, and P. to be a, that between
P. and P, to be §, and that between P,, P, to be y. |

Let the distance between AC be 0, then a=7r6, B=(n—r)0, y=n0.

There are three cases, according as the roots of z?—Az+1=0 are real, equal or
imaginary, that is, according as X\>=< 2.
8 8
In the first we may put A=2cosh>, z=p*, z*=p * where k is some constant; then

tad 76 Zz

zZ—zZ7=p*—p *=2sinh7.

Hence ee
sinh 7 sinh 7 sinh |.

In the third we may put X=2cos6, z=p", z*=p%

?
rb _ 7h a
2 —z7=p*—p * = Asin=

ke?

so that Po pee Se i

Lastly, when z=27?=1,
2 — 27 =(Z—2") (4 2+ to) =H (2 — 2")

in the limit = “(2 —2z");

and therefore I

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 81

These equations are the same as those found before.

The identity of this method with Klein’s is seen by considering the points that will
remain unaltered by a translation.

We must put 2Aty,C =2(2,At+y,0),
or 2n,+y,=0,
—%,+(z—rA)y,=0;
so that Z—’Az+1=0,
the same equation for determining z as before.
In the first case then there are two real points that remain unaltered by a translation.
These are ae: C and Cp.

If H be the latter point then H is in AC produced, and its position is given by

sinh CHP 2
sinh dP *?
Ci) AGH
pep 4 CH-AH
or pit — pan P k=p k S

but this is the case when CH and AH are infinite, so that H is a point at infinity:
This clearly ought to be the case, since the distance of all points at a finite distance
is diminished.

20

)
The number of times H is taken is J +p *—2p F cosh © = 0.

In the third case there are two imaginary null points at an infinite distance.

In the second or intermediate case there is one null point C—A at an infinite distance.

The connection between the quantities p, g, 7 in the equation pP+qQ=rh and
the distances PR, RQ is then determined. Whether these relations be considered to

give the distances in terms dopa! or, ou the other hand, to give f, d in terms of the

Pp?

distances, will depend on which we are considered to know originally. In the case of
addition of points on a plane or on a sphere, it is the distance that may be supposed
to be immediately known. But supposing pP, qgQ represent portions of different fluids
that mix without condensation and p, q be the volumes of the fluids, then r=p+gq,
the second of the three possible laws of combination. We can define then the distance
p

pry
plied by the same constant. This kind of manifold might be represented by a finite
straight line, the two extreme points of which would be the two unmixed fluids By

multi-

PR to be ey multiplied by some constant, and the distance R@ to be

82 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

taking three fluids and mixing them in all proportions a manifold would be obtained
which might be represented by the portion of a plane enclosed within a triangle.

If again, according to Young’s theory, all colours be made up of three fundamental
colours, we shall have a manifold bounded by a triangle. The quantities p, g, r would
naturally be taken to mean the intensities of the different colours. But the idea of
distance cannot always be introduced. For we found that 7, p, g are not independent
but are connected by the relation

= pi + gq’ + 2Cpq,
where C is some quantity independent of p and gq, and constant for the same two points.
That the relation must be of this form can be proved directly.

For assume it to be r*°=¢(p, q) where (p,q) is a homogeneous function of degree
n is p and gq, and involves besides only the distance between P and Q.

Put P=2,A+y,B, Q=2,A+y,B,
and let (z,y) be a funetion corresponding to ¢@ for A and B;
then ¥ (ay) =1, ¥ (ay) = 1;
and if rR = pP+qQ = (px, + 9x,) A+ (py, + Wy.) B,
then m= (px, + Gry PY, + Wo)
=p"+n p™a(%, a +Y, 4 +...4+9".
The qu, , tities i ie +o must be independent ‘of L,Y,, LY, except in so far as

they involve one single Nicolae the distance. This can only be the case if n is 2, for
otherwise we should get a new relation between 2,y,, 7,y,.

Therefore r= p+ ¢ + 2epq.

If the relation between r, p, g is not of this form, no meaning can be attached to
the distance. The theorems then that have been proved before, about intersection of Iines,
anharmonics, ratios, conics, &c. are so general that they apply not only to the cases
distinguished but even to manifolds where all measurement of distance is impossible.

If P=rA,A+y,B, Q=rAA+t+yu,B then a was defined to be the anharmonic ratios

fms b
of APBQ. With the first form of measurement of distance this becomes

AIP PO
sinh a sinh ~7—
. PB .
sinh oe sinh a
We will assume that lengths along all lines in a manifold of two dimensions (or

three-point manifold) are measured in the same way. We will consider only the first
case, and put the constant k=1. :

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 83

The expression for any point was P= 2d +yB+2C.
AP intersects BC in a point D which is a multiple of yB+2C,
therefore y sinh BD =z sinh DC;
PB intersects CA in HL where z sinh CE =azsinh LA,
CP intersects AB in F where «sinh AF = y sinh FB.

sinh BD.sinh CE.sinh AF _
sinh VC .sinh LA .sinh FB

It follows that 1.

This is the necessary and sufficient condition that AD, BE, CF meet in a point,
but regard must be paid to the signs.

In particular the lines drawn from the opposite angles of a triangle to bisect the
sides meet in a point which is a multiple of d +B+C.

A straight line whose equation is lx+my+nz=0 will cut BC in a point D for
sinh BD _ sinh DC

which my = — nz, or
ne —n
Similarly BC cuts CA in E where a =,
AB in F where sinh AF’ _ sinh FB
—m
therefore sinh BD. sinh CE. sinh AF _ ab

sinh DC .sinh HA. sinh FB
and this is the necessary and sufficient condition that D, H, F lie on a straight line.

Similarly other theorems, such as Carnot’s theorem, can be adapted.

MULTIPLICATION OF POINTS.

Taking any point P and raising it to successive powers P, P’, P®, P*... we must
either have an infinite series or else we must come to some power which can be expressed
numerically in terms of the preceding powers. That is, there must be some relation of
the form P"+aP"'+...=0. We will take the latter hypothesis and assume as the
simplest relation

122 = GPs (8p
We will assume also that multiplication is distributive, or that
A(B+C)=A.B+A.C.

Indeed, any operation which did not satisfy this latter condition could not properly
be spoken of as a multiplication.

84 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Now square the equation pP+qQ@=rk.
We have p(aP+8)+¢(@Q+8)+pq(P. Q+Q.P) =r (aR+8),
or pg (P. 0+ Q.P) = (PF -p-—P)B +4 {(2p—p") P+ (rg — 9°) Q}.

Now if the multiplication be uniform the relation between P.Q, Q.P, P and Q
must be independent of the quantities p and g. But this can only be the case if a = 0.

Supposing then @ the distance PQ we shall have
P= G = F=8,

and P.Q@+Q.P = 28 cosh 6;
or 1D Oae (Ode = P)sp
or PQO-= QO) P= 28icosig,

according as the first, second, or third law of addition holds,
This then is the most general law of multiplication consistent with giving P* a real
meaning.

It has been seen that the two conditions P?=Q@=—8, P.Q+Q.P=28cosh@ are
sufficient to ensure that. R’=8 where R is any other point, but it has to be proved
that they are sufficient to ensure that P’. Q'+ Q'. P’ =28cosh¢, where P’, Q’ are two
other points, the distance between which is ¢. Let PP’=a, PQ'=y, then $=y—a.

We are supposing, of course, that the first law of addition holds.

sinh (@—«) P+ sinh oe sinh (@—y) P+sinh yQ

abn is sinh @ Q= sinh 6
and PY = — 7; {sinh (@ — z) sinh (6 — y) P? + sinh (@ — 2) sinh y. PQ
+ sinh # sinh (@—y) QP + sinh « sinh y Q};
hence P’'Q'+Q'P — {sinh (@ — «) sinh (@- y) + sinh (@ —2) sinh y cosh 0
+ sinh & sinh (6 — ¥) cosh @ + sinh z sinh y}
a sinh? 0 [sinh (@ — a) {sinh (@ — y) + sinh y cosh 6}
+ sinh {x sinh y + sinh (6 — y) cosh 6}]
9
= oe 7 {sinh (@ — z) cosh y + sinh x cosh (@ — y)}

= 28 (cosh « cosh y — sinh « sinh y) = 28 cosh ¢.
The laws P=(F= 6, PQ+ QP = 2£ cosh 8,

are thus proved to be true for all points if they are assumed for any two; and the same
result can be obtained in the other two cases.

At this point several distinct assumptions can be made.

Ist. We can put B=0, P= =O, PQ=- OP.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 85

This applies to all three addition laws, With the first we shall have

LY = r sinh (.— x) sinh y — sinh z sinh (0 — y)} P. Q

sinh? 6

Q

= aang (sinh y cosh w — sinh w cosh y).

ed) Se Oe
Therefore sinh ~ sinh 6’

or, for any two points on the same straight line the product P.@ is proportional to the
hyperbolic sine of the distance between P and Q.

We can put P.Q=sinh @ x some constant peculiar to the line,
In the second case P.@Q=6@ x some constant peculiar to the line.
In the third P.Q=sin @ x some constant peculiar to the line.

The second and third cases are what Grassmann calls the outer (dussere) multiplication
of points and strokes (strecken). He has not considered the first case.

2nd. We can put I= (P= il. ROO —eoshd-
in the 2nd case J = OF = il, EQ =(QIP = ile
» ord case Peale JAD) (IP = eos),

This is in the third case what Grassmann called the inner (innere) multiplication of
strokes. He dismisses the inner multiplication of points as useless.
3rd. We can combine these two forms of multiplication and obtain the following
general form,
I =(UP=/31 PQ =f cosh@+y¥ sinh 6, QP = B cosh 6 — ysinh @,
where 8 and y are constant for the special line under consideration. 8 must indeed be

constant for all lines since it is the square of a point.

4th. We obtain a special form of multiplication by assuming—Ist, the associative
principle; 2nd, that 8 is a mere number.

Then multiplying PQ and QP we have

8’ = 8° cosh’ 6 — y sinh’ 6;

therefore [s =o77

We will introduce a quantity « peculiar to the line under consideration such that
“=1. Then we may put y=, and therefore

PQ = B (cosh @ + e sinh 8), QP = B (cosh 6 F esinh 6).
Dividing the second equation by 8 or P” we have

(P* = cosh 6 F ce sinh 6.
Wor XSi, Parr I. 12

86 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S
Now QP may be considered to be the operation of transferring P to Q. If then
the direction PQ be considered positive we will put
QP* = cosh @+csinh 8,
and therefore PQ=8' (cosh @ —«csinh 8), QP = f° (cosh @ + sinh 6).
In the second case P*= Q’= 8, PQ =P+78, QP = 8-8;
uf =-—7e-
Hence y*=0, and we may put y=+Ae with the condition ?’=0.
With the same convention as before,
. PQ = 8’ (1—18), QP = 6 (1+ 16), QP*=1+4+0.
In the third cae P?=@=B8, PQ=B+ysnd, QP=Bcosd—ysiné.
8° = B’ cos? 0 — ¥ sin’ 8,
&=—7'.
We may put y=+ Au where ° =—1, and with the same convention
PQ=8 (cos —csin 8), QP = B (cos 64 csin 8), and (P™*=cos @ + csin 0.
Comparing the three cases we have,
in the first QP*=cosh 6+ csinh6, ie
second QP*=1+10, i =U);
third QP*=cos0+vcsin8@, v=—l1.
In all three cases QP? = e* (?=1, 0, —1 respectively).

This result might have been arrived at directly by assuming—Ilst, that QP” is the
operation of transferring P to @ and is the same for any two points at the same distance
on the same line; 2nd, that multiplication is associative.

If then we put QP* =f (6), SEO = 77),
where @ is the distance between P and Q, ¢ between @ and R.
Then f(0)f ($) = RQ”. QP" = RP“ =f(0 +4).

This functional equation gives f(@) =e”, where v is some constant connected with the
special line.

We need not in this method assume the addition formule, but can deduce it. Assuming
only that if R be the middle point of P and Q, then P+Q=some multiple of R, we
have since P= Re”, Q= Re”, if O=distance PR, e”+e-”=some number.

Since this is true for all values of @ it follows that v? is a number.

The imaginary or complex numbers of algebra may be excluded, since we are dealing
with real points, and they must be reserved for cases of non-intersection of curves, &c.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 87

1

It follows that either v’ is positive = E where & is a number ;
v’ is null = 0.
aus , 1 :
v’ is negative = — 7, where & is a number.

t
ps3,

In all three cases we may put Views,

(c= 0Mor ah):

Then, in the first case,
fi) 1]

P4+Q=(p§ +p *) R=2Qeosh 4. R.

More generally, if pP+qQ=rf, and a be as before the distance from P to R, £ the
distance from FR to Q, y the distance from P to Q,

a ce
pe * +9p*=r.

This gives the equations cosh 5+ cosh 8 — i
g 4 P ppl i

p sinh i= q sinh? ‘

Squaring and adding these we have, since

a. [Sia eres Y
cosh 7 cosh a sinh 7 sinh ae cosh I?

r= p+ @+2pqcosh”,
and these are the same equations as those obtained before.

In the same way the suitable formulae for the other two cases can be found.

Thus, omitting the previous section, the conception of distance might have been derived
from the division of points. This naturally raises the question, Why should distance be
connected with QP rather than with Q@—P? Or, in other words, why should QP” be
supposed to be the same for two points on the same line at the same distance, when
Q—-—P is not supposed to be the same ?

We may try the result of the latter supposition. If, then, again, we suppose the
distance between P and Q to be @, and that between Q and & to be ¢, and put

Q-P=f(%), R-Q=f(%),
then F(A) + F(¢) =F (0+ 4)
Therefore f(@) = v0, where vy is some constant depending on the particular straight line.
Hence we have with the notation used a little while before, if pP+qQ =rh,
p(R v2) + q(B +08) =r;
meet) ai
px = gB.

88 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

This is the second of the three systems before found, and is that of points in
ordinary plane space.

We may say then,—There are three uniform systems in which distance depends on
division; there is only one uniform system in which distance depends on subtraction, and
it is the special or intermediate case of the three former systems.

If in the third system we put P?=Q@=@=-1,
we have PQ=-—cos@+csin 0.

This is the Quaternion multiplication. It can be, as Grassmann has shewn in the
Mathematische Annalen (Die Ort der Quaternionen in der Ausdehnunglehre), derived from
the general form PQ=8cos@+vysin 6, by introducing the associative principle.

The corresponding forms for the other two systems will be

PQ =— cosh 6+ sinh 6,
and PQ =-1+ 06.

It remains only to prove the distributive principle. That is, to prove if pP+qP=rh
and O be any point on the same line,

pO.P+q0.Q=r0.R.
We will take the first system and the most general form of multiplication ; that is, if
distance Mi = Ge PTE, = (2). RQ = ¢, PQ=%,
then we will put O.P = Bcosh(o—6)+ysinh(¢—8), and so on.
It has to be shewn then that
p cosh (« — 8) + q cosh (« + d) = r cosh a,
p sinh (c — 6) + qsinh(o + $) = rsinho.
But these equations follow at once from the equations
p cosh 6 + q cosh ¢ = 7,
and p sinh 6 = qsinh ¢.

DETERMINATION OF ANGLES.

If we consider the multiplication of points not merely in a straight line but in a
plane every point of which is a multiple of e4+yB+2z2C, we shall have a number of
different quantities « connected with the different lines that can be drawn in the plane,

Now all the lines (/, m,n) that can be drawn through a point a, y, 2 satisfy the equation

le+ my +nz= 0,

and therefore they are singly infinite in number.

We will now use Hamilton’s multiplication and notation.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 89

Thus, if PQ =— cosh 6+ sinh 6, we will put SPQ=—cosh@, and call it the scalar
part of PQ, and VPQ=vcsinh @, and call it the vector part of PQ. Then VPQ=-—VQP,
so that Hamilton’s vector multiplication is for two points identical with Grassmann’s outer
multiplication. What immediately follows will apply to all three kinds of geometry.

If «,, 4, be the quantities corresponding to the lines joining

wtA+yB+2zC to 71 A+y,B+z20 and 2,4+y,B+z,C respectively,
then t,=some multiple of V (#A+yB+2C) (1,A+y,B+z,C)
=some multiple of (yz,—zy,) VBC + (zx, — xz,)(VCA + (xy, —yz,) VAB
=some multiple of (VBC +m,VCA+n,VAB,
if 1m,n, be the coefficients of the equation to the line.
Similarly +,=some multiple of 1,VBC+m,VCA+n,VAB.

And if «4, be another line passing through the point (we may say line instead of
quantity connected with line) /m,n, coefficients of its equation,

4, = some mult. of 1.VBC+m,VCA +n,VAB.

Now (l,m,n,), (l,m.n,), (l,m,n,) are connected by the equation

lL, M, 7,
Lm, 1, |= 0)
L, Mm, Nz

and therefore we may put /,=Al,+ yl,, m,=Am,+pm,, n,=An, +n, It follows that we
may put ,=pi,+q,, so that all the limes passing through a point are lineally connected
and form a system similar to that of points on a line.

We will assume then that we may introduce the conception of distance for straight
lines, and we will call the quantity corresponding to the distance between two points the
angle between two straight lines.

Consider the point O(«, y,z) to be within the triangle ABC, so that 2, y, z are all
positive. Then the equation lc+my+nz=0 requires that one of the quantities 1, m, n
should be negative if 1, m, n be the coefficients of the equation to a line passing through 0.
This line will cut the sides in points given by my+nz=0, c=0; nz+le=0, y=0;
lx+ny=0, z=0. That is, in points nB—mC, 1C—nA, mA—IB,

If 7, be negative, the second and third points must lie between C and A and A and B
respectively. So that every line drawn out from the point O in either direction cuts one
of the sides of the triangle.

If then we start with the line OA, and draw lines from O to successive points of
AB, then to successive points of BC, then to successive points of CA; the line OA will
have returned to its old position. It follows that the equations connecting p, g, 7 with the
angles must contain only periodic functions of the angles, and therefore the third kind of

90 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

addition laws must hold. We have then, if @ be the angle between +, and «4, 8 between
«, and «, and y between ¢, and «,, while mm, = pi,+ qt,

ri = p'+q¢ +2pq cos y,
p sina = sin 8.
The angle a line has turned through when it has come back to its old position is said

to be four right angles. The uniformity supposed to exist in the plane requires us to
consider all right angles equal.

MULTIPLICATION OF LINES MEETING AT A POINT.
This will follow the same laws as multiplication of points on a line in the third
kind of geometry.

We will consider for the present only the quaternion form of multiplication. Then,
if p, « be two lines making an angle 6,

po =o°(cos 0—TJ sin 8),
where J is a quantity whose square is —1.

Suppose 0 to be the point where the lines meet; then we may identify the quantity
T with the point O, and write po =o" (cos @— Osin 6).

In the first kind of geometry o*=1,
therefore po = cos 6 — O sin @.
Therefore Spo = cos 8, Vpo =— Osin 8, TVpo = sin 6.
In Grassmann’s notation plo = cos 8, po =O sin @.
In the second kind o*=0,
therefore pa = 0, Spo =0, Vpo = 0.

In the third kind o?=—1,
therefore po = — cos 6 + O sin 8,

Spo = — cos 8, Vpo = O sin 8, TVpo = sin 0.
In Grassmann’s notation pic = cos 6, po =O sin 0.

The inner multiplication of Grassmann corresponds (neglecting sign) to the scalar
multiplication of quaternions; and the outer multiplication to the vector multiplication of
quaternions.

If p and o be lines at right angles, we have in the first case

po =— 0, cp = O.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 91

Therefore rae or O may be interpreted as the operation of turning a line through

a right angle in the positive direction. Multiplying by o and by p we have also the
equations
Op =a, pO=-a,

Oc =—p, c0=p.
If A be another point on the line o and p be the distance OA, then

= = cosh p + csinh p,

or A =O cosh p + p sinh p.

Let B be a third point at a distance s from O, then we have
B= Ocoshs +p sinhs.
This gives
A sinh s — B sinh p = O sinh (s ~ p) = Osind, if 6=s—p,
an equation equivalent to those already found, and also

A cosh s — Beosh p = —p sin 6.

The first equation assigns a real point for 7X4 —yB for all values of = from 0 to

~* The point gradually moves from A to infinity along the line BA produced. When

“1 &

is greater than e~ the first equation ceases to give a value for X\A—wB,

sinh Cee oe oe
Pe - <e° when p and s are positive.

snhs eé—e

since

But the second equation gives a value for XA —pB, since

coship e? He? 5
coshs e'+¢é"*

1 ; 7 ‘
ash S° A —pB is a line perpendicular to AB, and

this line moves from infinity in BA produced up to A. When the line passes 4,

As then = increases from e~ to

p becomes negative, but the sign of . is unchanged. When f=1, then

cosh $ (A — B) =~ psin8, or A-B=-2 sinh 3,

where p is a line drawn through the middle point of AB at right angles to it and on the

positive side of rotation. As E becomes greater than 1, we can consider the expression

Xr

w#B-—XA and the ratio ae. and everything will occur as before, but in inverse order.
be

92 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

: : : : ; : x
The line will be at an infinite distance in AB produced when i = Grr t= é.
The expression will then represent a point which will move up to B and reach it when
x eS
rea or >=.

Therefore the expression 1A —pB, as f varies from 0 to «, represents first a point

which moves from A to infinity in BA produced; then a line at right angl s to AB
which moves from infinity in BA produced to infinity in AB produced; then a point
which moves from infinity in AB produced to B.

Since sinh* p + sinh*s — 2 sinh p sinh s cosh 6 = sinh’ 6,

2 cosh p cosh s — cosh’ p — cosh’ s = sinh’ 6,

the magnitude of »4—pB when it represents a point is ,/A?+p?—2dpcosh 6, and when

it represents a straight line /2\m cosh 6—2*— p?, so that it can never represent a real point
and straight line at the same time.

Returning to the equation
A=O cosh p+p sinh p,
and multiplying by o we have, if we call p’ the perpendicular to OA at A,
p' =p cosh p + Osinh p
= (cosh p+o sinh p) p,
so that E = cosh ptesinhp, as ought to be the case, since p’p* represents the operation

of transferring a line along o in the positive direction.

Also Sp'p = Spp’ = cosh p,
Vp'p=— Vpp' =c sinh p.
Now if the lines p, p’ were to intersect at an angle @ we should have, by what has
gone before, Spp'=cos@, but it is impossible that cos@=cosh p, since cos@ is always less

than 1 and coshp always greater than 1. It follows that two lines at right angles to
the same line can never intersect.

If the equation A =O coshp+p sinh p be multiplied by p, we find
SAp = SpA = sinh p,
VAp = — VpA=c cosh p.
In words,——The scalar of the product of a point and a line is the hyperbolic sine
of the perpendicular from the point on the line, and the magnitude of the vector part

is the hyperbolic cosine of the perpendicular; supposing always the point to be on the
positive side of the straight line.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 93

If we take a point C at a distance ¢@ along p, then
C = (cosh ¢ + p sinh ¢) O
= 0 cosh ¢ —c sinh ¢.
If p’ be the distance of C from p’, then, since C is on the negative side of p’,
— sinh p’ = SCp’= S(O cosh ¢ — o sinh g) (O sinh p + p sinh p)
=— cosh ¢ sinh p.

Therefore sinh p’ is greater than sinhp and p’ greater than p, or the distance of C
from p’ is greater than the distance from 0.

Hence lines drawn at right angles to the same line diverge from one another.

In the third kind of geometry, if p and o be at right angles, we shall have equations

pa = O,; op =-— 0,
Op =a, pO=-o,
c0 =p, Oc =—p;

and also, if the letters have the same meaning as before,
A sins—Bsin p= Osin6,
A coss—Bcosp=—psin6.

But if O' be a point at a distance za from O in the positive direction
A sin (s — 477) — Bsin (p—477) =O sind.
It follows that O’=p.

Now o may be any line perpendicular to p, therefore all these lines pass through a
point O' at a distance $7 from p.

This may be confirmed by the other equations analogous to those obtained above.
Thus sin p’=cos¢sinp shews that p’ is less than p and the perpendiculars converge, and
also that p’=0 when $=5 whatever p may be; so that all perpendiculars to the same
straight line pass also through the same point. The point and line are called pole and

polar, and considered as symbolic quantities may be identified.

Now, if p, p’ be the perpendiculars and @ their angle of intersection, then
Spp'=cosp and Spp’=cos6;
so that @=p, if the signs be properly chosen.

That is to say, the distance between two lines along their common perpendicular is
equal to the angle at which they intersect.

Vou. XIII. Parr II. 13

94 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

If at points distant $2, perpendiculars be raised to the line joming them, these per-
pendiculars will intersect at right angles, and we shall have a triangle all of whose sides
and angles are equal to }7.

If i, j, & be the points or sides of this triangle they may be identified (except in
sign) with O, p, ¢, and it follows that
je=t, Wj=-i% M=y, h=-j7, Y=k ji=—k.
These are the symbols of Quaternions.

In the intermediate kind of geometry 0, p, « may be taken to satisfy the follow-
ing laws

pa =0, op =0, Op=c, pO=—c, Oc=—p, c0=p.

The equation = =l+op
gives A=O+ pp.
We have also B-A=pé,

but if p’ be the perpendicular at A B-A=p'6.

Therefore all lines drawn at right angles to the same line are to be considered
identical.

If C be a point distant ¢ from O, and on the line drawn through O perpendicular
to «, and D be a point at the same distance on the line through A, then

C=0-co¢,
D=A-—c¢.
Therefore D— C=A— O=pp.

But if p’ be the imaginary quantity perpendicular to the line CD, and p’ be the
length CD, then D— C=p'p’.

Therefore PP =pp, p =p. p =p.

It follows that lines perpendicular to the same line are everywhere equidistant.
Also CD must be perpendicular to OC and AD, since p’ coincides with p, the quantity
belonging to OC and AD.

The figure OCDA has then its angles, right angles, and its opposite sides equal.
It is said to be a rectangle and the lines perpendicular to the same line are said to
be parallel.

Since in general z4+y0O represents x+y times a point P such that cAP=yPO,
A-—O will represent a small point at an infinite distance. But D—C=A-—O so that
parallel lines may be said to intersect at infinity.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 95

GENERAL EXPRESSION FOR Ratios OF LINES AND POINTS.

We obtained for the first kind of geometry in the preceding section three points
OneA. C expressed literally in terms of O, p, o. But by definition any point in the
plane can be expressed lineally in terms of three points O, A, C. Therefore any point
in the plane can be expressed lineally in terms of O, p, o. So also can any line,
since any line is the vector part of the product of two points. To find these expres-
sions let P be the point, a@ the distance OP, and a the angle OP makes with the
line p, then the imaginary quantity corresponding to the line OP will be pcosa+osina,

and PO™* = cosh a + (pcos a+osina) sinh «.
Therefore P= Ocosha—co sinhacos «+p sinh asin a.

If then zp+yo+z20 be identified with any multiple rP of a point P we must
have z=rsinhasina, y=—rsinhacosa, z=rcosha.

These equations give

r=ez—-x7—-y, tana=—_-, tanha=

y

It is always possible then to find values of r, a, a in terms of 2, a, y if 2—a°—y*
be positive.

x ete
=

Next take a line drawn through P perpendicular to OP and let X be the cor-
responding imaginary quantity. The imaginary quantity for a line at O perpendicular

to OP is o cos a—p sina.
Therefore dX = {cosh a + (p cos a+ co sin a) sinh a} (o cos a — psin a)

=— Osinha +p cosh a cos a—p cosh a sina.

If then zp+ yo+2z0 be identified with rn,
z=-—recoshasina, y=rcoshacosa, z=—rsinha.
So that P=c+y—2, tana=—~, tanhha=—;.
y v (a +y’)
It is always possible to find values of r, a, in terms of 2, y, z, if z’+y° —2 be positive.

It follows then zp+yo+z0O is a point on a line according as 2 ><a’ +y’.

Tf zap+yo+zO0 be a point then wt+ap+yo+z0=w+rP may be identified with
some multiple of the ratio of two lines meeting at the point P. Let @ be the angle
at which they meet, s the multiple of their ratio. Then s (cos @ + P sin @) =w+rP,

scosd=w, ssind=r, s=w' +7", tan ==.

Substituting the value of 7?

S=w+2—a—y’.
13—2

96 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

If zp+yo+z20 be a line then we may identify w+ap+yo+z0 either with the
ratio of two points on the line or else with the ratio of a point to a line. In the
first case we must put

w+up+yo+z20=w+rr=s (cosh 0 +2dsinh 8).

Therefore w=scosh@, r=ssinh@,

tanhO=—, S=w—r=w+2—-a’—y’.

a
w
The condition that this substitution may be possible

is w'>7* or w+ 2°—2*—y’ > 0.
In the second case we must put

w-+1TX=s (sinh @ + dX cosh 6)
Ww 2 2 2 2 2 2 2
tanh@=—", S=P-w=a+y—u 2’.

This will be possible if 7°>w* or #+y?—w*—2>0.
Thus a meaning can always be found for w+ap+yo +20.

If we form the product of two such expressions
(w+ap+yo +20) (w'+2'p +y'co + 2'0)
= ww’ + aa! + yy’ — 22’
+ (wa' + w'a + yz —y'z) p
+ (wy'+ w'y + za’ —2'2) o
+ (w2' +w'z + yx’ — yx) O,
then since the product of the magnitudes of two ratios is the magnitude of their product,
(w+ 2—a?—y’) (w? + 2"— 2? —y?)
= (ww' + va’ + yy — 22’)? + (we! + w'z + ya! — x’)?
— (wa + w'x + ys —y'2) — (wy +w'y + 2a’ — 22)’,
a formula analogous to Euler's for the product of two sums of four squares.

Since the multiplication of the quantities 0, p, o is associative and all others can
be formed lineally from them, multiplication will be always associative.

All the terms of Quaternions such as conjugate, tensor, versor, can be employed.

In the third kind of geometry every point can be expressed in terms of the three
points 7, 7, k.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 97

If we put rP=ai+yj+zk and call a, 8, y the distances of P from the points
2, j, k, then multiplying by 7, and taking scalars

rcosa=—rSPi=2,
rcos B= Y,
rcos y= Ly
Also squaring the equation rP=xi+ yj + 2k,
P=er+y +2,
and therefore cos’ a+ cos’ 8 + cos’ y=1.

If w+ait+yj+zk be the ratio of two points at a distance 0, we may put
w+ at+yj+2k =w+rP =s (cos 6+ Psin@),
where P is the pole of the line on which the points are.
Then SHwWt+Pawte+ty+Z,
Lf
w

tan @ =

We arrive at Euler’s formula in the way shewn above, as is pointed out in Hamilton’s
Elements of Quaternions.

FORMULAE IN COORDINATES.

If OX, OY be any lines at right angles, P any point in their plane, then the position
of P may be determined by the three quantities

x=sinhOPcosPOX, y=simnhOPsn POX, z=cosh OP.
These quantities are not independent, but are connected by the equation
e—e—y=l.

If p, « be the imaginary quantities corresponding to the lines OX, OY, then, by the

preceding section,
P=2z0—20 + yp.

Now if PL, PM be the perpendiculars from P on OX, OY,
sinh PL = SPp = y, sinh PM = —SPo = xz.
Since smh PZ =sinh OP sin POX we have the theorem,—In a right-angled triangle the

hyperbolic sine of the perpendicular divided by the hyperbolic sign of the hypothenuse is
equal to the sine of the angle.

If OQ be the perpendicular from the origin on any line, then the position of the
line may be determined by the three quantities

2 = sinh OQ, m = cosh OQ cos QOX, n = cosh OQ sin QOX.

98 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

They are connected by the relation

P+n?—-n=1.

If X be the imaginary quantity corresponding to the line

r~=—nO+lo —yp.

Supposing PR to be the perpendicular from P on the line A,
sinh PR =— SPr = la + my — nz.

If P be on the line, PR = 0 and therefore its equation is
le + my —nz = 0.

The distance between two points P = z0—ac—yp, P =2'0—2'c+y/p is given by
cosh 6 = — SPP’ = zz’ — xa’ — yy’.

The angle or shortest distance between two lines is given by
cos 0 = SAX = Il’ + mm’ — nv’,

or cosh 6 = Il’ + mm’ — nn’.

These are all the formulae that are required for questions concerning distances and
angles in this kind of geometry. As an example the following problem may be taken:
To find the locus of two lines meeting at right angles and passing through two fixed
points.

If (sinh ¢, 0 cosh ¢) (— sinh ¢, ocosh@) be the points, so that they are at equal dis-
tances @ on either side of O, (a, y, 2) the points whose locus is required; then, if
(1, m, n), (U, m’, n’) be the lines

if m n

y cosh zsinhd—xcoshg ysinhd¢’

bi m' —n

ycosh —zsmh¢—acoshd ysinhd’

and, since ll’ + mm — nn’ =0,
y’ cosh® @ + (a cosh ¢ — zsinh ¢) (« cosh ¢ + z sinh $) + y sinh’ d = 0,
x cosh’ + ¥* cosh2¢ — z* sinh*d = 0.

The formulae for the third kind of geometry will be identical with those of spherical
geometry referred to ordinary rectangular co-ordinates.

In the second or intermediate kind, if P be any point we can put
P=0+4+ai+ yj,
where, if PL be the perpendicular on OX, then OL =a, PL=y.
It follows that L-O=x, P-M=y.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 99

If r be the length of OP then es0 represents a unit line perpendicular to PO;

so that if OP makes an angle @ with OX, then

PHO ~ 0050 +; sin 8,
Therefore “=r cos G; y=r sin 6,
and hence ey,

relations connecting the angles and sides of the right-angled triangle POL.

If @ be the angle OPZ, then in the same way
2—asin\e/, y=recos @,
so that cos 8 = sin 0’, sin 6= cos 0’.

The angles @, @' therefore together make a right angle, and it follows that every
straight line cuts parallel straight lines, so that the alternate angles are equal.

The distance between two points is found in the usual way (which assumes «nothing
not already proved) to be («—w’)’+ (y—y')’.

This is not at all a convenient system for deriving properties of the ordinary plane,
but it is the one which offers the closest analogy with Quaternions for the sphere.

RELATIONS BETWEEN THE SIDES AND ANGLES OF A TRIANGLE.

Let A, B, C be the points of a triangle in the first kind of geometry;
a, B, y the corresponding angles;
p, o, 7 the imaginary quantities belonging to the opposite sides ;

a, b, ¢ the length of these sides.

Then CB* = cosha+ p sinh a, BC = cosh a — p sinha,
AC*=coshb + csinh b, CA™= cosh 6 — co sinh J,
BA™ = coshe + Tsinhe, AB” = cosh c—7rT sinh c,
Therefore cosh a — p sinh a = (coshe + 7 sinh ¢) (cosh b + o sinh 8).
Take the scalar parts: then, since Sto = cos(7—a) =— cosa,

cosh a@ = cosh b cosh c — sinh b sinh ¢ cos a.

Take the vector parts:
—psinh a = 7sinhe¢ coshb +o sinhb coshe + A sinh b sinh e sin z,

since Vro = Asin(m—a) = Asina.

100 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Multiply by A and take scalars,

SAp.sinha = sinh 0 sinhe sina.

Now, if p be the perpendicular from A on BO, SAp = sinh p.

Also, SAp sinha = SAVBC = SABC, since S(ASBC) = 0
= SpA sinha = SVBC.A=SBCA = SBVCA
= SBo = SCr.

Hence the quantity SABC is unaltered by the interchange in cyclic order of A, B, C,
and we have the equations

sinh 6 sinh e¢ sina = sinhc sinha sin 8 = sinha sinhd siny

= sinha sinh p = sinh 6 sinhg = sinhe sinhr

= ,/sinh*b sinh*c — (cosh b cosh ¢ — cosh a)"

=,/1+2 cosh a cosh b cosh ¢ — cosh’ a — cosh’ 6 — cosh’ ¢,
by the preceding equation.

We have thus three independent equations between a, b, c, a, 8, y, and can determine
any three in terms of the others. The equation giving a in terms of 8, ya can however
be obtained directly in the same form as that giving a in terms of 8, ¢, a.

For To =—cosa+Asina, ot '=—cosa— Asina,

pt =—cosB+B sinB, tp =—cos8— Bsin£,
cp’ =—cosy+ Csiny, po *=—cosy—Csiny.

Therefore —cosa— A sina=(—cosy+C sin y) (— cos 8+ B sin B).

Taking scalars cos @ = — cos B cosy + sin 8 sin y cosh a.

From this last equation it is seen that
cos (7 — a) <cos(8+¥y),
aw—a>B+y,
T>at+B+y;

or, the sum of the angles of a triangle is always less than two right angles.

Suppose C to be a right angle, then
cosh ¢c = cosh a cosh 6,
cos 4 =sin 8 cosh a,

sin
cot a = = si

— cosh a = sinh b cosh a.
In a

Now let B move off to infinity while A is unaltered, then AB will be said to be
parallel to CB, and cota = sinh 8,

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 101

This gives the angle the line drawn from any point parallel to a given line makes
with the perpendicular on that line. This angle is less than a right angle, and continually
decreases as the length of the perpendicular increases. From the symmetry of any line
it is obvious that two parallels can be drawn from any point, one on either side of the
perpendicular and making equal angles with it. An infinite number of lines can be drawn
not meeting the given line. The equations found are sufficient to determine all properties
of this kind of geometry. The trigonometrical relations of the other two kinds will be
those of the sphere and the ordinary plane.

THE DIFFERENT KINDS OF UNIFORM SPACE.

We have by a purely analytical method. arrived at three different kinds of space
relations, and as no assumptions have been made except those necessary to ensure
uniformity, these are the only possible uniform relations. We may now bring together
the chief distinctive properties.

In the first case.
The angles of a triangle are together always less than two right angles.
Lines perpendicular to the same line diverge from one another.

From any point outside a line, two lines can be drawn meeting that line at infinity
and an infinite number of lines not meeting it.

In the second case.
The angles of a triangle are together always equal to two right angles.
Lines perpendicular to the same line are always equidistant.

From any point outside a straight line only one line can be drawn to meet that
line at infinity, and every other line will meet it at a finite distance.

In the third case.
The angles of a triangle are together always greater than two right angles.
Lines perpendicular to the same line approach one another.

From any point outside a straight line no lines can be drawn to meet that line at
infinity. All lines meet at a finite distance.

The first kind of geometry is the imaginary or non-Euclidean geometry of Gauss,
Lobatschewsky and Bolyai.

The second is the geometry of the ordinary plane.

The third kind may be divided into two sub-cases according as we treat A and
—A as distinct or identical points,

In the first sub-case since if the equation la+my+nz=0 is satisfied by (a, y, 2)
Vou XIII. Parr IL. 14

?

102 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’'S

it is also satisfied by (-—2, —y, —2); all the lines drawn through a given point pass
also through another point which may be called the opposite point. The distance from
t to j is $7, from 7 to —i% 47, from —2 to —j 3x, from —j to z 2; so that the whole
length of a line returning into itself is 27. Every line, for instance that joining J, f,
divides the space symmetrically, and it is impossible to pass from a point in one half
to a point in the other without cutting the line. This is the geometry of the sphere.

The other sub-case gives a geometry which has been considered by Simon Newcomb
(Crelle, Vol..83, for the year 1877, p. 293); Killing (Crelle, Vol. 86, for the year 1879,
p. 72); and Frankland “On the simplest continuous Manifoldness of two dimensions” in
Nature, 1878.

The distance from z to 7 is }7, and from 7 to —7 is 47, and therefore as —7 coincides
with i the length: of all straight lines is 7. As in the sphere, all the points distant
17 from z lie on a straight line.

The surface may be represented by the figure if it be remembered that opposite

points on the circle are identical. If a man were to start from 7 and walk along the
line jk till he returned to j, he would then be in the position Pj, that is to say
upside down, and he would have to complete the line twice to return to his original
position. For this reason, the line jk may be considered a double line, and just as in
the sphere every line has two poles through which the same lines pass, so we may say
in this geometry every point has two polar lines on which the same points lie. The
properties are therefore reciprocal to those of the sphere, as is pointed out by Killing.
Any two points P, P’ for instance can be joined without cutting jk, so that the straight
line does not divide the plane.

This kind of surface could not like the sphere exist in a non-Euclidean or Euclidean
space of three dimensions.

It is obvious that the ordinary trigonometry can be obtained from the imaginary
or spherical trigonometry by putting (1, a) for (cosha, sinha), or (cosa, sin a) respectively.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 103

Or else reintroducing the constant & we have only to put in the equations

k=c after having multiplied by & to obtain the corresponding plane equations. Similarly
those of imaginary trigonometry can be obtained by making & imaginary. Thus the
three kinds of relations are those of a sphere of imaginary, infinite and real radius
respectively. This could have been seen originally from the fact that the other two .
addition equations can be derived from the third by making k imaginary or infinite.

Only lear manifolds have been considered, but a non-linear manifold such as
that of the pomts p*A+pqgB+q°C will obviously lie in a linear manifold 2d +yB+4+2C

of higher dimensions.

The assumptions that must be made to identify our actual space with the second
of the three kinds here considered seem to be the following. They are arranged in the
order in which they have been successively introduced.

1. There exists a continuous line determined in a single way by any two points
on it and capable of being drawn between any two points.

2. There exists a continuous surface determined in a single way by any three
points on it.

3. If the line be called a straight line, then a straight line can be moved along
itself. To determine the motion it is sufficient to know the new position of any one
point, then that of any other will be known.

4. Keeping a point fixed a straight line can be moved in only one way while
remaining in the same plane (if the surface be called a plane), so as to coincide with
any other line in that plane passing through the point.

5. The line equidistant from a straight line, is itself a straight line.

6. Space is of three dimensions.

The first two assumptions are equivalent to the definitions that a line consists of
all the points pA +qB and a plane of all the points pA+qB+rC. The third is requisite
to enable us to measure distance. It is verified every time a distance is measured in
the usual way. In fact the ordinary measure of distance coincides, as Klein points out,
with the definition given in a former section. It must be observed that the lines spoken of
are actual physical lines, yard measures, stiff wires, &c. We find the properties nearly
true of these actual lines, and that as far as our knowledge goes they can be approximated
to, indefinitely.

Geometry of course merely expresses in abstract language a special class of relations
between existing bodies. Moreover, as Riemann observes, our observation is limited just
as much on the side of the very small as on the side of the very great. In the same
way as the conception of rays of light is found to give a very accurate explanation of
shadows, images, &c., but a closer observation shews a different class of optical phenomena ;

14—2

104 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

so it might be that if our microscopes were powerful enough, we should discover different
geometrical relations among the particles of matter.

The fourth axiom shews how to measure angles and compare distances on different
straight lines, for the preceding axiom only shews how to compare them on the same
straight line. The fifth is one of the many forms in which an axiom distinguishing
our space from the other two kinds of uniform space may be stated. The sixth requires
no comment.

It is difficult in such an enumeration to be sure of having included all the axioms
and of not having combined two distinct propositions in one. But at least it is clear
that the number of axioms must be limited. Grassmann’s results refute the view that
geometry contains an indefinite number of distinct synthetic propositions. When once the
axioms have been assumed, all other geometrical truths can be derived by the mere rules of
calculation.

IMAGINARY GEOMETRY OF THREE DIMENSIONS.

We proceed next to construct a calculus analogous to Quaternions for the imaginary
geometry of Bolyai, when it is in three dimensions,

All the points equidistant from a given point form a closed uniform surface identical
in all its properties with the ordinary sphere.

Draw from the point O three lines at right angles, and let 7, 7, & be the correspond-
ing imaginary quantities. Let also J denote the operation of turning the line 7 round O
till it coincides with k, and J, K the operations of turning k& round to 7 and 7 to j.
If we consider the points where 7, j, & cut the sphere, then these operations will move
the points along three arcs, distances equal to right angles. J, J, K then must be
identical in their properties with the quantities before found for the spherical calculus.

Therefore JK=I, KI=J, V=K, I?=-1, J?=-1, K?=-1.
But since J turns j to & and therefore k to — j,
Ij=k, Ik=-},
and therefore since j=h?=1, kj=J, gh=-I, jl =—-k, kI =).
Similarly Jk=1, kJ=-i, M=—-k, W=k, tk=J, hM=-J,
'Ki=j, iK=-—j, Kj=-i, jK=i, pi=K, y=-K.

The equations may be written thus

I=JK=hy, i=jK=Jk,

J=KI=tk, j=kIl=Ki,

K=l=j, k=WJ=f,

P=J*=K'=—-1, el hf

remembering always that the products change sign when the order of the factors is inverted.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 105

Multiplying the equations i=jK=Jk by J and assuming the associative principle
Ti=kK=Kk.
Therefore the quantities Li, Jj, Kk, if, jJ, kK must be all equal.

The equation I=—jk shews that iJ=— ik.
Put Ue; i[=—v, then 7? =1, (PS Sil,
vl=%, wuw=—-f, v=-l1,
and since M=—v2, Iv =4, w=—T.

The quantity v is therefore commutative with each of the six quantities J, J, K, 4, j, k.

A unit line through the origin will be + mj+nk if (l, m, ) be its direction-
cosines. A rotation through a right angle about this line will be /J+mJ+nK. The
product (i+ mj+nk) (U+mJ+nK) is equal to v as it ought to be.

If r times the line li+mj+nk be transferred through a distance @ along a line at
right angles whose direction-cosines are (J, m’, n’) we shall obtain the line

r {cosh 6 + (Ut + mj + wk) sinh 6} (li+ mj + nk)
= rl cosh 61 + rm cosh 6j + rn cosh 6k
+r (mn‘ —m'n) sinh I +r (nl' —n'l) sinh 6J +r (Im' —I'm) sinh 6K.

If we write this Xi+Vj+Zk+LlI+MJ+ NK,
then XL+YM+ZN =0,
and X?+ Y?+ Z*—-L*— M’?— N* =r’ a positive quantity,

since (mn’— m'n)* + (nl — n'l)’ + (Im' —Um)? = (? +m? + n°) (1? +m? + n) — (UU + mm + nn’) = 1.

If these two conditions hold Xi+ Yj+Zk+LZI+MJ+NK can always be identified
with a translation along a certain line, for there will be five independent equations to
determine five quantities fixing the line. We shall have, in fact,

= : 1? + M?+ N?
= 72 2 ay 2 __ 2 a
i) Oa Fe Ne, © tank = ya
BY ZM —YN

v=

ae (exes 577)? V(Xi+ 47) V(b 4 e+)’
When r becomes 0 and 6 infinite while X, Y, Z LZ, M, WN still keep finite values,
X?+ Y?4+Z?—L?— M’— N’*= 0,
and Xi+ Yj+Zk+LI+MJ+NK represents a null translation along a line at an infinite
distance,

If r times the rotation 1J+mJ+nkK be transferred through a distance @ along the
line whose direction-cosines are (l’, m’, n') where Jl'+mm’'+nn'=0, we shall obtain the

rotation
r {cosh 0+ (i+ mj + nk) sinh 6} {1+ mJ +nK}

= rl cosh 6J + rm cosh 6J + rn cosh 0K
+r (m'n — mn’) sinh 01 +r (n'l— nl’) sinh 6j + r ('m —U'm) sinh 6k.

106 Ms COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

We may put this equal to
—Li-Mj-Nk+XI+ YJ+ ZK,
and if this be multiplied by v, it gives the corresponding line as it ought to do. It

follows that, if
AL+YM+ZN=0,

then Ni+ ¥j+Zk+LI+ MJ+ NK
represents a line or rotation according as
X?+Y*+Z°-L’— M*—N* > or < 0.
Take a unit rotation, or as it will perhaps be better to call it, couple
+rit+ pj t+vk+ pl+oJ+7K.

This may be written (9p +XV—1)I + (o+pV—1)J+ (r+vV—1K,
for the quantity v used above has all the properties of the algebraic V—1, and may be
identified with it.

The two conditions pt+ot+r—-N-p-v=1
and pry+ou+7v=0
are equivalent to (op +XV—1)?4+ (c+ pV—1) 4 (r+yV—-1P% =1.

It follows that a unit couple in imaginary geometry is identical with a unit bivector

in the sense Hamilton gives that word in his Quaternions. A force along the same line

will be identical with a bivector of magnitude or tensor V—1. Hence a force F along a
line and couple G@ about that line may be identified with a bivector of magnitude
G+F/—1. Now any expression

Xi+ Yj+ Zk+LI+ MJ+ NK =(L4+XV-1)14+ (M+ YV-1)J4+(N4ZV-1)K
may be equated to
(G4 FV=1) {((ptav—1)I4+(o+ pV¥—-1)J 4+ (7 +0V-1)K}.
For we have only to solve the equations
L4+XV-1=(@+FV-1) (p+rAV-D),
M+ YV-1=(G4+FV-1)(¢+pV-D,
N+ZV-1=(G@G+FV-1)(r+vV-1),
and this is analytically the same problem as in ordinary space finding the length and

direction-cosines of a line whose rectangular co-ordinates are given. Squaring and adding
the three equations, we have

(4+ FV—-1f = (L+XV—-1)?+ (M+ ¥V-1)4+(N4+ZV-1);

and this determines 7’, @ in a single way if we agree that the sign of J’ shall always

be positive. Moreover Xi+ Y¥j+Zk+LI+MJ+ NK can represent the sum of any number
of forces and couples.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 107

Therefore we have, as in ordinary geometry, the proposition that any number of
forces and couples are equivalent to a force along a certain line and a couple about that
line. We may call this line the Central Axis, X, Y, Z may be called the components
of the resultant force at the point O, and LZ, M, N the components of the resultant
couple. Then it is found from the above equations, that

XP + VY? 42? = F? (p? +0? 47°) + P(t p+ v’).
Now the method by which the expression for a line was obtained shews that, if @ be
the distance of line (pot, Awv) from the origin,

cosh*@ = p?+o°+7’, sinh* 6 =X? + p? +27.
Therefore X?+ Y?+ Z? = F’ cosh? 6 + G sinh? 6.

This shews, first, that X?+ Y°+2Z" is always greater than #”, and therefore the
resultant force is least at any point on the central axis; secondly, that X?+ Y?+ Z? for
given values of F and G depends only on 6, and therefore the resultant force is the same
on all points of cylinders described about the central axis.

The square of the resultant couple at O is
TI? + M*+ N* = F? sinh? 6 + G? cosh’ 6,
and the same theorems are true with respect to it.

Let a, 8 be two unit lines and let @ be their shortest distance, @ the angle the
planes passing through either and that shortest distance make with one another. Then
the operation of transferring a to PB is equivalent to turning a through an angle ¢ about
the shortest distance, and moving it through a distance @ along the shortest distance.

If & be a unit line along the shortest distance, D unit rotation about the shortest
distance, then we may put

Ba* = (cosh @ + 6 sinh @) (cos ¢ + D sin $)
= (cos 0,/—1+ Dsin 0,/—1) (cos + D sin ¢)
= cos (6+ 0./—1) + Dsin(¢+6/—1).
This is a biversor with angle $+0/—1.

If we identify
P+Ql+RJ+ SK +p J-1+qit+rj+sk
= (P+pJ/—-1)+(Qt+qJ/-DI+(R+rJ—-1)J+(S+s/-) K,
with n times the ratio of two lines, we must have
ncos (b—6/—1) = P+pJ—1,
and n° sin? (6 — 0, /—1) = (Q4+qJ—1) + (R+7J—1) + (Sts /—1)¥.

108 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Therefore n= (P+p/—1)+ (Q+¢qJ/—1)+ (R+rJ/—1)?+ (8 + s/—1)%,

giving v= P+ P+ R4+ VS -—p—g-r-s’,
Pp + Qq+ Rr + Ss = 0.

Only then, if this last condition holds, can the general expression be identified with
the ratio of two forces or of two couples or of a couple to a force. Otherwise it will
be the ratio of a force and a couple to a couple; that is, some multiple of the ratio of
a screw to a couple.

If 8, a be two rotations, we still have
Ba*=cos($+6,/—1) + Dsin(¢+6/—1);
and, as with real quaternions,
SaB = —cos(¢+6,/—1).
But if (p, o, 7, A, w, v) (p’, o, 7’, Vy w, v’) be the co-ordinates of the lines a, 8
S28 = — (p+ XJ =1)(p' +N J=1) - (+p =1) o +p J=1)- (7 +0N=1) (7 + =D).

Equating the real and imaginary parts in the two expressions Szf,

cosh 8 cos @ = pp +0’ + 77 — AN — py’ — wy’,

sinh 6 sin = pX top’ +7 +pX+ow+Tv.
More generally, if the system of forces (X, Y, Z, LZ, M, N) reduce to a force F and

a couple G, while (X’, Y’, 7, L', M’, N’) reduce to a force F” and a couple G’, and ¢, @
have the same meaning as before, and are referred to the central axes

(G@+FJ—1) (G+ F’ J/—1) cos ($+6/—1)
=(L4+X J/-1) (+ X' J-D4+(M+Y J-1) (M4 Y'/=1) 4+ (N42 J/-1)(W' +7’ f=).

Therefore (FF’ — GG’) cosh 6 cos ¢ — (F'G’ + F’G) sinh @ sin b
= XX'+ YY'’+ZZ'-— LL'- MM’ - NN’,
(FG' + F'G) cosh 6 cos $ + (FF’ — GG’) sinh @ sinh
= XLI'+ YM'+ ZN + LX'+ MY’ + NZ.
These are the simplest expressions for the two invariants of a system of forces. The

second is identical with that called by Prof. Ball, in his “Theory of Screws,” the virtual
coefficient of two screws.

If O be the origin and J, m, n the direction-cosines of the line OP, @ the length of
OP, then i+my+nk is the imaginary quantity corresponding to the line OP; we have

PO™ = cosh 6 + (li + mj + nk) sinh 6
=w+aityj+zk
we may say, where w—e—y—-2=1,

(w, ©, y, 2) may be taken for homogeneous co-ordinates of the point P.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 109
Also, OP™ = cosh 6 — (i+ mj + nk) sinh 0
=w—ai— yj — zk.
If (w’, 2’, y’, 2’) be the co-ordinates of any other point Q,
QO* = w+ eit yj 4+ak.

Therefore QPt=(w' + a+ y9+ 2k) (w — at — yj — zk)

= ww’ — xa’ — yy — 22

+ (wa — w'x)i + (wy! — w'y)) + (w2' — w'2) k

— (yz — yz) I (za — za) J — (ay — a'y) K.
But if (p, o, 7, A, », v) be the co-ordinates of the line PQ, w its length,

QP* = cosh wp + (pi+ oj + tk — AI —dXJ—vK) sinh yp.

Therefore cosh yr = ww’ — aa’ — yy’ — zz,
po oy Rg,
sinh yp sinh

= (yz — y'2)'— (ex! — £0) (ay — ay)
We may use these formulae to find the equation to the cylindroid.

Suppose (X, 0, 0, L, 0, 0) (0, Y, 0, 0, M, 0) to be two screws about axes at right
angles to each other, then the cylindroid is the locus of the axes of screws that are obtained
by adding different multiples of these two screws together. (X, 0, O, Z, O, O) will represent

any multiple of a screw about the axis of w, but the ratio = must be taken constant, and
we will put it equal to p, in accordance with Prof. Ball’s notation. Also, — Pp-

If F, G, be the force and couple of the resultant screw, (p, o, 7, X, “, v) its co-ordinates,
Xan =a 1 Ge Ely) al) (pte a) =e),
M+Y/-1=(@4+F /-1)(¢+pJ/-)),
0=(44+F/-1)(r+0/—)).

The last equation shews that r=0, v=0, or the axis meets the axis of Z at right
angles.

Multiplying the first by p—2,/—1,

(L+X J—1) (p—aJ/—1) = (4+ F J=1) (p? + 2’).

GNA a pep
Therefi poe seis
aan fF Xp-InN p—par
Similarly G@ _ Yu+Mo _pbt+ppo

F Yo—Mu o— Pop
Wor, ALE Parr IL. 15

110 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Now, if w, 2, y, 2, be any point on the axis of the screw, w’, 0, 0, 2 the point
where it meets the axis,

p=—w2e, =-wy, N92, w=—az.

P ; wm 2 5 : :
But t=w2'—w'z=0, so that [= 5, and as only the ratios of p, o, A, are required,
wm 2

we may write p= wz, o=Wwy, rA=—-YZ, fe = 22.

—YZ+pPwr «wZ+ pgwy
wet PayZ wy — Pavz-

Hence

That is
(Pa— Dp) (w* — 2") wy = (1 + papa) (a* + 9?) we.
This is the equation to the cylindroid and it is a surface of the fourth degree as

has already been shewn by Lindemann.

SPHERICAL GEOMETRY OF THREE DIMENSIONS.

In this geometry we shall have three imaginary quantities, 7, j, & represent transla-
tions along lines, such that ?=j?=/*=—1, and three quantities J, J, K representing
rotations about the lines, such that

P=f=K*=—), JK—i, Kid
Also i=Jk=-kJ, j=Ki=-ik, k=Tj=— I.
We write the equations thus
t=Jk=jK, I=JK=jk,
Ga he = hela
k=Tj=W, K=I1J =y.

It follows that i= Jj = Kk=i1 =jJ =kK=— o say,
and Ov
Hence ol =i, oJ] =), ok=k.

The general vector expression is
(X + Lw) [+ (¥+ Mo) J+(N+ Ko) K.

This is a different sort of bivector, and has been considered by Prof. Clifford in
the Proceedings of the London Mathematical Society and the American Journal of
Mathematics.

Results exactly similar to those of the last section can be obtained by using this

imaginary quantity #, which is commutative with J, J,.K in the place of /—1. It
must be noticed that a rotation about a line is always equivalent to a translation along

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. dela

another real line which may be called the conjugate line. In fact, the rotation J moves
all points in the plane 7, & along circles with O for centre. One of these circles, namely,

that of radius =

5» 1s a straight line, so that this straight line moves along itself, and the

distance of all points from it remains unchanged.

The equation to the cylindroid is
(Pa — Pa) (w* + 2") wy = (1 — pa pe) (x* +") we.

ORDINARY GEOMETRY OF THREE DIMENSIONS.
In this case v=pP=kh=0, P=JP=K*?=-1.
We may take a quantity such that w°=0, and put
ol=i, oJ =), oK=k.
As has been shewn already for two dimensions, all parallel translations will have to
be considered equal when they are of equal magnitude. A rotation about a line through

the origin whose direction-cosines are (J, m, n) will be expressed by 1J+mJ+nK. If
this be transferred to a point a, y, z along a perpendicular line we have the rotation

(1+2el+yoJ+zoK) (I+mJ+nKk)
=l1+mJ+nK +o {(yn — zm) I + (2l — an) J+ (al— ym) K}.
In general the quantity
(X + Lo) [+ (¥Y+ Mo) J+ (7+ No) K
will represent a rotation F’ and a translation G, or we may say a rotation of magnitude
F+ Gu.

Putting p=l, c=m, tT=n, XR=yn—zm, w=z2l—xn, v=al—ym where (I, m, n) are
the direction-cosines of the rotations and z, y, z a point on its line of application, we
may call p+2w, +o, t+vw the direction-cosines of the line considered in position
as well as magnitude.

Since the two equations

ptot+r=1, pAtonp+7v=0
are equivalent to the equation
(p +A)’ + (o + po)? + (7 + vo)? = 1.

To find then the magnitudes of # and @ and the lines about which they act, we
have to solve the equations

X + Lo =(F£ + Go) (p+ ro),
Y+ Mo=(F + Go) (c+ po),
Z+No=(F+ Go) (t+ v0).

=
o1
bo

112 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

We see then that in all three kinds of geometry the properties of lines in space
can be derived from those of lines meeting at a point if we write X+JZ, &c., for
the co-ordinates of the extremity of the line, p+ 2, &c., for its direction-cosines, F+ Go
for its length. where in the first kind of geometry #*=—1, in the second w?=0, and
in the third #*=1.

In spherical geometry if (p, o, tT, A, mw, v) are the co-ordinates of a unit force along
a line, (A, w, v, p, o, T) will be the co-ordinates of a unit couple about that line. But
it is easy to see that these are the co-ordinates of a line conjugate to the original line
with respect to the sphere at infinity w°+a*+y?+2=0.

Hence decomposing a system of forces into a force along a line and a couple about
that line is a particular case of decomposing the system into forces along conjugate lines
with respect to a given surface of the second degree. This is also true in imaginary
geometry and ordinary geometry, though in the one case the conjugate line becomes
imaginary and in the other passes off to infinity. Now in all three kinds of geometry
the equation to any surface of the second degree may be written

w+ a? +y?+ 2?=0.
We shall have then the following theorems:

Any system of forces may be decomposed into forces along conjugate lines with respect
to a given surface of the second degree.

If two systems of forces be compounded in different proportions, and the resultant
systems be decomposed into pairs of conjugate lines, the locus of these conjugate lines
is a surface of the fourth degree.

When the given surface degenerates into the imaginary circle at infinity then the
surface of the fourth degree degenerates into a surface of the third degree, the ordinary
cylindroid.

SPACES OF HIGHER DIMENSIONS.

The units of an imaginary space of three dimensions were derived from three inde-
pendent units 7, j, k.
They could be considered as the commutative product of two systems
1B Aah ip
1, Uke;
which may be written more shortly
We Ba dee
Tee ie

where 1, 7, j, K form an imaginary plane system and v* =-— 1,

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 113

The units of the imaginary space of four dimensions are got by multiplying these
units by a new unit J or instead by w= yl.

pw’ =—1 and p is commutative with 7, 7 and therefore with k but not with v. In fact
pv = ylyk=lyyk = —lk=kl=—vp.

Hence v, », vw form a quaternion system.

The system of four dimensions is therefore the commutative product of the two systems
1, %, j, &,
I, », B, vp,

the one an imaginary system of two dimensions, and the other a spherical system of two
dimensions.

The system of five dimensions 1s obtained by multiplying this by a new unit m
or else by w =aklm,
w* = yklmijklm = ijklmmijhl = ijklijkl = — klk = — ijkijk =1,

w is commutative with each of the units 7, 7, k, J and therefore with their products.

Hence the system of five dimensions is the commutative product of
ROR a
1, v, », vp,
1, w.
The system of six dimensions is obtained by multiplying the system of five by a
new unit » or else by m=ijkln.

m™=1 and w is commutative with each of the units formed out of 7, 7, k, l but not
with w. In fact
wr =Uyklmigkin = myklijkin = mn = — nm =— Tw,

so that (wr)? =—wrrw =—-1.
Hence w, 7, wma form an imaginary plane system, and the system of six dimensions
is the commutative product of the three systems
ile, Jonles
1, v, mw, vp,
1, w, 7, wr,

or of two imaginary systems of two dimensions and one spherical system of two dimensions.

The system of seven dimensions is formed by multiplying this by a new unit o or by
vy’ = 1klmno,
¥ = yklmnoijklmno = jklmnorjklmno = jklmnoklmno = — klmnojklmno = — 1,

and v’ is commutative with all the former units 7, j, &, l, m, n and their products.

114 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Hence the system of seven dimensions is the product of the systems
Te ae a
I) v, p, vp,
1, w, 7, wr,
ible
If p be another unit and y’=ijklmnp, the system of eight dimensions will be the
product of 174,95,
1, v, Pp, Ups
1, w. 7, wr,
US 1, Date
that is of two imaginary systems and two quaternion systems.

We arrive then at the following results:

The system of 4m dimensions is the product of m spherical systems and m imaginary
system of two dimensions.

The system of 4m+1 dimensions is the product of the system of 4m dimensions by
the system 1, w, where w*=1. °

The system of 4m+2 dimensions is the product of m spherical systems and m+1
imaginary systems.

The system of 4m+38 dimensions is the product of the system of 4m+2 dimensions
by the system 1, »v where v= —1.

The only difference in the spherical systems is that the squares of the fundamental
units are —1 instead of 1. It is easily seen that the following laws will hold:

The system of 4m dimensions is the product m spherical and m imaginary systems.

The system of 4m+1 dimensions is the product of the system of 4m dimensions by
1, v where Y=—1.

The system of 4m+2 dimensions is the product of m+1 spherical systems and m
imaginary systems.

The system of 4m+3 dimensions is the product of the system of 4m +2 dimensions
by 1, » where wo’ =1.

The most general quantity in a system of the n™ degree will contain 2” terms.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 115

GRASSMANN’S AUSDEHNUNGSLEHRE. THE OUTER MULTIPLICATION.

We return to the point we left off at in the first section of the former part, and
put aside for the time all considerations respecting distance. We have to determine a
multiplication which will be independent of these considerations. If we assume that the
square of a point is always the same, we have

(pA + qB)’ = A’ = B? for all values of p and q.

Hence (p+ q@—1)4°+ pq (A. B+ B.A) =0.

This can only be the case if A*=0, B’=0, and A.B+B.A=0.

That this multiplication is the only one which can include all three laws of distance
is seen by recalling what were shewn in the former part to be the most general laws of
uniform multiplication.

They were in the three cases

A.B+B.A = 2 cosh 6. A’,
A.B+B.A= 26°,
A. B+-B.A= 2 cos 6..A’,

and these can only be collected in one by putting
A.B+B.A=0, A’ = 0.

This multiplication, called by Grassmann “the outer multiplication,” is therefore the
proper one for treating descriptive theorems, as it involves no ideas of distance*.

If there be three points A, B, C, we must put

A?= B= C0? =0, BC =- CB, CA =—-AC, AB=—- BA.
Assuming the associative principle to determine the products of three factors, we have
ABC =— ACB=- BAC = BCA=-— CBA = CAB,

or the product is the same when the cyclical order is unchanged.

The P=cA+yB+20, Q=av7A+yYB4 720, R=a"At+y’B+7C
be any three other points in the plane of A, B, C, then, by the distributive law of multiplication,

P=@7=f7=0, QR=-RQ RP=—-PR, PQ=—-QQP,

and IKMi == WARS on =|, 2
zy, 2 ABC.
Ge MW a’
If P lie on the line QR, then P=AQ+ypR,
and therefore PQR =r’QV?R — nw PR’Q = 0.

* Die Ausdehnungslehre, Hd. of 1862, pp. 6—30. ‘‘Die verschiedenen Arten des Produktbildung.”’

116 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Hence the equation to the line QR is | 2, y, 2
i 8 5 | =\05
Gi ys 2”

The product of any two points may be called a line, and every line in the plane may
be expressed as the sum of three lines, BC, CA, AB.

For QR = (y'2" — yz) BC + (¢2" — 22) CA + (2’y” — yx”) AB.
Conversely, every expression of the form 1BC+mCA+nAB may be considered a line,

for we may write ; (LB- mA) (lC—nA).

By multiplying two new points on the same line we only obtain a different multiple
of the line, since
(\Q+pR) (VQ + wR) = Ap’ —Ny) QR.
The sum of any number of lines in a plane is itself a line in the plane, for it must
always be of the form IBC +mBA+nAB.

If there be four independent points, all the points derived from them form a space of
three dimensions, any point of which is represented by P= 2A+yB+zC+wD.

If four points P, Q, R, S be in the same plane, PQRS=0, and hence the equation

to the plane is
2. ¥, B

Zs Yp 2% V,
Z, Yor 2a, We
Zs, Ys» Zs, Ww,

The product of any three points may be called a plane, since only the degree of
multiplicity is altered by taking different points in the same plane. The general expression

for a plane will be
IBCD +mCDA+nDAB+rABC.

Conversely, every expression of this form is the product of three points; for

IBCD+mCDA = CD(\B+mA), DAB =" DA(IB+ mA),

singe A=, [BCD +mCDA + nDAB = D (nA —IC) (1B + mA),
rABC = - A (nA —1C) (1B+ mB).

Hence IBCD+mCDA+nDAB+rABC= (ID—7rA)(nA—1C)(lB+ mA).

It follows that the sum of any number of planes in a four point space is itself
a plane.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 117

Any line or product of two points in a four-point space may be represented by
F=XAB+ YAC+ZAD+LCD+ MDB+ NBC,

where X, Y, Z, Z, M, N are numbers. But such an expression need not represent a line;
for it involves (disregarding multiples of the same expression) five constants, and a straight
line only involves four.

The condition that it should be a line is found by putting F?=0 to be

LX+MY+NZ=0.

In general, let ¢,¢,¢,¢,-..e€, be m unconnected points, then the expression for an
(n —1)-point space is
Pp, 6,0, +++ G, + PaO, 0, --+ &, + D0, 0,8, .2- EF oo;

and, conversely, such an expression will always be the product of (n—1) points. For

Pyeq +++ On + Poles sjee) On (DP, + p,e,) 504 <-- Cn

— Ps
| OK RR a (9,0, + Poe,) Cn€ -+» Ens

1
[Pils o> €;, A- D301 €n,0-0 0, 1 DiC,€, Cp 00. C= a (p,€, + Pye,) (Poly + Pals) Cy +++ Ens
2
and so term after term may be joined together.
The general expression of the form

a,,_ 2, @ oven é.. = Dy 41 Oo Pg see Cas Tees

2—-1)...n—r+1 : F
( : 5 ie ;: nak nt 1 constants, and a r-point space in

; 5 : ON maa
involves (disregarding multiplicity)
an n-point space requires r(n—r) constants to determine it. Therefore

n(n—1)...(n—r+1)
ieee Te

—r(n—r)—1

conditions are necessary that the two may coincide. For instance, in a 5-point space, if
F = 4,,€,0, + A,,€,05 + ++
é : é : 5.4 a an
is equivalent to a single line, we must have —~—2x8-—1=8 conditions satisfied. In

1.2
fact, putting #*=0 we obtain the five equations,

Any Ugy + yy Ugg + Ay, A, = O,

A544, str As5Ay4 ate Ay, A; = 0,

(a5 aia 4.)

U5A 0 =F gy As, a OyMey = 0,

D5, Ang T Aga Mg + Ugg Ap = 0,

Uy Ae4 a A139 st: D4 Ag = 0,

of which it is easily seen only three are independent.

Vou. XIII. Parr IT. 16

118 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

APPLICATIONS TO SYSTEMS OF Forces AND LINEAR COMPLEXES.

Although the general expression for the sum of any number of lines in space of three
dimensions |
F= Xe,e,+ Ye,e, + Ze,e, + Le,e, + Me,e, + Ne,e,
cannot be reduced to a line, it can be reduced to the sum of two lines in an infinite
number of ways.

For if 1 be any line and X a number the equation
(F—2l)?=0, or F?—AFI=0, since ?=0,

always gives a value of for which F’—Al is a straight line.
If wm be this straight line (where w is a number) then
F' =I + pm.

Thus we can in general take any straight line and find another corresponding to it,
such that # will be the sum of the two. The exception is when Fl=0, for then the
equation gives no finite value of >. All the lines satisfying the equation /l=0 form
‘what is called by Pliicker a linear complex of the first degree, and it is easy to see that
this equation is equivalent to the most general linear equation between the co-ordinates of
a straight line.

If #*=0 and F be consequently a straight line, the equation Fl = 0 represents all the
lines meeting this line; for the product of two intersecting lines is 0, since it is equal
to the product of four points in the same plane.

Decomposing /” into two lines so that F =X’ + pm’, the equation FI will be satisfied
if 17=0, ml=0; and, conversely, if FI=0 and JJ=0, then also mJ=0, so that the
complex consists of all the lines that can be drawn intersecting any pair of corresponding
lines. It follows that the lines of the complex which can be drawn through any point lie
in a plane, and this is seen directly by putting 1=ay where x, y are two points, then
keeping a fixed the equation Fxry=0 is satisfied by all the points y which lie on the
plane Fr. To every point there exists therefore a polar plane, and the polar planes of all
the points lying on a straight line pass through another straight line, since

F(A,@,+,2,) = 0,Fo,+2,Fa,,
and A,v,+A,x, is any point on the line joining a,, x,, while \¥z,+2,Fx, is a plane pass-
ing through the planes Fz,, Fz,.

If Ff, F,, Ff, F,, F,, F, be any six given sums of lines not connected by any
linear relation, then the quantities ¢,¢,, e,e,... may be expressed in terms of I, F,, &e.;
and therefore any system of line /’ may be expressed in the form

F= VF AF, + AF, +P, +A P+ AF,

6?

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, pS

and this only: in one way. All the quantities of F may then be said to form a 6-point
space or space of five dimensions. If the co-ordinates of F' satisfy any linear relation, then
F may be derived from five independent line-systems satisfying that relation; just as all
the points in a linear n-point space satisfying an equation of the first degree belong to a
(n—1)-point space. The most general relation of the first degree between the coefficients
of # may be expressed by the equation #”#’=0 where f” is another line-system.

If CK ee Opi oe 9 ee Sere f= X ee, +...+L'e,6,+...,
then FF =F F=XU'+YM+2ZN' + LX'+MY'+ NZ,
since Cyl, Cp€g = Cyl x01» 0,60, = 0.

When F’F=0 either line-system may be said to be reciprocal to the other or the
line-systems may be said to be co-reciprocal.

Thus all the line-systems reciprocal to a. given line-system can be derived from five
systems reciprocal to the given system.

Similarly, all the systems reciprocal to two, three, or four systems can be derived
from four, three or two reciprocal systems, and there always exists one system of lines
reciprocal to five given systems. Hence we may choose Ff, F,... F,, so that each is
reciprocal to the rest. The lines of a complex #1=0 belong to the screws (we may call
the quantities /’ screws for shortness) reciprocal to F.

The lines common to two complexes F,J=0, F,J=0 are said to form a congruence.
If F,, F, are real, the congruence will always contain real lines. For, taking any real line
l not belonging to it, #, may be put equal to Al+ pl and F, to Xl+ypl’, so that a line
intersecting the line /, 1’, /’ will belong to the congruence, But an infinite number of real

lines can be drawn to intersect three real lines.

From this may be proved that if F,, F,.../, be co-reciprocal, three of the quantities
F7F?... FZ must be positive and three negative. For the lines of the congruence #/ = 0,
F,1=0 can be expressed in the form

NeHyt MEA Nels + MaMa
since they are a particular case of screws conjugate to two given screws.

Squaring this last expression, and remembering that F,F',=0, &c.,
De +22R? +A7F? de Nie = 0.

But this last equation can only be satisfied by real values of X,, &c. if one at least of
the quantities 77, Ff, &c. is different in sign from the others, and so with any other four.
In the language of mechanics, If six screws be co-reciprocal three must be right-handed
and three left-handed. It is understood all along, of course, that the word “real” is used in
an algebraical sense, and means “expressed by co-ordinate not involving the algebraic /—1.

16—2

120 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

The lines belonging to two complexes F,J=0, F,1=0 belong also to any complex of
the system (A,F,+A,F,)1=0.
Now VF +1,F, is a straight line when
(AF, +4,F,)?=0, or 027+ 2105 F,

Sonal a tA, ly = 0.
Xz
ie?

1

This equation gives in general two real or imaginary values of and thus in general

& congruence consists of all the lines intersecting two given lines.

The lines belonging to three complexes F,J=0, F,J=0, F,J=0 are singly infinite
in number. They consist in fact of all the lines that can be made by combining three
screws of the reciprocal system F,, F,, F. Now if A,7,+2,F,+A,/, be a straight line

Ve! og ar a We i a6 AE =F 20,0, LF, ar 20, FF, ap 2n,\, 2, F, =0 ;

and this equation leaves one of the ratios me, we arbitrary.
4

All these lines intersect the lines contained in the expression A,F,+),/,+A,%.

These latter lines therefore lie on the surface formed by the former lines, and are a
second set of generators.

The lines belonging to three complexes, although not the same, generate the same
surface as the lines belonging to three reciprocal complexes. Through any point on the
surface can be drawn one generator of each set. In fact the lines that can be drawn
through @ are the intersection of the planes F.w=0, F,x=0, F,e=0, and these must
intersect in a straight line if z lie on the surface.

There are two lines common to four complexes F,J=0, F,1=0, F,/=0, F,l=0.
They are in fact the two lines included in the expression A,F,+A),/,, where F,, F, are
reciprocal screws to F\F,F,F,.

“fag ae : : s
The ratio = is obtained from the equation A77?+2\1,4,F,+A2F7=0. In particular
6
if F, be a straight line, it is seen that there are two lines on the surface consisting

of the lines common to FJ=0, F,J=0, F,J=0 which also intersect a given line; or in
other words that any straight line cuts that surface in two points and it is therefore a
surface of the second degree. If J=X,F,+2,F, gives only one line, that is to say, if
the equation A F7+2),7F,+\7F?=0 has equal roots, we must have

rv, F, " +A,FF, = 0, V+ =0,
or F.J=0, FJ=0. Hence J can be derived from the screws F,F,F,F,, and we may put

VF +0, EF +0,F, + 0,7, = 1.
Substituting in the equations
Fl=0, Fl=0, F=0, FJ =0

we have VF + AF PF, +A, 2 F, + 0, FF, =0,

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 121

with three others. Hence
eee aie, LEE,

teeny, i582.
ied ua JT Spi ie

2° 8? 2

HOR, F3, FF,

2 3? 8?

Pi. PREP. Fe

2 4? gs 4?

is the condition that four complexes should have only one line in common. In particular

the condition that the four lines J/,, /,, J,, 7,, should only be intersected by one line, reduces to

Stl, Sil, £ SUL, JUL, + tt, Jil; = 0.

If the five complexes FI=0, F=0, FJ=0, FJ =0, #J=0 have a line in common,
then since / is reciprocal to itself, it must belong to the screws derived from the five

Hence VF +ALL +A th, +A, 2, =1,
and since £1 =0 NV +A LE, +0, +N EF, +,F Ff, = 0,
with four others. Eliminating 2,A,\,\,\.,
LE: Le eleegel! Wis Maio eh Bie |
Eee ee, Hel WWE, |

TB HHS Gab es Hs, AE

1 3) 2 3?

Hn ron hens Et, eT
EF, del its Weve LF, Be

Lastly the condition that there should be a linear relation among the screws

EFF EF FF, is found by putting
VF +A, +A FP, +L, +2, 2, + VF, = 9,

and multiplying by #, &e. to be
Boa Me 081 ea Od DM et Deora Db! genead J) J

ya Pe Fy; EE EE, Els PE,
Le Lael jee IS vase PEs IIE

Bia h Meet, Hite. ak Wa FR

p Ney i 2 4) 3” 4? 4 5? 6

FLT Ip TET Le 0 ea 1 Re a OP

Tt 5? 2° 5? 3 5) 5?

AUAEERIT DY A GN 0 OPT OY en pe

1 6? 2 6? 3 6? 4 6? 5 6?

If F?, F.F,, &e. be interpreted in ordinary geometry these are identical with the
equations Prof. Ball has given in his Theory of Screws, in generalisation of those of

Profs. Cayley and Sylvester*.

* Sturm ‘‘Sulle Forze in Equilibrie” in the Annali di Matematica.

122 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

THE REGRESSIVE MULTIPLICATION.

If there be only three units e,, e,, e,, or in other words if only points in a plane
are being considered, we may put ee,e,=1, since no higher products can occur. With
this supposition the product of any three points will be a number. It may be noticed
that there is a reciprocity between the line and the points, in so far that just as the
sum of any two points is a point on the same line, so the sum of any two lines is a
line through the same point. Again any point can be expressed in the form
r=2,e,+2,e,+2,¢,, and so if we put L,=e,¢,, H,=e,,, E,=e,¢,,
any line can be expressed in the form

X =X,E, + X,H,+X,E,.

We may carry this reciprocity further, and introduce a multiplication of line exactly
corresponding to the multiplication of points.

We may put #,24,=e,=—E,E,, EE, =¢,=—E,E,, E.E,=e,=—-L£,E,,

EL? =0, H2=0, H3=0.

It must be observed that the associative principle cannot hold when the two kinds
of multiplication are combined.

For ee, . €¢,=e, and e,e,"e,=0, since e,7=0.

The product of two lines X= X,H,4+X,H,+X,H,, Y= Y,E,+ Y,E,+ Y,£, is defined
by XY = (X,£,+ X,E,+ X,E,) (Y,£,+ Y,£,+ Y,£,)

ms (x, ie oa x, Y,) LE, oz (X,Y, an x, Y,) EE, an (X,Y, = x, Y)) ELE,
and this definition involves the distributive principle,

or X(V¥+Z)=XV+XZ.

Now the equation —#,H,=e, may be written ee, . ee,=e,=(e,¢,¢,)e,, and we will
shew that if z, y, z be any poiuts
zy . a2 = (xyz) a,
or that just as the product of any two points is some multiple of the line joining
them, so the product of any two lines is some multiple of their point of intersection.

For if L=LE,+LC,+L Cy Y= Ye, + Yolo + Yglgs 2 = 2,0, +20 + Z C5;
y2= XE. + XE, +X,E,, za= YE,+ YE,+ Y,E,, cy=Z E+ ZE,+ ZE,,
then X,, Y,, &c. are the minors of
Tas Yaa
on PALI

Ls, Yar 7,

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE 123
and 22. LY = (Y,4Z, ig Y,Z) e+ (Y,Z, my Y,Z,) @, + (Y,Z, a YZ) e,
@,, Yp> Z,
= Zz, Yo» 2, (,e, a7 Ley a C5)
Ts Ya 25
= (xz) x

Just as the multiplication of not more than three points is associative, so the
multiplication of not more than three lines is associative.

For ya (Ze. ry)=ayz . yer =(axyz)’,
since xyz is a number
(yz . zz) ay=(ayz . 2) cy=(ayzy:
The multiplication of points to make lines is called by Grassmann progressive; the

multiplication of lines to make points, regressive. These results may be applied to find
the equation of any locus generated by linear constructions.

For example, the three sides of a triangle pass through fixed points and two of
the angles lie on fixed straight lines; to find the locus of the third angle. If 2 be the
point whose locus is required, a, b, ¢ the fixed points, A, B the fixed straight lines,
then awa is the line joining 2 and a, and therefore one of the sides of the triangle,
aaA is the point where za intersects A and one of the angles, 2aAb another side,
za AbB the next angle, zaAbBc the third side, and as this must pass through 2,

caAbBex =0,
and this is therefore the equation to the locus, since the product of three points in a
straight line must vanish.
This curve is cut by a straight line in two points since if Aw+py be a point on
the line joining «, y and also a point on the locus
(vw + wy) aAbBe (Ax + py) = 0,
or {ra AbBex} + rp {xa AbBey + ya AbBex} + p’{yaAbBey} = 0.

The coefficients of X7, Aw, mw? are numbers, and this equation gives two values of “
if, fe q 5

determining the points of intersection. It is clear that in general the number of times «
appears in an equation will represent the number of times the curve can be cut by a
straight line, and therefore its degree in the usual sense. The equation in y,

caAbbey + yaAbBeu= 0,

will represent the tangent to the curve in a, and from the symmetry of this equation
in # and y follows the theory of poles and polars*.

* This example and the proof of Pascal’s theorem are 1864, p.226. The generation of a cubic is given in Crelle’s
given in the Ausdehnungslehre of 1862, p. 195, or in that of Journal.

124 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

The condition that the opposite sides of the hexagon formed by a, a, b, ¢, d, e
should intersect in points lying on a straight line is
(wa . cd) (ab . de) (bc . ex) =0,
and the equation shews that the locus of 2 is a curve of the second degree.
Moreover the curve passes through the points a, b, c, d, e.
For it is obvious that it passes through @ and e since a*=0, e?=0, and using the sign
= to mean, is congruent to, is a multiple of
(ba . cd) (ab . de)=ab, (be . eb)=b,
and ab. b= 0,
ca. cd=c, bc. ec=c,
c(ab . de)c=0,
da .cd=d, (ab.de) (bc. ed) = ed,
d . ed=0.
Therefore it passes through 8, c, d also.
As a curve of the second degree is determined by five points, this is the most
general form of its equation and Pascal’s theorem is thus proved. Again, from a

variable point, lines are drawn through fixed points to meet fixed straight lines and the
points of intersection lie on a straight line. Find the locus of the variable point.

If a, b, ¢ be the fixed points, A, B, C the fixed straight lines, its locus is
(aa . A) (ab . B) (ac . C)=0,
and is therefore a curve of the third degree.

It obviously passes through the points a, b, c. It also passes through BC, CA, AB;
since BCLB= BC, BCcC= BC, and lastly through the points be . A, ca. B,ab.c.

For if x be any point on the line be, 2b =be, xc=be,
(ab . B)(ac . C)= (be . B) (be . C)h=be,
and aa.A. be=0.
za, A, be must pass through a point or x must be on the intersection of A and
be. The cubic therefore passes through nine points, but these nine points are not

arbitrarily situated. However as a, b, c, A, B, C involve twelve constants a cubic can
be generated by this construction in an infinite number of ways.

All these results are included by Grassmann in the following general theorem:

The locus of any point determined by linear constructions leading to the condition
that three points should lie on a straight line or three lines pass through a point,
is an algebraical curve whose degreé can be found by mere counting. The degree is
equal to the number of times the variable point is introduced in the construction.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 125

If a, Aa+pb, 6, Na+w'b are four points on a straight line their anharmonic ratio
x a : : :
was defined to be are Similarly the anharmonic ratio of four lines A, AA + pB,

B, NA+HB is defined to be # -

If o be any other point the lines joining it to a, rAa+pb, b, Nat+pb are
oa, oa+pob, ob, Noa+p'ob, and therefore the anharmonic ratio of any range of four
points is the same as that of the pencil of lines joining these points to any given point.

If we put c=Aa+pyb, d=Na+yb, then the anharmonic ratio of the points a, c, b, d

tage a where it must be remembered that = = are numbers, and that ac, db are

cb * ad
not to be multiplied together. If a, b, c, d be any four points in a plane, we may
put a+b+c+d=0, since the proper multiples can be included in the symbols for the
points.
Let ab, cd intersect in e, ac, dd in f, and ad, be in g, then since
(a +c) (6+ d) =— (a+b)? =0,
ab +cd =—(ad+bc)=some multiple of line eg since it passes through e and g;
again (a+ d) (b+c¢)=0,
ab — cd =— (ac + db) =some multiple of ef.
Hence since the lines ea, ef, eb, eg are multiples of ab, ab—cd, cd, ab+cd, they form
an harmonic pencil.
The anharmonic ratio of the four lines A, B, C, D will be
AB DO
BEL AD
but if these be the lines joining z to four points a, 0, c, d
LAB 20 co (cab) ere —\ (0). ace) =| (0c) ac

azab adc

and the ratio becomes —— .—.
abe ° xad

If this be constant
(xab) (adc) = k (xbe) (xac),
a curve of the second degree passing through a, 8, ¢, d.

This proof is really identical with that given in the first section of the former part
but Grassmann’s notation enables it to be written more concisely.

Tt must be noticed that though the product ab corresponds to Va in Quaternions,
the product abe corresponds to SaBy.

Wore Lil, Parr De 17

126 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

The theorem corresponding to ab . ac= (abc) a is
V . VaBVay=—aSaBy,
but in Quaternions «, 8, y could, if taken to be points, only be points on a sphere.
In Grassmann’s system if a, b, ¢ were points on an ordinary plane, abe is double the
area of the triangle they enclose. The Quaternion expression is VBy+ Vya+ Va, so
that there is no correspondence except for points on a sphere. The proof of Pascal's
theorem, given in Tait’s Quaternions, applies directly to spherical conics, or the cones
joining them to the centre of the sphere, and thus indirectly to plane conics. If p, 2, B,

y, §, €, be. vectors SVVpaVy5, VVaBVse, VVByVep=0
is the equation the expression of that theorem leads to, and it is identical with
(wa . cd) (ab . de) (be . ex) =0.

Grassmann’s method however proves the theorem, independently of all metrical assumptions.

We will consider now the regressive multiplication generally in an n-point space.
So long as the number of points multiplied together is not greater than m and so long
as the points are comprised in a space of lower dimensions, the laws of the progressive
multiplication will hold. Thus the product of three points in a straight line, or of four
points in a plane will vanish if the space be of higher dimensions than a three-point
space. It is seen then, that though when we are considering only points in a plane,
the product of two lines is their point of intersection, yet when we are considering points
in space the product of two intersecting lines vanishes. This can give rise to no confusion
any more than the fact that the reciprocal of a point is different in the two cases.

If eg,...e, be nm points then since no product can have a higher term than

n

€,€,..-€, we may put e¢,...eé,=1.

If follows that the product of any other n points unconnected by a linear relation
is some number different from 0.

If E, F, @ be different products formed from e,, @,,...e,, Such that HFG contains
all the points ¢,e,...¢, without repetition and is therefore +1, we will say that

EF . EG=(EFG)E.

For example €.e, - 66,...€. = (€6,.... €,) €, =6

CB vcs 6.6.5, or0l8,*s O05 0 €,8, e —(Gie, °. 16.) 6:65.

r-st1 °°°

This assumption, with the distributive principle, will form the definition of the
regressive multiplication,

*We will now prove that if Z’, F’, G’ be products of any points such that E’F’G@’
is a number E'F’. E'@=(£F'G)E£.

For this purpose let H=e,e,... €,;
the points ¢,, ¢,...€, Into d,, @,...d,.

n
.

* This proof is given in nearly the same form in Ausdehnungslehre of 1862, p. 68,

PSO 008, F=E,,, os,

n?

and change successively

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 127

Let a,=a,e,+27,¢,+...@,e, then

nn

Ua
YEON cot Qa Cod DCD. (Dina 2.000 a SA CUO COCO PEA Cea An

',

since Clan CHO CLO C10) OCC,

la
Pi = "oe EAB A), 6.6) 200.6, 200 4 D0, 2. Cny

my 1 2) * 8

'
JOG td CP HL BC ae CCR Gb CE ERO EOC nen CMO y Ey cone

r+. -r+i

r-sti n
Now EF . EG=(EFG) FE
LM & Qukoos OG 1,600 C= AO 600 OOinr, c0aG, 6 Gt Gycoo GGrmn.cc0 Os

=~ 6.4% CEs, 49 é, e410, CO 41 ci
= = (Fn(Fs 000 CAG. 000 @.)) Org B00 G
= (CP, 00 @.)) rcGyo00 Gn

Similarly PTS 0050 0.0, Shona CL —1(C.O.aciG, CL, ashe

Also CxO sce Gs 6 CHE SO Proce@, & GCs ond are coal

Co41%0 Cray 220 Os » Cn41% €,€ 10.40 en

== (G2) 400 OBE 000 GO) Cx sOy060

== (Channa) Cae aorta

and @,,,€,+--€, + €4,€, +++ Cr€,4; +++ &, = 0, since one of the points e is wanting.

Hence collecting all the terms

E'F . E’'G=z, (ee, ... €,) {2 B, + Le Orailn <0» Op Hove FL Cnly -+» 6} = (EEG) BE’,

‘T+172 °°

since L,C,0y +» Cn = (L,E, + 0,0, + ..+) Ay +++ C, = A,0, «+. 0, = HFG.

n

Likewise a point in F for instance e,,, may be changed into q,,,.

1

If Op = EO G6 wae ny Orgy ome
then LEW 1B Or ani O2 (iC 30) oy, rans Lplg) Crag >> Cy
te Dil =f eC Cuan 6.4 Caiy on stab ane t- PaCsGn nee C0. Cann oon Cy,
but C0, «++ Clnas Cry v0» Oy © C:0n +++ C,Cy4 oe G, =O, since e@,,, 18 wanting.
Therefore EF". EG =«,,,EF. EG =1,,,(EFG) E =(EF'@) E;

and the same reasoning will apply to a point in G.

Thus point by point all the points in ZH, &, G may be changed, and we shall have
generally E’P". E’G' =(E’F'G’) E’, or in words, the product of two spaces is the space
common to both, provided always the two spaces cannot be included in any space lower
than the n-point space.

Let JE SG ooo G- EE, = — 0,0, +++ Cn» b= OOC roa. p= ar OG ccc
where the signs of /,, H,... are taken so that
ef, =1, ef, =1, e,H, =1, &e.
(2

128 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Then Ee B= 0r6, «++ 6,0, «GG, +0 C6, = (6,6, «22 C16: 0:)16,6) 001)
=1(6 16, wos @,)1G:6, oC, = 6.6) a0 6,

n?

EE, . Er, = (0:64 +++ Gq) ©,0c0y 202 On = E Gy 00» 00g sy woe C60. = + (6, «+» €,650,6) G, «0+ @,

and EE, = — €,6,0, --» &, - C68, -+» &

=— (6,6, ..- 66,6) 6,6, «++ 6,

= (6,2,0,6, << ©.) 6.6, ws 6, = 6,6, 0» 6,
E,. BE, = 6,0,0,.0++ 6,0 0,6, 10 Op + &, ose

145 (Qi con OOO dna (GAG 005 G4) oon

Sen ey EB

Thus the multiplication of the quantities E,, Z,, EZ, is associative. Proceeding in the same
way EE, ... E,=e Os

r41°°° “n

We can arrange the quantities ee... so that ¢,¢e,...=1, where e,e, are any two of
them; then in the same way
L_E,=e,...= product of all other units so arranged that ¢e,.#,.H,=1
And for any three of the quantities LEZ...
E,. E, E,= E,E,. E,= product of all the units not containing ¢,¢e,, so arranged that
¢,¢,¢,. HEE, =1.

It follows that any product of three terms is associative, so long as the multipli-
cations involved are either all progressive or all regressive. This was assumed originally
for the progressive multiplication. For the regressive let A, B, C be the products of
n—r, n—s, n—t points, then they can be expressed as the product of 7, s, ¢ quantities Z.
None of the quantities H can be equal for they correspond to all the e’s which are
absent from A, B, C and if the same e were absent from two of these quantities, either
AB or ABC would be enclosed in a space of lower than n dimensions and would
vanish. Again AB consists of the product of the points which occur in A and B,
therefore the degree of AB is n—r+n—s—n=n—r—s. Similarly the degree of ABC
is n—r—s—t, and hence r+s+¢ is less than n. Therefore there are less than n quan-
tities H and their product is therefore associative. A product in general including both
progressive and regressive multiplications will not be associative.

Grassmann writes the quantity # in the form |e, and calls it the complement (Ergiin-
zung) of e,. In general if A be any product of the units e,, e,...,|A is the product of
the remainining units so arranged that A| 4=1.

If B=| A, then AG=1 and therefore BA=+1, and | B= +A.

BA can only be —1 when B and A both contain an odd number of points, and
therefore n is even.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 129

For example in a two-point space, if ee,=1, ee,=—1, e,=|e e=—le,

If A=ege,...e, then it has been shewn that |4=ZE,... EL, let also
B=e |e s8 Sey eae Jp

&

then [4|B=2Z,... HE.,,... E.=|e¢,...¢,=| AB.

This is for the case when the product AB is progressive and does not include all
the factors, but the same result is shewn by Grassmann to be true in the other cases.
(Ausdehnungslehre, p. 64.)

We may apply this general theory to multiplication in space of three dimensions or
four-point space. We will write down for comparison the definitions of progressive
multiplication along with the definitions of regressive multiplication.

The product of two points is the line joining them.

The product of two planes is their line of intersection.

The product of three points is the plane containing them.

The product of three planes is their point of intersection.

The product of four points is a number.

The product of four planes is a number.

The product of a line and point is a plane.

The product of a line and plane is a point.

The product of three points on the same straight line is zero.

The product of three planes passing through the same straight line is zero.
The product of four points on the same plane is zero,

The product of four planes passing through the same point is zero.
The product of two intersecting straight lines is zero.

These results may be used as in the case of plane multiplication for proving
descriptive theorems.

Thus the equation to the surface generated by a line which meets three given lines

A, B, Cis cABCz=0.

For «A is the plane containing x and A, #AB the point where it meets B, so
that the line joining z to AB meets both A and B: and eABCr=0 expresses that
zAB, C, « lie in the same plane so that the line joining 2 to zAB meets C. Hence z
lies on a line meeting A, B, and C. It might seem at first as if we could write the
equation to the surface in the form tA.a2B.x2C0=0, as this expresses that the three
planes have a line in common, and that therefore the surface must be of the third
degree.

130 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

But 2wd.2B.aC is always some multiple of the point a=mzx say, where m is a

number of the second degree in @.

Therefore the equation #d.2B.x2C=0 reduces to m=0 again an equation of the
second degree in a. In fact in every equation employed the left-hand side must, if @
were unrestricted, be a number. If it were a point or a plane, equating it to 0, would
give four equations and in general only determine special points.

To ensure the left-hand side being a number, we have only to add up the number
of points that are altogether multipled together and see that it is divisible by four.
Thus in eA BCxr, x and «x each give one point, A, B, C two points each; so that the
whole number of points is eight and this is divisible by four.

Pascal's theorem may be stated thus: If from a variable point lines be drawn through
two fixed points to meet two fixed straight lines and the line joining the points of
intersection passes through a fixed point, then the locus of the variable point is a conic.

In this form the corresponding theorem in space is

If from a variable point lines be drawn through two fixed points to meet two fixed
planes and the line joining the points of intersection, intersects a fixed straight line then
the locus of the variable point is a surface of the second degree.

In fact if a, b be the fixed points, A, B the planes, Z the straight line, its equation is
(ca A) (xbB) L = 0.
This surface, as is easily seen from the equation, passes through the points a, b, and the

points ZA, ZB and it contains the line AB.

Since passing through two fixed points involves two conditions and containing a
given generator involves three more, and this, together with the four constants in JZ,
makes nine, enough to determine the surface; we may say:—If through any two fixed
points on a surface of the second degree and through a variable point on the surface
lines be drawn to meet any two planes passing through a generator the line joiming the
points of intersection will always meet a certain fixed line.

The equation (aA) (xbB) (xcC) (xdD) = 0 gives a generation of a quartic surface
analogous to that for cubic curves, but it does not involve enough constants to generate
any quartic surface.

THE INNER MULTIPLICATION.
If @= 0,0, +00, + Ae, +--+ + O,En;
where a,a,...are numbers and e,...¢, a system of points such that e,e,...e,=1, then
| a is defined to be a, | e,+a,| ¢,+4,|¢,+...+4, | @,.

It is clear that | Gb+e)=|b4+]c.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 131

Now Gal eral wess| ey Tadic:
Gal Ce) AC Cairn Ox —Orney || 62 — ON .e0.5

therefore a|a=a'+a7+...+4,7.

It will be assumed that when a@ is a simple point and not a multiple of a point

a |a=1 just as e, | e, =1, &e.

This assumption limits the spaces treated of to those in which distance is possible.

A system of points e,, e,...e, such that e,[e=1, e,|e,=0 is called by Grassmann

2
a normal system.

If f,, f,.--f, form another normal system, and if
uA a ay e te Bio Ce ar ate Diner ?

Sa = Foy Oy + Mop Oy + pe a

&e., &e.
then since ap lA= MN Greig tee +35 = 1,
and since eee. io On ones a a ee a 0,
&e., &e
From these equations Ce al ree secs cea

R
Q
R

and we will take it equal to +1.

Then peewee

Now with reference to the system f,, f,,--.f,
lh =Sifs tne

But fy heh a NCP es

= 4%. | Cy eee FH, | C,>

and this is the meaning of | f, referred to the system ee, ... é,.
of |f,, &c. are identical.

Now if x =2,f,+2,f,+. £2, Fs
then |c=a,\fi+e,|A+-..-4+2, |

and this is the same whether 2 be referred to the system eg, -..

Hence | does not depend on any special set of points, but
normal system.

Similarly the meanmgs

Ca (0G wiOajafarscif oe:

is the same for any

132 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S

As any point can be expressed in terms of e,, @,...@, 80 any quantity of the
rth order can be expressed in terms of the products e@,...€,, @0,-+-@, 7... If

A,, A,....A,, be these products and -

2°?

A=6,A,+¢,A,+...+4,45,
then | A is defined to be a,| 4,+ 0, | A,+... +2, | A,.
Let | B= B, | 4,4+8, | 4,+---+8, | 4a:
Then | A | B=@A,—2,8,) | 4, | 4,4+-.....
= (4,8, —2@,8,) | A,A,+......
=| AB,

since it was shewn in the last section that | A, | A,=|A,A, where A,, A, are products
of the original units.

And this is also true when A, B are not of the same order.
It follows that | A|B|C &.=| ABC, &.
and in particular | eer Sle eel es

and therefore | f,f,...f, is independent of the particular normal system to which it is
referred since | f,,| f,-.. |, are independent. Since any quantity A of the rth order can
be expressed in terms of the quantities fif,...f, ff, .--f,,, &e it follows as before
that | A or the complement of A is independent of the special normal system to which
it is referred.

Since | (B+C)=|B+|¢,
A|(B+C)=A/Bt+AlGC,
and therefore the quantity A |B may be considered to result from a new kind of mul-
tiplication between A and B.
If A,, A,, A,... A, be products of the rth order formed from the original units
e,e,-..@, then by the definition of | 4,, | A,, &e.
AS AL=", A || A= &e,

and if B= CON Oy. AsO stb
so that | A,="F¢.6,,--- @,
then Al) AH 60, ..% G60... cant y—O;

since e, is repeated twice and e,,, is missing; and similarly with all the other products.

If then A and B be of the same order, and
A=4,A,+4,4,+...+@,4,,
B=8,A,+8,A,+...+8,A4,,,
A| B=4,8,+4,8,4+...+4,8,, = B| A,
so that the inner multiplication of two quantities of the same order is commutative and
the product is a number.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE, 133

Take two quantities C105 +++ C04, +-+€,, €,0,-..€, Which ate not of the same order,

s r

then C0, = C6riy 0-8, | 6,0, 32. 6,
HG, te O,Cragnn- C, « Cop! sav €, «ov
ae ergy ainw O,0,09 20 Ens Cpe, one Oy 0.s 6,

=a (Cpe AAG RAC AC ICR

8

= (6,0, +++ Cy) Crug ore Cy = Cpyy vee Gy

so that #| F when £# contains F is equal to the product of the remaining limits so
arranged always that
in |e Ye

Again, C10, +++ Cy | C10, r0e Cy = Cyl, 0 COs va Oy

= product of units except e,,, ... ¢,
res r(e-r)
aa (e 1) | C41 cee @,

= (ioc

That is to say if £ be of higher order than F and s, r be their orders
F|B=(-1y""" | (E| FP),

From the distributive principle it follows that this result must be true for any
quantities A, GB of orders s, r.

If both # and F contain quantities which do not occur in the other, then H | F=0
for | F will contain quantities which occur in FE since F does not contain all the quan-
ties in H, hence in L| F some points will occur twice over; also # | F will not contain
all the points, for since ZH does not contain all the points in # there will be some
points neither in # nor in | F.

If, in the equation F{H| F}=E£, for F we substitute any quantity B of the same
order where B=£,/,+ 8,F,+... and F,, F, are only made up of points contained in £, then

since F{E| F}=£, F,(E| FJ =£,
and it is easily seen that Fi {z | #}=0, FL {EH | F}=0,
we have BiE| B}=(6F,+8,F,+...} (CF | F+B,E| B+ see

(Gi, tf --.) BB || BY B.
Also with the notation before used for F and £,
F | H=e¢,...¢¢ e
ELF | B}=e,e,... 6... « ee... &€ é.,
e.= Ff,

=6,6,...

if # be entirely contained in the points of #. But if not, and F= F’+F",
then Ff” | H=0, so that F| H=F’ | £,
and E{F| E}=E{F | E}=F’.

Wor, XII, Parr I. 18

134 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Hence when £ is a product of normal points and F any quantity of lower degree,
E{F| E} expresses the part of F which is contained in #. If EH were some multiple
of a product instead of the product itself, we should have #{F'| H}={E| EL} F’.

For example, if «= pe, + qe,+7e,, where p, q, 7 are numbers,
then x | €,e, = (ge, +7.) | C0, = Ge, +7€,%>
and €,¢, {v | €,€,3 =e, + 7e,-

The quantity F’ may be called the projection of F on #, Also @ | ee,=ae, passes
through the point « and is normal to e,e,, since we, | ¢,e,=xee,= 0. It may therefore be
ealled the perpendiculars from a on e,e,; or the inner product of a point into a line
represents the perpendicular from the point on the line. Again e,e,| a is the point
where the complement of x or its polar intersects ¢,¢,.

If a,a,...a, be any points and « a point belonging to the system

a,a,...a, | x=(a, | z) (a,@,... @,) — (a, | 2) a,4,... a, + (a, | ©) (a,0,0,... @,) — ce.

For let e,e,...e, be a normal system including a,a,...a4, and therefore a.

n

Let G, = G0, +250, +o. Binns
&e. &e.
L=7,e, + UC, .-. + 2,2, 5
and let A,,,-d,,. be the minors of D=| @,,, @y_5 --» %,, |
ig oe eee I.
An Eno» Cnn |
Then OG oA. ile ct At | ekej. aed en,

AN [AC Reet Bn oP

Thus the coefficient of x, |e, on the right-hand side is 4,,4,,+4,,4,,+...=, and of
w,|¢, is 4,A,+4,,4,,+0¢,,4, +... =0.
Hence the right-hand side become D {z, | e,+ 7, | e+ -..}=4@,a,...@, | #,
since GG. — 1, end «aa lve a eect
As particular cases ab | c=(a|c)b—(b| c)a,
abe | d=(a | d) be+ (b | d) ca+(c | d) ab,
abed | e=(a | e) bed—(b | e) cda + (c | e) dab—(d | e) abc*.
From this first equation and two similar
be | a+ca | b+ab|c=0.
Hence, as a | be=— | bc | a}, a|be+b|ca+c| ab=0.
This equation expresses that the perpendiculars from the angles of a triangle on the

opposite sides meet in a point.

* Ausdehnungslehre, pp. 131, 134.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 135

Again, bed | a—cda | b+dab | c—abe | d=0,
and since a | bed = | {bed | a},
a | bed -b | cda+c | dab—d | abc =0.
That is to say, the four perpendiculars from the angles of a tetrahedron on the opposite

faces are lines such that forces along them can be in equilibrium and they are therefore
generators of a surface of the second degree.

The locus of the third of three points forming a normal system, in a three point
space, 1s a conic when the other two points lie on fixed straight lines. For if « be
the variable point, 4, B the fixed straight lines, then A|2a, B
points and since these are normal

aw, are the other two

(A | x) | (B| 2) =0,

an equation of the second degree*. This is MacCullagh’s theorem.

If a, b,c... a,b, ¢,..: be two sets ef points of equal number,
then ON BOC..—|ale, "ab, @ | tor.
Galiae (OMG (Osler.
@\\a ONS Cll 'Gcee

For we may write
G=4,6,1+4,6,+...+aeé,,

b=Be,+8,e,+...+8,¢,,

, ' , / ’
a =0,6,+ Oe Crain O Cate ese H Gon Coy:
Then Dine =

Gis Gan ie ilh
B,, B+ B,
sadeeelddacece OC Ceo
vb'c' =| a,’a, ae |
BYBy + By
Weoresoarocsecos €,€, «+. @, +
Hence ee POC. —\eme wo | aay ae
B,B,--- B, | | BB, --- By |
=| aa/+a,,'+..., @,8, +46, +... |
Ba, +8a,+-., B,B/+B8, +. |
=|a la, alB
eoulrarm. eOn re
In particular ab | ab’ =(a| a) (b| B')—(a| 8) (6 | @),
‘and therefore ab | cd+ac | db+ad | be=0.

* Tait’s Quaternions, p. 146.
18—2

136 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

MEASURE OF DISTANCE.

If a, b be two simple points, we will put a|b=cos@, and call @ the distance between
a and b.

Since ab | ab=(a | a) (b| b) —(a | b)? =1 —cos’O=sin’#,
we may put ab=sin @ee,, where ee, are two normal points on the same line. The dis-

: : 7
tance between two normal points is always 5

9°?
a

since e, | e =0=cos =
1 a. xfer 9°

7

Again if A, B be two intersecting lines of length 3 we will put A| B=cos¢, and

call @ the angle between A and B.

If o be the point of intersection of A, B; a, b, the points distant ad from o, so that oa= A,

=

ob = B, then
cos d = 0a | ob =(0 | 0) (a | b) — (0 | a) (0 | b) =a | b;

and thus ¢ is equal to the distance between @ and b, or the angle between two lines is
equal to the distance between the points where they cut the polar of their point of inter-
section.
Again if a, b be the poles of A and B in a three point space
ANB (a5 \b—|\faoi— ya,
or the angle between two lines is the distance between their poles. Since
AB| AB=(A | A) (B| B)—(A | B) (B| A) =1 —cos’p= sin’;
it follows that AB=sind.o, where o is their point of intersection. If a, b, ¢ be points
of a triangle whose sides are a, B, y and angles 6, $, x, then

ab | ac=sin PB sin y cos 6,

but ab | ac =(b | c) —(a| b) (a | c) = cosa—cos B cosy,
so that cos a=cos 8 cos y + sin 8 sin y cos 6*.
Again ab.ac=sin §sin ¥ sin 6a,
so that abe = sin 8 sin ysin @=sin y sinasin ¢=sin asin f sin y,

sin@ sing siny
sna sinf siny’

and hence
The locus of a point such that the product of the cosines of its distances from two
fixed points is constant, is

(x | a) (x | b) = const. = cx | a,

* When 6=7, a=f+y, and this justifies the use of the term distance.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 137

and it is therefore a curve of the second degree. It is equivalent too to the most general
equation, since the two points and the constant distance make up five constants.

We may also say that in any conic the product of the sines of the perpendiculars on
two straight lines is constant, the straight lines being the polars of the former points, and
then in the usual way it may be shewn that the portions of any line cut off between
the curve and these lines are equal.

The locus of the point such that lines drawn from it to two fixed points are at right
angles is given by
xva\xb=0, or («| 2) (a|b)—(@|a) (|b) =0,

and is a particular case of the former curve.

° . Tv . ae
If a, b, be any two points distant > from c and d in a space containing more than

=

three independent points, then
(2a + 8b)| (ye + 6d) = 0,

and therefore every point in the line ab is distant = from every point in the line cd.

Also ca | cd = (c| c) (a| d) —(c\ a) (¢|d) = 0,
or ca is at right angles to cd.

Again, ca|cb = a\b = da| db,
or the angle between two lines ca and cb is equal that between two lines da and db.
This angle may be defined to be the angle between the planes acd, bed, since it is the

angle between lines drawn in the respective planes at right angles to the line of inter-
section from any point in that line.

If 6 be this angle, and ¢ be equal to the distance cd, then

abed|abed =| 1, ab, 0, c = sin’ @ sin*® ¢,
GubecelemaOl cen |
Onaice la clan
| @ -@=e@lles ul

so that in a four-point space, where abcd is a number,
abed = + sin @ sin ¢.

But this is the product of the lines ac, bd with its sign changed. Hence the product of

two lines of length = is equal to the product of the sines of the perpendicular distances

between them.

If ac, bd, instead of being of lengths 7 be of lengths y, x, then

9 ,

acbd = + siny siny sin @ sin ¢,

138 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

or the product of two lines is equal to the product of the sines of their lengths multiplied
by the sines of their perpendicular distances.

Ee

Making again ac, bd of lengths

9?

b

ac\|bd =| a|b, 0 |= cos @ cos ¢,
| 0, eld
and thus the inner product of two lines of length ul is the product of the cosines of their
perpendicular distances.

If L, L’ be two lines the equation ZL’=0 expresses that they meet, and when also
L| L’=0, they must meet at right angles.

If F be any sum of lines it can always be expressed as the sum of two conjugate
lines; that is to say, we can put F=L+a| TL, where a is a number called the pitch of F’.

For |F=|L+al.
Hence F-a|F=(l-2)L;
and therefore (F—a|F)?=0,
or F?(1+a°)—22F | F=0.

This gives a quadratic equation for determining @ the roots of which are a, —. The
a

meaning of this is that if Z+a|Z be one solution, then a|L+2\(a\L) is an identical
solution.
Either Z or |Z may be called the axis of F.

Suppose we have two screws ¢,¢,+ 9¢,¢,, ,e, form a normal

system, so that e,e,=|e,e,, ¢,e,=|¢,e,, and wish to find the cylindroid or the locus of

e,e, + Bee,, where e,e,e

the axis of a screw compounded of these.
Put L+ y L=x (2, €, + ae, €,) + Ke (Ge, oF Be,e,),
then e,e,L + ye,e,| L =a,

e,e,| L + ye,e,L =»,

since L\e,e,=¢,e,|L and |e,e,| L =| Le,e, = Le,e,.
Hence e,e,L — ae,e,| L = ¥ (ae,e,L — e,e, | L) ;
similarly e,e,L — Be,e,| L =(Be,e,L — e,e,| L).

Eliminating y,
(c,e,L.e,e,L —e,e,| L.e,e,| L) (a—f) = (a8 +1) (e,¢,L.e,e,| L —e,e,| L. e,e,L).

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 139

Now, since Le,e,=0, Le,e,=0, L meets the lines e,e,, e,e,, and we may put

2°3?

I = oxe,+ 7xe, where « is any point on L,
Le,e,=0 gives — ox|e,-+7x/\e, =0,
and the previous equation gives
(7°a | e,.2|€,+ o°@|e,.a|€,)(@—B8) = (a8 +1) {(e|e,)°+ (cle,)*} ot.
And therefore
(28) (#|e,) (x\e,) {(x| e,)° + (| e,)") = (28 +1) (| e,) (w| e,) {(a | e,)? + (a e,)?}.
The product of two screws L+y\L, M+6|M is
LM (1 +8) + (y+6) Z| M.

If M meets Z at right angles, or if it meets it, and has an opposite pitch, the
product vanishes. Any three screws on the cylindroid, F,, F,, F,, are connected by a linear
relation, and therefore, if the product of a screw into any two of them vanishes, the
product of the same screw into the third will vanish. Now from a point of the cylindroid

draw a line M perpendicular to the line passing through that point. It will also meet at
right angles the polar line, and it will meet two other limes of the cylindroid in addition.

We may suppose a screw with pitch equal and opposite to one of ‘these last two
lines and with axis M. Then M+ 6 JM is reciprocal to two screws of the cylindroid, and
therefore to all.

Hence, if L+y,\Z be the last line it meets since LM = 0,
(yt+6)L|M=0 and y=—6,

that is, a line perpendicular to a line of the cylindroid meets it again in two lines
corresponding to screws of equal pitch*.

If x be any point, r+ 5 dt its consecutive position,
dx : : :
then ua dt is a small portion of its path,
dx . :
and oy its velocity ;
d (« ) = rye is the acceleration
GN at) —~ oF ett a

If F be the force acting, m the mass,
2

Mx = = 7

* Ball, Theory of Screws.

140 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S
and for a system of particles, if F be the system of forces acting

d?x
Smer— =
dma Te F,

and this includes the six equations required to determine the motion.

For a particle /= al, since # must pass through 2,

” Ca _ 1
me Te = wl.
: dx
Multiply by |« aie then
eit) pak Bye da _ ,\dx dx _dx| Px
FEA Fi # ae \ at = \dt * ae =F dt’
S poi Rae 0, and =1,
since dg = 8 and «| | a
dz &. d
and therefore m = | ae = i ,
da | dx
2m 7 iE fl\ dz.

If 1 peeeade only on 2, this is the equation of vis viva.
For a mass 13m — Ee = f(l| da, +1,|da,+...).
Suppose 1’, 1’ the parts of J,, 1, which arise from the mutual actions of a, and a,.
Then, if aa,z, be this mutual action where z does not depend on the time,
lL = az,, LY = aa,
L' | dx, +1,'| dx, = a (x,| dx, + x7,|dz,) = 0,
so that the mutual action disappears.

Considering only the external forces,

and this is the equation of vis viva, which is therefore true for any space allowing the
measure of distance.

The motion of a particle under the action of a central force is given by

ma os = Pza, if P be a number.

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 141

Hence mas — =

daz
max —- = const.

dt

If v be velocity, » perpendicular on tangent,

mv sin p = h, a constant.

IMAGINARY AND FLAT GEOMETRY.

Instead of putting a|b=cos6, we may put a|b= Hee. where & is any number;
and we shall have three distinct cases according as & is real, infinite or imaginary.

If k be imaginary, we may take it equal to the algebraic /—1, and put

a|b=cosh 6;

then ab | ab = 1 — cosh? 8 = — sinh’ @.

Take a line £ of length y on ab such that sinh y = 1;
then ab =sinh@.#, and #|H=—1.

For any two lines £,, £, of length y we may put

E,| E, = — cos ¢,
where ¢ is the angle between them. Then
E,E,=sin ¢.a, if a is the point of intersection.
For lines ab, ac of any lengths 6@,, 0,,
ab. ac = sinh 6, sinh @, sin ga,

and therefore abe = sinh @, sinh @, sin ¢.

The points | a, | 6 are imaginary.

In ordinary geometry, making & infinite, a@|b =1, and therefore this quantity does not
give a measure of distance.

But if a = Bb +4¢,
the equation a\a= Ba\b+yale
gives 1=8+y,
and hence B(a—b) =y(c—a).

For points on the same line therefore a—b, e—a are always in a numerical ratio,
and we may take a—d proportional to the distance between a and b.

Vou. XIII. Parr II. 19

142 Mr COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN’S

Since ab = a(b—a), ac = a(c—a),

ab, ae are also proportional to these distances.

The quantity b—a is called by Grassmann a stroke (streike), and the quantity ab a
line (Linientheil).

A stroke may also be considered a point at an infinite distance, for since Bb + ye
represents a point dividing 0c internally, so that the distance ab is to ae in the ratio
y to 8; c—b will represent outside bce, whose distance from ¢ is equal to its distance
from }, and therefore a point at an infinite distance.

Hence strokes are equal when they lie on equal and parallel lines, and a stroke in-
volves magnitude and direction but not position. Since the inner multiplication for points
leads to no results, Grassmann discards it and introduces an independent inner multipli-
cation for strokes. If a, b be strokes of unit length making an angle @, a,|b,=cos 6, so

that a, a,=1, b,/b,=1. Therefore a,b,|a,b,=sin’@; and, if a,b, be of any length, and

U be the product of two unit strokes in the same plane at mght angles,
a,b, = «8 sin 6U,

so that the outer product of two strokes is always proportional to the magnitude of the
parallelogram they involve. This product may be called a parallelogram, and it is the
same for all parallel planes, since strokes are unaltered by being moved parallel to themselves.

A parallelogram thus depends only on its magnitude and the direction of the normal
to the plane-in which it lies,

A line involves ‘position as well as direction, and if cd, ab be equal and parallel lines,
cd — ab = c(d—c) —a(b—a) = (c—a) (b —a),
(since d—c=b—a); = parallelogram of which cd, ab are opposite sides.

The product of three points
abe = a(b—a) (c—a) = b(b—a) (c—a) = c (b—a) (c—a)

= product of any point in the plane into a parallelogram double the area of the triangle
enclosed by the points.

This quantity is called by Grassmann a plane.

The product of three strokes is a parallelepiped with the strokes for edge.

The difference of two planes is a parallelepiped with the planes for opposite faces.

If a,, b,, ¢, be three strokes forming the sides of a triangle so that c,=b,—a,, then

¢,|¢, =a,|a, +0, | 4, — 2a,|b,;

AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 143

or, if a, b, ¢ be the sides in magnitude, a, 8 the angles,
ec =a’? +b? —2ab cose.

Calculations with strokes will be identical with calculations with vectors as far as only
the product of two quantities is concerned. The chief differences between Grassmann’s
system and Quaternions are that Grassmann takes into account the outer multiplication
of points, the regressive multiplication, and the position of lines as well as their direction*.

* A great portion of the ‘““Ausdehnungslehre” is also analogous to Hamilton’s linear and vector functions, and
deyoted to the algebraical multiplication of different quan- called by Grassmann fractions, are considered.
tities which has not been mentioned here: also expressions

4,
RM Oe

hill aeeen wht ifie ahh ob @ Py y ie a
| , ee
ida) ah, Vite! My Aitde mewtialuie
oon ie weeny on ee
oneal tab on Sine iia. by a
eee nee

IV. Table of the Descending Exponential Function to Twelve or Fourteen Places
of Decimals. By F. W. Newman, Emeritus Professor of University College,
London,

[Read December 4, 1876.]

My object in constructing these tables was to facilitate the calculation of those Anti-
eyclic functions which become prominent in Elliptic Integrals,

Mode of Formation. This must be explained, in order to give confidence in tbeir
use. First, a table was constructed of e* (to 16 decimals) with 2 an integer, by a
method which afforded a systematic verification of the results. Put generally

hte h’
= rth __ —z
eg eres ea+rsties
in which we may make A=1, or h='l, or h=‘01, or A=-001 or any convenient small
fraction, At starting, take h=1. To find ¢** when we know e™

the separate terms of
as tf Ze 1 A i. &
: TURES Oo Tp Aaa ee aa

each term being derived by a simple division, Add in separate sums the odd terms
(=M) and the even terms (=N). If the work is done correctly, it ought to yield
M+N=e7" a quantity already known; and if (in working with 18 decimals) the last
equation is found to have 16 decimals correct, it is almost certain that our WZ and N
are attained accurately for those 16, Any small error in the two last decimals (for some
must exist) will be less in M—WN than in M+WN. We may then trust the result
e**=M—N. Thus from ¢* we calculate ¢*, by making #=1, and the test of accuracy
is M+N=e=1. For the next step we make «=2, and seek for e* from e”, and our
new test is M+N=e'. This being fulfilled, we attain e*=d—N; and so on con-
tinually,

+ &e.}

and e”, calculate

After this first skeleton table was finished in 37 entries, from ¢* to e” (since ¢*
does not yield any digits to 16 decimals), I proceeded to make h=‘1, so as to calculate
ée*, e*, €&*... up to e, with the same method of verification. But I found it con-
venient first to halve the intervals, by making h=4, then from e” I deduced simul-
taneous e~*-? and e-*+4, For instance, having deduced e** and e~*° from e™, I proceeded
to deduce e** and ¢** from e“, and compared the two values thus obtained for e°*. If
they agreed to 16 figures, I trusted my work. When disagreement was found, I searched

out the error.

After thus halving the intervals, I made h= 1, and had a new check after every
five steps, so that any error was sure to be discovered.

Vou. XIII. Parr III. 20

146 Mr F. W. NEWMAN'S TABLE

Since I worked with 18 decimals, in order to get 16 always accurate, the first idea
was to correct the 16th by the two which follow, and drop the two last. But this
leaves it uncertain whether the figure presented as 16th is too small or too large; and
if further deductions are needed (as in interpolation) we no longer have 16 figures
accurate, For this reason I have left the complete 18, which can be dealt with as
pleases him who uses the table.

3

Problem. Given e** and e** to find e®*, true and verified.

We shall divide e** by 10, 20, 30,... successively. Put M for sum of odd rows,
and N for sum of even rows. Then I+ N ought to make e”*®* accurately, if we are to
trust M—N=e™.

The error in M+WN is naturally greater than in 1/—N; hence M+WN varying from

e** (as given) by only 4 in the 18th place, we have just confidence that M—WN gives
é** correct to the 16th place.

0907 1795 3289 4124 98 =e**

"1002 5884 3722 8037 29 =e

10) 907 1795 3289 4125 00

20 ) —90 7179 4828 9412 50

30 ) 4 5358 9766 4470 62

40 ) — 1511 9658 8815 68

50 ) 37 7991 4720 39

60) —7559 8294 40

70) 125 9971 57

80) —1 7999 59

j trustworthy for 16 places.

90) 224 99
100) —2 49
9

M=911 7192 1173 3512 59

N= 90 8692 2549 4524 66
M+N='1002 5884 3722 8037 25=e°%°, which is sufficiently correct.
M-—N= 820 8499 8623 8987 93 =e7°°*, which was sought.

A third step was, to make h=‘01, and limit the new table to 12 decimals. This
is compact enough, and occupies barely 26 pages of a full sized quarto copy book. In fact
e€** yields only zero to the first 12 decimals. But this table remains in MS., being
superseded by a fourth (also carried to 12 decimals), in which h, the increment of 2,
is only ‘001. The entries in this great table are all made by interpolation into the pre-
ceding; and the interpolation is in all cases conducted by the same perfectly accurate
formula, the series for e*. Thus error can scarcely exist without detection, except errors
of copying out and errors of printing.

It has been my good fortune to find in Mr J. W. L. Glaisher, a mathematician who
has given his time and valuable superintendence to the differencing of this large table;
whereby he has exterminated errors of. copying or printing, not at all numerous on the
whole, yet enough to have hurtfully infused suspicion. I feel much indebted for his
zealous co-operation, so critically important.

OF THE DESCENDING EXPONENTIAL. 147

From not having retained the 13th decimal in my third table, I have had occasional
hesitation as to the accuracy of the 12th in my large table founded upon it. Though
I believe it could only affect a unit in the last place, it sometimes gave me much
trouble of recalculation, until I reached about 2=43, after which I fell back on my
second table, and worked from it with 14 decimals, checking myself from the third table
after every ten entries, and by my second after every hundred.

Mode of using the tables. When «x does not exceed 15349, and the decimal part
contains only tenths, hundredths and thousandths, the value of e* will be found in its
own place, somewhere in the first 77 pages of the table (pp. 151—227).

But on p. 228 a Second Part of the table begins, carried to 14 decimals, and from
x=15°350 to c=17-298 the values of « which end with an odd digit, are omitted. But
we get the intermediate values of e%, true to 12 decimals, by taking an arithmetic mean.
Thus to find ¢**", we have ¢&*°*°=1662 1229; e&**?=1658 8020 (each to 14 decimals),
sum = 3320 9249, half sum =1660 4624. Therefore e, true to 12 decimals, is 166046
(stz zeros must be prefixed). Proof of this rule. Given consecutive entries A=e”,
B=e*”, to find the intermediate U=e~* which is not in (this part of) the table. Here
h=-001, A and B begin with 6 zeros, therefore each is less than 10°; so is h®, Now

2 F 2
Ane*.e=U (14475), oe
since Uh’ is less than 10%%, or does not affect the 15th decimal; therefore
4(4+B)=U(1+4h’), whence U=4(A + B)(14+43h’)*=3(44+B) (1-3 4+4h4+ &).
But h? being <10°, }(A+B).4h? is less than $10. Thus, true to 12 decimals, V=4(A+B),
as was asserted. To obtain accuracy to 14 decimals, we must take account of the small
factor 4H?. For this, divide the half sum by 2.10° and add the quotient to the half sum.

After «=17-298, z increases by ‘005 at each step, as 17300, 17305, 17310, 17315...
and the new question rises how to find an intermediate a. Its last figure may be,
first, either 9 or 1, on the two sides of a zero; next, 4 or 6, on the two sides of a 5,
giving to our z the form —a+h; where e~® is in the table, and h=‘001. Or again
the « may end in 2 or 3, else in 7 or 8, so as to have the form —a+2h. Thus
either by assuming y=x+-‘001, else by assuming u=x#+ 002, y or wu will be in the
series of the table (ending either in 0 or in 5), and e” or e™ will be known to us by
the table. Hither then c=yFh, and e*“=e"%(1 Fh), else c=uF 2h, e*=e "(1 F 2h). Each
will be true to 16 decimals,

Hence the Rute. If z is a unit greater than y, subtract from e” its thousandth
part, to find e*; or add, if z is less than y. But if x be two units greater than y,
subtract from e” its five hundredth part to find e*, or when « is less than y, add the
five hundredth part.

Examples. Given in the table e¥=104 6740, e"=104 1519, where y=18'375. It
is required to fill up for the intervals between 0 and 5. Divide by 1000, and we get 1046
and 1041. Subtract the former from ¢” and add the latter to e””, then &*°"°=104 5694;
ee —104 2560; else, divide by 500, which yields 2093 and 2083. Subtract the former
from e”, and add the latter to ¢~, then e&%*”=104 4647, ¢&***=104 3602. In these
four new results, only the last (14th) figure is in. error.

Bae*.e*=U(1-

20—2

148 Mr F. W. NEWMAN’S TABLE

Table of e* to eighteen decimal places (sixteen exact).

x e* x e x ex
r | 9048 3741 80359 59545 || 5°I 60 9674 65655 15637 | 101 4107 95552 25302
2 | 8187 3075 30779 81848 5°2 55 1656 44207 60774 || I0°2 3717 03186 84128
3 | 7408 1822 06817 17871 53 49 9159 39069 10218 || 10°3 3363 30951 85721
4 | 6703 2004 60356 39307 54 45 1658 09426 12670 || 10°4 3043 24830 08403
5 | 6065 3065 97126 33423 | 5°5 40 8677 14384 64068 || 10°5 2753 64493 49746
6 | 5488 1163 60940 26441 5°6 36 9786 37164 82931 || 10°6 2491 60097 31501
7 | 4965 8530 37914 09523 || 5°7 33 4596 54574 71272 | 10°7 2254 49379 13206
8 | 4493 2896 41172 21599 58 | .30 2755 47453 75813 || 10°8 2039 95034 11166
"9 | 4065 6965 97495 99120 | 5°9 27 3944 48187 68370 || 10'9 1845 82339 95777
ro | 3678 7944 11714 42321 || 6:0 24 7875 21766 66358 || 11'0 1670 17007 90246
rr | 3328 7108 36980 79553 | 61 22 4286 77194 85802 || 11°1 I51r 23238 19857
I°2 | 301I 9421 19122 02096 || 6:2 20 2943 06362 95735 || 11°2 1367 41960 65685
1°3 | 2725 3179 39340 12603 6°3 18 3630 47770 28910 || 11°3 1237 29242 61791
1'4 | 2465 9696 39416 06475 6°4 16 6155 72731 73937 || 11°4 111g 54848 42595
I'5 | 2231 3016 01484 29829 || 6°5 15 0343 91929 77572 || 11°5 1013 00935 98631
1°6 | 2018 9651 79946 55407 || 6°6 13 6036 80375 47893 || 11°6 916 60877 36245
17 | 1826 8352 49527 34648 || 6:7 12 3091 19026 73481 || 11°7 829 38191 60755
18 | 1652 9888 82215 86535 68 II 1377 51478 44802 || 11°8 750 45579 15075
I'9 | 1495 6861 92226 35054 6°9 10 0778 54290 48510 || 11'9 679 04048 07381
| 20 | 1353 3528 32366 12691 7'0 9g 1188 19655 54515 || 12°0 614 42123 53327
| 21 | 1224 5642 82529 81909 er 8 2510 49232 65905 || 12°1 555 95132 41665
| 2:2 | 1108 0315 83623 33881 7°2 7 4658 58083 76681 || 12:2 503 04556 o7114
2°3 | 1002 5884 37228 03731 || 7°3 6 7553 87751 93846 || 12°3 455 17444 63084
2°4 907 1795 32894 12500 14 6 1125 27611 29574 || 12°4 411 85887 07538
2°5 | 820 8499 86238 98791 || 7°5 5 5308 43701 47832 | 12°5 372 66531 72085
2°6 | 742 7357 82143 33876 || 7°6 5 0045 14334 40611 | 12°6 337 20152 34153
27) 672 0551 27397 49761 Wel 4 5282 71828 86790 || 12°7 305 11255 58050
2°8 | 608 1006 26252 17961 78 4 0973 49789 79781 || 12°8 276 07725 72053
2°9 | 55° 2322 00564 07225 || 7°9 3 7974 35404 59080 | 12°9 249 80503 25884
| 3°0 | 497 8706 83678 63943 8:0 3 3546 26279 o2501 || 13°0 226 03294 06997
31 | 45° 4929 23935 57806 81 3 0353 91380 78857 || 13°71 204 52306 24491
a 407 6220 39783 66216 8:2 2 7465 35699 72135 || 13°2 185 O601I 97553
nats 368 8316 74012 40006 83 2 4851 68271 07947 || 13°3 167 44932 09446
Ly rane 333 7326 99603 26081 84 2 2486 73241 78844 || 13°4 I5I 51441 12156
35 | 301 0738 34223 18502 8'5 2 0346 83690 10644 | 73°5 137 09590 86393
36 | 273 2372 24472 92561 || 86 1 8410 57936 67577 | 13°6 124 04950 79965
3°7.| 247 2352 64703 39390 8-7 I 6658 58109 87632 | 13°7 II2 24463 65241
3°38 | 223 7077 18561 65595 || 88 I 5973 39759 95474 | 13°8 Tor 56314 71020
3°99 | 202 4191 14458 04390 || 8:9 1 3638 89264 82008 || 13°9 gt 89813 57913
4°0 183 1563 88887 34179 || 9’0 I 2340 98040 86675 || 14'0 83 15287 19119
41 | 165 7267 54017 61246 g'I 1 1166 58084 gorII || 14°1 75 23982 99227
4°2 149 9557 68204 77705 || 9°2 I 0103 94018 37091 || 14'2 68 07981 34408
4°3 135 6855 go122 00932 || 9°3 9142 42314 78171 || 14°3 61 60116 26191
4°4| 122 7733 99030 68440 9°4 8272 40655 56631 || 14°4 55 73903 69323
4'5 | 111 0899 65382 42396 | 9°5 7485 18298 87702 || 14°5 5° 43476 62588
46 | 100 5183 57446 33583 || 9°6 6772 87364 90855 | 14°6 45 63526 36810
47 | 9° 9527 71016 95819 || 9°7 6128 34950 53224 || 14°7 41 29249 41607
48 | 82 2974 70490 20030 || 9°8 5545 15994 32180 | 14°8 37 36299 38007
49 | 74 4658 30709 24342 || 9°9 5017 46820 56176 | 14°9 33 80743 48400
5° | 67 3794 69990 85467 || I0°0 4539 99297 62485 | 15°0 30 59023 205T9 |

19'0
1g'I

193

20°0

Table of e” to eighteen decimal places (sixteen exact).

OF THE DESCENDING EXPONENTIAL.

Io

NNwWwWwW HANNA Wn COW

sa HSN ND

+s = mt

65864
37241
12790

57575 ||

36271

53993 ||

96759
72802
02371
74726
36937
08336
07883
83479
33757

13351

13920 ||

13478

87708 ||
77202

05575
43271

12954 ||

32429
91571

59942

22310 ||

39278
30706

79752

55555
52792
46525
60750
49702

90136
84148 ||
71049 ||

47706
96459
19869
81754
53683
66761
67815
79877
65367
98715

27°37

53619

|

Io

NNWwWWwW BRANNAN DN OO

He eH NN

Hee

ater
eR
313

150 Mr F. W. NEWMAN'S TABLE.

TABLE, OF ¢ 75.
Part I. From 2=0 to x=15'349 at intervals of 001 to twelve decimal places;
pp. 151—227.

Part II. From 2=15°350 to c=17:298 at intervals of 002 and from x=177300 to
x =27-635 at intervals of 005, to fourteen decimal places; pp. 2285—241.

[-coo—199] MR F. W. NEWMAN’S TABLE OF THE DESCENDING EXPONENTIAL. 151

| | ; [i ]
xv e-% fe e-2 eae e-t x e-e

900 |1’0000 0000 0000 |] ‘a50 | ‘9512 2942 4501 || ‘10g | ‘9048 3741 8036 || 150 | 8607 0797 6425
‘00T | “9990 0049 9833 || 051 02 7867 0533 || ‘ror 39 3303 2886 || "151 | “8598 4769 8659
"002 80 o199 8667 || 052 | ‘9493 2886 6843 || ‘102 30 2955 1669 || 152 89 8828 0741
003 70 0449 5504 || 053 83 8001 2482 || ‘103 21 2697 3481 || "153 81 2972 1811
004 60 0798 9344 || 054 74 3210 6502 || "104 I2 2529 7421 || "154 2 7202 IOIr

| 005 5° 1247 9193 || 055 64 8514 7953 || ‘105 03 2452 2586 || "155 64 1517 7483
‘006 49 1796 4054 || ‘056 55 3913 5890 |} "106 | ‘8994 2464 8076 || 156 55 5919 0371
‘007 3° 2444 2933 || ‘057 45 9406 9366 || “107 85 2567 2990 || 157 47 0405 8817
008 20 3191 4837 || 058 36 4994 7437 || :108 76 2759 6430 || 158 38 4978 1968
=e 10 4037 8773 || 059 27 0676 9157 || ‘tog 67 3041 7498 || "159 29 9635 8969
‘O10 00 4983 3749 || 060 17 6453 3584 || ‘I10 58 3413 5296 || 160 21 4378 8966
O11 | *9890 6027 8774 || ‘061 08 2323 9777 || ‘111 49 3874 8928 || 161 I2 9207 1107
‘O12 80 7171 2861 || ‘062 | 9398 8288 6792 || “112 40 4425 7500 || ‘162 04 4120 4540
O13 70 8413 5018 || 063 89 4347 3690 || "113 31 5066 o1rs || "163 | *8495 9118 8414
"C14 60 9754 4261 || ‘064 80 0499 9531 || 114 22 5795 5882 || "164 87 4202 1880

‘O15 51 1193 9603 || ‘065 70 6746 3378 || "115 13 6614 3906 || "165 78 9370 4088
016 41 2732 0054 || 066 61 3086 4292 || "116 04 7522 3297.|| 166 70 4623 4189
‘O17 31 4368 4635 || 067 51 9520. 1337 || 117 | *8895 8519 3163 || 167 61 9961 1337
018 21 6103 2358 || 068 42 6047 3578 |} 118 86 9605 2615 || 168 53 5383 4685
O19 II 7936 2244 || 069 33 2668 0079 || ‘119 78 0780 0763 || °169 45 0890 3386

“020 ot 9867 3307 || ‘070 23 9381 9906 || “120 69 2043 6717 || 170 36 6481 6596

*o2I | *9792 1896 4570 || ‘071 14 6189 2128 || ‘121 60 3395 9593 || ‘171 28 2157 3471
"022 82 4023 5051 || ‘072 05 3089 5812 || ‘122 51 4836 8503 || "172 19 7917 3168
023 72 6248 3773 || 073 | °9296 0083 0026 || 123 2 6366 2561 || "173 II 3761 4844

024 62 8570 9758 || 074 86 7169 3842 || "124 33 7984 0883 || °174 02 9689 7658
"025 53 0991 2028 || 075 77 4348 6329 || "125 24 9690 2585 || "175 | *8394 5702 0769
026 43 3508 9608 || 076 68 1620 6560 || ‘126 16 1484 6784 || 176 86 1798 3336
027 33 6124 1524 || 077 58 8985 3607 || "127 07 3367 2598 || "177 77.7978 4522
028 23 8836 6801 || 078 49 6442 6544 |] "128.| °8798 5337 9145 || 178 69 4242 3488
"029 14 1646 4466 || ‘079 49 3992 4446 || "129 89 7396 5546 || "179 61 0589 9396

030 04 4553 3548. || ‘080 31 1634 6388 || ‘130 80 9543 og21 || ‘180 52 7021 141
‘031 | 9694 7557 3°75 || 081 21 9369 1446 || “131 72 1777 4392 || ‘181 44 3535 8695
032 85 0658 2079 || ‘082 12 7195 8697 || "132 63 4099 5080 || ‘182 36 0134 0415
033 75 3855 9588 || 083 Og) Ril Gap ||) 3 54 6509 2108 || 183 27 6815 5737
034 65 7150 4637 || 084 | 9194 3125 6096 || "134 45.9006 4603 || 184 19 3580 3827
035 56 0541 6257 || ‘085 85 1228 4402 || 135 37 1591 1688 || ‘185 II 0428 3852
036 46 4029 3483 || ‘086 75 9423 1221 || "136 28 4263 2488 || "186 02 7359 4982
037 36 7613 5349 || ‘087 66 7709 5634 || "137 1g 7022 6131 ||187 | 8294 4373 6385
| 038 27 1294 o8g1 || ‘088 57 6087 6724 || "138 10 9869 1745 || ‘188 86 1470 7232
039 17 5070 9146 || 089 48 4557 3575 || 139 02 2802 8458 | ‘189 77 8650 6694

“040 07 8943 9152 || ‘ogo 39 3118 5272 || "140 | 8693 5823 5399 || "190 69 5913 3943
041 | ‘9598 2912 9947 || ‘ogt 30 1771 Ogoo |} "141 84 8931 1699 || ‘191 6r 3258 8151
‘042 88 6978 0572 || ‘092 2I 0514 9546 || ‘142 76 2125 6487 || ‘192 53 0686 8491
043 79 1139 0067 || ‘093 II 9350 0297 || °143 67 5406 8896 || 193 44 8197 4139
‘044 69 5395 7473 || ‘094 02 8276 2242 || "144 58 8774 8061 | "194 36 5790 4268
045 | 59 9748 1833 || 095 | -9093 7293 4469 || “145 3465 8056
046 50 4196 2191 || ‘096 84 6401 6069 || 146 AI 5770 3185 || “196 20 1223 4678
‘047 40 8739 7590 || 097 75 5600 6134 || "147 2 9397 7417 || 197 II 9063 3312
048 31 3378 7077 || 098 66 4890 3754 || 148 24 ne 4942 198 i 03 e985 3137
zoe 4 c I Il 48¢ or 81 2

049 ZU OL 2 ao 099 57 4270 8024 || "149 5 6911 4599 99 95 4999 333

or
fo}
N
iS}
N
ios)
ra
H
4
4
No}
an
iS)
[oe]

MR F. W. NEWMAN'S TABLE [-200—-399]

NNNN WN

oe

NNN NN
tw Me ee
Oo On nn + WwW

NNN

| f
e-2 x | CSe | w i v e-*
3075 3078 || 250 ‘7788 0078 3071 || "300 | "7408 1822 0682 || "350 | "7046 8808 9719 | #
1243 1553 || “251 80 2237 1559 | *301 00 7777 2747 || 351 39 8375 3856
9492 7942 || 252 72 4473 8069 || 302 | 7393 3806 4890 || °352 32 8012 1977
7824 1425 || 253 64 6788 1823 || 303 85 9909 6371 || "353 25 7719 3378
6237 1187 || ‘254 56 9180 2046 || ‘304 78 6086 6451 || "354 18 7496 7356

4731 6411 || 255 49 1649 796r || “305 7 2337 4392 || 355 II 7344 3209
3307 6282 || -256 41 4196 8792 || °306 63 8661 9456 || "356 04 7262 0236
1964 9987 || °257 33 6821 3765 || °307 56 5060 0907 || 357 | 6997 7249 7735
0703 6711 |] ‘258 25 9523 2106 || -308 49 1531 8009 || 358 9° 7307 5007
9523 5643 || 259 18 2302 3043 || “309 41 8077 0026 || 359 83 7435 1352

8424 5970 || ‘260 IO 5158 5803 || ‘310 34 4695 6224 || °360 76 7632 6071
7406 6881 || ‘261 02 8091 9614 || *311 27 1387 5869 || °361 69 7899 8467
6469 7567 || °262 | *7695 1102 3707 || 312 19 8152 8228 || °362 62 8236 7842
5613 7217 || ‘263 87 4189 7310 || 313 12 4991 2569 || 363 55 8643 3499
4838 5023 || *264 79 7353 9656 || 314 05 1902 8159 | "364 48 9119 4743

4144 0177 || '265 72 0594 9975 || 315 | ‘7297 8887 4269 || 365 41 9665 0878
3530 1873 || 266 64 3912 7501 || 316 9° 5945 0167 || °366 35 0280 1209
2996 9305 || ‘267 56 7307 1465 || °317 83 3075 5126 |) °367 28 0964 5044
2544 1666 || °268 49 0778 1103 || °318 76 0278 8415 || "368 21 1718 1688
2171 8153 || 269 41 4325 5648 || 319 68 7554 9306 || "369 14 2541 0450

1879 7962 || ‘270 33 7949 4337 || 320 61 4903 7074 || °370 ©7 3433 0637
1668 0290 || ‘271 26 1649 6405 || 321 54 2325 0990 || ‘371 00 4394 1558
1536 4334 || "272 18 5426 1090 || °322 46 9819 0330 || °372 | 6893 5424 2524
1484 9294 || 273 To 9278 7629 || 323 39 7385 4368 || °373 86 6523 2843
1513 4369 || ‘274 03 3207 5261 || "324 32 5024 2380 | "374 79 7691 1828

1621 8759 || :275 | °7595 7212 3225 || °325 25 2735 3642 || 375 72 8927 8790
1810 1665 || 276 88 1293 0761 || °326 18 0518 7432 || °376 66 0233 3041
2078 2290 || -277 80 5449 7111 || -327 10 8374 3027 || °377 59 1607 3895
2425 9835 || 278 72 9682 1514 || 328 03 6301 9705 || ‘378 52 3050 0665
2853 3505 || 279 65 3990 3215 || 329 | 7196 4301 6747 || 379 45 4561 2667

3360 2503 || ‘280 57 8374 1456 || "330 89 2373 3432 || “380 38 6140 g212
3946 6035 || 281 50 2833 5481 || 331 82 0516 go4o |] “381 31 7788 9620
4612 3307 || 282 42 7368 4534 || °332 74 8732 2854 || 382 24 9505 3205
5357 3524 || ‘283 35 1978 7861 || °333 67 7019 4156 || °383 18 1289 9286
6181 5895 | *284 27 6664 4707 || °334 60 5378 2227 || 384 II 3142 7179

7084 9628 | 285 20 1425 4319 || °335 53 3808 6352 || "385 04 5063 6204
8067 3932 || 286 12 6261 5946 || °336 46 2310 5816 || °386 | *6797 7052 5680
9128 8018 || 287 05 1172 8837 || °337 39 0883 9903 || 387 92 g109 4926
0269 1094 || 288 | *7497 6159 2239 || 338 31 9528 7898 || 388 84 1234 3263
1488 2373 || 289 9° 1220 5402 || °339 24 8244 9089 || "389 17 3427 0013

2786 1066 || ‘290 82 6356 7578 || °340 17 7032 2763 || ‘390 7° 5687 4498
4162 6388 || -2o1 75 1567 8018 || 341 10 5890 8206 || *391 63 8015 6039
5617 7551 || -292 67 6853 5974 || "342 03 4820 4709 || 392 57 O411 3960
7151 3771 || 293 60 2214 0697 || °343 | °7096 3821 1560 || °393 50 2874 7586
8763 4262 || -294 52 7649 1443 || "344 89 2892 8049 || 394 43 5405 6240
0453 8241 | -295 45 3158 7466 | “345 82 2035 3468 || 395 36 8003 9249
2222 4925 |\° 8020 || *346 75 1248 7107 || °396 30 0669 5937
4069 3530 || ‘297 3° 4401 2362 || “347 68 0532 8258 || 397 23 3402 5633
5994 3277 | "298 | 23 0133 9748 | -348 60 9887 6215 || °
7997 3384 | ‘299 15 5949 9435 | 349 | 53 9313 0270 || "399 09 9070 1353

is)
Oo
QO
Ww
xn
co
a
aS
nN

[-400—"599] OF THE DESCENDING EXPONENTIAL. 153

x Ce x e-# x e-= x e-% |

Ver. Xi, Parr III. 21

‘600 | +5488

“601 82
"602 77
603 71
604 66
"605 60
606 55
607 49
*608 44
*609 38
“610 33
“O11 28
612 22
613 17
614 II
615 06
616 Or
617 | 5395
618 go
619 84
620 79
621 74
622 68
62 63
624\| 57
625 52
626 47
"627 41
628 36
629 31
630 25
631 20
632 15
633 °9
634 04
635 | °5299
636 94
637 88
638 83
639 78
640 42
641 67
642 62
643) 57
644 51
645 46
646 41
647 36
648 30

6094
8772
9714
8370

4194 || °

6639
5159
9207
8238
1709

9°74
979°
3314
9103
6615

5309
4644
4080
3076
1094

7595
2040
3892
2613
7668

8519
4632
5471
0593
9293
1007
5413
1879
9872
8862
8318
77°99

6506 ||
4179 |"

o201

4043
5178
3°77
7216
7068

2107
1809

5649

3103. || *
3648 ||

MR

F. W.

NEWMAN'S TABLE

e-2

8530
8896
9312

9778
0292

0857
1471
2134
2846
3608

4419
5279
6189
7147
8154
g2I1
0316
1470
2673
3925

5225
6574
7972
9418
0913

2456
4048
5688
7377
gr114

0899
2732
4613
6542
8520

0545
2619
4740
6909
9126

1391
3794
6064
8471
0927
3429
5980
8578
1223
3915

3792
6697
5692
o281
9967

4256
2653
4666
9800

7562

7461
9004
1699
5057
8586

1796
4199
5395
4626
1673

5960
6999
4304
7388
5767
8955
6468
7821
2531
O1I5
0090
1974
5285
9543
4267

8975
3189
6429
8216
8073

5521
0083

1282
8642
1686

994°
2928
0176
1209

5556

v

Wa2
‘751
‘752
‘753
754

155
756
157
758
759

4723
18
14
09
04
[oje)

"4695

[‘600o—"799]

6655
9442
2276
5157
8086
1061
4083
7153
0269
3432
6642
9899
3202
6553
9949
3393
6882
0419
4002

7631
1306
5028

8796

26011
6471
0378
4339
8329
2374
6464
o6or
4783
QoIl
3285
7604
1970
6381
0837

5339
9886

4479
gIl7
3801
8530
3304
8123
2987
7897
2852
7851

2741
2293

3739
6608

0429,
473°
9943
2896
5820
7348

7010)

-4338

8865
O25

7650

0974
9633
3160
1092
2963
8311
6672—
7584
0583
5208

0999
7493
4230
Oo751
6595

1305
4420.
5483
4037
9623
1785
0067
4013
3168
7°75
5282
7334
2776
1157

2023

4922
9403
5°13
1302
7820

[-800—-999] OF THE DESCENDING EXPONENTIAL. 155

156

x | oe xv = x ere x Ce
f |

1'000 | +3678 7944 1871 || 1:050|°3499 3774 gIII || 1'100]°3328 7108 3698 || 1°150| "3166 3676 9379
I‘oor 5 1174 5608 || 1051 5 8798 6272 || I°101 5 3837 8995 || 1151 3 2029 0875
1002 | I 4441 7557 || 1052 2 3857 3022 || I'102 2 0600 6830 || 1'152 © 0412 8691
1°003 | 3667 7745 6651 || 1:053| -3488 8950 gor0 | 1103 | 3318 7396 6870 || 1153 | 3156 8828 2512
I"004 4 1086 2522 || 1°054 5 4079 3887 | I°104 5 4225 8785 || 17154 3 7275 2021
I°005 © 4463 4804 || 1055 I 9242 7306 | 105 2 1088 2242 || 1'155 © 5753 6903
1'006 | *3656 7877 3130 || 1:056| °3478 4440 8917 || 1°106/ "3308 7983 6910 || 1°156|°3147 4263 6842
1'007 3 1327 7136 || 1°057 | 4 9673 8372 || I°107 5 4912 2458 || 1°157 4 2805 1525
1008 | *3649 4814 6454 || r°058 I 4941 5324 || 1'108 2 1873 8555 || 1158 I 1378 0635
| 17009 5 8338 o721 || 1°059 | *3468 0243 9426 || 1109 | "3298 8868 4871 || 1°159 | °3137 9982 3859
| toro 2 1897 9571 || 1'060 4 5581 0330 || I°r10 5 5896 1075 || 1°160 4 8618 0883
beans 3638 5494 2640 || 1'06r | I 0952 7690 || I°rr1 2 2956 6839 || 1161 I 7285 1393
I’oI2 4 9126 9565 || r:062| *3457 6359 1159 || 1112] °3289 C050 1832 || 1'162|°3128 5983 5075
I°013 I 2795 9980 || 1°063 4 1800 0392 |] I°'113 5 7176 5725 || 17163 5 4713 1618
T'014 | °3627 6501 3524 || 1:064 © 7275 5043 || 1114 2 4335 8191 || 17164 2 3474 0708
Tors 4 0242 9832 || 1°065 | +3447 2785 4767 || 1115 | 3279 1527 8900 || 1165 | *3119 2266 2033
1016 © 4020 8543 || 1:066 3 8329 9219 || 1116 5 8752 7524 || 1166 6 1089 5280
T°0r7 | °3616 7834 9295 || 1:067 © 3908 8054 || 1117 2 6010 3735 || 1°167 2 9944 0138
1018 3 1685 1724 || 1:068 | -3436 9522 0928 || 1:118 | °3269 3300 7207 || 1'168| "3109 8829 6296
I'01g | *3609 5571 5471 || 1:069 3 5169 7497 || 1119 6 0623 7612 |) 17169 6 7746 3442
17020 5 9494 0173 || r:070 © 0851 7419 || 1120 2 7979 4623 || 1'170 3 6694 1266
1'o21 2 3452 5470 3426 I°121 | °3259 5367 7914 | 1171 © 5672 9456
1'022|°3598 7447 1002 1122 6 2788 7158 || 1:172|] "3097 4682 7703
1'023 5 1477 6408 I°I23 3 0242 2031 || 1°173 4 3723 5697
1°024 I 5544 1329 I°I24 | °3249 7728 2206 || 1°174 I 2795 3129
1'025 | °3587 9646 5406 | I°I25 6 5246 7358 || 1°175|°3088 1897 9688
1'026 4 3784 8279 1126 3 2797 7163 || 1176 5 1031 5066
1027 © 7958 9590 1127 © 0381 1296 || 1177 2 0195 8955
1°028 | *3577 2168 8980 1°128 | 3236 7996 9433 || 1°178 | “3078 9391 1046
1'029 3 6414 6093 | 1129 3 5645 1249 || 17179 5 8617 1030
17030 © 0696 0569 I'I30 © 3325 6422 || 1°180 2 7873 8601
T°031 | *3566 5013 2052 I°I31 | °3227 1038 4628 || 1181 | °3069 7161 3451
1°032 2 9366 0186 | 1°132 3 8783 5545 || 17182 6 6479 5272
1033 | "3559 3754 4613 1133 © 6560 8850 || 1183 3 5828 3758
1°034 5 8178 4978 | I°I34|°3217 4370 4220 || 1184 © 5207 8602
1035 2 2638 0925 1135 4 2212 1334 || 1185 | “3057 4617 9499
17036 | 3548 7133 2098 1°136 I 0085 9870 || 17186 4 4058 6141
1037 5 1663 8142 1°137 | "3207 7991 9507 || 17187 I 3529 8225
17038 I 6229 8704 1°138 4 5929 9924 || 17188 | “3048 3031 5443
17039 | 3538 0831 3427 1°I39 I 3900 o8or || 1°189 5 2563 7492 |
1"040 4 5468 1959 1'r40| 3198 1902 1816 || r°190 2 2126 4067
1‘O4I I O140 3945 I'I4t 4 9936 2650 || 1°191 | *3039 1719 4863
| 1°042| °3527 4847 9033 1°42 I 8002 2984 || 1°192 6 1342 9576
1°043 3.9590 6870 1°143 | *3188 6100 2498 || 1°193 3 0996 7902
1°044 © 4368 7102 I'I44 5 4230 0873 || 1-194 © 0680 9539
1°045 | "3516 9181 9378 I'I45 2 2391 7790 || 1°195 | °3027 0395 4182
17046 3 4030 3346 | 1°146 | *3179 0585 2931 || 1°196 4 O140 1530
1'047 | °3509 8913 8654 I'I47 5 8810 5978 || 1'197 © 9915 1278
| 17048 6 3832 4952 1'148 2 7067 6613 || 1198] "3017 9720 3126
1'049 | 2 8786 1887 Ts 3169 1 9555 6771

MR F. W. NEWMAN’S TABLE

[1:000—1 199]

[1:200—1'399]

OF THE DESCENDING EXPONENTIAL.

157

a“ e-#

1°200 | "3011 9421
I*201 | °3008 9316
I'202 5 9242
I'203 2 9198
1'204 | "2999 9184

I'205 6 9199
1'206 3 9245
1°207 © 9321
1°208 | °2987 9427
I'209 4 9562
I‘2I0 I 9727
I'21I | ‘2978 9923

6 0148
1213 3 0402
I'214 ° 0687
1°215|°2967 1oor
1216 4 1345
T2077 I 1718
1°218|°2958 2121

alg 5 2554
I'220 2 3016
1'22I|°2949 3508
1'222 6 4029
1'223 3 4580
I'224 © 5160

1°225 | 2937 577°
4 6408
I 7°77
1228 | ‘2928 7774
1°229 5 8501

1'230 2 9257
1°231 © 0043
I°232|°2917 0857
1-233 4 1701
1234 I 2574

1235 | "2908 3476
1°236 5 4407
1237 2 5367
1°238 | *2899 6356
1239 6 7374
I'240 3 8421

° 9497
1°242 | *2888 0602
1243 5 1736
I'244 2 2899

1'245 ' "2879 4090
6 5311
I°247 3 6560
° 7837
1'249 | ‘2867 9144

1912
8247
5475
3296
1409

9513
7310
4499
o781
5858
943°

1199
0868

8138
2713

4294
2585
7290
8112

4755

6924
4323
6657
3631
4951

0323
9453
2047
7811
6453

7681
1201
6721
S952
2596
2368
2975
4125
5529
6897

7939
8365
78386
6213
3957
8131
1145
1813
9846
4957

e-z

2738
6
3
Co)
2728

0479
1843
3235
4657
6106

7584
gogr
0626
2190
3782

5402
7051
8728
0433
2167

3929
5719
7538
9384
1259
3162
5093
7051
9038
1053

3096
5167
7266
9393
1547

373°
5940
8178
0443
2737

5058
7497
9783
2187
4619
7078
9564
2079
4620
7189
9786
2410
5061
774°
0446

e7-#

‘2725 3179
2

5939

"2719 8727

7 1542
4 4384

I 7253

"2709 0149

6 3°73
3 6023
© goor

2698 2005

5 5°37
2 8095
o 1180

"2687 4293

4 7432
2 0598

‘2679 3791

6 7oIO
4 0257
T 353°
8 6830
6 0156
33599
° 6889
8 0295
5

2

°

3728
7188

, 0674

‘2647 4187

4 7726
2 1291

2639 4883

6 8501
2146

4
I 5817
8 9514
6 3238
3 6988
I 0764

2618 4566

5 8395
3 2250
6130

)
"2608 0037

5 397°
2 7929
© 1914

2597 5925

4 9963

3934
7462
4149
2823
3212

5046
8052
1959
6498
1396

6385
1194
5553
9193
1845

3239
3109
1184
7197
0880
1966
0187
5277
6968
4994
9089
8987
4422
5130
0844
1300
6233
5379
8474
5255
5456
8816
5°71
3958
5215
8580
3791
0586
8703
7881
7860
8379
9176
9993
0570

x“

1°350
I°351
1°352
1353
17354

1355
1°356
1357
1358
1359
1°360
1°361
1°362
1°363
1°364
1°365
1°366
1°367
1°368
1'369

1°370
1°371
1°372
1373
1314

1°375
1°376
1377
1°378
1379
1°380
1°381
1°382
1°383
1°384
1°385
1°386
1°387
1°388
1°389

1°390
1°391
1°392
1393
1394

1395
1°396
1397
1°398

5
3
1)
"2508 2494 8373
5
3
Oo

e-%

2592 4026 0646
2589 8114 9962

7 2229 8260
4 6370 5279
2 0537 0763

"2579 4729 4452

8947 6088
4 3191 5414
7461 2171

I
"2569 1756 6104

6 6077 6953
4 0424 4464
4796 8379

I
poe 9194 8442

3618 4397

3 8067 5988
I 2542 2960

"2548 7042 5057

6 1568 2025
3 6119 3608

I 0695 9553

2538 5297 9604

5 9925 35°09
3 4578 1013
© 9256 1862

"2528 3959 5805

5 8688 2587

3) 3442) £955
o 8221 3659

"2518 3025 7444

7855 3060
2710 0254

7589 8776
7424 8795

2379 9792
7360 1112

"2498 2365 2506

5 7395 3724
3 2450 4516

7530 4632

°
2488 2635 3823

5 7765 1841
3 2919 8437
© 8099 3362

2478 3303 6368

5 8532 7207
3 3786 5631
© 9065 1393

1°399 | 2468 4368 4246

158 MR F. W. NEWMAN’S TABLE [1'400—1'599]

zx ea wv Cau wv Cae xv Ce
ee ee ees
1400] °2465 9696 3942 || 1450] °2345 7028 8094 || 1°500 | *2231 3016 o148 || 1°550| ‘2122 4797 3827
I°401 3 5049 0235 | 1451 3 3583 5052 || 1501) 2229 0714 1516 || 1551 © 3583 1942
I°402 I 0426 2879 || 1°452 I o161 6346 || 1502 6 8434 5791 || 1°552| "2118 2390 2093
1°403 | °2458 5828 1628 || 1:453| °2338 6763 1741 || 17503 4 6177 2750 || 1°553 6 1218 4067 |
"404 6 1254 6234 || 1454 6 3388 1004 || 1504 2 3942 217% |} 1554 4 0067 7654
| 1°405 3 6705 6453 | 17455 4 0036 3901 |] 1°505 © 1729 3832 |} £°555 1 8938 2641
1°406 I 2181 2039 || 1456 I 6708 o199 || 1°506]*2217 9538 7510 || 1°556| ‘2109 7829 8818
1°407 |°2448 7681 2747 || 1457 | ‘2329 3402 9663 || 1°507 5 737° 2983 || 1°557 7 6742 5973
1408 6 3205 8332 || 1-458 7 0121 2062 |] 17508 3 5224 0030 |] 1°558 5 5676 3896
1°49 3 8754 8549 || 1459 4 6862 7161 || 1°509 I 3099 8429 |} 1559 3 4631 2375
I°410 I 4328 3153 || 1460 2 3627 4730 ||1°510| ‘2209 0997 7959 || 1560 I 3607 1201
t'411 | ‘2438 9926 Igor || 1°461 © O415 4535 || 1511 6 8917 8399 || 1°561| "2099 2604 0163
| t°412 6 5548 4548 || 1-462 #2317 7226 6343 || 1512 4 6859 9529 || 1562 7 1621 go5t
| 1°413 4 1195 0850 || 1°463 5 4060 9925 || 1°513 2 4824 1127 || 1°563 5 0660 7655
| I'414 I 6866 0565 || 1464 3 0918 5046 || 17514 © 2810 2973 |} 1°564 2 9720 5765
I°415 | 2429 2561 3448 || 1-465 © 7799 1477 || 1°515| ‘2198 0818 4847 || 1°565 o 8801 3173
1°416 6 8280 9257 || 1°466| 2308 4702 8986 || 17516 5 8848 6530 || 1°566) :2088 7902 9669
1-417 4 4024 7749 || 1467 6 1629 7342 || 1517 3 6900 78or || 1567 6 7025 5044
1418 I 9792 8681 || 1°468 3 8579 6315 || 1518 I 4974 8441 || 1°568 4 6168 gogo
| 1419 | 2419 5585 1811 || 1469 I 5552 5673 || 1519/2189 3070 8231 || 1569 2 5333 1597
| 1°420 7 1401 6897 || 1°470|°2299 2548 5187 || 1520 7 1188 6952 || 1°570 © 4518 2357
| 1-421 4 7242 3697 || 1-471 6 9567 4626 || 1521 4 9328 4384 || 1°571|°2078 3724 1163
| 1422 2 3107 1969 || 17472 4 6609 3761 || 1°522 2 7490 0310 || 1°572 6 2950 7805
1°423 | ‘2409 8996 1472 || 1°473 2 3674 2362 || 1523 © 5673 4511 || 1573 4 2198 2078
1°424 7 4909 1966 || 1474 © 0762 0200 || 17524] 2178 3878 6768 || 1°574 2 1466 3772
| 1°425 5 0846 3208 || 1°475 |°2287 7872 7045 || 1°525 6 2105 6865 || 1°575 © 0755 2681
| 17426 2 6807 4959 || 1°476 5 5006 2670 || 1526 4 0354 4582 || 1°576|-2068 0064 8598
| 1°427 © 2792 6978 || 1477 3 2162 6844 || 1527 1 8624 9703 ||1°577 5 9395 1315
| 1°428|°2397 8801 go25 || 1-478 © 9341 9340 || 1°528} "2169 6917 2010 || 1°578 3 8746 0626
1°429 5 4835 0860 || 1479 |°2278 6543 9929 || 1°529 7 5231 1287 || 1°579 1 8117 6325
| 1°430 3 0892 2244 || 1°480 6 3768 8384 |] 1'°530 5 3566 7316 || 1°580| ‘2059 7509 8205
| 1°431 © 6973 2936 || 1°481 4 1016 4476 || 1°531 3 1923 9880 || 1581 7 6922 6060
1°432 | 2388 3078 2608 || 1482 1 8286 7979 || 1°532 I 0302 8764 || 1°582 5 6355 9684
1433 5 9207 1291 || 1°483 | "2269 5579 8665 || 1°533| ‘2158 8703 3751 || 1583 3 5809 8872
1°434 3 5359 8476 || 1484 7 2895 6306 | 1°534 6 7125 4625 | 1584) x _ 5284 3418
1°435 I 1536 4014 || 1°485 5 0234 0676 || 1535 4 5569 1170 | 1°585 | 2049 4779 3117
1°436 | ‘2378 7736 7668 || 1°486 2 7595 1549 || 1°536 2 4034 3171 || 1°586 7 4294 7764
1°437 6 3960 g200 || 1°487 © 4978 86098 || 1°537 © 2521 O42 || 1'°587 5 3830 7153
1°438 4 0208 8371 || 1°488]°2258 2385 1896 || 1538] 2148 1029 2678 || 1588 3 3387 1081
1439 1 6480 4944 | 1°489 5 9814 0919 || 1°539 5 9558 9755 || 1589 I 2963 9343
1°440 | '2369 2775 8682 || 1490 3 7265 5539 || 1540 3 8110 1427 || 1°590| 2039 2561 1734
1°441 6 9094 9347 || 1491 I 4739 5532 || 1541 I 6682 7481 || 1°59 7 2178 8051
1°442 4 5437 6704 || 1'492|°2249 2236 0673 || 1°542| ‘2139 5276 7701 || 1592 5 1816 8090
1443 2 1804 0515 || 1°493 6 9755 9736 || 1543 7 3892 1874 || 1593 3 1475 1647
1'444 | 2359 8194 0545 || 1494 4 7296 5497 || 1544 5 2528 9786 || 1594 I 1153 8519
| 1445 7 4607 6556 | 1°495 2 4860 4730 || 1°545 3 1187 1223 || 1°595 | "2029 0852 8503
1°446 5 1044 8313 || 1°496 © 2446 8212 || 1546 © 9866 5972 || 17596 7 0572 1395
1°447 2 7505 5581 || 1°497|°2238 0055 5719 || 1°547| ‘2128 8567 3820 || 1°597 5 0311 6993
1448 © 3989 8123 || 17498 5 7686 7026 || 1548) 6 7289 4554 || 17598 3 0071 5094
1'449 "2348 0497 5706 || 1'499 3 5340 1911 || 1'549 4 6032 7960 || 1'599 © 9851 5495
ie eee eee Be RRS 827 2

[r*600—1'799] OF THE DESCENDING EXPONENTIAL. 159

av Cau av Cae av GAY xv Cre
1*600 | "2018 9651 7995 || 1°650] ‘1920 4990 862rx || 1°700| "1826 8352 4053 ||1°750| 1737 7394 3450
1601 6 9472 2392 || 1°651| ‘1918 5795 4705 || 1°70L 5 0093 1840 | 1°751 6 0025 6365
1602 4 9312 8483 || 1°652 6 6619 2647 || 1°702 3 1852 2128 || 1-752 4 2674 2879
1603 2 9173 6067 || 1°653 4 7462 2256 || 1°703 I 3629 4735 || 1°753 2 5340 2821
1604 © 9054 4944 || 1°654 2 8324 3339 || 1°704| ‘1819 5424 9478 || 1°754 © 8023 6016
1°605 |°2008 8955 4911 || 1°655 © 9205 5705 || 1°705 7 7238 6175 || 1°755 | ‘1729 0724 2291
1°606 6 8876 5767 || 1°656| "1909 o105 g164 || 1°706 5 9070 4645 || 1°756 7 3442 1474
1°607 4 8817 7312 || 1°657 7 1025 3523 || 1°7°7 4 0920 4706 || 1°757 5 6177 3391
1°608 2 8778 9345 || 1°658 5 1963 8593 || 17708 2 2788 6175 || 1°758 3 8929 7870
1609 o 8760 1666 || 1659 3 2921 4183 || 1°709 © 4674 8872 || 1°759 2 1699 4738
1610] ‘1998 8761 4075 || 1660 I 3898 oror || 1°710| "1808 6579 2617 || 1°760 © 4486 3823
1611 6 8782 6371 || 1661 | ‘1899 4893 6159 || 1°71 6 8501 7227 || 17761 | 1718 7290 4953
1612 4 8823 8356 || 1°662 7 5908 2166 || 1°72 5 0442 2522 || 17762 7 OLII 7956
1613 2 8884 9828 || 1663 5 6941 7932 || 1°713 3 2400 8322 || 1°763 5 2950 2660
1614 © 8966 0590 || 1°664 3 7994 3267 || 1714 I 4377 4446 || 1-764 3 5805 8893
1615 | "1988 9067 0441 || 1°665 I 9065 7982 || 1°715|°1799 6372 0713 || 1°765 1 8678 6485
1616 6 9187 9182 || 1°666 o 0156 1888 || 1°716 7 8384 6944 || 1°766 0 1568 5263
1617 4 9328 6616 || 1°667| ‘1888 1265 4795 || 1°717 6 0415 2959 || 1°767|°1708 4475 5057
1618 2 9489 2543 || 1°668 6 2393 6516 || 1°718 4 2463 8578 || 1°768 6 7399 5696
1619 © 9669 6765 || 1669 4 3540 6860 |} 1°719 2 4530 3622 || 1-769 5 0340 7009
1°620| 1978 9869 9083 || 1°670 2 4706 5639 || 17720 o 6614 7911 || 1°770 3 3298 8825
1621 7 0089 9301 || 1°671 © 5891 2666 || 1°721/ -1788 8717 1267 || 1°771 I 6274 0975
1622 5 0329 7219 || 1°672|°1878 7094 7751 || 1°722 7 0837 3509 ||1°772| ‘1699 9266 3287
1623 3 0589 2640 || 1°673 6 8317 0708 || 1°723 5 2975 4460 ||1°773 8 2275 5591
1°624 1 0868 5368 || 1°674 4 9558 1347 || 1°724 3 5132 3941 || 1°774 6 5301 7719
1625 | "1969 1167 5204 || 1°675 3 0817 9482 || 1°725 I 7305 1773 || 1°775 4 8344 9499
1626 7 1486 1952 || 1°676 I 2096 4925 |} 1°720) -1779 9496 7778 || 1-776 3 1405 0763
1627 5 1824 5414 || 1°677| ‘1869 3393 7490 || 1°727 8 1706 1778 || 1°777 I 4482 1341
1628 3 2182 5395 || 1°678 7 4709 6988 || 17728 6 3933 3595 || 1°778| ‘1689 7576 1064
1°629 I 2560 1698 || 1°679 5 6044 3233 ||1°729 4 6178 3052 || 1°779 8 0686 9763
1°630 | "1959 2957 4127 || 1°680 3 7397 6039 || 1730 2 8440 9970 || 1°780 6 3814 7269
1°631 7 3374 2485 || 1°681 I 8769 5219 |} 1°731 I 0721 4173 || 1°78 4 6959 3413
1°632 5 3810 6576 || 1°682 © o160 0587 || 1°732| 1769 3019 5483 || 1-782 3 0120 8026
1°633| 3 4266 6206 || 1-683) +1858 1569 1956 |/1°733) 7 5335 3723 || 1°783 I 3299 0940
1634 I 4742 1179 || 1684 6 2996 9141 || 1°734 5 7668 8716 || 1°784| 1679 6494 1988
1635 | 1949 5237 1299 || 1°685 4 4443 1956 || 1°735 4 0020 0286 || 1'785 7 9706 1000
1°636 7 5751 6371 || 1°686 2 5908 o215 ||1°736 2 2388 8257 || 1°786 6 2934 7810
1637 5 6285 6201 || 1°687 © 7391 3734 || 1°737 © 4775 2451 || 1-787 4 6180 2249
1638 3 6839 0594 || 1688] +1848 8893 2326 || 1°738|-1758 7179 2693 || 1°788 2 9442 4150
1°639 I 741 9355 || 1°689 7 0413 5807 || 1°739 6 g600 8807 || 17789 I 2721 3345
1°640 | ‘1939 8004 2291 || 1°690 5 1952 3993 || 1°740 5 2040 0617 || 1°790|°1669 6016 9667
1641 7 8615 9206 || 1-691 3 3509 6698 || 1°741 3 4496 7947 || 1°791 7 9329 2949
1642 5 9246 9908 || 1692 I 5085 3738 || 1°742 I 6971 0623 || 1°792 6 2658 3025
1643 3 9897 4202 || 1°693| 1839 6679 4929 || 1°743| "1749 9462 8468 || 1°793 4 6003 9728
1°644 2 0567 1895 || 1°694 7 8292 0087 ||1°744 8 1972 1307 || 1°794 2 9366 2890
1645 © 1256 2794 || 1695 5 9922 9028 | 1°745 6 4498 8967 || 1-795 I 2745 2346
1646 | 1928 1964 6705 || 1°696 4 1572 1568 || 1°746 4 7043 1271 || 1°796|°1659 6140 7930
1647 6 2692 3436 || 1697 2 3239 7524 || 1°747 2 9604 8046 || 1°797 7 9552 9475
1648 4 3439 2794 || 1°698 © 4925 6712 ||1°748 I 2183 9117 || 1°798 6 2981 6816
1649 2 4205 4586 || 1699 | “1828 6629 8949 || 1°749| 1739 4780 4310 || 1°799 4 6426 9788

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MR F. W. NEWMAN’S TABLE

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161

[2:000—2'199] OF THE DESCENDING EXPONENTIAL.

| wv Cae wv Cm x“ Qe v Cmts
2°000 | 1353 3528 3237 || 2°050| 1287 3490 3588 || 2°100| "1224 5642 8253 || 2°150| ‘1164 8415 74773
2°00% 2 O00% 5599 || 2°051 6 0623 3030 || 2*101 3 3403 3032 || 2°151 3 6773 1838
2°002 o 6488 3161 || 2°052 4 7769 1079 || 2*102 2 1176 0146 || 2°152 2 5142 2271
| 2°003| °1349 2988 5788 | 2°053 3 4927 7605 || 2°103 © 8960 9471 | 2°153 I 3522 8955
2°004 7 9502 3344 || 2°054 2 2099 2481 || 2°104) "1219 6758 0886 || 2-154 © 1915 1774
2°005 6 6029 5696 || 2°055 © 9283 5578 || 2105 8 4567 4269 | 2°155| "1159 0319 0613
2'006 5 2570 2708 || 2°056| ‘1279 6480 6767 || 2°106 7 2388 9497 || 2°156 7 8734 5354
2°007 3 9124 4246 || 2°057 8 3690 5921 || 2107 6 0222 6449 || 2°157 6 7161 5883

“| 2°008 *2 5692 0175 || 2°058 7 2913 2913 || 2°108 4 8068 5004 || 2°158 5 5600 2084
2°009 I 2273 0361 || 2°059 5 8148 7613 || 2"109 3 5926 5039 || 2°159 4 4050 3841
2°010| 1339 8867 4669 || 2°060 4 5396 9895 || 2:110 2 3796 6433 || 2°160 3 2512 1038
2°01 8 5475 2967 || 2:061 3 2657 963x || 2-117 | I 1678 9066 || 2°161 2 0985 3560
2012 7 2096 511g || 2:062 I 9931 6693 || 2°112)| -1209 9573 2815 || 2°162 © 9479 1292 |
2°013 5 8731 0992 | 2063 © 7218 0955 | 2113 8 7479 7560 || 2°163| "1149 7966 4119
2°O14 4 5379 0452 || 2°064| 1269 4517 2280 | ZENE 7 5398 3180 || 2°164 8 6474 1926
2°015 3 2040 3366 || 2°065 8 1829 0568 | 2-115 6 3328 9553 | 2°165 7 4993 4597
2°016 1 8714 9601 || 2°066 6 9153 5665 || 2°116 5 1271 6560 || 2°166 6 3524 2018
2'017 © 5462 9023 || 2°067 5 6490 7454 || 2°117 3 9226 4080 || 2°167 5 2066 4074
2°018 | *§329 2104 1499 || 2°068 4 3840 5808 || 2118 2 7193 1992 || 2°168 4 0620 0652
2°019 7 8818 6895 || 2°069 3 1203 o601 || 2-119 I 5172 0176 || 2°169 2 9185 1635
2°020 6 5546 5080 || 2°070 1 8578 1705 || 2°120 © 3162 8511 || 2°170 I 7761 6910
2021 5 2287 50921 || 2°071 © 5965 8995 || 2°r21| “1199 1165 6879 || 2-171 © 6349 6363
2°022 3 9041 9285 |!2°072| 1259 3366 2345 || 2-122 7 9180 5158 || 2°172) “1139 4948 9879
v | 2023 2 5809 5039 | 2°073 8 0779 1629 || 2-123 6 7207 3229 ||2°173 8 3559 7345
2024 E 2590 3050 || 2°074 6 8204 6720 || 2°124 5 5246 0972 || 2°174 7 2181 3647
2°025| “1319 9384 3188 || 2°075 5 5642 7493 || 2°125 4. 3296 8267 || 2°175 6 0815 3670
2°026 8 6191 5320 || 2°076 4 3093 3823 || 2-126 3 1359 4995 || 2°176 4 9460 2301
2°027 7 391L 9313 || 2°077 3 9556 5584 || 2°127 I 9434 1037 |) 2°177 3 8116 4428 |
2°028 5 9845 5037 || 27078 I 8032 2650 || 2°128 © 7520 6273 || 2:178 2 6783 9935 |
2°029 4 6692 2360 || 2'079 © 5520 4897 || 2°129| 1189 5619 0585 | 2°179 I 5462 8710
2°030 3 3552 1149 || 2°080| “1249 3021 2199 || 2°130 8 3729 3852 || 2180 © 4153 0640
2°031 2 0425 1273 || 2°081 8 0534 4431 || 2-131 7 185i 5957 || 2°18r| 1129 2854 5611
2°032 O 7311 2602 || 2:082 6 8060 1468 || 2°132 5 9985 6781 || 2182 8 1567 3511
2°033 | “1309 4210 5005 || 2083 5 5598 3186 || 2°133 4 8131 6204 || 2°183 7 0291 4226
2°034 8 1122 8349 || 27084 4 3148 9460 || 2°134 3 6289 4109 | 2184 5 9026 7645 |
2°035 6 8048 2504 || 2°085 3 0712 0166 || 2°135 2 4459 0376 | 2-185 4 7773 3654
2°036 5 4986 7340 || 2°086 rt 8287 5178 || 2°136 I 2640 4889 || 2°186 3 6531 2140
2°037 4 1938 2726 || 2°087 © 5875 4374 || 2°137 © 0833 7527 || 2°187 2 5300 2992
27038 2 8902 8531 || 2°088 | “1239 3475 7628 || 2°138] 1178 9038 8174 || 2°188 I 4080 6097
2°039 I 5880 4625 || 2089 8 1088 4817 || 2°39 7 7255 6712 || 27189 © 2872 1343
27040 © 2871 0878 || 2:090 6 8713 5817 || 2°140 6 5484 3022 || 2190] ‘1119 1674 8617
2°041 | “1298 9874 7160 || 2091 5 6351 0504 | 2°141 5 3724 6987 || 2-191 8 0488 7808
2°042 7 6892 3341 || 2-092 4 4000 8755 || 2:142 4 1976 8489 || 2°192 6 9313 8804
27043 6 3920 9290 ||2°093 3 1663 0446 || 2°143 3 0240 7A4II || 2°193 5 8150 1493
27044 5 0963 4879 || 2094 I 9337 5453 || 2°144 1 8516 3635 | 2°194 4 6997 5764
2°045 3 8018 9977 | 2°095 © 7024 3654 | 2°145 © 6803 7044 | 2°195 3 5856 1505
2°046 2 5087 4456 ||2°096|-1229 4723 4925 || 2°146|-1169 5102 7521 || 2°196 2 4725 8604
2°047 1 2168 8185 || 2097 8 2434 9143 || 2-147 8 3413 4950 || 2°197 I 3606 6951
| 2°048 1289 9263 1036 || 2°098 7 0158 6186 (2148 7 1735 9213 | 27198 © 2498 6433
| 2"049 8 6370 2880 ||27098 5 7894 5930 || 2"149 6 0070 0193 | 2°199|/"I109 1401 6941
Vout. XIII. Parr TIL 22

2

162 MR F. W. NEWMAN'S TABLE

x e-= x | e-% x | e-@

|} 2°200 | "1108 0315 8362 2°250 | “1053 22 4562 || 2°300] ‘1002 5884 3723
| 2°201 6 9241 0587 || 2-251 2 9387 8020 | 2-301 | I 5863 4992
(27202, 5 8177 3504 2252) rt 8863 6771 || 2°302 © 5852 6419
| 2°203 | 4 7124 7003 || 2°253 © 8350 o711 | 2°303!) 999 5851 7906
2°204) 3 6083 0973 | 2°254 “1049 7846 9734 | 2°304 8 5860 9350
| 2°205 2 5052 5304 27255) 8 7354 3736 | 2305) . 7 5880 0654
| 2°206 | I 4032 9886 || 2°256 7 6872 2612 || 2°306 6 5909 1716
| 2°207 © 3024 4608 || 2°257 6 6400 6256 || 2°307 5 5948 2437
| 2-208 "1099 2026 9360 || 2°258 5 5939 4565 | 2°308 | 4 5997 2718
27209 | 8 1040 4032 || 2°259 4 5488 7432 || 2°309 | 3 6056 2458
2°210 | 7 0064 8515 || 2-260 3 5048 4755 || 2°310 2 6125 1560
2-211 | 5 9100 2698 || 2°261 2 4618 6428 | 2-311 I 6203 9922
2°212 4 8146 6473 || 2°262 I 4199 2347 || 2°312 © 6292 7447
2°213 3 7203 9729 || 2°263 © 3790 2409 | 2°313| 989 6391 4034
2°214 | 2 6272 2357 || 2°264|-1039 3391 6508 || 2°314 8 6499 9586
2-215 | I 5351 4248 || 2°265 8 3003 4541 || 2°315 7 6618 4002
| 2-216 © 4441 5292 | 2°266 7 2625 6404 || 2°316 6 6746 7185
| 2°217 | "1089 3542 5381 | 2°267 6 2258 1994 || 2°317 5 6884 9035
j2218 8 2654 4405 || 2°268 5 Igor 1206 || 2°318 4 7032 9454
sis} 7 1777 2256 || 2269 4 1554 3937 || 2°319 3 7190 8343
| 2"220 6 og10 8825 || 2°270 3 1218 0083 || 2°320 2 7358 5604
| 27221 | 5 0055 4003 || 2°271 2 0891 9542 || 2°321 I 7536 1139
2°222 | 3 9210 7681 || 2°272 I 0576 2210 || 2°322| © 7723 4849
| 2"223 | 2 8376 9751 (2273 © 0270 7983 || 2°323| 979 7920 6636
| 2-224 | I 7554 O105 || 2°274|*1028 9975 6759 || 2°324 8 8127 6403
2°225| © 6741 8635 || 2°275 7 9690 8435 || 2°325 7 8344 4051
2°226| "1079 5940 5232 || 2°276 6 9416 2908 || 2°326 6 8570 9482
2°227| 8 5149 9789 || 2°277 5 9152 0075 || 2°327 5 8807 2599
2°228 | 7 437° 2197 || 2°278 4 8897 9834 || 2°328 4 9953 3304
2°229 6 3601 2348 || 2°279 3 8654 2082 || 2°329 3 9309 I500
2°230 5 2843 0136 | 2°280 2 8420 6716 || 2°330 2 9574 7089
2°231 4 2095 5452 || 2°281 1 8107 3634 | 2°331 t 9849 9974
2°232 31358 8189 || 2°282 © 7984 2734 || 2°332 I 0135 0057
2°233 2 0632 8240 || 2°283| ‘1019 7781 3914 || 2°333 © 0429 7241
(2 234 © 9917 5497 ||2 284 8 7588 7072 | 2°334 969 0734 1430
| 2°235 ‘1069 9212 9853 || 2°285 7 7406 2106 || 2°335 8 1048 2526
| 2°236 | 8 8519 1201 || 2°286 6 7233 8914 || 2°336| 7 1372 0432
2°237 | 7 7835 9435 || 2°287 5 7°71 7394 ||2°337| 6 1705 5053
2°238 6 7163 4447 || 2°288 4 6919 7445 || 2°338 5 2048 6290
2°239 5 6501 6130 || 2°289 3 6777 8966 | 2°339 | 4 2401 4048
2°240 4 585° 4379 || 2 290 | 2 6646 1854 || 2°340 3 2763 8230
2°241 3 5209 9086 || 2291 1 6524 6008 || 2°341 2 3135 8740
2°242 2 4580 O145 || 2°292 © 6413 1328 || 2°342 I 3517 5480
2°243 I 3960 7450 || 2°293| ‘1009 6311 7712 || 2°343 © 3908 8356
2°244) © 3352 0895 || 2294) © 8 6220 5059 || 2°344.| 959 4309 7272
2°245 “1059 2754 0373 | 2°295 7 6139 3268 | 2°345| 8 4720 2130
2°246 8 2166 5779 || 2°296 6 6068 2239 || 2°346 7 5140 2835
2°247 | 7 1589 7006 || 2°297 5 6007 1870 || 2°347 6 5569 9292
2°248 | 6 1023 3950 || 2°298 4 5956 2062 || 2°348 5 6009 1405
2°249 5 0467 6503 | 2299| 3 5915 2713 | 2°349 4 6457 9078

[2'200—2 399]

Xo)
is
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2°400—2'599| OF THE DESCENDING EXPONENTIAL. 163

we e-% x Ce% xv Pet v Cae

= =|

2400] 907 1795 3289 ||2°450| 862 9358 6499 ||2°500| 820 8499 8624 || 2°550| 780 8166 6oor
2°401 6 2728 0679 || 2°451 2 0733 6045 || 2°50 © 0295 4654 || 2°551 © 0362 3363
2°402 5 3669 8697 || 2°452 I 2117 1798 ||2°502| 819 2099 2687 ||2°552) 779 2565 8728
2°403 4 4620 7252 | 2°453 © 3509 3673 || 2°503 8 3911 2641 || 2°553 8 4777 2019
2°404 3 5580 6253 || 2°454) 859 4910 1582 | 2°504 7 5731 4435 | 2554, 7 6996 3158
27405 2 6549 5609 || 2°455 8 6349 5441 | 2'505 6 7559 7985 ||2°555| 6 9223 2067
2°406 I 7527 5231 | 2456 7 7737 5163 || 2506 5 9396 3212 ||2°556| 6 1457 8669
2°407 © 8514 5029 || 2°457 6 9164 0662 || 2°507 5 124i 0032 |] 2°557 5 3700 2884
2°408) 99 9510 4911 | 2458 6 0599 1853 || 2508 4 3093 8364 || 27558 4 5950 4637
2°409 9 9515 4789 | 2°459 5 2042 8650 || 2°509 3 4954 8128 ||2°559| 3 8208 3849
2°410 8 1529 4572 || 2°460 4 3495 0967 || 2'510 2 0823 9241 | 2560, 3 0474 0443
2°41 7 2552 4170 || 2°46 3 4955 8719 || 2°51 I 8701 1622 || 2°561 | 2 2747 4342
2-412 6 3584 3494 || 2°462 2 6425 1821 || 2°512 I 0586 5191 || 2°562 I 5028 5469
2°413 5 4625 2453 || 2°463 I 7903 0187 || 2°513 © 2479 9865 || 2°563 © 7317 3746
2°414 4 5675 0959 | 2°464 © 9389 3732 ||2°514| So9 4381 5564 | 2°564| 769 9613 9096
2°415 3 6733 8921 || 2°465 © 0884 2371 || 2°515 8 6291 2207 || 2°565 9 1918 1442
2°416 2 7801 6251 |} 2°466| 849 2387 6org || 2°516 7 8208 9713 || 2°566 8 4230 0707
2°417 I 8878 2859 || 2°467 8 3899 4591 || 2°517 7 0134 8001 || 2°567 7 6549 6815
2°418 © 9963 $656 || 2°468 7 5419 8002 || 27518 6 2068 6990 || 2°568 6 8876 9688
2°419 © 1058 3552 || 2469 6 6948 6167 || 2°519 5 4010 6600 || 2°569 6 1211 9250 |
2°420| 889 2161 7459 || 2°470 5 8485 goor | 2°520' 4 5960 6750 || 2°570 5 3554 5424 |
2°421 8 3274 0288 || 2-471 5 0031 6420 || 2°521 3 7918 7360 |] 2°571 4 5904 8134
2°422 7 4395 1949 || 2°472 4 1585 8340 | 2°522 2 9884 8349 || 2°572 3 8262 7303
2°423 6 5525 2354 || 2°473 3 3148 4676 || 2°523 2 1858 9636 || 2°573 3 0628 2854
2°424 5 6664 1415 || 2°474 2 4719 5343 || 2°524 I 3841 1142 || 2°574 2 3001 4712
2°425 4 7811 9042 || 2°475 I 6299 0257 || 2°525 © 5831 2787 || 2°575 I 5382 2799
2°426 3 8968 5147 || 2°476 © 7886 9334 || 2°526| 799 7829 4490 || 2°576 © 777° 7040
2°427 3 0133 9642 || 2°477| 839 9483 2490 || 2°527 8 9835 617% || 2°577 © 0166 7360
2°428 2 1308 2438 || 2°478 9 1087 9641 | 2°528 8 1849 7751 ||2°578| 759 2570 3680
2°429 I 2491 3448 || 2°479 8 2701 0703 | 2°529 7 3872 9149 || 2°579 8 4981 5927
| 2°430 © 3683 2582 || 2-480 7 4322 5592 || 2°530 6 5902 0286 || 2°580 7 7400 4023
2°431| 879 4883 9753 || 2°481 6 5952 4224 | 2°531 5 7940 1082 || 2°581 6 9826 7894
2°432 8 6093 4873 || 2°482 5 759° 6516 || 2°532 4 9986 1457 || 2°582 6 2260 7462
2°433 7 7311 7854 || 2°483 4 9237 2383 ||2'533 4 2040 1333 || 2°583 5 4702 2653
2°434 6 8538 8608 || 2°484 4 0892 1743 | 2°534 3 4102 0628 || 2°584 4 7I5I 3392
2°435 5 9774 7048 || 2°485 3 2555 4512 | 2°535 2 6171 9265 || 2°585 3 9607 9601
2°436 5 1019 3085 || 2°486 2 4227 0606 || 2°536 1 8249 7163 || 2°586 3 2072 1207
2°437 4 2272 6632 || 2487 T 5906 9943 || 2°537 I 0335 4244 || 2°587 2 4543 8134
2°438 3 3534 7603 || 2°488 © 7595 2439 | 2°538 © 2429 0429 || 2°588 I 7023 0306
2°439 2 4805 5908 || 2°489} 829 g291 8orr | 2°539| 789 4530 5637 || 2°589 © 9509 7648
2°440 I 6085 1462 || 2°490 9 og 46 6575 || 2°540 8 6639 9791 || 2°590 © 2004 0085
2°441 © 7373 4176 || 2-491 8 2709 8049 || 2°541 7 8757 2811 || 2°591| 749 4505 7543
2°442| 869 8670 3964 || 2-492 7 4431 2351 | 2°542 7 0882 4619 || 2°592 8 7014 9945
2°443 8 9976 0739 || 2°493 6 6160 9397 || 2°543 6 3015 5136 | 2°593 7 9531 7218
2°444 8 1290 4414 || 2°494 5 7898 gt05 || 2°544 5 5156 4282 | 2°594 7 2055 9286
2°445 7 2613 4902 || 2-495 4 9645 1392 || 2°545 4 7395 198r || 2°595 6 4587 6074
2446 6 3945 2115 | 2°496 4 1399 6175 || 2°546 3 9461 8152 | 2°596 5 7126 7509
2°447 5 5285 5969 | 2°497 3 3162 3372 || 2°547 3 1626 2718 || 2°597 4 9673 3514
2°448 4 6634 6375 || 2°498 2 4933 2901 || 2°548 2 3798 5601 | 27598 4 2227 4017
2°449 3 7992 3247 2-499 I 6712 4679 || 2°549 I 5978 6721 || 2°599 3 4788 8942

22—2,

164 MR F. W. NEWMAN'S TABLE [2°600—2"799}

|

xv | Ca” wv (At? xv (G3 } wv Cs”
2600! 742 7357 S214 || 2°650| 706 5121 3060 |/2°700] 672 0551 2740 | 2750! 639 2786 1207
2°601 I 9934 1760 || 2°651 5 8059 7161 || 2°70r I 3834 0818 ||2-751 8 6396 5299
2°602 I 2517 9506 || 2°652 5 1005 1843 || 2°702 © 7123 6035 | 2°752 8 0013 3255
2°603 © 5109 1376 |} 2°653 4 3957 7034 || 2°703 © 0419 8324 || 2°753 7 3636 Sori
|2°604| 739 7797 7298 2°654 3 6917 2665 || 2°704] 669 3722 7616 | 2-754 6 7266 0504
| 2°605 9 0313 7197 || 2°655 2 9883 8665 || 2°705 8 7032 3846 2-755 6 ogo1 9669
| 2°606 8 2927 0999 || 2°656 2 2857 4964 | 2°706| 48 0348 6947 || 2°756 5 4544 2443
| 2°607 7 5547 8631 || 2°657 1 5838 1492 || 2°707 7 3671 6850 || 2-757 4 8192 8763
| 2°608 6 8176 oo17 || 2°658 o 8825 8178 || 2°708 6 7oor 3491 |} 2°758 4 1847 8564
| 2°609 6 o811 5086 || 2°659 © 1820 4952 || 2°709 6 0387 6801 || 2°759 3 5509 1784 |
2°610 5 3454 3763 | 2°660| G99 4822 1745 | 2°710 5 3680 6715 |] 2760 2 9176 8360
2-611 4 6104 5974 || 2°661 8 7830 8485 || 2°711 4 7030 3166 || 2-761 2 2850 8227
2°612 3 8762 1647 || 2°662 8 0846 5105 || 2°712 4 0386 6086 || 2762 I 6531 1323
2°613 3 1427 0707 || 2°663 7 3869 1532 | 2°713 3 3749 5411 || 2°763 I 0217 7583
2°614 2 4099 3081 || 2°664 6 6898 7698 || 2°714 2 711g 1074 || 2°764 © 3910 6946
2°615 16778 8696 || 2°665 § 9935 3533 || 2775 2 0495 3007 || 2°765| 629 7609 9348
2°616| _ © 9465 7479 || 2°666 5 2978 8968 || 2°716 I 3878 1146 || 2°766 9 1315 4727
2°617 © 2159 9356 || 2°667 4 6029 3932 || 2°717 © 7267 5423 || 2°767 8 5027 3018
|2°618| 729 4861 4256 || 2°668 3 9086 8357 || 2°718 © 0663 5773 || 2°768 7 8745 4159
| 2°619 8 7570 2104 || 2°669 3 2151 2172 ||2°719| 659 4066 2130 || 2-769 7 2469 8089
2°620 8 0286 2827 || 2°670 2 5222 5309 || 2°720 8 7475 4426 || 2°770 6 6200 4742
| 2°621 7 3009 6354 || 2°671 1 8300 7698 || 2°721 8 o891 2598 || 2°771 5 9937 4058
| 2°622 6 5740 2611 || 2°672 I 1385 9271 || 2°722 7 4313 6579 || 2°772 5 3680 5973
| 2°623 5 8478 1525 | 2°673 © 4477 9957 ||2°723| 6 7742 6303 || 2°773 4 743° 0425
2°624 5 1223 3023 ||2°674| 689 7576 9688 |) 2°724 6 1178 1705 || 2°774 4 1185 7352
2°625 4 3975 7934 || 2°675 9 0682 8394 || 2°725 5 4620 2718 || 2°75 3 4947 6690
2°626 3 6735 3485 || 2°676 8 3795 6008 || 2°726 4 8068 9278 || 2°776 2 8715 8377
2°627 2 9502 2303 || 2°677 7 6915 2459 || 2°727 4 1524 1318 || 2°777 2 2490 2352
2°628 2 2276 3417 || 2°678 7 0041 7680 || 2°728 3.4985 8773 || 2°778 1 6270 8552
2°629 I 5057 6752 || 2°679 6° 3175 60x || 2°729 2 8454 1578 | 2°779 I 0057 6914
2°630 © 7846 2239 || 2°680 5 6315 4154 || 2°730 2 1928 9668 || 2:780 © 3850 7377
2°631 © 0641 9804 || 2°681 4.9462 5270 || 2°731 I 5410 2977 ||2°781| 619 7649 9879
2°632| 719 3444 9375 || 27682 4 2616 4881 || 2°732 o 8898 1441 || 2°782| . 9 1455 4357
2°633 8 6255 0881 || 2°683 3.5777 2918 || 2°733 © 2392 4993 || 2°783 8 5267 0750
2°634 7 9972 4250 | 2°684 2 8944 9312 || 2°734| 649 5893 3568 || 2-784 7 9084 8995
2°635 7 1896 9409 || 2°685 2 2119 3996 || 2°735 8 9400 7104 | 2°785 7 2908 9031
2°636 6 4728 6286 || 2-636 I 5300 6gor || 2°736 8 2914 5533 | 2°786 6 6739 0796
| 2°637 5 7567 4812 || 2°687 © 8488 7959 || 2°737 7 6434 8792 | 2°787 6 0575 4229
| 2°638 5 0413 4913 || 2°688 © 1683 7103 || 2°738 6 9961 6814 | 2°788 5 4417 9267
| 2°639 4 3266 6518 || 2°689| 679 4885 4263 || 2°739 6 3494 9536 || 2-789 4 8266 5850
| 77640 3 6126 9556 || 2°690 8 8093 9372 || 2°740 5 7034 6893 || 2°790 4 2121 3915
2°641 2 8994 3955 || 2°691 8 1309 2361 || 2°741 5 0580 8820 || 2°791 3 5982 3402
2°642 2 1868 9644 || 2°692 7 4531 3164 || 2°742 4 4133 5254 || 2°792 2 9849 4248
2°643 I 4750 6552 || 2°693 6 7760 1712 || 2°743 3 7692 6129 ||2°793|, 2 3722 6392 |
2°644 © 7639 4607 || 2°694 6 0995 7938 || 2°744 3 1258 1380 || 2-794 I 7601 9774
2°645 © 0535 3739 | 2°695 5 4238 1774 || 2°745 2 4830 0944 | 2°795 1 1487 4332
2°646| 709 3438 3876 || 2°696 4 7487 3152 || 2°746 r 8408 4757 | 2°796 © 5379 9005
2°647 8 6348 4948 || 2°697 4 0743 2005 || 2°747 I 1993 2753 ||2°797| 609 9276 6732
2°648 7 9265 6883 || 2°608 3 4005 8266 || 2°748 © 5584 4870 | 2°798 9g 3180 4451
\ecianes 7 2189 9611 || 2°699 2 7275 1866 ||2°749| 639 9182 1042 | 2-799 8 7090 3103
as) a ee | : Jes :

[2'800—2'999|

HRwwbhn Oana

OOHwN

RPNWwWP BODAAT WHMO O OHH NW WHEN D Ar OW

= Se a Se ee ea a ee ee ee ee ee Se ee eee eee

2625
2957,

4038
5809 |

8207

1172

4644
8563

2867
7497

2392
TAQ2
2737
8067

3421
8740

3964 |

032 |

3886
8464
2709
6559
9956
2839
5150
6829
7817

8054 ||

7481
6039

3669

0312 ||

59°99
O401
3728
5833
6657
6141
4225
0853

5964
9502
1406
1619
0083

6740
1531

4398 |

5283
4128

o1
sr
ao

COOH NN WHWHhABN DAQYS

HNWWE HAN AD 31 HCO O

OHHNN WHHEMNMNM DWA~IANI CO OO OOF

DESCENDING EXPONENTIAL, 165
ms =
C= | & coz le pay, |
4432 0875 I 900] 550 2322 0056]|2°950] 523 3970 5948
8650 5467 ||2°901} 549 6822 4338) 2-951 2 8739 2403
2874 7845 || 2°902 9 1328 3589] 2°952 2 3513 1146
7104 7952 || 2°903 8 5839 7753)| 2°953 1 8292 2124
1340 5730 || 2°904 | 8 0356 6775|!| 2°954 I 3076 5284
5582 II2I || 2°905 7 4879 0601'| 2°955 © 7866 9575
9829 4068 | 2°906 6 9406 9176)| 2°956 © 2660 7945
4082 4513 | 29907; 6 3940 2445//2°957| 519 7460 7342
8341 2399 | 2°908 5 8479 0353] 2°958 9 2265 8713
2605 7669 | 2°909 | 5 3023 2846]| 2-959 8 7076 2007
6876 0265 || 2910) 4 7572 9869) 2°960 8 1891 7172
II52 0130 |] 2-911 | 4 2128 1368) 2-961 7 6712 4156
5433 7206 || 2°912 | 3 6688 7288)| 2-962 7 1538 2907
9721 1436 | 2°913 3 1254 7575| 2°963 6 6369 3373
4014 2764 || 2 914 | 2 5826 2175|| 2°964 6 1205 5502
8313 1132 || 2°915 2 0403 1033]|| 2°965 5 6046 9244
2617 6483 || 2°916 I 4985 4095) 2°966 5 0893 4547
6927 8760 || 3°917 2 9573 1307) 2°967 4 5745 1358
1243 7906 || 2°918} © 4166 2614]| 2°968 4 o601 9627
5565 3865 | 2°919| 539 8764 7963] 2°969 3 5463 9302
9892 6580 || 2920} 9 3368 7300]| 2°970 3 0331 0331
4225 5993 || 2°921 | 8 7978 0571|| 2°971 2 5203 2664
8564 2049 || 2°922 8 2592 7721\| 2°972 2 0080 6249
2908 4690 |2°923/ 7 7212 8697] 2°973 I 4963 1035
7258 3861 ||2°924| 7 1838 3446] 2°974 © 9850 6970 |
1613 9504 || 2°925 6 6469 1912) 2°975 © 4743 4004
5975 1563 || 2926 6 1105 4044|/2°976| 509 9641 2085
0341 9982 | 2°927 5 5746 9786|| 2°977 9 4544 1163 |
4714 4704 || 2°928 5 0393 9086]| 2°978 8 9452 1186
9092 5674 || 2°929 4 5046 1890]| 2°979 8 4365 2104
3476 2834 || 2°930 3 9703 8145]|| 2980 7 9283 3865
7865 6129 || 2°931 3 4366 7797) 2°981 7 4206 6419
2260 5503 || 2932 2 9035 0792) 2°982 6 9134 9715
6661 ogoo0 |! 2°933 2 3708 7077|| 2°983 6 4068 3703
1067 2263 || 2°934 I 8387 6600)| 2°984 5 9006 8331
5478 9536 || 2°935 I 3071 9306) 2°985 5 395° 3549
9896 2665 || 2°936 © 7761 5143]| 2°986 4 8898 9307
4319 1593 |2°937| 9 2456 4058) 2°987 4 3852 5554
8747 6263 || 2°938| 529 7156 5998|| 2°988 3 8811 2240
3181 6621 || 2°939 9g 1862 ogog|| 2°989 3 3774 9313
7621 2611 || 2°940 8 6572 8738|| 2°990 2 8743 6724
2066 4178 || 2°941 8 1288 9434|| 2991 23717 4422
6517 1265 || 2-942 | 7 GoIo 2942]|| 2-992 1 8696 2358
0973 3817 || 2 943 | 7 0736 9210|| 2°993 I 3680 0481
5435 1779 || 2°944 6 5468 8186) 2-994 o 8668 8741
9902 5095 || 2°9045 6 0205 9816] 2°995 © 3662 7087
4375 3710 || 2°946 5 4948 4048] 2°996| 499 8661 5470
8853 7568 || 2-947 4 9696 0830) 2°997 9 3665 3839
3337 6616 || 2:948 4 4449 O10} 2:998 8 8674 2146
7827 0797 || 2949 3 9207 1833} 2°999 8 3688 0339

166 MR F. W, NEWMAN'S TABLE [3000 —3'199]
= - = ae eee e

12s ea x Cs wv Cnt xv Cr

| 3000! 497 8706 8368 | 3°050| 473 5892 4391 |/3°100| 450 4920 2393 ||3°150| 428 5212 6867
3001 7 3730 6185 | 37051 3 1158 9138 || 3°01 © 0417 5708 || 37151 8 0929 6159
| 37002 | 6 8759 3739 || 3°052 2 6430 1197 ||3°102) 449 5919 4027 ||/3°152 7 6650 8260
| 3°003| 6 3793 0981 || 37053 2 1706 0520 || 3°103 9 1425 7305 || 3°153 7 2376 3128
37004 5 8831 7860 || 3:054 1 6986 7060 || 3104 8 6936 5497 | 3°54 6 8106 0720
37005 5 3875 4328 |} 3:055 I 2272 0770 || 3°105 8 2451 8559 || 3°155 6 3840 0992
| 37006 4 8924 0335 || 3°056 © 7562 1603 || 3°106 7 797% 6445 || 37156 5 9578 3903
| 3°07 4 3977 5831 || 3:057 © 2856 9511 || 3°107 7 3495 Q9EIX |) 3157 5 5329 94To
37008 | 3 9036 0767 || 3°058| 469 8156 4448 || 3:108 6 9024 6512 || 3°158 5 1067 7470
3°009 3 4299 5093 || 3°059| 9 3460 6367 || 3109 6 4557 8603 || 3159 4 6818 8041
3010} 2 9167 8760 || 3°060 8 8769 5220 || 3°110 6 0095 5340 || 3°160 4 2574 1080
3011 | 2 4241 1719 || 3°061 8 4083 o961 || 3°111 5 5637 6678 || 3:16 3 8333 6545
3012 I 9319 3920 || 37062 7 9401 3542 || 3112 5 1184 2572 || 3162 3 4°97 4393
| 37013 | I 4402 5315 || 3°063 7 4724 2918 || 3°113 4 6735 2978 || 3°163 2 9865 4582
314 © 9490 5853 || 3064 7 0051 9041 || 3114 4 2290 7851 || 3164 2 5637 7069
3015 © 4583 5487 || 3°065 6 5384 1864 || 3115 3 7850 7147 || 37165 2 1414 1813
|3°016| 489 9681 4166 || 3:066 6 0721 1342 || 3°116 3 3415 0822 || 3166 I 7194 8772
3017 9 4784 1842 || 3:067 5 6062 7426 || 3°117 2 8983 8831 || 3167 I 2979 7902
37018 8 9891 8466 || 3°068 5 1409 0071 || 3°118 2 4557 1130 || 3°168 o 8768 9162
37019 8 5004 3989 || 3:069 4 6759 9230 || 3°119 2 0134 7674 || 3°169 © 4562 2510
| 37020 8 or21 8362 || 3:070 4 2115 4857 || 3120 I 5716 8420 || 3170 © 0359 7903
| 37021 7 5244 1536 || 3°071 3 7475 6905 || 3°121 I 1303 3322 | 3°171| 419 6161 5300
3°022 7 0371 3463 || 3072 3 2840 5328 || 3122 © 6894 2338 || 37172 9 1967 4658
3'023 6 5503 4093 || 3°073 2 8210 0079 || 3°123 © 2489 5423 || 3173 8 7777 5937
| 3°024 6 0640 3378 || 3°074 2 3584 1112 |/3°124| 439 8089 2533 | 3°174 8 3591 9093
3025 5 5782 1270 | 3°075 1 8962 8381 | 3-125 9 3693 3623 || 3°175 7 9410 4084
37026 5 0928 7720 || 3:076 I 4346 1840 |] 3°126 8 9301 8651 || 3°176 7 5233 0870
| 3°027 4 6080 2679 || 3:077 © 9734 1442 || 3°127 8 4914 7571 || 3°177 7 1059 9409
[ses 4 1236 6098 || 3°078 © 5126 7142 || 3°128 8 0532 0341 || 3178 6 6890 9657
37029 3 6397 793° || 3079 © 0523 8893 || 3'129 7 6153 6916 || 3179 6 2726 1575
“yr030 3 1563 $126 || 3080] 459 5925 6649 || 3°130 7 1779 7253 || 3°180 5 8565 5121
| 37031 2 6734 6638 || 3081 9 1332 0364 || 3°131 6 7410 1307 || 3°181 5 4409 0251
| 3°032 2 I9IO 3417 || 3:082 8 6742 9993 || 3°132 6 3044 9035 || 3°182 5 0256 6926
| 3°033 I 7090 8415 | 3°083 8 2158 5489 || 3133 5 8684 0394 || 3°183 4 6108 5104
| 3°034 I 2276 1584 | 3°084 7 7578 6807 || 3134 5 4327 5340 || 3°184 4 1964 4742
' 37035} © 7466 2875 || 37085 7 3903 3900 || 3°135 4 9975 3829 || 3185 3 7824 5801
3036 | © 2661 2242 || 3:086 6 8432 6724 || 3°136 4 5627 5818 || 3°186 3 3688 8238
|3°°37| 479 7860 9635 || 3:087 6 3866 5231 || 3°137 4 1284 1263 || 3°187 2 9557 2011
3°038| = 9 3065 5007 || 3°088 5 9304 9378 || 3°138 3 6945 ora || 3°188 2 5429 7079
3'°039| 8 8274 8309 || 37089 5 4747 9118 | 37139 3 2610 2348 || 3°189 2 1306 3402
3°040 8 3488 9494 || 3°090 5 ©195 4405 || 3°140 2 8279 7902 || 3°190 r 7187 0939
3/041 | 7 8707 8514 || 3°091 4 5647 5194 | 3°141 2 3953 6738 | 3°191 I 3071 9647
3°042| = 7 3931 5321 || 37092 4 1104 1439 || 3°142 1 9631 8814 | 37192 o 8960 9486
3043) 6 9159 9867 || 3°093 3 6565 3096 || 3°143 I 5314 4086 | 3°193 © 4854 0414
| 37044 6 4393 2105 | 3°094 3 2031 o118 || 3°144 I IOOI 2511 || 3°194 © 0751 2391
3°045 5 9631 1987 || 3°095 2 7501 2460 | 3°145 © 6692 4047 | 3°195| 409 6652 5376
37046 5 4873 9465 ||3'096 2 2976 0078 || 3°146 © 2387 8649 | 3°196 9 2557 9327
3°047 5 C121 4493 || 3°097 x 8455 2925 ||3°'147| 429 8087 6275 || 3°197 8 8467 4204
3°048 4 5373 70°21 || 37098 I 3939 0957 || 3148 9 3791 6882 || 3°198 8 4380 9965
3°049 4 0639 7003 || 37099 © 9427 4128 || 3'149 8 9500 0427 | 3°199 8 0298 6570

[3'200—3°399] OF THE DESCENDING EXPONENTIAL. 167

|

| 2 Cae Zz Cm | v Cm He Cae

3200] 407 6220 3978 ||3°250] 387 7420 7832 ||3°300] 368 8316 7401 eee 350 8435 4101
3201 7 2146 2149 || 3°251 7 3545 3005 || 3°30r 8 4630 2669 || 3°351 © 4928 7284
3202 6 8076 rogr || 3°252 6 9673 6913 || 3°302 | 8 0947 4783 | 3°352 © 1425 5515
3°203 6 4010 0613 || 3°253 6 5805 9518 || 3°303) 7 7268 3707 | 3°353| 349 7925 8761
3'204 5 9948 0826 || 3-254 6 1942 078r || 3°304 | 7 3592 9404 | 3°354 9 4429 6986
3°205 5 5890 1638 || 3°255 5 8082 0663 || 3°305 6 9921 1836 | 3°355 9 0937 0155
3206 5 1836 3009 || 3°256 5 4225 9127 || 3°306 6 6253 0968 | 3°356 8 7447 8234
3°207 4 7786 4899 || 3°257 5 0373 6132 || 3°307 6 2588 6762 | 3°357 8 3962 1187
3°208 4 3740 7266 || 3°258 4 6525 1642 || 3°308 5 8927 9182 || 3°358 8 0479 8980
3°209 3 9699 0071 |) 3°259 4 2680 5616 || 3°309 5 5270 8192 || 3°359 7 7001 1578
37210 3 5061 3272 || 3°260 3 8839 8017 || 3°310 5 1617 3754 || 3°360 7 3525 8945
Beer 3 1627 6831 || 3261 3 5002 8807 || 3°311 4 7967 5832 || 3°361 7 0054 1048
37212 2 7598 0705 || 3°262 3 1169 7947 || 3°312 4 4321 4390 || 3°362 6 6585 7851
3°213 2 3572 4856 |) 3°263 2 7349 5398 || 3°313 4 0678 9391 || 3°363 6 3120 9320
3°214 I 9550 9242 || 3°264 23515 1123 || 3°314 3 7°49 0799 | 3°364 5 9659 5421
3°215 I 5533 3824 || 3°265 I 9693 5083 || 3°315 3 3404 8577 | 3°365 5 6201 6118
3'216 I 1519 8561 || 3°266 I 5875 7240 || 3°316 2 9773 2690 || 3°366 5 2747 1377
3217 © 7510 3413 || 3°267 I 2061 7556 || 3°317 2 6145 3100 || 3°367 4 9296 1164
3218 © 3504 8341 |] 3°268 © 8251 5993 || 37318 2 2520 9771 || 3°368 4 5848 5444
3219] 399 9503 3304 || 3°269 © 4445 2512 || 3°319 I 8900 2668 | 3°369 4 2404 4182
37220 9 5505 826r || 3°270 © 0642 7075 || 3°320 I 5283 1754 || 3°37¢ 3 8963 7343
3°221 9 1512 3174 13271] 379 6843 9645 || 3°321 1 1669 6993 || 3°371 3 5526 4895
3°222 8 7522 8001 || 3:272 9 3049 0183 || 3°322 © 8059 8348 || 3°372 3 2092 6802
3°223 8 3537 2704 || 3°273 8 9257 8652 || 3°323 © 4453 5784 || 3°373 2 8662 3030
3°224 7 9555 7242 || 3°274 8 547° 5013 || 3°324 © 0850 9264 || 3°374 2 5235 3545
3225 7 5578 1576 |] 3°275 8 1686 9229 || 3°325] 359 7251 8753 || 3°375 2 1811 8312
3°226 7 1604 5666 || 3°276 7 7907 1262 || 3°326 9 3656 4215 || 3°376 1 8391 7297
3°227 6 7634 9472 || 3277 7 4131 1074 || 3°327 9 0064 5613 | 3°377 I 4975 0465
3°228 6 3669 2954 ||3°278 7 0358 8627 || 3°328 8 6476 2912 || 3°378 IT 1561 7783
3°229 5 9707 6073 || 3°279 6 6590 3884 || 3°329 8 2891 6975 | 3°379 o 8151 g218
3°230 5 5749 8788 || 3°280 6 2825 6807 || 3°330 7 9319 5067 || 3°380 © 4745 4734
3231 5 1796 1062 || 3-281 5 9064 7358 || 3°331 7 5732 9853 || 3°381 © 1342 4298
3232 4 7846 2853 || 3°282 5 5307 5500 || 3°332 7 2159 0396 | 3°382]| 339 7942 7875
3°233 4 3900 4123 |] 3°283 5 1554 1195 || 3°333 6 8588 6660 || 3°383 9 4546 5431
3234 3 9958 4832 || 3°284 4 7804 4405 || 3°334 6 5021 8610 | 3°384 9 1153 6933
3°235 3 6020 4940 || 3°285 4 4058 5093 || 3°335 6 1458 6211 || 3°385 8 7764 2346
3°236 3 2086 4409 || 3°286 4 0316 3222 || 3°336 5 7898 9426 || 3°386 8 4378 1636
3°237 2 8156 3198 || 3°287 3 6577 8754 || 3°337 5 4342 8220 || 3°387 8 0995 4771
3°238 2 4230 1269 || 3°288 3 2843 1652 || 3°338 5 0790 2558 || 3°388 7 7616 1716
3239 2 0307 8583 || 3°289 2 gt12 1879 | 3°339 4 7241 2403 || 3°389 7 4240 2436
3°240 I 6389 5099 || 3°290 2 5384 9396 || 3°340 4 3695 7721 || 3°390 7 0867 6899
3241 I 2475 0779 || 3°291 2 1661 4168 || 3°341 4 0153 8476 || 3°391 6 7498 5071
3242 © 8564 5584 || 3°292 I 7941 6155 || 3°342 3 6615 4633 || 3°392 6 4132 6918
3°243 © 4657 9475 || 3°293 I 4225 5323 || 3°343 3 3080 6156 || 3°393 6 0770 2406
3°244 © 0755 2412 || 3°294 I 0513 1633 || 3°344 2 9549 3009 || 3°394 5 7411 1502
3°245| 389 6856 4357 || 3°295 © 6804 5047 || 3°345 2 6021 5158 | 3°395 5 4055 4171
3°246 9 2961 5271 || 3296 © 3099 5530 || 3°346 2 2497 2567 || 3-396 5 0703 0382
3°247 8 9070 5114 || 3°297| 369 9398 3044 || 3°347 t 8976 520% || 3°397 4 7354 ©1100
37248 8 5183 3847 || 3°298 9 5709 7552 || 37348 I 5459 3025 || 3°398 4 4008 3291
3°249 8 1300 1433 || 3°299 9 2006 9017 || 3°349| I 1945 6003 || 3°399 4 0665 9922

168 MR F. W. NEWMAN'S TABLE [3°400—3'599]
We | c-* 2 e-% | x e-# 2 Cae |
3400} 333 7326 9960 || 3:450| 317 4563 6378 | 31500 301 9738 3422 || 3°550| 287 2463 9654
3401 3 3991 3371 | 3°451 7 1390 6609 | 3°501 I 6720 1133 || 3°551 6 9592 9372
3"402 3 0659 o122 || 3-452 6 8220 8554 || 3°502 I 3704 9010 || 3°552 6 6724 7785
3°403 2 7330 o18o |} 3°453 6 5054 2182 | 3°503 I 0692 7025 || 3°553 6 3859 4866
3°44 2 4004 3510 || 3-454 6 1890 7459 || 3°504 © 7683 5146 || 3°554 6 0997 0586
37405 2 0682 0081 | 37455 5 8730 4356 || 3°505 © 4677 3344 || 3°555 5 8137 4916
37400 1 7362 9859 | 3°456 5 5573 2840 || 3°506 © 1674 1589 || 3556 5 5280 7827
37497 1 4047 2811 | 3457 5 2419 2880 || 3°507| 299 8673 9851 || 3557 5 2426 9291
3°408 I 0734 8903 | 3°458 4 9268 4444 || 3°508 9 5676 8100 | 3°558| 4 9575 9279
| 3409 © 7425 8102 3-459 4 6120 7500 || 3°509 g 2682 6305 || 3°559 4 6727 7762
| 3410 © 4120 0376 || 3°460 4 2976 2018 |] 3510 8 9691 4437 || 3°560 4 3882 4714
3,411 © 0817 5690 | 3-46r 3 9834 7966 || 3511 8 6703 2466 || 3°561r 4 I040 O104
3412] 329 7518 4073 || 37462} 3 6696 5312 || 3°512 8 3718 0362 || 3562 3 8200 3904
S413 9 4222 5311 | 3°403} 3 3561 4025 |) 3°513 8 0735 8095 || 3°563 3 5363 6086
3414 699929)2554 gee 3 0429 4073 | 3514 7 7756 5636 | 3°564 3 2529 6622
3415 8 7640 6701 | 3465; 2 7300 5426 | 3°515 7 4780 2954 || 3565 2 9698 5484
3°416 8 4354 6727 || 3°466 2 4174 8052 || 3516 7 1807 0020 || 3°566 2 6870 2642 |
1 3°417 8 1071 9597 || 3°467 2 1052 1920 | 3°517 6 8836 6804 || 3°567 2 4044 8069 |
3415 7 7792 5277 | 3'408| | 1 7932 6998 | 3518 6 5869 3277 || 3568 2 1222 1737 |
3419 7 4516 3735 || 3'469 r 4816 3256 | 3°519 6 2904 9408 || 3°569 1 8402 3616 |
3°420 7 1243 4939 | 3°47° I 1703 0661 | 3°520 5 9943 5168 | 357° 1 5585 3680 |
37421 | 6 7973 8854 || 3471 o 8592 9184 || 3521 5 6985 0528 | 3571 I 2771 1899
3422 6 47°97 545° | 3°472 © 5485 8792 | 3°522 5 4029 5457 || 3572 © 9959 8247
3°423 6 1444 4693 | 3°473 © 2381 9456 | 3523 5 1076 9927 || 3573 © 7151 2694
3424 5 8184 6550 | 3474) 309 9281 1143 | 3°524 4 8127 3908 || 3°574 © 4345 5212
3'425| 325 4928 0989 || 3-475 9 6183 3823 || 3°525| 294 5180 7369 | 3°575 © 1542 5774
3420| 5 1674 7977 || 3°476 9 3088 7465 || 3526 4 2237 0283 | 3°570| 279 8742 4351
3°427| . 4 8424 7482 || 3-477 8 9997 2038 || 3°527 3 9296 2619 || 3577 9 5945 0916
3°428 4 5177 9471 | 3'478 8 6908 7511 | 3528 3 6358 4348 ||3°578; 9 3150 5440
3429 4 1934 3912 | 3°479 8 3823 3853 || 3529 3 3423 5440 |) 3°579 9 0358 7896
3°43°| 3 8694 0773 || 3480 8 0741 1033 || 3°530 3 0491 5867 || 3°580 8 7569 8255
3431 | 3 5457 0020 || 3°481 7 7661 go20 || 3°531 2 7562 5599 || 3581 8 4783 6490
3°432 | 3 2223 1622 || 3-482 7 4585 7785 | 3°532 2 4636 4607 || 37582 8 2000 2573
3433, 2 8992 5546 | 3°483 7 1512 7295 || 3'533 2 1713 2860 || 3583 7 9219 6476
hoa 2 5765 1760 | 3°484 6 8442 7520 | 3'534 1 8793 0331 || 3°554 7 6441 8171
(3435) 2 2541 0232 | 37485 6 5375 8429 || 3°535 1 5875 6990 || 3°585 7 3666 7630
3°436 I 9320 0929 || 3°486 6 2311 9993 || 3°536 1 2961 2807 || 3°586 7 0894 4826
|3°437 | I 6102 3819 | 3°487 5 9251 2179 || 3°537 1 0049 7754 ||3°587| 6 8124 9731
| 3°438 1 2887 8871 || 3°488 5 6193 4958 || 3538 © 7I4I 1802 || 3588 6 5358 2317
3'439 © 9676 6051 || 3°489 5 3138 8299 || 3°539 © 4235 4921 || 3°589 6 2594 2557
| 3°440 © 6468 5328 || 3°490 5 0087 2171 || 3540 © 1332 7082 || 3590 5 9833 0423
3441 © 3263 6670 | 3491 | 4. 7038 6544 || 3°541| 289 8432 8257 || 3°591 5 7974 5887
3442 © 0062 0044 || 37492 4 3993 1388 | 3°542 9 5535 8416 | 3'592 5 4318 8922
3443) 319 6863 5419 || 3493 4 0950 6671 | 3°543 9 2641 753° || 3°593 5 1565 9500
|3°444| = 9 3668 2762 | 3°494 3. 7911 2364 || 3544 8 975° 5571 | 3594 4 8815 7594
| 3°445 9 0476 2043 || 3°495 3 4874 8436 | 3°545 8 6852 2500 | 3°595 4 6068 3176
3446, 8 7287 3228 || 3-496 3 1841 4857 | 3°546 8 3976 8316 || 3°596 4 3323 6218
3°447 | 8 4101 6286 || 3°497 2 8811 1597 || 3547 8 1094 2963 || 3597 4 0581 6694
3448 8 og19 1185 | 3°498 2 5783 8624 || 3°548 7 8214 6421 || 3598 3 7842 4576
| 3°449 7 7739 7893 || 3°499 2 2759 5909 | 3549 7 5337 8661 | 3'599}~ 3 5105 9836 |

[3°600—3"799] OF THE DESCENDING EXPONENTIAL. 169

wv Cat av GX a“ page xv Q-
3600] 273 2372 2447 || 3°50) 259 g112 8779 || 3°700| 247 2352 6470 |3°75°|| 235 1774 5856
3601 2 9641 2382 || 3°651 9 6515 0642 || 3°70 6 988r 5301 |3°751 4 9423 9865
3°602 2 6912 9613 || 3°652 9 3919 8469 || 3°702 6 7412 8831 | 3°752 4 7075 7368
3603 2 4187 4114 || 3°653 9 1327 2236 || 3°703 6 4946 7036 | 3°753 4 4729 8342
3604 2 1464 5856 || 3°654 8 8737 1916 || 3°794 6 2482 9889 | 3°754 4 2386 2764
3°605 1 8744 4813 || 3°655 8 6149 7483 || 3°795 6 o02t 7368 | 3°755 4 0045 o609
3606 I 6027 0957 || 3°656 8 3564 S8gr2 || 3°706 5 7562 9446 |3°756 3 7706 1855
3°607 I 3312 4262 || 3°657 8 0982 6177 || 3°7°7 5 5106 6101 | 3°757 3 5369 6478
3°608 I 0600 4700 || 3°658 7 8402 9252 || 3°708 5 2652 7306 |3°758 3 3035 4454
3°609 © 7891 2244 || 3°659 7 5825 8110 || 3°799 5 o201 3038 | 3°759 3 0703 5761
3610 © 5184 6866 || 3°660 7 3251 2726 ||3°710 4 7752 3272 |3°760 2 8374 0375
3611 © 2480 8541 || 3°661 7 0679 3076 | 3°711 4 5305 7983 | 3°761 2 6046 8273
3°612| 269 9779 7240 || 3°662 6 8109 9132 || 3°712 4 2861 7147 | 3°762 2 3721 9431
3613 9 7081 2937 || 3°663 6 5543 0869 || 3°713 4 0420 0741 | 3°763 2 1399 3826
3614 9 4385 5605 || 3664 6 2978 8261 || 3°714 3 7980 8738 | 3°764 I 9079 1435
3°615 9 1692 5217 || 3°665 6 0417 1284 || 3°715 3 5544 1115 | 3°765 I 6761 2235
3°616 8 goo2 1746 || 3°666 5 7857 9910 || 3°716 3 3109 7848 | 3°766 I 4445 6203
3°617 8 6314 5165 || 3°667 5 530L 4115 ||3°717 3 0677 8912 | 3°767 iP Dig Sells
37618 8 3629 5447 || 3°668 5 2747 3873 || 3°718 2 8248 4282 | 3°768 © 9821 3549
3619 8 0947 2565 || 3669 5 0195 9159 || 3°719 2 5821 3935 | 3°769 © 7512 6881
3620 7 8267 6493 || 3°670 4 7646 9947 || 3°720 2 3396 7846 | 3°770 © 5206 3287
3621 7 5590 7203 || 3°671 4 5100 6211 || 3°721 2 0974 5991 |3°771 © 2902 2746
3°622 7 2916 4669 || 3672 4 2556 7926 || 3°722 1 8554 8346 | 3°772 © 0600 5234
3623 7 0244 8865 || 3°673 4 0015 5006 || 3°723 1 6137 4887 |3°773|| 229 8301 0728
3624 6 7575 9763 || 3674 3 7476 7607 || 3°724 I 3722 5588 | 3°774 g 6003 9205
3625 | 266 4909 7336 | 3°675| 253 4940 5523 || 3°725 I 1310 0427 | 3°775 9 3709 0642
37626 6 2246 1559 || 3°676 3 2406 8787 || 3°726 © 8899 9380 | 3°776 9 1416 5016
3627 5 9585 2405 | 3°677 2 9875 7377 || 3727 © 6492 2421 |3°777) 8 9126 2305
37628 5 6926 9846 || 3°678 2 7347 1264 || 3:728 © 4086 9527 | 3°778| 8 6838 2484
3'629 5 4271 3856 | 3°679 2 4821 0425 || 3°729 © 1684 0673 | 3779 | 3 4552 5532
3°630 5 1618 4409 | 3°680 2 2297 4835 ||3°739| 239 9283 5837 | 3°780| 8 2269 1426

: 4 8968 1479 || 3°681 I 9776 4467 || 3°731 9 6885 4994 | 3°781 7 9988 o142

4 6320 5037 || 3°682 I 7257 9298 || 3°732 9 4489 8119 | 3°782 7 7709 1658
4 3675 5060 || 3683 I 4741 9300 || 3°733 9 2096 5190 | 3°783 | 7 5432 5951
4 1033 1519 || 3°684 I 2228 4451 || 3°734 8 9705 6181 | 3°784 7 3158 2998
3 8393 4388 || 3°685 © 9717 4723 || 3°735 8 7317 1069 | 3°785 7 0886 2777
3 5756 3641 || 3°686 © 7209 0093 || 3°736 8 4930 9831 | 3°786 6 8616 5265
3 3121 9252 || 3687 © 4703 0534 || 3°737 8 2547 2441 | 3°787 6 6349 0439
3 0490 1194 || 3688 © 2199 6023 || 3°738 8 0165 8878 | 3°788 6 4083 8276
2 7860 9441 || 3°689| 249 9698 6534 | 3°739 7 7786 9116 | 3°789 6 1820 8755
2 5234 3966 || 3°690 9 7200 2042 || 3°740 7 5410 3131 | 3°790 5 9560 1851
2 2610 4744 || 3691 9 4704 2522 |13°741 7 3036 ogor | 3°791 5 7301 7544
I 9989 1748 || 3°692 9 2210 7949 || 3°742 7 0664 2402 |3°792|/ - 5 5045 5809
I 7379 4952 || 3°693 8 9719 8298 || 3°743 6 8294 7609 | 3°793 || 5 2791 6624
I 4754 4329 || 3°694 8 7231 3544 || 3°744 6 5927 6499 | 3°794 5 2539 9968
I 2140 9854 || 3°695 8 4745 3662 || 3°745 6 3562 9048 | 3°795 4 8290 5817
© 9530 1501 || 3°696 8 2261 8628 || 3°746 6 1200 5233 | 3°796 4 6043 4149
© 6921 9243 | 3°697 7 9780 8417 || 3°747 5 8840 5030 | 3°797 4 3798 4941
© 4316 3054 || 3°698 7 7302 3003 || 3°748 5 6482 8415 |3°798 4 1555 8171
© 1713 2908 || 3°699 7 4826 2363 || 3°749 5 4127 5365 |3°799 3 9315 3817

Wow, OUNL 12a 0 23

170 MR F. W. NEWMAN’S TABLE [3°800—3'999]

ee eee
| x | e-% tage C=% x er* | HD ‘. @-%

13800] 223 7077 1856 | 3°850] 212 7973 6438 || 39900] 202 4191 1446 | 3°50] 192 5470 1775
3°Sor 3 4841 2266 || 3°851 2 5846 7338 || 3°90 2 2167 9652 || 3°951 2 3545 6698
| 3'802 | 3 2607 5024 || 3°852 2 3721 9496 | 31992 2 0146 8080 || 3°952 2 1623 0855
| 3803 3 0376 o10g || 3°853 2 1599 2892 || 39903 1 8127 6709 | 3°953 I 9702 4229
3804 2 8146 7497 | 3°854 I 9478 7504 || 3:904 I 6110 5519 || 3954 1 7783 6801
3805] 2 5919 7166 || 3°855 I 7360 3310 || 3°905 I 4095 4491 || 3°955 1 5866 8549
3°806 | 2 3694 9095 | 3°856 I 5244 0290 || 3°906 I 2082 3604 || 3:°956 I 3951 9457
3807 2 1472 3261 || 3°857 I 3129 8422 || 3907 I 0071 2837 || 3°957 I 2038 9504
| 3808 | I 9251 9641 | 3°858 I 1017 7686 || 3°908 © 8062 2172 || 3°958 I 0127 8672
|3'809! x 7033 8214 || 3859 © 8907 8060 | 3°909 © 6055 1586 | 3°959 © 8218 6940
| 3°810 1 4817 8957 || 3°860 © 6799 9523 || 3°90 © 4050 1062 || 3960 © 6311 429%
3811 1 2604 1849 || 3°86 © 4694 2054 || 3911 © 2047 0578 || 3961 © 4406 0705
3812 I 0392 6866 || 3°862 © 2590 5632 || 3912 © 0046 o114 || 3°962 © 2502 6164
3°813 o 8183 3987 | 3863 © 0489 0236 ||3°913}| 199 8046 9651 || 3°963 © o601 0647
(3814 © 5976 3190 || 3864] 209 8389 5845 | 3°914 9 6049 9168 | 3°964) 189 8701 4136
3815 © 3771 4453 || 3°865 9 6292 2437 || 3°915 9 4054 8645 | 3°965 9 6803 6612
3°816 © 1568 7754 || 3°866 9 4196 9993 || 3°916 9 2061 8064 || 3°966 9 4907 8057
3°817| 219 9368 3070 || 3°867 9 2103 8490 || 3°917 9 0070 7403 || 3°967 9 3013 8450
3818 9 7170 0381 || 3°868 9 0012 7909 || 3°918 8 8081 6642 || 3968 Q I12I 7774
3819 9 4973 9662 || 3869 8 7923 8227 || 3°919 8 6094 5763 || 3°969 8 9231 6008
3°820 g 2780 0894 || 3°870 8 5836 9425 || 3°920 8 4109 4744 || 3°970 8 7343 3135
| 3°821 9 0588 4053 || 3°871 8 3752 1481 || 3°92 8 2126 3567 || 3°971 8 5456 9136
3822 8 8398 9119 || 3°872 8 1669 4375 || 3°922 8 o©45 2211 || 3°972 8 3572 3991
3°823 8 6211 6068 || 3°873 7 9588 8085 || 3°923 7 8166 0655 || 3°973 8 1689 7682
3824 8 4026 4879 || 3°874 7 7510 2592 || 3°924 7 6188 8882 || 3°974 7 9809 o189
3°825| 218 1843 5531 | 3°875| 207 5433 7873 || 3°925| 197 4213 687z |13°975| 187 7930 1495
3826 7 9662 8001 || 3°876 7 3359 3909 || 3°926 7 2240 4602 || 3°976 7 6053 1580
3°827 7 7484 2268 || 3°877 7 1287 0679 || 3927 7 0269 2055 || 3°977 7 4178 0425
3°828 7 5307 8309 || 3°878 6 9216 8161 || 3°928 6 8299 g2i1 || 3°978 7 2304 8013
3829 7 3133 6104 | 3°879 6 7148 6336 || 3°929 6 6332 6050 || 3°979 7 0433 4323
3°830 7 0961 5630 || 3880 6 5082 5182 || 3930 6 4367 2553 || 3980 6 8563 9338
3831 6 8791 6865 || 3°881 6 3018 4678 || 3°93 6 2403 8699 || 3°98 6 6696 3038
3832 6 6623 9789 || 3882 6 0956 4805 || 3932 6 0442 4469 || 3982 6 4830 5405
3833 6 4458 4378 || 3883 5 8896 5542 || 3°933 5, 8482 9843 || 3°983 6 2966 6421
3834 6 2295 0613 || 3°884 5 6838 6868 || 3°934 5 6525 4803 || 3°984 6 1104 6066
3835 6 0133 8470 || 3°885 5 4782 8762 | 3°935 5 4569 9327 || 3°985 5 9244 4323
3°836 5 7974 7929 || 3886 5 2729 1203 || 3°936 5 2616 3398 || 3°986 5 7386 1171
3°837 5 5817 8967 || 3°887 5 0677 4172 | 3°937 5 0664 6994 || 3987 5 5529 6594
3°838 5 3663 1563 || 3888 4 8627 7648 | 3°938 4 8715 0097 || 3°988 5 3675 0572
| 3°839 5 1510 5696 || 3889 4 6580 1610 | 3°939 4 6767 2687 || 3°989 5 1822 3087
|3°840| 4 9360 1345 || 3°890 4 4534 6038 | 3°940 4 4821 4745 | 3°99°0 4 9971 4120
3°841 | 4 7211 8486 | 3°891 4 2491 OQII || 3°941 4 2877 6251 || 3991 4 8122 3653
3°842 4 5065 7100 || 3892 4 0449 6209 || 3°942) 4 0935 7186 || 3°992 4 6275 1667
3843 4 2921 7165 || 3°893 3 8410 1912 | 3°943 8 8995 753° || 3993 4 4429 8143
3344 4 0779 8659 | 3894) 3 6372 7999 | 3°944 3 7957 7265 || 3°994 4 2586 3064
3°845 3 8640 1561 || 3°395 3 4337 4449 || 3°945 3 5121 6369 || 3°995 4 0744 6411
3°346| 3 6502 5849 || 3°896| 3 2304 1243 || 3°946 3 3187 4826 || 3°996 3 8904 8165
3°847 3 4367 1502 | 3°897 3 0272 8360 | 3°947 3 1255 2613 || 3°997 3 7066 8308
3°848| 3 2233 8499 | 3°898 2 8243 5780 | 3°948 2 9324 9714 || 3°998 3 5230 6822
3°49} 3 o102 6818 || 3°899 2 6216 3482 | 3:949 2 7396 6108 || 3°999 3 3396 3689

[4:000—4'199]

xv GL
4000) 183 1563 8889
ay ieten 2 9733 2405
4°002 2 7904 4218
4°003 2 6077 4310
4/004 2 4252 2663
4/005 2 2428 9259
4006 2 0607 4079
4°007 t 8787 7105
4/008 1 6969 8319
47009 E.5153 77/02
47010 T3339 5237
4011 I 1527 0go5
4°012 © 9716 4690
4013 © 7907 6570
4014 © 6100 6530
4015 054205 4592
4°016 © 2492 oO615
4/017 © 0690 4704
4'018| 179 8890 6799
4°019 9 7092 6884
4020 9 5296 4939
4/021 9 3502 0948
4°022 9 1709 4891
4°023 8 9918 6752
4°024 8 8129 6512
4°025| 178 6342 4153
4°026 8 4556 9658
4027 8 2773 3008
4/028 8 ogg1 4186
47029 7 9211 3174
47030 7 7432 9954
4°031 7 5056 4508
4°032 7 3881 6819
4°033 7 2108 6868
4°034 7 9337 4639
4°035 6 8568 o113
4°030 6 6800 3273
4°037 6 5034 4101
4038 6 3270 2579
4°039 6 1507 8690
4'040 5 9747 2416
4°041 5 7988 3739
4/042 5 6231 2642 |
4°043 5 4475 9108
4°044 ee 722 63
4°045 5 0970 4656
4°046 4 9220 3703
4'047 4 7472 0243
4048 45725 4257 |
47049 4 3980 5728 |

OF THE DESCENDING EXPONENTIAL. W/L
%

“we (GAY wv Cm a“ Ee-2
4050] 174 2237 4639 ||/4100] 165 7267 5402 || 4-150] 157 6441 6485
4/051 4 0496 0973 || 4101 5 5611 Toro || 4°151 7 4865 9948
4°052 3 8756 4712 || 4°102 5 3956 3174 || 4152 7 3291 9160
4053 3 7018 5838 || 4103 5 2303 1878 || 4153 7 1719 4104
4°05 4 3 5282 4334 || 4°104 5 0651 7105 || 4°154 7 0148 4766
4°055 3 3548 0183 || 4°105 4 goor 8838 || 4°155 6 8579 1130
4/056 3 1815 3368 || 47106 4 7353 7062 || 47156 6 7OII 3179
4057 3 0084 3871 || 4°107 4 5797 1759 || 4°157 6 5445 0898
4°058 2 8355 1674 || 4°108 4 4062 2913 || 47158 6 3880 4272
4059 2 6627 6762 || 4°109 4 2419 0507 || 4°159 6 2317 3284
4°060 2 49OI QII5 || 4°10 4 0777 4526 || 4°160 6 0755 7920
4061 2 3177 8718 || 47111 3 9137 4953 || 4°161 5 9195 8163
4062 2 1455 5552 || 4112 3 7499 1771 || 4°162 5 7637 3998
4°063 I 9734 9601 || 4°113 3 5862 4964 || 4°163 5 6080 5410
4°064 I 8016 0847 || 4°114 3 4227 4515 || 47164 5 4525 2382
4'065 I 6298 9273 || 4°115 3 2594 0409 || 4°165 5 2971 4900
4°066 I 4583 4863 || 4°116 3 0962 2629 || 4°166 5 1419 2947
4°067 I 2869 7598 || 4°117 2 9332 1159 || 4°167 4 9868 6509
4°068 I 1157 7462 || 4°118 2 7703 5981 || 4°168 4 8319 5569
4069 © 9447 4437 || 4119 2 6076 7081 || 4°169 4 6772 0113
4°070 © 7738 8507 || 4°120 2 4451 4442 || 4°170 4 5226 o124
4071 © 6031 9655 || 4121 2 2827 8047 || 4171 4 3681 5587
4'072 © 4326 7862 || 4122 2 1205 7880 || 4172 4 2138 6488
4'073 © 2623 3113 || 4°123 T9585 3926 ||4°173 4 0597 2809
4074 © 0921 5390 || 4°124 I 7966 6167 || 4°174 3 9957 4537
4075] 169 9221 4677 ||4:125| 161 6349 4588 ||4-175| 153 7519 1655
4076 9 7523 0956 || 4°126 I 4733 9173 || 4176 3 5982 4148
4°077 9 5826 4209 || 4127 I 3119 9904 || 4177 3 4447 2002
4078 9 4131 4421 || 4°128 I 1507 6767 || 4°178 3 2913 5199
4°079 9 2438 1575 | 4°129 © 9896 9745 || 4°79 3 1381 3726
4°080 9 0746 5653 || 4°130 © 8287 8823 || 4°180 2 9850 7567
4081 8 9056 6638 || 4°131 © 6680 3982 || 4°181 2 8321 6706
4°082 8 7368 4513 || 4°132 © 5074 5209 || 4182 2 6794 1128
4°083 8 5681 9263 || 4°133 © 3470 2487 || 4°183 2 5268 0818
4°084 8 3997 0869 || 4°134 o 1867 5799 || 47184 2 3743 5761
4085 8 2313 9315 || 4°135 © 0266 5130 || 4°185 2 2220 5942
4°086 8 0632 4585 || 47136] 159 8667 0463 || 4186 2 0699 1344
4087 7 8952 6661 || 4°137 g 7069 1784 || 4°187 I 9179 1954
4088 7 7274 5526 || 4:138 9 5472 9074 || 4188 I 7660 7755
4089 7 5598 1164 | 4°139 9 3878 2320 || 4°189 1 6143 8733
4/090 7 3923 3558 || 4:140 9g 2285 1504 || 4°190 It 4628 4873
4/091 7 2250 2691 || 4°141 9 0693 6612 || 4*19% I 3114 6159
4°092 7 0578 8547 || 4°142 8 9103 7626 || 4:192 I 1602 2576
4°093 6 8909 1109 || 4°143 8 7515 4531 | 4°193 I O09 4109
4°094 6 7241 0360 || 4°144 8 5928 7312 || 4194 © 8582 0742
4°095 6 5574 6283 || 4°145 8 4343 5952 || 4°195 © 7074 2462
4/096 6 3909 8862 || 4°146 8 2760 0434 || 4°196 © 5567 9252
4°097 6 2246 8080 || 4147 8 1178 0745 || 4°197 © 4063 1098
| 47098 6 0585 3920 || 4°148 7 9597 6868 | 4:198 © 2559 7985
4/099 5 8925 6366 || 4°149 7 8018 8786 | 4'199 © 1057 9898

172 MR F. W. NEWMAN’S TABLE [4:200—4'399]
| x Cu x OF xv Ca x Cm
4°200] «149 9557 6820 || 4250] #142 6423 3909 || 4°300) 135 6855 goT2 | 4°350|| 129 0681 2580
| 4201]. )4,9 8058 8739 || 4°251 2 4997 6805 || 4301 5 5499 7235 |4°351 8 9391 2219
4202 9 6561 5638 || 4252 2 3573 3951 || 47302 5 4144 901r3 | 4°352 8 8102 4752
4°203 9 5065 7502 || 4253 2 2150 5332 || 4393 5 2791 4333 |4°353 8 6815 0165
4°204 9 3571 4318 || 4254 2 0729 0935 || 4°304 5 1439 3180 | 4°354 8 5528 8447
4°205 9 2078 6069 || 4°255 I 9309 0745 | 4°305 5 0088 5542 | 4°355 8 4243 9584
| 4°206 g 0587 2741 || 4°256 1 7890 4749 || 47306 4 8739 1404 | 4°356 8 2960 3564
| 4:207 8 9097 4319 || 4°257 I 6473 2931 | 4°3°7 4 7391 0754 |4°357 8 1678 0373
4/208 8 7609 0787 || 4°258 I 5057 5278 | 47308 4 6044 3578 | 4°358 8 0396 9999
4°209 8 6122 2132 || 4°259 I 3643 1776 || 4°309 4 4698 9863 | 4°359 7 9117 2428
4°210 8 4636 8338 || 4260 I 2230 2410 | 4°310 4 3354 9594 |4°360 7 7838 7649
4211 8 3152 9390 || 4°261 I 0818 7166 || 4°311 4 2012 2759 | 4°361 7 6561 5649
4212 8 1670 5274 || 4°262 © 9408 6031 |} 4°312 4 0670 9344 | 4°362 7 5285 6414
4213 8 o189 5975 || 4°263 © 7999 8990 || 4°313 3 933° 9336 | 4°363 7 4010 9932
4°214 7 8710 1477 |} 4°264 © 6592 6028 || 47314 3 7992 2721 |4°364 7 2737 6190
4°215 7 7232 1767 || 4265 © 5186 7133 || 4°315 3 6654 9486 | 4°365 7 1465 5175
4216 7 5755 6829 || 4°266 © 3782 2289 || 4°316 3 5318 9618 | 4°366 7 0194 6875
4217 7 4280 6648 |}-4°267 © 2379 1484 || 4°317 3 3984 3102 | 4°367 6 8925 1277
4/218 7 2807 1211 || 4°268 © 0977 4702 || 4°318 3 2650 9927 | 4°368 6 7656 8368
4°219 7 1335 0501 || 4°269| 139 9577 1930 || 4319 3 1319 0078 | 4:369 6 6389 8136
| 4-220 6 9864 4505 || 4'270 g 8178 3153 || 4°320 2 9988 3542 | 437° 6 5124 0568
| 4°221 6 8395 3207 || 4°271 9 6780 8359 || 4321 2 8659 0307 | 4°37! 6 3859 5651
4°222 6 6927 6593 || 4272 9 5384 7532 || 4322 2 7331 0357 |4°372 6 2596 3372
4°223 6 5461 4649 || 4-273 9 399° 0659 || 4°323 2 6004 3681 | 4373 6 1334 3720
4°224 6 3996 7359 || 4°274 9 2596 7726 || 4324 2 4679 0265 |4°374 6 0073 6681
4°225 146 2533 4709 || 4°275 139 1204 8719 || 4°325] 132 3355 0096 |4°375 125 8814 2242
4/226 6 1071 6685 || 4°276 8 9814 3624 || 4°326 2 2032 3160 |4°376 5 7550 0392
4°227 5 9611 3271 || 4277 8 8425 2427 || 4°327 2 0710 9446 | 4°377 5 6299 ILI7
| 4228 5 8152 4454 || 4278 8 7037 5114 || 4328 I 9390 8938 | 4378 5 5043 4406
4°22 5 6695 0217 || 4°279 8 5651 1672 || 4°329 I 8072 1623 |4°379 5 3789 0244
4°230 5 5239 0548 || 4280 8 4266 2086 || 4°330 I 6754 7490 | 4°380 5 2535 8621
| 4°231 5 3784 5432 || 4°281 8 2882 6343 || 4°331 I §438 6524 |4°381 5 1283 9523
4°232 5 2331 4853 || 4282 8 1500 4429 || 4°332 1 4123 8713 | 4382 5 0033 2938
4°233 5 0879 8797 || 4283 8 org 6330 || 4°333 I 2810 4042 | 4°383 4 8783 8853
4234 4 9429 7250 || 4°284 7 874° 2032 || 4°334 1 1498 2500 | 4°384 4°7535 7256
4°235 4 7981 0198 || 4°285 7 7362 1521 || 4°335 1 0187 4073 |4°385 4 6288 8134
4°236 4 6533 7625 || 4°286 7 5985 4784 || 4°336 © 8877 8748 | 4°386 4 5043 1475
4°237 4 5087 9518 || 4°287 7 4610 1807 || 4°337| © 7569 6511 | 4°387 4 3798 7267
4°238 4 3643 5861 || 4°288 7 3236 2576 || 4°338 © 6262 7350 |4°388 4 2555 5497
4239 4 2200 6641 || 4°289 7 1863 7077 || 4°339 © 4957 1252 |4°389 4 1313 6152
4°240 4 0759 1843 || 4°290 7 0492 5297 || 4°340 © 3652 8203 | 4°390 4 0072 9220
4241 3 9319 1453 || 4'291 6 g122 7222 || 4°341 © 2349 S8rgr | 4391 3 8833 4689
| 4°242 3 7880 5455 || 4292 6 7754 2838 || 4°342 © 1048 1203 | 4°392 3 7595 2547
4°243 3 6443 3837 || 4293 6 6387 2132 || 4°343| 129 9747 7224 |4°393 3 6358 2780
(4244 3 5007 6583 || 4294 6 5021 5089 || 4°344 9 8448 6244 |4°394 3 5122 5377
| 4°245 3 3573 3679 || 4°295 6 3657 1697 || 4°345 9 7150 8248 | 4°395 3 3888 0325
| 4°246 3 2140 5111 || 4°296 6 2294 1941 || 4°346 9 5854 3223 | 4396 3 2654 7612
| 4°247 3 0709 0864 || 4°297 6 0932 5809 || 4°347 9 4559 1157 |4°397 3 1422 7226
| 4°248 2 9279 0924 | 4'298 5 9572 3285 || 4°348 9 3265 2036 | 4°398 3 OQ 9154
4°249 2 7850 5277 | 4°299 5 8213 4358 | 4349! 9 1972 5849 |4°399 2 8962 3384

[4'400—4'599]

wv

4°400
4°401
4°402
4°403
4°404
4°405
4-406
4°407
4-408
4-409
4°410
4411
4412
4°413
4414
4415
4-416
4417
4418
4419
4°420
4421
4422
4°423
4°424
4°425
4-426
4°427
4-428
4°429
4°430
4431
4°432
4°433
4434

4°435
4°436
4°437
4°438
4439
4°440
4°441
4°442
4°443
4444

4°445
4°446
4447
4°448
4°449

s = & NN

HHH HH

Leal
HH

MoOonmnanm MBMMMODHO ONUWWOHO OWO000 O0OO0d000

Ons Ss

OF THE

DESCENDING EXPONENTIAL.

e7-e

7733
6506
5280
4056
2832

1610
0389
9169
7951
6733
5517
4302
3089
1876
0665

9455
8246
7038

-=

=)
oO fFHHRAH HPAHAYN AUMNUNUN UNUM AD DAnaagn

ce
et

SH AAR AR Re SH HE NHDH NHNNNDND N NO OO OW Go Go Go

173

x Oma x Ca
4500] 111 0899 6538 | 4°550] 105 6720
4°501 © 9789 3094 || 4°551 5 5664
4/502 © 8680 0748 || 4°552 5 4609
4503 © 7571 9489 || 4°553 5¥3555
4504 © 6464 9306 | 4°554 5 2501
4°505 © 5359 0187 || 4°555 5 1450
47506 © 4254 2122 || 47556 5 0399
4°507 © 3150 5099 || 4°557 4 9349
47508 © 2047 9108 || 4°558 4 8300
4°509 © 0946 4137 | 4°559 4 7252
4°510|} 109 9846 0176 || 4560 4 6205
4511 9 8746 7213 |] 47561 4 5160
4°512 9 7648 5238 || 4°562 4 4115
4513 9 6551 4239 | 4°563 4 3071
4514 9 5455 4205 || 4564 4 202
4°515 9 4360 5127 | 4°565 4 0987
4516 9 3266 6992 || 4-566 3 9947
4517 9 2173 9789 || 4°567 3 $908
4°518 9 1082 3508 || 47568 3 7869
4519 8 9991 8138 || 4°569 3 6832
4°520 8 8902 3668 || 4°570 3 5795
4521 8 7814 0087 || 4°571 3 4760
47522 8 6726 7385 || 4°572 3 3726
4°523 8 5640 5549 || 4°573 3 2693
4°524 8 4555 457° || 4°574 3 1661
4°525| 108 3471 4436 || 4°575| 103 062

| 47526 8 2388 5137 || 4576 2 9599
4°527 8 1306 6662 || 4°577 2 8570
47528 8 0225 goor || 47578 2 7542
4°529 7 9146 2141 | 4°579 2 6515
4°530 7 8067 6073 || 4°580 2 5489
4°531 7 6990 0785 || 47581 2 4464
4°532 7 5913 6267 || 4°582 2 3440
4°533 7 4838 2509 | 4°583 2 2417
4°534 7 3763 9499 || 4584 2 1395
4°535 7 2690 7226 | 4°585 2 0374
4°536 7 1618 5681 || 4°586 I 9355
4°537 7 0547 4851 || 4°587 r 8336
4538 6 9477 4728 || 4°588 I 7318
4°539 6 8408 5298° || 4°589 I 6301
4°540 6 7340 6553 || 4°590 r 5285
4541 6 6273 8482 || 4°591 I 4271
4°542 6 5208 1073 | 4°592 I 3257
4°543 6 4143 4316 || 4°593 I 2244
4°544 6 3079 8201 || 4°594 I 1232
4°545 6 2017 2716 || 4°595 I 0222
4°546 6 0955 7852 || 4596 © 9212
4°547 5 9895 3597 | 4°597 2 8203
4°548 5 8835 9941 || 4°598 © 7195
| 4°549 5 7777 6873 || 4°599 ° 6189

174 MR F. W. NEWMAN'S TABLE [4°680-—-4-790]

a }
x2 | e-z 2 e- xv Cm | v Ca
‘en Broo 5183 5745 | 4°650 95 6160 1930 || 4°700 9° 9527 7102 | 4°750 86 5169 5203
4°601 | © 4178 8933 | 4651 5 5204 5108 || 4°701 © 8618 6371 | 4°751 6 4304 7832
4°602 | © 3175 2163 || 4°652 5 4249 7837 || 4°702 © 7710 4726 || 4°752 6 3440 9104
4°603 | © 2172 5425 || 4°653 5 3296 o1og || 4°703 © 6803 2158 || 4°753 6 2577 gorr
4°604 | © 1170 8709 || 4°054 5 2343 1914 || 4°704 © 5896 8658 4754 6 1715 7544
4°605 | © O170 2005 || 4°655 5 1391 3242 || 4°705 © 4991 4218 | 4°755 6 0854 4693
47606 | 4.499 9170 5302 || 4°656 5 0440 4084 || 4°706 © 4086 8827 | 4°756 5 9994 0451
4°607| 9 8171 8591 || 4°657 4 949° 4431 || 4°7°7 © 3183 2477 | 4°757 5 9134 4809
4°608 9 7174 1861 || 4°658 4 8541 4272 || 4°708 © 2280 5159 || 4°758 5 8275 7759
47609 9 6177 5104 || 4°659 4 7593 3599 || 4°709 © 1378 6864 || 4°759 5 7417 9291
4°610 0099 sri 8308 || 4°660 4 6646 2402 || 4°710 © 0477 7582 || 4°760 5 6560 9397
4611 9 4187 1464 || 4°661 4 5700 0671 | 4°711 89 9577 7306 || 4°761 5 5704 8069
4612 9 3193 4562 || 4°662 4 4754 8397 || 4712 9 8678 6025 || 4°762 5 4849 5208
4613 9 2200 7591 || 4°663 4 3870 5571 || 4°713 9 7780 3731 | 4°763 5 3995 1076
4614 9 1209 0543 || 4°664 4 2867 2183 || 4°714 9 6883 0414 | 4°764 5 3141 5393
4615 9 0218 3407 || 4°665 4 1924 8223 || 4°715 9 5986 6067 4765 5 2288 8242
4°616 8 9228 6173 || 4666 4 0983 3683 || 4°716 9 5091 0679 || 4°766 5 1436 9614
4°617 8 8239 883r || 47667 4 0042 8553 || 4°717 9 4196 4242 || 4°767 5 0585 9500
4°618 8 7252 1372 || 4°668 3 9103 2823 || 4°718 9 3302 6748 | 4°768 4 9735 7892
4619 8 6265 3785 | 4°669 3 8164 6484 || 4°719 9 2409 8186 || 4°769 4 8886 4782
| 4°620 |0098 5279 6061 | 4°670 3 7226 9527 || 4°720 9 1517 8548 477° 4 8038 o160
[452% 8 4294 8190 || 4°671 3 6290 1942 || 4°721 9 0626 7826 || 4°771 4 7190 4018
4°622 8 3311 0162 || 4°672 3 5354 3720 || 4°722 8 9736 6010 || 4°772 4 6343 6349
4°623 8 2328 1966 | 4°673 3 4419 4851 || 4°723 8 8847 3091 || 4°773 4 5497 7143
4°62< 8 1346 3594 || 4674 3 3485 5327 || 4°724 8 7958 g06r || 4°774 4 4652 6392
4°625 |< 998 0365 5036 || 4°675| 93 2552 5138 ||4°725| 88 7071 3910 || 4°775| 84 3808 4087
4°626 7 9385 6281 | 4°676 3 1620 4274 || 4°726 8 6184 7630 || 4°776 4 2965 0221
4°627 7 8406 7320 | 4°677 3 0689 2726 || 4°727 8 5299 0212 || 4°777 4 2122 4784
4628 7 7428 8143 || 4°678 2 9759 0485 || 4°728 8 4414 1646 || 4°778 4 1280 7768
4°62 7 6451 8740 || 4°679 2 8829 7542 || 4°729 8 353° 1925 || 4°779 4 0439 9166
4°630|997 7 5475 9102 || 4°680 2 7901 3887 || 4°730 8 2647 1040 || 4°780 3 9599 8967
| 4°631 7 4500 9219 || 4°681 2 6973 9511 || 4°731 8 1764 8980 || 4°781 3 8760 7165
4°632 7 3526 go8r || 4°682 2 6047 4405 || 4°732 8 0883 5739 || 4°782 3 7922 3750
4°633 7 2553 8678 | 4°683 2 5121 8559 || 4°733 8 0003 1306 || 4°783 3 7084 8715
4°634 7 1581 8000 || 4°684 2 4197 1965 || 4°734 7 9123 5673 || 4°784 3 6248 2050
4°635 7 0610 7038 || 4°685 2 3273 4612 || 4°735 7 8244 8832 || 4°785 3 5412 3748
4°636 6 9640 5783 || 4°686 2 2350 6492 || 4°736 7 7367 0773 || 4°786 3 4577 3800
4°637 6 8671 4223 || 4°687 2 1428 7596 || 4°737 7 6490 1487 || 4°787 3 3743 2197
4°638 6 7703 2351 || 4688 2 0507 7914 || 4°738 7 5614 0967 || 4°788 3 2909 8933
4°639 6 6736 0156 || 4°689 1 9587 7437 || 4°739 7 4738 9202 | 4°789 3 2077 3997
| 4°640 0096 5769 7627 || 4°690 1 8668 6156 || 4°740 7 3864 6185 || 4°790 3 1245 7382
4°641 6 4804 4757 || 4691 I 7750 4062 || 4°741 7 2991 1907 || 4'791 3 0414 9079
4°642 6 3840 1535 || 4692 1 6833 1145 | 4°742 7 2118 6359 | 4°792 2 9584 9081
4643 6 2876 7951 || 4693 I 5916 7397 || 4°743 7 1246 9531 || 4°793 2 8755 7378
4°644 6 1914 3995 | 4°694 I 5001 2807 || 4°744 7 9376 1417 || 4°794 2 7927 3963
4°645 6 0952 9660 || 4°695 1 4086 7368 || 4°745 6 9506 2006 || 4°795 2 7099 8828
4646| 5 9992 4933 || 4°696 I 3173 1069 || 4°746 6 8637 1290 || 4°796 2 6273 1963
4647) 5 9032 9807 || 4°697 1 2260 3903 | 4°747 6 7768 9260 | 4°797 2 5447 3361
4°648 | 5 8074 4270 || 4°698 1 1348 5859 || 4°748 6 6901 5908 | 4°798 2 4622 3013
4°649 5 7116 8315 || 4°699 1 0437 6928 | 4°749 6 6035 1225 || 4°799 2 3798 og12

[4:800—4'999] OF THE DESCENDING EXPONENTIAL. 175

| 1

| v Oe x (Bac | & a x e-%
|a800 82 2974 7049 || 4°850| 78 2837 7549 || 49900) 74 4658 3071 | 4:950| 70 8340 8929
- 2 2152 1415 || 4°851 8 2055 3084 || 4°901 4 3914 0210 | 4°951 © 7632 go6r
oe 2 1330 4093 ee : 1273 re pk 4 3170 aie 4°952 ° ee oa

; _2 0509 4805 3 0492 760 03 4 2427 6798 | 4°953 © 6219 054
4804 I 9689 3811 || 4°854 7 9712 6583 || 4°904 4 1685 6232 | 4:954 © 5513 1885
4805 t 8870 ror4 | 4°855 7 8933 3353 | 4°9°5 4 0944 3083 | 4°955 © 4808 0280
tion] {ass ga [asay| faut cee [taey| gue gta toes) |S Sok a

: I 7233 9979 j 0255 | 4°90 3 9463 9006 | 4°957 © 3399 820
4°808 I 6417 1724 || 4°3858 7 6600 0370 || 4°908 3 8724 8063 || 4:958 © 2696 7723
4809 I 5601 1633 | 4859 7 5823 8251 | 4909 3 7986 4507 | 4°959 © 1994 4268
4810 I 4785 9698 | 4860 7 5048 3891 | 4:910 3 7248 8331 || 4'960 © 1292 7833
4811 I 3971 5910 | 4861 7 4273 7281 | 4:91 3 6511 9528 | 4'961 © o591 8410
4812 I 3158 0263 | 4°862 7 3499 8414 | 4°912 3.5775 8090 | 4°962| 69 9891 5993
4813 I 2345 2747 | 4863 7 2726 7282 | 4913 3 5040 4009 || 4°963 9 9192 0576
4814 I 1533 3355 fame 7 1954 3877 hes 3 4305 7279 | 4°964 9 8493 2150
4815 I 0722 2078 | 4°865 7 1182 8191 | 4915 3 3571 7892: | 4°965 9 7795 2709
ME 2 | gear | pies scar cect lace in eee eee

. 9102 3837 ae Oe | 3 4°967 9 64 754
4818 © 8293 6858 | 4 8872 7375 | 4918 3 1374 3716 || 47968 9 5704 822
4819 © 7485 7961 || 4°869 6 8104 2491 | 4°919 3 0643 3628 | 4°969 9 5009 4655
4820 © 6678 7139 | 4°870 6 7336 5288 | 4:920 2 9913 0847 || 4°970 9 4314 8035
a ° Sp 4384 ne 2 ee 5758 | 4921 2 9183 5364 || 4°971 9 3620 8357
4822 © 5066 9 4872 03 3894 | 4°922 2 8454 7174 | 4°972 9 2927 561
4823 © 4262 3042 || 4°873 6 5037 9688 | 4°923 2 7726 6267 || 4°973 9 2234 9804
4824 © 3458 4439 | 4874 6 4273 3132 | 4°924 2 6999 2639 | 4°974 9 1543 0914
4°825 80 2655 3870 || 4°875 76 3509 4219 | 4°925 72 6272 6280 || 4°975 69 0851 8939
4826 o 1853 1328 || 4°876 6 2746 2941 || 4°926 2 5546 7184 || 4°976 9g o16r 3874
soe © I05I 6805 a - 1983 929° [4927 2 4821 5343 4977 gay 5709
4°62 © 0251 0292 || 4°07 1222 3260 | 4°92 2 4097 0750 || 4°97 752 4440
4829} 79 9451 1782 || 4879 6 o461 4841 | 4°929 2 3373 3399 | 4°979 8 8094 0058
4°830 9 8652 1266 | 4°880 5 9701 4028 || 4°930 2 2650 3281 || 4:980 8 7406 2557
Ee lisa) 2 aes stele co tte ee lel! eee ae
4°83 9 7056 4 4 5 8183 5183 | 4°932 2 4718 || 4°982 32 8171
4833 9 6259 7605 || 4883 5 7425 7138 || 4°933 2 0485 6258 || 4983 8 5347 1272
4834 9 5463 8988 || 4°884 5 6668 6667 | 4°934 I_ 9765 5003 || 4°984 8 4662 1226
4835 9 eee $325 455 5 ee 3762 4988 I gene 0946 a5 2 3977 eer
4°83 9 3874 5609 || 4 5 5156 8417 | 4°93 I 8327 4079 || 4°98 3294 1668
EE TN A eee] eee] Pemmeer ee a [pace Deep
4°83 9 2288 3984 || 4 5 3648 0373 | 4°93 I 6892 1888 || 4°9 1928 9442
4839 9 1496 5060 || 4889 5 2894 7659 || 4°939 1 6175 6549 || 4°989 8 1247 3561
a 9 Brag; 4e52"| sa 5 2142 a 4°940 It 5459 8372 || 4°990 8 aoe 4492
4°841 9915 0950 || 4°89r 5 1390 4812 | 4°941 I 4744 735° || 4°991 7 9886 2229
ae : oe $747 a92| 5 039 4663 | 4-942 I a5 3475 | 4°992 7 oe ones
4°843 336 8436 | 4°893 4 9889 2020 | 4°943 I 3316 6741 || 4993 7 8527 8094
4844 8 7548 9007 || 4894 4 9139 6876 | 4944 I 2603 7139 || 4°994 7 7849 6207
4°845 8 6761 7455 || 4°895 4 8390 9224 | 4°945 r 1891 4664 || 4°995 7 7172 1099
ee 3 5975 3770 pie 4 is 9055 4°946 I le oe | 47996 7 6495 2763
4847 5189 7945 | 4°897 4 6895 6363 | 4°947 I 0469 1063 || 4°997 7 5819 1191
4848 8 4404 9971 | 4898 4 6149 1140 | 4°948 2 9758 9923 || 4:998 7 5143 6378
4849 8 3620 9842 | 4°899 4 5403 3379 | 4°949 © go49 5881 | 4°999 7 4468 8316

176

na annnn Nn

Non tN

Oo0000
bh NNN
On nur

Se)

MR F. W.
Cre xv ea
67 3794 6999 || 5:050| 64 0933
7 3121 2420 || 5:o5r 4 0292
7 2448 4572 | 5:052 3 9652
7 1776 3448 | 5:053 3 9013
7 T1O4 9043 | 5°054 3 8374
7 0434 1348 | 5°055 3 7736
6 9764 0358 || 5:056 3 7°99
6 9094 6065 || 5057 3 6462
6 8425 8463 || 5058 3 5826
6 7757 7546 | 5°59 3 5190
6 7090 3306 || 5060 3 4555
6 6423 5737 || 5061 3 3921
6 5757 4832 || 5062 3 3288
6 5092 0585 || 5063] 3 2655
6 4427 2989 || 5°064 3 2022
6 3763 2037 || 5065 3 1391
6 3099 7723 || 5:066 3 0760
6 2437 0039 || 5:067 3 0129
6 1774 8980 || 5:068 2 9499
6 III3 4539 || 5°069 2 8870
6 0452 6709 || 5:070 2 8242
5 9792 5484 || 5071 2 7614
5 9133 0856 || 5:072 2 6986
5 8474 2820 || 5:073 2 6360
5 7816 1368 || 5:074 2 5734
65 7158 6495 ||5°075| 62 5108
5 6501 8193 || 5:076 2 4483
5 5845 6456 || 5°077 2 3859
5 5190 1278 || 5:078 2 3236
5 4535 2652 || 5079 2 2613
5 3881 0570 || 5:080 2 1990
5 3227 5028 || 508: 2 1369
5 2574 6018 || 5:082 2 0748
5 1922 3534 | 5083 2 O12
5 1270 7569 || 5084 I 9507
5 0619 8117 || 5085 1 8888
4 9969 5171 || 5086 1 8270
4 9319 8724 || 5°087 I 7652
4 8670 8771 5088 I 7034
4 8022 5304 || 5-089 I 6418
4 7374 8318 || 5090 1 5801
4 6727 7806 || 5091 I 5186
4 6081 3760 || 57092 I 4571
4 5435 6176 || 5093 I 3957
4 479° 5046 || 57094 I 3343
4 4146 0364 | 5°095 I 2730
4 3502 2123 |] 57096 I 2118
4 2859 0317 | 5°097 I 1506
4 2216 4940 || 5:098 I 0895
4 1574 5985 || 5°099 I

NEWMAN’S TABLE

3446 || 5"100
7316 | 5101
7589 || 5"102
4259 | 5°103
7319 || 5-104
6762 || 5105
2583 || 5"106

2 4775 || 5197

3331 || 5°108
8246 || 5"109

9513 || 5°tr0
7125 || 5°111
1076 || 5112
1361 || 5°13
7972 || 5114
0903 || 5°I15
o148 || 5"116
5790 || 5°L17
7554 || 5118
5793 || 5119
OI4I || 5°120
o86r || 5121
7857 || 5122
1123 || 5123
0653 || 5124
6440 || 5°125
8478 || 5°126
6761 || 5°127
1282 || 5°128
2036 || 5°129

gor6 || 5"130
2216 || 5131
1629 || 5132
7250 || 5°133
9973 || 5-134

IO yeh)
1297 || 5136
1686 || 5°137
8251 || 5°138
0987 || 5°139

9887 || 5*140
4945 || 5141
6155 | 5142
3521 || 5143
7006 || 5°144

|
6635 | 5°145
2391 || 5°146
4268 || 5147
2260 || 5°148

0284 ‘6361 || 5149

60

fo) (ol fej {oy to) fo}. fo) fo). fe)

o0000

ri on
PCOOmODMmD HDHOHDMHM HDHMMHMH COMOWY wouusd wovovovse wonunow Oo

[5:000—5"199]

[5"200—5°399] OF THE DESCENDING EXPONENTIAL. 177
ve (Ga xv Cn xv Ge xv Gee
5'200 55 1656 4421 || 5-250 52 4751 8399 || 5"300 49 9159 39°97 || 535° 47 4815 0999
5°201 5 1105 0614 || 5:251 2 4227 3503 || 5°301 9g 8660 4808 || 5-351 7 4340 5222
5°202 5 0554 2318 || 5-252 2 3703 3850 || 5°302 9g 8162 0696 || 5°352 7 3866 4187
5°203 5 9003 9527 |\5 253 2 3179 9434 || 5303 9 7664 1565 || 5°353 7 3392 7892
5204 4 9454 2237 || 5°254 2 2057 0250 || 5°304 9 7166 7411 || 5°354 7 2919 6330
5'205 4 8905 0441 || 5:255 2 2134 6292 || 5°305 9 6669 8228 || 5°355 7 2446 9497 |
5°206 4 8356 4134 | 5256 2 1612 7555 || 5°306 9 6173 4013 || 5°356 7 1974 7389 |
59207 4 7808 3311 || 5-257 2 IO9I 4035 || 5°397 9 5677 4759 ||5°357 qf UbfeR) Cele
5°208 4 7260 7966 || 5:258 2 0570 5726 || 5308 9 5182 0461 ||5°358 7 1031 7328
5°209 4 6713 8093 || 5259 2 0050 2622 || 5°309 9 4687 1116 ||5°359 7 0560 9365
5°210 4 6167 3688 || 5-260 I 9530 4719 ||| 5-310 9 4192 6718 || 5°360 7 0090 6107 |
5211 4 5621 4744 || 5:261 I QOII 201I || 5311 9 3698 7261 || 5°36 6 9620 7551 |
5°212 4 5076 1256 || 5-262 I 8492 4493 || 5°312 9 3205 2741 || 5°362 6 9151 3691
5213 4 4531 3219 || 5263 I 7974 2160 || 5°313 Q 2712 3154 || 5°363 6 8682 4522
5214 4 3987 0628 || 5:264 I 7456 5007 || 5314 9 2219 8494 || 5°364 6 8214 co4o
5°25 4 3443 3476 || 5:265 t 6939 3028 || 5°315 9 1727 8755 || 5°365 6 7746 0240
5216 4 2900 1759 || 5266 1 6422 6219 || 5°316 9 1236 3934 || 5°366 6 7278 5118
5°217 4 2357 5471 || 5267 I 5906 4574 || 5°317 9 0745 4026 || 5°367 6 6811 4669
5218 4 1815 4606 || 5:268 I 5390 8088 || 5318 9 0254 9025 ||5°368 6 6344 8887
5°219 4 1273 9160 || 5:269 1 4875 6756 || 5°319 8 9764 8926 || 5°369 6 5878 7769
5220 4 0732 9126 || 5-270 I 4361 0573 || 5°320 8 9275 3725 ||5°37° 6 5413 1310
5°221 BeOrgz ASCO 5 277 T 3846 9533 | 5:322 8 8786 3417 || 5°371 6 4947 9505
57222 3 9652 5276 || 5'272 I 3333 3932 || 5°322 8 8297 7997 || 5'372 6 4483 2350
5°223 3 9113 1448 || 5-273 r 2820 2864 || 5°323 8 7809 7459 || 5°373 6 4018 9839
5°224 3 8574 3011 || 5°274 I 2307 7225 ||5°324 8 7322 1800 || 5°374 6 3555 1968
5'225| 53 8035 9960 |\5:275| 5x 1795 6708 || 5°325| 48 6835 rorg |/5°375| 46 3091 8733
5°226 3 7498 2289 || 5:276 I 1284 1310 || 5-326 8 6348 5096 || 5°376 6 2629 o129
5°227 3 6960 9993 || 5°277 I 0773 1024 || 5°327 8 5862 4042 || 5°377 6° 2166 6152
5°228 3 6424 3067 || 5-278 1 0262 5846 || 5"328 8 5376 7847 | 5378 6 1704 6796
5°229 3 5888 1505 || 5°279 © 9752 5779 | 5°329 8 4891 6505 || 5°379 6 1243 2056
5°230 3 5352 5303 || 5*280 2 9243 0793 | 5330 8 4407 o0t2 || 5°380 6 0782 1930
| 5°231 3 4817 4453 || 5°281 © 8734 0907 || 5°331 8 3922 8363 || 5°381 6 0321 6411
| 5:232 3 4282 8952 || 5:282 o 8225 6109 || 5°332 8 3439 1554 || 57382 5 9861 5496
5°233 3 3748 8793 || 5°283 © 7717 6393 || 5°333 8 2955 9579 || 5°383 5 9401 9179
5°234 3 3215 3972 || 5:254 © 7210 1755 | 5°334 8 2473 2433 || 5°384 5 8942 7456
5°235 3 2682 4484 || 5:285 © 6703 2188 || 5°335 8 1991 o1f2 || 5°385 5 8484 0322
57236 3 2150 0322 || 5°286 © 6196 7688 || 5°336 8 1509 2611 || 5°386 5 8025 7773
5°237 3 1618 1481 || 5°287 © 5690 8251 | 5°337 8 1027 9925 || 5°387 5 7567 9805
5238 3 1086 7957 || 5°288 o 5185 3870 || 5°338 8 0547 2050 || 5388 5 7110 6412
5239 3 9555 9744 || 5289 o 4680 4542 || 5°339 8 0066 8980 || 5°389 5 6653 759%
5°240 3 0025 6836 || 5:290 © 4176 0260 || 5°340 7 9587 0710 || 5°390 5 6197 3336
5241 2 9495 9228 || 5:291 © 3672 1019 || 5°341 7 9107 7237 ||5°391 5 5741 3643
5°242 2 8966 6915 || 5292 o 3168 6816 || 5°342 7 8628 8554 || 5°392 5 5285 8507
5°243 2 8437 9892 || 5°293 © 2665 7644 || 5°343 7 8150 4658 || 5°393 5 4830 7924
5°244 2 7909 8154 || 5°294 © 2163 3499 || 5°344 7 7672 5543 | 5°394 5 4376 1889
5°245 2 7382 1694 || 5°295 o 1661 4375 || 5°345 7 7195 1205 | 5°395 5 3922 0399
5°246 2 6855 0509 || 5°296 © I160 0268 | 57346 7 6718 1639 | 5°396 5 3468 3447
5°247 2 6328 4592 || 5°297 © 0659 1173 || 5°347 7 6241 6840 | 5°397 5 3015 1030
5°248 2 5802 3938 || 5°298 © 0158 7084 || 5°348 7 5765 6804 || 5°398 5 2562 3144
5°249 2 5276 8542 |5°299| 49 9658 7997 ||5'349 7 5290 1525 || 5°399 5 2109 9783
Vou. XIII. Parr III. 24

178 MR F. W. NEWMAN'S TABLE [5-400—-5°599]

x | e-* ee e-@ 2 e-% x e-#
nee L

5400, 45 1658 0943 | 5450) 42 9630 4691 | 5500} 40 8677 1438 || 5550] 38 8745 7243 |
5401 5 1206 6619 || 5451 2 9201 0533 || 5°50 © 8268 6710 |] 5°551 8 8357 1729
5-402 | 5 0755 6808 || 5-452 | 2 8772 0668 ||/'5*502 © 7860 6064 || 5°552 8 7969 0099 |
5403] 5 0305 1504 || 5-453| 2 8343 5091 || 5°593 © 7452 9496 | 5°553 8 7581 2348 |
5°404 4 9855 ©7093 || 5°454 2 7915 3797 || 5°54 © 7045 7003 || 5°554 8 7193 8473
5405 4 9405 4401 || 5°455| 2 7487 6782 || 5°505 © 6638 858r | 5°555 8 6806 8470
5°406 4 8956 2593 | 5°456 2 7060 4042 15 506 © 6232 422 57556 8 6420 2335
5°407 4 8507 5274 || 5°457 2 6633 5572 || 5°5°7 © 5826 3931 || 5°557 8 6034 0064
5"408 4 8059 2441 || 5458 2 6207 1369 || 57508 © 5420 7695 || 5°558 8 5648 1653
| 5*409 4 7611 4088 |! 5-459 2 5781 1428 || 5°509 © 5015 5514 || 5°559 8 5262 7099
| 5410 4 7164 o21r | 5'460 2 5355 5745' || 5510 © 4610 7383 || 5°560 8 4877 6398
5 ‘411 4 6717 0806 || 5461 2 493° 4315 || 5511 © 4206 3208 || 5:561 8 4492 9545
5412 4 6270 5868 || 5"462 2 4505 7135 || 5512 © 3802 3255 || 5°562 8 4108 6537
5°413 4 5824 5393 || 5463 2 4081 4199 || 5513 © 3398 7250 || 5°563 8 3724 7371
57414 4 5378 9376 || 5464 2 3657 5505 || 5°514 © 2995 5279 || 57564 8 3341 2041
5415 4 4933 7813 || 5°465 2 3234 1047 || 5°515 © 2592 7338 || 5°565 8 2958 0545
5°416 4 4489 0699 || 5°466 2 2811 o821 || 5°516 © 2190 3423 || 5°566 8 2575 2879
5°417 4 4044 8030 || 5°467 2 2388 4824 |15°517 o 1788 3530 || 5°567 8 2192 9038
57418 4 3600 g8or || 5-468 2 1966 3050 || 5°518 © 1386 7655 || 5°568 8 1810 go20
5°419 4 3157 6009 || 5°469 2 1544 5496 | 5°519 ° 0985 5794 || 5°569 8 1429 2819
5°420 4 2714 6648 || 5-470} 2 1123 2158 || 5°520 © 0584 7942 ||5°570 8 1048 0433
5°421 4 2272 1714 || 5°471 2 0702 3031 || 5°521 © 0184 4096 || 5°571 8 0667 1857
5°422 4 1830 1203 || 5°472 2 0281 81Io || 5'522 39 9784 4252 |15°572 8 0286 7088
542 4 1388 5110 || 5473 1 9861 7393 || 5°523 9 9384 8406 || 5°573 7 9906 6121
5°424 4 0947 3431 || 5°474 I 9442 0874 | 5°524 9 8985 6554 | 5°574 7 9526 8954
5°425 44 0506 6162 || 5°475 41 9022 8550 || 5°525 39 8586 8692 || 5°575 37 9147 5582
5°426 4 0066 3297 || 5-476 1 8604 0416 |} 5°526 g 8188 4816 || 5:576 7 8768 6002
| 5°427 3 9626 4834 || 5°477 18185 6468 | 5°527 9 779° 4921 || 5°577 7 8390 0209
57428 3 9187 0766 || 5°478 i 7767 6701 |} 57528 9 7392 9004 || 57578 7 8011 8200
5°429 3 8748 rogr | 5°479 I 7350 I113 | 5°529 9 6995 7062 || 5°579 7 7633 997%
5°430 3 8309 5803 || 5'480 I 6932 9698 || 5°530 9 6598 9089 || 5°580 7 71256 5519
5°431 3 7871 4808 | 5481 I 6516 2452 || 5°531 g 6202 5082 || 5-581 7 6879 4839
5°432 3 7433 8371 || 5°482 1 6099 9372 | 5°532 g 5806 5038 || 5°582 7 6502 7928
5°433 3 6996 6219 | 5°483 I 5684 0452 || 5°533 9 5410 8951 || 5583 7 6126 4782
5°434 3 6559 8437 || 5°484 1 5268 5689 | 5°534 9 5015 6818 || 5°584 7 575° 5397.
5°435 3 6123 S502x || 5°485 t 4853 5079 || 5°535 9 4620 8636 || 5:585 7 5374 9770
5°430 3 5687 5966 || 5°486 r 4438 8618 || 5°536 9 4226 4400 || 5°586 7 4999 7896
5°437 3 5252 1268 || 5°487 I 4024 6301 || 5°537 9 3832 4106 || 5°587 7 4624 9773
| 5°438 3 4817 0922 || 5°488 1 3610 8124 | 5°538 9 3438 7750 || 5588 7 425° 5395
5°439 3 4382 4924 | 5°489 1 3197 4083 || 5°539 9 3045 5329 |/5°589| 7 3876 4761
5°440 3 3948 3271 || 5°49° 1 2784 4174 | 5°540 9 2652 6838 || 5°590 7 3502 7865
5°441 3 3514 5956 || 5°491 I 2371 8393 || 5°541 g 2260 2274 || 5591 7 3129 4704
5°442 3 3081 2977 || 5492 I 1959 6736 | 5°542 g 1868 1632 || 5-592 7 2756 5274
5°443 3 2648 4329 || 5°493 I 1547 9198 | 5°543 9 1476 4909 || 5°593 7 2383 9572
5°444 3 2216 0007 || 5°494 1 1136 5777 | 5°544 g 1085 2101 || 5°594 7 2011 7594
5°445 3 1784 0008 || 5°495 1 0725 6465 | 5°545 9 0694 3204 || 5°595 7 1639 9335
5°446 3 1352 4326 || 5496 I 0315 1262 || 5°546 9 0303 8214 || 5596 7 1268 4794
5°447 3 0921 2957 || 5°497 © 9905 0162 | 5°547 8 9913 7126 || 5°597 7 0897 3965
5448 3 0490 5898 || 5°498 © 9495 3160 | 5°548 8 9523 9938 || 5°598 7 0526 6845
|5°449| 3 0060 3144 |15'499| 0 9086 0254 | 5°549| 8 9134 6645 || 5599] 7 0156 3430

[5°600—5799]

OF THE

DESCENDING EXPONENTIAL.

ae e-% G e-# Ay
5°600 6 9786 37156 | 5°650 35° 1751 6775 || 5-700
5601 6 9416 7701 || 5°651 5 1400 1016 | 5-7or
5602 6 9047 5380 || 5°652 5 1048 8772 || 5-702
5603 6 8678 6749 | 5°653 5 0698 0038 || 5°793
5604 6 8310 1805 || 5654 5 0347 4810 | 5-704
5605 6 7942 0544 | 5°655 4 9997 3087 || 5"705
5°606 6 7574 2963 | 5°656 4 9647 4863 || 5-706
5°607 6 7206 9057 || 5°657 4 9298 0136 5°707
5°608 6 6839 8823 | 5°658 4 8948 8902 || 5-708
5609 6 6473 2258 || 5°659 4 8600 1157 || 5709
5610 6 6106 9358 | 5660 4 8251 6898 || 5-710
5611 6 5741 o118 | 5°661 4 7903 6122 | 5-711
5612 6 5375 4536 | 5662 4 7555 8825 | 5°712
5613 6 5010 2608 || 5°663| 4 7208 5003 || 5°713
5614 6 4645 4330 || 5°664 4 6861 4654 | 5714
5615 6 4280 9698 || 5°665 4 6514 7773 | 5°75
5616 6 3916 8709 | 5666 4 6168 4357 || 5°716
5617 6 3553 1359 | 5°667 4 5822 4403 || 5°717
5618 6 3189 7645 | 5°668 4 5476 7907 || 5°718
5619 6 2826 7563 | 5°669 4 5131 4866 | 5719
5620 6 2464 1109 || 5°670 4 4786 5276 || 5:720
5621 6 2101 8279 || 5671 4 4441 9134 || 5°721
5622 6 1739 9071 || 5672 4 4097 6437 || 5°722
5°623 6 1378 3480 || 5673 4 3753 7180 || 5°723
5624 6 1017 1503 || 5674 4 3410 1361 || 5°724
5625 6 0656 3136 || 5°675 34 3066 8976 || 5-725
57626 6 0295 8375 || 5°676 4 2724 0022 || 5°726
5627 5 9935 7218 || 5°677 4 2381 4495 || 5°727
5628 5 9575 9660 || 5°678 4 2039 2392 || 5°728
5629 5 9216 5697 || 5°679 4 1697 37099 || 5°729
5630 5 8857 5327 || 5680 4 1355 8443 || 5°730
5°631 5 8498 8546 || 5°681 4 1014 6591 || 5°731
5°632 5 8140 5349 || 5682 4 0673 8149 | 5°732
5°633 5 7782 5734 || 5683 4 0333 3114 || 5°733
5°634 5 7424 9696 || 5°684 3 9993 1482 || 5°734
5°635 5 7067 7233 || 5°685 3 9653 3250 || 5°735
5°636 5 6710 8341 || 5686 3 9313 8414 |] 5-736
5°637 5 6354 3015 || 5°687 3 8974 6972 || 5°737
57638 5 5998 1253 || 5688 3 8635 8919 | 5°738
5°639 5 5642 3052 || 5°689 3 8297 4253 |15°739
5°640 5 5286 8406 || 5*690 3 7959 2969 || 5°740
5641 5 4931 7314 || 5°691 3 7621 5066 || 5°741
5642 5 4576 9770 || 5692 3 7284 0538 || 5°742
5643 5 4222 5773 || 5693 3 6946 9383 | 5°743
57644 5 3868 5318 || 5694 3 6610 1598 | 5°744
5645 5 3514 84or || 5*695 3 6273 7179 || 5°745
57646 5 3161 5020 || 5696 3 5937 6123 || 5°746
5647 5 2808 5170 || 5°697 3 5601 8426 | 5-747
5648 5 2455 8848 || 5698 3 5266 4085 | 5-748
5°649 5 2103 6051 || 5699 3 4931 3097 || 5°749

w

NWWWW WWW wo Wo Wo Ww Oo

N NHN N NN NNN NhwNNN Ny nN WN iS eS eS ©) NRNNN ND

a = SA S&S eS

179
Yam | «& Ca
4596 5457 |5°750| 31 8278 0796
4262 1164 |15°751 I 7959 9606
3928 0214 || 5°752 I 7642 1596
3594 2603 || 5°753 I 7324 6762
3260 8328 || 5-754 I 7007 5102
2927 7385 ||5°755 1 6690 6611
2594 9772 |! 5°756 1 6374 1287
2262 5484 | 5°757 I 6057 9127
1930 4520 || 5°758 I 5742 0128
1598 6874 | 5°759 I 5426 4286
1267 2545 | 5°760 I 5111 1598
0936 1528 || 5:761 I 4796 2062
o605 3821 | 5°762 I 4481 5673
0274 9419 | 5°763 I 4167 2429
9944 8321 || 5764 I 3853 2327
9615 0522 | 5°765 I 3539 5364
9285 6019 || 5°766 I 3226 1536
8956 4808 || 5°767 I 2913 0840
8627 6888 || 5°768 I 2600 3273
8299 2254 || 5°769 I 2287 8832
7974 0902 || 5°77° T1975 7514
7643 2831 |15°771 I 1663 9316
7315 8036 || 5°772 I 1352 4234
6988 6514 || 5°773 I 1041 2266
6661 8261 || 5°774 I 0730 3409
6335 3276 |5°775| 31 0419 7659
6009 1554 || 5°776 I O109 5013
5683 3092 || 5°777 © 9799 5468
5357 7887 || 5778 © 9489 go2r
5032 5935 || 5°79 © 9180 5668
4797 7234 || 5°780 o 8871 5408
4383 1779 || 5°781 0 8562 8237
4058 9569 || 5°782 © 8254 4151
3735 2599 || 5°783 © 7946 3147
3411 4867 || 5-784 © 7638 5223
3088 2368 || 5°785 © 7331 0376
2765 3101 || 5°786 © 7023 8602
2442 7061 || 5°787 © 6716 9898
2120 4246 || 5°788 © 6410 4261
1798 4651 | 5°789 © 6104 1688
1476 8275 || 5°790 © 5798 2176
1155 5114 | 5°791 © 5492 5723
0834 5164 || 5°792 o 5187 2324
0513 8422 | 5°793 o 4882 1977
0193 4886 || 5°794 © 4577 4079
9873 4552 | 5°795 © 4273 0427
9553 7416 | 5°796 © 3968 9217
9234 3476 || 5°797 © 3665 1047
8915 2728 | 5°798 © 3361 5914
8596 5169 | 5°799 © 3058 3814

180 MR F. W. NEWMAN’S TABLE
|
xr e-% gv e7- Wh e-r

15800} 30 2755 4745 || 5°850| 28 7989 9158 || 5*900 7 3944 4819
5 ‘Sor © 2452 8704 || 5°851 8 7702 0698 || 5‘90r 7 3670 6743
: 5*So2 © 2150 5687 || 5852 8 7414 5116 || 5:g02 7 3397 1404
5803 © 1848 5691 || 5°853 8 7127 2407 || 5°903 7 3123 8799
5804 © 1546 8714 || 57854 8 6840 2570 || 5:904 7 2850 8926
5°805 © 1245 4753 || 5°855 8 6553 5601 || 5°905 7 2578 1781
5806 © 0944 3804 || 5°856 8 6267 1498 || 5:906 7 2305 7361
5807 © 0643 5864 || 5°857 8 5981 0257 || 5°907 7 2033 5665
5808 © 0343 0931 || 5°858 8 5695 1876 || 5-908 7 1761 6689
5"809 © 0042 goo2 || 5°859 8 5409 6352 || 5"909 7 1490 0431
5810 29 9743 0072 || 5°860 8 5124 3683 || 5‘910 7 1218 6887
5811 9 9443 4140 || 5°861 8 4839 3864 | 5911 7 2947 6056
5812 9 9144 1203 || 5°862 8 4554 6894 || 5912 7 0676 7934
5°813 9 8845 1257 || 5°863 8 4270 2769 || 5°913 7 0406 2519
5814 9 8546 4299 || 5°864 8 3986 1487 | 5°914 7 0135 9808
5°815 9 $248 0327 || 5°865 83702 3045 || 5915 6 9865 9799
5816 9 7949 9338 || 5°866 8 3418 7440 || 5°916 6 9596 2488
5°817 9 7652 1328 || 5°867 8 3135 4670 || 5917 6 9326 7873
5818 9 7354 6294 || 5868 8 2852 4730 || 5918 6 9057 595%
5819 9 7957 4234 || 5869 8 2569 7619 || 5919 6 8788 6720
5820 9 6760 5145 || 5°870 8 2287 3334 || 5920 6 8520 0177
5821 9 6463 9023 || 5°87 8 2005 1872 || 5921 6 8251 6319
5822 9 6167 5866 || 5'872 8 1723 3229 || 5922 6 7983 5143
5823 9 5871 5670 || 5°873 8 1441 7404 || 5-923 6 7715 6648
5°824 9 5575 8433 || 5°874 8 1160 4394 || 5°924 6 7448 0829
5825 29 5280 4152 || 5°875 28 0879 4194 || 5°925 6 7180 7685
5826 9 4985 2824 || 5°876 8 0598 6804 || 5-926 6 6913 7213
5827 9 4690 4446 || 5°877 8 0318 2220 || 5927 6 6646 9410
5828 9 4395 9014 || 5°878 8 0038 0439 || 5928 6 6380 4273
5829 9 4101 6527 || 5°879 79758 1458 || 5°929 6 6114 1800
5°830 9 3807 6980 || 5°880 7 9478 5275 || 5°930 6 5848 1989
5°831 9 3514 0372 || 5°881 7 9199 1887 || 5°931 6 5582 4836
5832 9 3220 6698 || 5882 7 8920 1290 || 5°932 6 5317 0338
5°833 9 2927 5957 || 5°883 7 8641 3483 || 5°933 6 5051 8494
5°834 9 2634 8146 | 5°884 7 8362 8462 | 5°934 6 4786 9300
5°835 9 2342 3260 || 5°885 7 8084 6225 || 5°935 6 4522 2754
5°836 9g 2050 1298 || 5°886 7 7806 6769 || 5°936 6 4257 8854
5°837 9 1758 2257 || 5887 7 7529 9091 || 5°937 6 3993 7596
57838 9 1466 6133 || 5°888 7 7251 6188 || 5:938 6 3729 8978
5°839 9 1175 2923 || 5°889 7 6974 5058 || 5939 6 3466 2997
57840 9 0884 2626 || 5°890 7 6697 6697 || 5°940 6 3202 9651
5841 9 0593 5237 || 5891 7 6421 1103 || 5°941 6 2939 8937
5°842 9 0303 0754 || 5892 7 6144 8274 || 5°942 6 2677 0852
5°843 9 0012 9175 || 5893 7 5868 8206 || 5"943 6 2414 5394
57844 8 9723 9495 || 5894 7 5593 0896 || 5944 6 2152 2561
5°845 8 9433 4713 || 5895 7 5317 6343 |) 5°945 6 1890 2348
5°846 8 9144 1825 || 5°896 7 5°42 4543 || 5°946 6 1628 4755
5°847 8 8855 1828 || 5897 7 4797 5493 || 5°947 6 1366 9778
5°848 8 8566 4720 || 5°898 7 4492 9191 | 5°948 6 1105 7415
5°849 8 8278 0498 || 5°899 7 4218 5634 | 5°949 6 0844 7662

[5*800—5"999]

ty
Amn AMIN AMnnNnN AMNnnN AN ADA
~I
aS
~
on

iS)

HhADRHR PHAM UUMUNUNN UUNUNYN UUMUNU
|
_
oo

0518
5980
4046
4711
7975
3835
2287
3329
6959
3174

1972
3350
73°04
3834
2936

4607
8846
5649
5014
6938

1419
8454
8041
0177
4860

2087
1855
4162
goo6
6383

6292
8730
3694
1182
I1QI

3719
8762
6320
6389
8966

4050
1637
1726
4313
9397

6974
7042
9599
4642
2169

[6:000—6'199] OF THE DESCENDING EXPONENTIAL. 181
x | Og xv Ca x (ae av (GA
a 24 7875 2177 ||6:050 23 5786 2006 || 6:100 2 4286 7719 || 6150 21 3348 1770
6'001 4 7627 4663 || 6051 3 5550 5322 || 6101 2 4062 5973 | 6-151 I 3134 9355
6-002 4 7379 9626 || 6:052 3 5315 0995 | 6'102 2 3838 6467 || 6-152 I 2921 9070
6°003 4 7132 7063 || 6053 3 5°79 9020 | 6'103 2 3614 9199 || 6153 I 2709 0916
6°004 4 6885 6971 || 6°054 3 4844 9396 || 6104 2 3391 4168 |) 6154 I 2496 4888
6:005 4 6638 9349 || 6:055 3 4610 2t2r | 6105 2 3168 1370 || 6°155 I 2284 0985
6°006 4 6392 4192 || 6°056 3 4375 7191 || 6106 2 2945 0804 || 6156 I 2071 9205
6°:007 4 6146 1499 || 6:057 3 4141 4605 || 6°107 2 2722 2468 || 6-157 I 1859 9546
6:008 4 5900 1268 || 6658 3 3907 4361 || 6°108 2 2499 6358 || 6°158 I 1648 2005
6:009 4 5054 3496 || 6059 3 3673 6456 || 6109 2 2277 2474 || 6°159 I 1436 6581
6010 4 5408 8180 || 6060 3 3440 0887 || 6110 2 2055 0813 || 6160 Ter225 3272
6:o1l 4 5163 5319 || 6:061 3 3206 7653 || 6-111 2 1833 1372 || 6°161 I 1014 2074
6012 4 4918 4909 || 6062 3 2973 6751 || 6-112 2 1611 4149 || 6162 I 0803 2987
6'013 4 4673 6948 || 6:063 3 2740 8179 || 6113 2 1389 9143 || 6163 I 0592 6007
6:014 4 4429 1434 || 6064 3 2508 1934 || 6114 2 1168 6350 || 67164 I 0382 1134
6:015 4 4184 8364 || 6:065 3 2275 8014 || 6115 2 0947 5769 || 6°165 I ot71 8364
6:016 4 3940 7737 || 6066 3 2043 6417 || 6°116 2 0726 7308 || 6166 © 9961 7697
6017 4 3696 9548 || 6:067 3 181r 7141 || 6-117 2 0506 1234 || 6167 © 9751 9128
6018 4 3453 3797 || 6:068 3 1580 o182 || 6118 2 0285 7275 || 6168 © 9542 2658
6-019 4 3210 0480 || 6'069 3 1348 5540 || 6119 2 0065 5518 || 6169 © 9332 $282
6'020 4 2966 9595 || 6:070 3 I1I7 3211 ||6°120 I 9845 5963 || 6°170 © 9123 6000
6:021 4 2724 1140 ||6°071 3 0886 3192 || 6-121 I 9625 8606 || 6171 © 8914 5809
6'022 4 2481 5112 ||6:072 3 0655 5483 || 6-122 I 9406 3445 || 6-172 © 8705 7708
6023 4 2239 1509 || 6:073 3 0425 oo081 || 6123 1°9187 0478 || 6°173 © 8497 1694
6:024 4 1997 0328 || 6°074 3 0194 6982 || 6-124 1 8967 9703 || 6174 o 8288 7764
6°025 24 1755 1567 || 6:075 22 9964 6186 | 6°125 1 8749 1118 || 6175 20 8080 5917
4 1513 5224 ||/6:076 2 9734 7689 || 6-126 I 8530 4720 || 6°176 o 7872 6151
4 1272 1296 || 6'077 2 9505 1490 || 6'127 I 8312 0508 || 6:177 o 7664 8464
4 1030 9781 || 6'078 2 9275 7586 || 6128 1 8093 8479 || 6:178 © 7457 2854
4 0790 0676 || 6'079 2 9046 5974 || 6129 I 7875 8630 || 6°179 © 7249 9318
4 0549 3979 || 6080 2 8817 6653 || 6°130 I 7658 o961 || 6180 © 7042 7855
4 0308 9687 || 6:081 2 8588 9620 || 6°131 I 7440 5468 || 6181 © 6835 8462
4 0068 7798 || 6082 2 8360 4873 || 6132 I 7223 2149 || 6182 © 6629 1137
3 9828 8310 || 6083 2 8132 2409 || 6°133 I 7006 1003 || 6'183 © 6422 5879
3 9589 1221 || 6:084 2 7904 2227 || 6°134 I 6789 2026 || 6184 o 6216 2684
3 9349 6527 || 6085 2 7676 4324 ||6°135 I 6572 5218 || 6°185 © 6010 1553
3 9110 4227 || 6:086 2 7448 8608 || 6:136 I 6356 0575 || 6°186 © 5804 2481
3 8871 4318 || 6°087 2 7221 5346 || 6137 I 6139 8096 || 6187 © 5598 5467
3 8632 6708 || 6:088 2 6994 4266 || 6°138 I 5923 7778 || 6°188 © 5393 0509
3 8394 1664 || 6089 2 6767 5457 || 6139 I 5707 9620 || 6°189 © 5187 7605
3 8155 8914 || 6:090 2 6540 8915 || 6140 I 5492 3618 || 6*190 © 4982 6753
3 7917 8545 || 6:o9t 2 6314 4638 || 6141 I 5276 9772 || 6:191 © 4777 7951
3 7680 0556 || 6:092 2 6088 2625 || 6:142 I 5061 8078 | 6:192 © 4573 1197
3 7442 4943 || 67093 2 5862 2872 | 6143 I 4846 8535 | 6°193 © 4368 6488
3 7205 1705 || 6-094 2 5636 5378 || 6-144 I 4632 1140 || 6"194 © 4164 3823
3 6968 0839 || 6:095 2 5411 O14 |/6°145 I 4417 5892 || 6°195 © 3960 3200
3 6731 2342 || 6:096 2 5185 7157 || 6°146 I 4203 2788 || 6:196 © 3756 4616
3 6494 6213 || 6097 2 4960 6426 || 6147 I 3989 1825 || 6°197 © 3552 8070
3 6258 2449 || 6:098 2 4735 7944 || 6148 I 3775 3003 || 6-198 © 3349 3559
3 6022 1048 || 6099 2 4511 1709 || 6149 I 3561 6319 || 6199 © 3146 1082

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MR F. W. NEWMAN'S TABLE [6:200—6°399]
wv Cm wv Cm wv Ca
6250} 19 3045 4136 | 6°300) 18 3630 4777 ||6°350| 17 4674 7136
6251 9 2852 4647 || 6-301 8 3446 9390 || 6°351 7 4500 1262
6°252 9 2659 7086 || 6°302 8 3263 5838 || 6352 7 4325 7133
6253 9 2467 1452 || 6°303 8 3080 4118 | 6°353 7 415% 4747
6°254 9 2274 7743 || 0304 8 2897 4229 || 6°354 7 3977 4103
6255 9 2082 5956 | 6305 8 2714 6169 | 6355 7 3803 5199
6°256 9 1890 6090 | 6°306 8 2531 9936 || 6-356 7 3629 8032
| 6°257 9 1698 8143 || 67307 8 2349 5528 || 6°357 7 3456 2602
| 6:258 9 1507 2113 || 6308 8 2167 2945 || 67358 7 3282 8906
(6°259 9 1315 7998 |6309] 8 1985 2182 | 6-359 7 3109 6944
| 6:260 9 1124 5797 || 6°310 8 1803 3239 || 6°360 7 2936 6712
| 6°261 9 0933 5506 || 6-311 8 1621 6115 || 6°361 7 2763 8209
| 6°262 9 0742 7125 || 6-312 8 1440 0806 || 6-362 7 2591 1435
6°263 9 0552 0651 || 6°313 8 1258 7312 || 6°363 7 2418 6386
6°264 9 0361 6083 | 6314 8 1077 5631 || 6°364 7 2246 3061
6°265 9 or7r 3418 || 6°35 8 0896 5761 || 6-365 7 2074 1459
6°266 8 9981 2655 || 6°316 8 0715 7699 || 6°366 7 1902 1578
6-267 8 9791 3792 || 6317 8 0535 1445 || 6-367 7 1730 3416
6°268 8 9601 6827 || 67318 8 0354 6995 || 6°368 7 1558 6970
6°269 8 9412 1758 || 6°319 8 0174 4350 || 6°369 7 1387 2241
6:270 8 9222 8583 || 6:320 7 9994 3506 ||6°370 7 1215 9225
6°271 8 9033 7300 || 6-321 7 9814 4462 || 6°371 7 1044 7922
6-272 8 8844 7908 | 6°322 7 9634 7217 || 6-372 7 0873 8329
6°273 8 8656 o404 || 6°323 7 9455 1767 || 6°373 7 2703 0445
6274 8 8467 4786 || 6-324 7 9275 8113 || 6°374 7 0532 4268
6°275| 18 8279 1054 ||6°325] 17 9096 6250 ||6°375| 17 0361 9796
6°276 8 8090 9204 || 6°326 7 8917 6179 ||6°376 7 O1QI 7027
6°277 8 7902 9235 || 6°327 7 8738 7898 || 6°377 7 0021 5961
6°278 8 7715 1145 || 6°328 7 8560 1403 || 6°378 6 9851 6595
6°279 8 7527 4932 || 9329 7 8381 6694 || 6°379 6 9681 8927
6280 8 7340 0594 || 6°330 7 8203 3769 || 6°380 6 9512 2957
6281 8 7152 8130 || 6°331 7 8025 2626 || 6°381 6 9342 8681
6°282 8 6965 7537 | 6°332 7 7847 3263 || 6-382 6 9173 6099
6:283 8 6778 8814 || 6°333 7 7669 5679 || 6°383 6 9004 5208
6°284 8 6592 1959 || 6°334 7 7491 9871 || 6°384 6 8835 6008
6°285 8 6405 6970 || 6°335 7 7314 5839 || 6°385 6 8666 8496
6'286 8 6219 3844 || 6°336 7 7137 3579 || 6°386 6 8498 2670
6°287 8 6033 2581 || 6°337 7 6960 3091 || 6°387 6 8329 8530
6°288 8 5847 3179 || 6°338 7 6783 4372 || 6°388 6 8161 6073
6-289 8 5661 5634 || 6°339 7 6606 7421 || 6°389 6 7993 5297
6'290 8 5475 9947 || 6°340 7 6430 2237 || 6390 6 7825 6201
6°291 8 5290 6114 || 6°341 7 6253 8816 || 6°39 6 7657 8784
G'292 8 5105 4134 || 6-342 7 6077 7159 || 6°392 6 7490 3043
6°293 8 4920 4005 || 6°343 7 5901 7262 || 6°393 6 7322 8977
6°294 8 4735 5725 || 6344 7 5725 9123 ||6°394 6 7155 6585
6°295 8 455° 9293 || 6°345 7 555° 2743 ||6°395 6 6988 5864
6°296 8 4366 4706 || 6°346 7 5374 8117 || 6°396 6 6821 6812
6°297 8 4182 1963 || 6°347 7 5199 5246 ||6°397 6 6654 9429
6298 8 3998 1062 || 6°348 7 5024 4126 ||6°398 6 6488 3713
6°299 8 3814 2000 || 6-349 7 4849 4757 ||6°399 6 6321 9661

[6-400—6'599] OF THE DESCENDING EXPONENTIAL. 183
wv (Ge xv GL xv Cue x“ e-£
6°400 16 6155 7273 || 6:450 15 8052 2169 || 6:500 15 0343 9193 || 67550 I4 301r 5598
6°401 6 5989 6546 || 6451 5 7894 2437 || 6501 5 0193 6505 || 6551 4 2868 6197
6"402 6 5823 7479 || 6452 5 7736 4283 || 6502 5 0043 5319 | 6°552 4 2725 8225
6403 6 5658 o07t || 6-453 5 7578 7707 || 6503 4 9893 5634 || 6°553 4 2583 1681
6404 6 5492 4319 || 6-454 5 7421 2707 || 6504 4 9743 7448 || 6554 4 2440 6561
6405 6 5327 0222 || 6-455 5 7263 9282 | 6°505 4 9594 0759 || 6°555 4 2208 2867
6406 6 5161 7778 || 6:456 5 7106 7428 || 6°506 4 9444 5506 | 6°556 4 2156 0595
6°407 6 4996 6986 || 6°457 5 6949 7146 || 6°507 4 9295 1867 | 6°557 4 2013 9745
6°408 6 4831 7843 || 6458 5 6792 8434 || 6°508 4 9145 9661 || 6558 4 1872 0315
6°409 6 4667 0349 || 6°459 5 6636 1289 || 6509 4 8996 8947 | 6°559 4 1730 2304
6°410 6 4502 4502 || 6:460 5 6479 5710 || 6510 4 8847 9723, || 6°560 4 1588 5710
6°41 6 4338 0300 || 6:461 5 6323 1697 || 6511 4 8699 1987 || 67561 4 1447 0532
6°412 6 4173 7741 || 6°462 5 6166 9246 || 6512 4 8550 5738 || 6°562 4 1305 6769
6°413 6 4009 6824 || 6463 5 6010 8358 || 6513 4 8402 0975 || 67563 4 1164 4418
6°414 6 3845 7547 || 6-464 5 5854 9029 || 6-514 4 8253 7696 || 6564 4 1023 3479
6°415 6 3681 9908 || 6-465 5 5699 1259 || 6515 4 8105 5899 || 67565 4 0882 3951
6°416 6 3518 3906 || 6-466 5 5543 5046 || 6516 4 7957 5584 || 6°566 4 0741 5831
6417 6 3354 9540 || 6°467 5 5388 0389 || 6517 4 7809 6748 || 6°567 4 0600 9119
6418 6 3191 6807 || 6-468 5 5232 7285 || 67518 4 7661 9390 || 67568 4 0460 3812
6°419 6 3028 5706 || 6'469 5 5°77 5734 || 6519 4 7514 3508 || 6569 4 0319 9910
6°420 6 2865 6235 || 6:470 5 4922 5733 || 6520 4 7366 gto2 || 6°570 4 0179 7412
6421 6 2702 8393 || 6471 5 4767 7282 || 6521 4 7219, 6170 || 67571 4 0039 6315
6'422 6 2540 2177 || 6472 5 4613 0378 || 6522 4 7°72 4709 || 6°572 3 9899 6619
6°423 6 2377 7588 || 6-473 5 4458 5020 || 6523 4 6925 4720 ||6°573 3 9759 8322
6°424 6 2215 4622 || 6°474 5 4304 1207 || 6524 4 6778 6200 || 67574 3 9620 1422
6°425) 16 2053 3278 || 6'475) 15 4149 8937 || 6°525) 14 6631 9147 ||6°575| 13 9480 5918
67426 6 1891 3555 || 6476 5 3995 8209 || 6526 4 6485 3561 || 67576 3 9341 1810
6-427 6 1729 545° || 6-477 5 3841 9020 || 6527 4 6338 9439 || 6°577 3 9201 9094
67428 6 1567 8963 || 6478 5 3088 1370 || 6528 4 6192 6781 || 6°578 3 9062 7771
6-429 6 1406 4092 || 6:479 5 3534 5257 || 9529 4 6046 5585 || 6°579 3 8923 7838
6°430 6 1245 0834 || 6480 5 3381 0679 || 6°530 4 5900 5850 || 6:580 3 8784 9295
6-431 6 1083 9190 || 6-481 5 3227 7635 || 6531 4 5754 7573 || 6581 3 8646 2139
6°432 6 0922 9156 || 6°482 5 3074 6123 || 67532 4 5609 0754 || 6°582 3 8507 6370
6°433 6 0762 0731 || 6'483 5 2921 6142 || 67533 4 5463 5391 || 6583 3 8369 1986
6°434 6 o601 3914 || 6484 5 2768 7691 || 6534 4 5318 1483 || 67584 3 8230 8986
6°435 6 0440 8702 || 6°485 5 2616 0767 || 6°535 4 5172 9028 || 6585 3 8092 7368
67436 6 c280 5096 || 6°486 5 2463 5369 || 6536 4 5027 8024 || 6586 3 7954 7130
6°437 6 0120 3092 || 67487 5 2311 1495 || 6°537 4 4882 8471 || 6°587 3 7816 8273
6°438 5 9960 2689 || 6-488 5 2158 9145 || 67538 4 4738 0367 || 6588 3 7679 °793
6°439 5 9800 3886 || 6-489 5 2006 8317 || 6°539 4 4593 3710 || 6589 3 7541 4691
6°440 5 9640 6681 || 6-490 5 1854 9008 | 6540 4 4448 8499 || 6590 3 7493 9964
6°441 5 9481 1072 || 6'491 5 1703 1218 || 67541 4 4304 4732 || 6591 3 7266 6610
6442 5 9321 7058 | 6-492 5 1551 4945 || 6542 4 4160 2409 || 6-592 3 7129 4630
6443 5 9162 4637 || 6-493 5 1400 0188 || 67543 4 4016 1527 || 6593 3 6992 4021
6444 5 9003 3808 || 6-494 5 1248 6944 || 6544 4 3872 2085 || 6594 3 6855 4781
6°445 5 8844 4569 || 6-495 5 1097 5213 || 67545 4 3728 4083 ||6°595| ~3 6718 6911
6°446 5. 8685 6918 || 6°496 5 0946 4993 || 6546 4 3584 7517 || 6°596 3 6582 0407
6°447 5 8527 0855 || 6497 5 0795 6283 | 6547 4 3441 2387 || 6597 3 6445 5269
6448 5 8368 6376 || 6-498 5 0644 9080 | 6°548 4 3297 8692 || 6°598 3 6309 1496
6449 5 8210 3481 || 6-499 5 0494 3384 || 6549 4 3154 6429 || 6°599 3 6172 9086

184 MR F. W. NEWMAN’S TABLE [6:%600—6'799]

7322 3723 || 6798 I 1600 4927

9661 2739 || 6698
7205 1086 || 6°799 I 1488 9480

9531 6775 | 6°699

|
v Ci? v Ca” wv Cm” v E-2
|
6°600 13 6036 8037 || 6°650 I2 9402 2105 || 6-700 I2 3091 1903 || 6°750 Ir 7087 9621
6601 3 5900 8349 || 6°651 2 9272 8730 || 6:701 2 2968 1606 || 6-751 1 6970 9326
| 6°02 3 5765 0020 || 6°652 2 9143 6647 || 6-702 2 2845 2539 || 6:752 I 6854 0202
6603 3 5629 3049 || 6653 2 go14 5856 || 6°703 2 2722 4700 |/ 6-753 I 6737 2246
6°604 3 5493 7434 || 6°654 2 8885 6355 || 6°704 2 2599 8089 | 6-754 I 6620 5457
| 6-605 3 5358 3174 || 6°655 2 8756 8143 || 6°705 2 2477 2704 || 6-755 6503 9834
67606 3 5223 0267 || 6°656 2 8628 1219 || 6°706 2 2354 8543 || 6°756 I 6387 5377
| 6°607 3 5087 8713 -|| 6°657 2 8499 5580 || 6°707 2 2232 5606 || 6-757 I 6271 2083
| 6°608 3 4952 8509 | 6°658 2 8371 1227 || 6°708 2 2110 3892 || 6°758 I 6154 9952
| 6°609 3 4817 9655 | 6°659 2 8242 8157 || 6°709 2 1988 3398 || 6°759 I 6038 8983
6°610 3 4683 2149 | 6°660 2 8114 6370 || 6°710 2 1866 4124 || 6°760 I 5922 9174
| 6-611 3 4548 5990 | 6°661 2 7986 5864 || 6°711 2 1744 6070 || 6-761 I 5807 0524
| 6°612 3 4414 1177 || 6°662 2 7858 6638 || 6:712 2 1622 9232 || 6°762 I" 5691 3032
| 6°613 3 4279 7708 || 6663 2 7730 8691 || 6°713 2 1501 3611 || 6763 I 5575 6698
6-614 3 4145 5581 Laas 2 7603 2020 || 6°714 2 1379 9204 || 6764 I 5460 1519
6°615 3 4011 4796 || 6°665 2 7475. 6626 || 6-715 2 1258 G6or2 || 6°765 I 5344 7494
6°616 3 3877 5351 || 6666 2 7348 2507 || 6716 2 1137 4032 || 6°766 I 5229 4623
6°617 3 3743 7245 || 6667 2 7220 9661 || 6-717 2 1016 3263 || 6-767 I 5114 2905
| 6°618 3 3610 0476 | 6°668 2 7093 8087 || 6718 2 0895 3705 || 6°768 I 4999 2337
6°619 3 3476 5043 | 6669 2 6966 7784 || 6°719 2 0774 53506 || 6°769 I 4884 2920
6620 3 3343 0946 || 6°670 2 6839 8751 || 6°720 2 0653 8214 || 6°770 I 4769 4651
| 6-621 3 3209 8181 || 6°671 2 6713 0986 || 6-721 2 0533 2279 ||6°771 I 4654 7530
6622 3 3076 6749 || 6°672 2 6586 4489 || 6°722 2 0412 7549 || 6°772 I 4540 1555
6°623 3 2943 6647 | 6°673 2 6459 9257 || 6°723 2 0292 4023 || 6°773 I 4425 6726
6°62 3 2810 7875 || 6°674 2 6333 5290 || 6-724 2 0172 1701 |] 6-774 I 4311 3042
| 6°625 13 2678 0431 || 6°675 I2 6207 2586 || 6-725 I2 0052 0580 || 6°775 II 4197 0500
6°626 3 2545 4314 || 6°676 2 6081 1144 || 6°726 I 9932 0659 || 6°776 I 4082 g1I00
6°627 3 2412 9522 ||6°677 2 5955 0963 || 6°727 I g812 1938 || 6°777 I 3968 8841
6°628 3 2280 6054 || 6°678 2 5829 2042 || 6°728 I 9692 4415 || 6:778 I 3854 9722
6°62 3 2148 3909 || 6°679 2 5703 4379 || 6729 I 9572 8089 | 6°779 I 3741 1742
| 6630 3 2016 3086 || 6°680 2 5577 7973 ||6°730 I 9453 2958 || 6°780 I 3627 4898
6°631 3 1884 3583 || 6°68r 2 5452 2823 || 6°731 I 9333 9022 || 6'781 I 3513 9191
6°632 3.1752 5398 || 6°682 2 5326 8927 || 6732 I 9214 6280 || 6°782 I 3400 4620
6°633 3 1620 8532 || 6683 2 5201 6284 || 6°733 I 9095 4729 || 6783 I 3287 1182
6°634 3 1489 2981 || 6684 2 5076 4894 || 6°734 I 8976 4370 || 6-784 I 3173 8877
6°635| 3 1357 8745 || 6°685 2 4951 4754 || 6°735 1 8857 5200 || 6-785 1 3060 7704
6°636 3 1226 5823 || 6°686 2 4826 5864 || 6°736 I 8738 7219 || 6°786 I 2947 7661
6°637 3 1095 4213 || 6°687 2 4701 8222 || 6°737 I 8620 0425 || 6°787 1 2834 8748
6°638 3 0964 3914 || 6688 2 4577 1827 || 6°738 I 8501 4818 || 6°788 I 2722 0963
6°639 3 0833 4925 || 6°689 2 4452 6678 || 6°739 I 8383 0395 || 6°789 I 2609 4306
6°640 3 0702 7244 || 6'690 2 4328 2773 || 6°740 I 8264 7157 || 6°790 r 2496 8774
6°641 3 0572 0870 || 6691 2 4204 oO112 || 6°741 1 8146 5101 || 6791 I 2384 4368
6°642 3 0441 5802 || 6:692 2 4079 8693 || 6742 It 8028 4226 || 6°792 I 2272 1085
6°643 3 0311 2038 || 67693 2 3955 8514" || 6°743 I 7910 4532 || 6°793 I 2159 8925
6°644 3 0180 9577 || 6°694 2 3831 9575 || 6°744 I 7792 6017 || 6°794 1 2047 7887
6°645 3 0050 8418 || 6695 2 3708 1875 | 6745 t 7674 8679 || 6-795 I 1935 7969
6°646 2 9920 8560 || 6:696 2 3584 5411 || 6°746 I 7557 2519 ||6°796 1 1823 9171
6°647 2 9791 coor | 6°697 2 3461 0184 | 6°747 I 7439 7534 || 6°797 I 1712 1490
2 2 I
2 2 I

[6:800—6'999] OF THE DESCENDING EXPONENTIAL.

x =F x Cax av 6-% xv
6°800 II 1377 5148 || 6850 TO 5945 5693 || 6900 10 0778 5429 || 6950
6801 I 1266 1929 || 6851 © 5839 6767 || 6-901 © 0677 8147 || 6-951
6°802 I 1154 9824 || 6852 © 5733 8899 || 6-902 © 0577 1872 || 6:952
6803 I 1043 8829 || 6853 © 5628 2089 || 6:903 © 0476 6603 || 6°953
6804 I 0932 8945 || 6°854 © 5522 6334 || 6:904 © 0376 2339 || 6954
6°805 I 0822 o171 || 6855 © 5417 1636 || 6:905 © 0275 9078 || 6955
6°806 I O7II 2505 || 6:°856 © 5311 7991 || 6°906 © 0175 6820 || 6956
6°807 I 0600 5946 | 6857 © 5206 5399 || 6907 © 0075 5564 || 6°957
6°808 I 0490 0493 || 6°858 © 5101 3860 || 6:908 ©9 9975 5309 || 6958
6°809 I 0379 6144 | 6859 © 4996 3371 || 6909 9 9875 6053 || 6°959
6810 I 0269 2900 | 6°860 © 4891 3933 || 6:910 9 9775 7796 || 6960
6811 I 0159 0758 | 6861 © 4786 5543 || 6-911 9 9676 0537 || 6961
6812 1 0048 9718 || 6:862 o 4681 8201 || 6912 9 9576 4275 || 6962
6813 © 9938 9778 || 6363 © 4577 1906 | 6-913 9 9476 9008 || 6-963
6814 0 9829 0938 || 6364 © 4472 6657 || 6-914 9 9377 4737 || 6-964
6°815 © 9719 3196 | 6:865 © 4368 2453 | 6915 9 9278 1459 || 6965
6°816 © 9609 6551 || 6°866 © 4263 9292 | 6916 9 9178 9173 || 6°966
6817 © 9500 1003 || 6°367 © 4159 7174 ||6°917 9 9079 7880 || 6967
6818 © 9390 6549 | 6°868 GS 4055 6097 | 6:918 9 8980 7577 || 6968
6819 © 9281 3189 || 6869 © 3951 6061 || 6919 9g 8881 8264 || 6:969
6°820 © 9172 0922 || 6°870 © 3847 7064 | 6°920 9 8782 9940 || 6-970
6821 © 9062 9747 || 6871 © 3743 9107 || 6921 9 8684 2604 || 6971
6°822 © 8953 9662 || 6872 © 3640 2186 || 6922 9 8585 6255 ||6:972
6°823 © 8845 0667 || 6°873 © 3536 6302 || 6:923 9 8487 0891 || 6°973
6824 © 8736 2761 || 6874 © 3433 1453 | 6924 9 8388 6513 || 6-974
6°825 10 8627 5941 || 6°875 IO 3329 7639 || 6:925 9 8290 3118 || 6°975
6°826 © 8519 0208 |! 6°876 © 3226 4857 || 6:926 9g 8192 0706 || 6°976
6°827 © 8410 5561 || 6877 © 3123 3109 || 6-927 9 8093 9276 || 6°977
6°828 0 8302 1997 || 6°878 © 3020 2391 6928 9 7995 8827 || 6978
6°829 eae SL etoa ie tee © 2917 2703 | 6929 9 7897 9358 || 6-979
6°830 © 8085 8118 | 6880 © 2814 4045 | 6°930 9 7800 0868 || 6-980
6°831 © 7977 7800 || 6°881 © 2711 6415 || 6931 9 7702 3356 || 6981
1 6°832 © 7869 8562 || 6°882 © 2608 9812 || 6°932 9 7604 6821 || 6982
6°833 © 7762 0402 || 6883 © 2506 4235 || 6°933 9 7507 1262 || 6:983
6°834 © 7654 3321 || 6884 © 2403 9683 | 6°934 9 7409 6679 | 6984
6°835 © 7546 7315 || 6885 © 2301 6155 | 6935 9 7312 3069 | 6985
6°836 © 7439 2386 || 6°3886 © 2199 3651 | 6936 9 7215 0432 || 6986
6°837 © 7331 8530 || 6°887 © 2097 2168 | 6°937 9 7117 8768 || 6987
6°838 © 7224 5748 | 6°888 © 1995 1706 || 6:°938 9 7020 8075 || 6988
6°339 © 7117 4038 | 6°889 o 1893 2264 | 6°939 9 6923 8351 || 6989
6°840 © 7OIO: 3400 | 6890 © 1791 3841 || 6°940 9 6826 9597 || 6:990
6841 © 6903 3831 || 6891 © 1689 6436 | 6941 9 6730 1812 | 6991
6°842 © 6796 5332 | 6°892 © 1588 0048 || 6942 9 6633 4993 | 6992
6°843 © 6689 7900 || 6°893 © 1486 4676 || 6943 9 6536 9141 | 67993
6844 © 6583 1536 || 6894 © 1385 0318 6°944 9 6440 4255 | 6-994
6°845 © 6476 6237 || 6895 © 1283 6975 || 6945 9 6344 0332 | 6:995
6°846 © 6370 2002 || 6°896 o 1182 4644 || 6946 9 6247 7374 || 6996
6°847 © 6263 8833 || 0-897 o 1081 3325 || 6947 9 6151 5377 | 6-997
6848 © 6157 6725 || 6°898 © 0980 3017 || 6948 9 6055 4343 |, 6:998
| 6-849 © 6051 5679 ||6'899| 9 0879 3719 || 6949 9 5959 4263 | 6999

Vou. XIII. Parr ILI.

OVOOOON OOOOH OOOO WO OOOOH CHWOUWKMOMW OOUWOUWHO DODODOO DONWUKUOH)O ONnUWUUYO wwowownso
ty ies}
on
No}
°

0000000000 COMM DHMH DH DHHDHD DHHHDDH HDHDHDHDH HDHDDMDDM MOWOO OOOOO D099

| 77095 |

MR F. W.

COCO COOH MH MONDHODH DHDHM DHMDHDMD DOHMH M HODHMDMHHMH DOHOMDDHDDMH OHDMDDMH HDHDMMHH DNDN NM

NEWMAN'S TABLE

8957

1982
5873
0630
6251

2737
0086

8296 |
7369 |

7301
8094

9745 ||
Bae
5620 |

9843
4920
0852
7638
5277
3767

3109
3301
4342
6232
8969

2554
6984
2259
8379

5342 |

3148

1796 |)
1284

1613

2781
4787 ||

7631

1312 ||

5828

1180

7365 ||

4385
2236

0920 ||

0434

°779
1952
3955
6784
0441

e-2

2510
2428
2345
2263
2181

2008
2016
1934
1853
1771
1689
1607
1526
1444
1363
1282
1200
IIIg
1038
0957
0876
°795
o715
0634
0553

0473
0392
0312
0232
O152
oo71
9991
ggit
9832
9752

9672
9592
9573
9433
9354

srs sr OT a TT OT TAIT OOOOH ODMDDNDNDMD WDOOHOMDDMDD OMDNAMDMD ODMDMDMH DHDWDOOH

[7:000—7"199]

ee ee eG SG GG ss es

e-#

8486
8407
8329
8251
8173

8094
8016
7938
7861
7783

7705
7627
755°
7472
7395

7317
7240
7163
7086
7009
6932
6855
6778
6701
6625
6548
6472
6395
6319
6242
6166
6090
6014

5938
5862

5786
5711
5635
5559
5484

5408
5333
5358
5183
5107
5032
4957
4882
4808

4733

[7'200—7°399]

DNNDNDN ANDNDADAADA DANDDAA ADDADAA ADDADAA ADDAADAAN ADAADAAN ADADAADAD AnDAD Ananag

8

8775

5196
2017

g5II

6516
6025
6204

7052
8569
©0755
3607
7227
1313
6164
1680
7861
4795
2211
0380
9210
8701
8853

9663
1133
3261
6046
9489
3587
8341
375°
9813
6530
3900
1922

0596

9921
9896
0521
1795
3717
6287
9504

3368

7877
3032
8331

OF THE DESCENDING EXPONENTIAL
C* x (Ga x
7 4658 5808 || 7-250 7 1017 4389 || 7:300
7 4583 9596 | 7°251 7 0946 4569 | 7°301
7 4509 4129 | 7°252 7 0875 5459 | 7°302
7 4434 9407 | 7°253 7 0804 7958 | 7°303
7 4360 5430 || 7°254 7 9733 9365 || 7-304
7 4286 2196 | 7°255 7 0663 2379 | 7°305
7 4211 9705 | 7256 7 9592 6100 | 7°306
7 4137 7956 ||7°257 7 0522 0527 | 7°307
7 4063 6949 || 7°258 7 0451 5659 | 7°308
7 3989 6682 || 7°259 7 0381 1495 | 7°309
7 3915. 7155 || 7260 7 0310 8036 | 7°310
7 3841 8368 || 7:261 7 0240 5279 || 7°311
7 3768 0318 || 7:262 7 0170 3225 || 7-312
7 3694 3007 || 7°263 7 0100 1872 | 7°313
7 3620 6432 || 7°264 7 0030 1221 || 7°314
7 3547 0594 | 7°265 6 9960 1270 | 7°315
7 3473 5491 || 7°266 6 9890 2018 || 77316
7 3400 1123 || 7°267 6 9820 3465 || 7°317
7 3326 7488 || 7°268 6 9750 5611 || 7-318
7 3253 4587 || 7'269 6 9680 8454 | 7°319
7 3180 2419 || 7°270 6 9611 1994 || 7°320
7 3107 0982 || 7°271 6 9541 6230 || 77321
7 3934 0277 || 7°272 6 9472 1161 || 7-322
7 2961 0301 || 77273 6 9402 6787 || 7°323
7 2888 1056 || 7°274 6 9333 3107 || 7°324
7 2815 2539 || 7°275 6 9264 of2t || 7°325
7 2742 4751 || 7276 6 9194 7827 || 77326
7 2669 7689 || 7°277 6 9125 6225 || 7°327
7 2597 1355 || 7278 6 9056 5314 | 7°328
7 2524 5746 || 7°279 6 8987 5094 || 7°329
7 2452 0863 || 7°280 6 8918 5564 || 7°330
7 2379 6704 || 7°281 6 8849 6723 || 7°331
7 2307 3270 || 7°282 6 8780 8570 || 7°332
7 2235 0558 || 7°283 6 8712 1105 || 7°333
7 2162 8568 || 7:284 6 8643 4328 || 7°334
7 2090 7300 || 7°285 6 8574 8236 || 7°335
7 2018 6753 || 7°286 6 8506 2831 || 7°336
7 1946 6927 || 7°287 6 8437 8110 | 7°337
7 1874 7819 || 7°288 6 8369 4074 || 77338
7 1802 9431 || 7°289 6 8301 0722 || 7°339
7 1731 1760 || 7°290 6 8232 8053 || 7°340
7 1659 4807 || 7°291 6 8164 6066 || 7°341
7 1587 8570 || 7°292 6 8096 4760 || 7-342
7 1516 3049 || 7°293 6 8028 4136 || 7°343
7 1444 8244 || 7°294 6 7960 4192 || 7°344
7 1373 4153 | 7°295 6 7892 4927 | 7°345
7 1302 0775 || 7°296 6 7824 6342 | 7°346
7 1259 811r | 7°297 6 7756 8434 | 7°347
7 1159 6159 || 7°298 6 7689 1205 || 7°348
7 1088 4913 | 7°299 6 7621 4652 | 7°349

IWNN NNN NN ONS SOS ST OSS SSO OO OSS OS

5274 ||7°

1

oO.

Oo on nn fWN

DADaAnninn anania
()

4

Da AO
DA £+wW hn

nad
© oO

Inn ~
NH O

as
vw

as
Oo om

aw
mw O

S)

WBWWWW WHWWW WHWWWW WHWWWW WUWWW WHWHAW WHWWWO8
co en
ow x

Cmommn ow
Oar dan

WW WWWWW WWW do
OD CODOO

396

NNNANAN ADANNNAN DANDDADAND ANDDADAA ANDDAAAGDA AQNngngn AQAnDAD AQAnngqnga ADADA Annan
nN
fo)
~
LS)

188 MR F. W. NEWMAN'S TABLE [7400—7'599]

' A

x | e-* | 7 e-* 2 e-* x |
| 7°400 6 1125 2761 | 7°450 5 8144 1612 || 7500 5 5308 4370 || 7°550 5 2611 0127
7°401 6 1064 1814 |i 7°451 5 8086 0461 || 7-501 5 5253 1562 || 7551 5 2558 4280
7402 6 1003 1477 |, 7°452 5 8027 9891 || 7*502 5 5197 9397 || 77552 5 2505 8958
7°493 6 0942 1751 || 7-453 5 7969 9901 || 7°503 5 5142 7603 || 7°553 5 2453 4162
|7'404| 6 0881 2634 es 5 7912 0491 || 7°504 5 5087 6451 | 7°554 5 2400 9890
7°405 6 0820 4125 || 7°455 5 7854 1660 || 7°505 5 5032 5850 || 7°555 5 2348 6142
7406 6 0759 6225 || 7456 5 7796 3407 || 7°506 5 4977 5799 || 77556 5 2296 2917
7°407| 6 0698 8933 | 7°457 5 7738 5733 || 7°5°7 5 4922 6298 | 7°557 5 2244 0216
7°408 6 0638 2247 | 7°458 5 7680 8636 || 7°508 5 4867 7347 ||7'558 5 2191 8037
7°409 6 0577 6168 | 7-459 5 7623 2115 || 7°509 5 4812 8944 || 7°559 5 2139 6380
| 7*410 6 0517 0694 || 7:460 5 7565 6171 || 7-510 5 4758 1089 || 7°560 5 2087 5244
| 7°411 6 0456 5826 || 7-461 5 7508 0803 || 7511 5 4703 3781 || 77561 5 2035 4629
77412 6 0396 1563 || 7°462 5 7450 6010 || 7512 5 4648 7021 || 7-562 5 1983 4534
7413 6 0335 7993 || 7463 5 7393 1791 |17°513 5 4594 0807 || 7°563 5 1931 4960
| 7°414 6 0275 4847 || 7°464 5 7335 8146 || 7°514 5 4539 5139 || 7564 5 1879 5904
7415 6 o215 2393 || 7-465 § 7278 5074 || 7°515 5 4485 0017 | 7°565 5 1827 7368
| 7416 6 0155 0542 || 7°466 5 7221 2575 || 77516 5 443° 5439 || 7°566 5 1775 9349
7417 6 0094 9292 || 7-467 5 7164 0649 || 7°517 5 4376 1406 || 7°567 5 1724 1849
77418 6 0034 8643 | 7-468 5 7106 9294 || 7*518 5 4321 7916 || 7°568 5 1672 4865
7419| 5 9974 8594 || 7-469 5 7049 8510 |} 7°519 5 4267 4970 | 7'569 5 1620 8399
7°420 5 9914 9145 | 7°470 5 6992 8297 || 7°520 5 4213 2566 || 7°570 5 1569 2449
7421 5 9855 0296 || 7°471 5 6935 8653 || 7°521 5 4159 0704 || 7571 5 1517 7014 |
7422 5 9795 2045 || 7472 5 6878 9579 || 7°522 5 4104 9384 || 7°572 5 1466 2094
77423 5 9735 4391 || 77473 5 6822 1074 || 7°523 5 4050 8605 || 7°573 5 1414 7689
7424 5 9675 7336 || 7474 5 6765 3137 || 77524 5 3996 8367 || 7°574 5 1363 3799
| 7°425 5 9616 0877 || 7°475 5 6708 5768 || 7°525 5 3942 8668 | 7°575 5 1312 0422
7°426 5 9556 5014 || 7°476 5 6651 8965 || 7°526 5 3888 9509 || 77576 5 1260 7558
| 7°427 5 9496 9746 || 7°477 5 6595 2729 || 7°527 5 3835 0889 || 7°577 5 1209 5207
| 7°428 5 9437 5°74 || 7°478 5 6538 7060 || 7°528 5 3781 2807 || 7°578. 5 1158 3367
| 7°429 5 9378 0996 | 7479 5 6482 1955 || 7°529 5 3727 5263 || 7°579 5 1107 2040
| 7°430 5 9318 7512 || 7°480 5 6425 7415 ||7°53° 5 3673 8257 || 7°580 5 1056 1223
7431 5 9259 4621 || 7-481 5 6369 3440 || 7°531 5 3620 1787 || 7581 5 1005 0917
| 7°432 5 9200 2322 || 7°482 5 6313 0028 || 7°532 5 3566 5853 || 7582 5 0954 I121
7°433 5 9141 0616 | 7°483 5 6256 7180 || 7°533 5 3513 0455 || 7583 5 0903 1835
7434 5 9081 9501 || 7484 5 6200 4894 || 7°534 5 3459 5592 || 7°584 5 0852 3057
7°435 5 9022 8977 || 7°485 5 6144 3170 || 7°535 5 3406 1263 || 7°585 5 0801 4788
7°436 5 8963 9043 || 7486 5 6088 2007 || 7°536 5 3352 7469 || 7°586 5 0750 7027
7°437 5 8904 9698 | 77487 5 6032 1406 || 7°537 5 3299 4208 | 7°587 5 0699 9774
7°438 5 8846 0943 | 7°488 5 5976 1364 || 7°538 5 3246 1480 || 7°588 5 0649 3028
7°439 5 8787 2776 | 7'489| 5 5920 1883 || 7°539 5 3192 9285 || 7°589 5 0598 6788
7°440 5 8728 5197 || 7°490 5 5864 2960 || 7°540 5 3139 7622 || 7°590 5 0548 1054
7°441 5 8669 8206 | 7°491 5 5808 4597 || 7°541 5 3086 6490 || 7591 5 0497 5826
| 7°442 5 8611 1801 | 7°492 5 5752 6791 || 7542 5 3033 5889 || 7°592 5 0447 1102
7°443 5 8552 5982 | 7°493 5 5696 9543 || 7°543 5 2980 5818 || 7°593 5 0396 6883
7444 5 8494 0749 | 7°494 5 5641 2852 | 7°544 5 2927 6277 || 7°594 5 0346 3168
| 7°445 5 8435 6100 | 7°495 5 5585 6717 || 7°545 5 2874 7265 || 7°595 5 0205 9957
7°446 5 8377 2036 | 7°496 5 5530 1138 || 7°546 5 2821, 8782 || 7°596 5 0245 7248
7°447 5 8318 8556 | 7°497 5 5474 6115 ||.7°547 5 2769 0827 |7°5971-" 5 0195 5042
| 7°448 5 8260 5659 | 7°498 5 5419 1646 | 77548 5 2716 3400 | 7°595 5 0145 3338
7°449 5 8202 3344 | 7°499 5 5363 7731 || 7549 5 2663 6500 | 77,99 5 0095 2135

[7°600—7°799] OF THE DESCENDING EXPONENTIAL. 189

| x e-% Hy GAe x é=* a e-#
|

| 77600 5 0045 1433 || 77650 4 7604 4129 || 7°700 4 5282 7183 || 7°750 4 3074 2541
7601 4 9995 1232 || 7°651 4 7556 8323 | 7°701 4 5237 4582 || 7°751 4 3031 2013
7°602 4 9945 1531 || 7°652 4 7509 2992 || 7°702 4 5192 2433 || 7°752 4 2988 1916
7603 4 9895 2329 || 7°653 4 7461 8137 || 7°703 4 5147 0737 || 7°753 4 2945 2249
7604 4 9845 3626 | 7°654 4 7414 3756 || 7°704 4 5101 9492 || 7°754 4 2902 3012
7°605 4 9795 5421 || 7°655 4 7366 9849 || 7°705 4 5056 8608 | 7°755 4 2859 4203
77606 4 9745 7715 | 77656 4 7319 6416 || 7°706 4 501t 8354 || 7°756 4 2816 5823
7°607 4 9696 0506 || 7°657 4 7272 3456 || 7°707 4 4966 846r || 7°757 4 2773 7871
7608 4 9646 3794 | 77658 4 7225 0969 || 7°708 4 4921 9017 |7°758 4 2731 0347
7609 4 9596 7578 | 7°659 4 7177 8954 || 7°709 4 4877 0023 || 7°759 4 2688 3251
7610 4 9547 1858 || 7°660 4 713° 7411 || 77710 4 4832 1477 || 7°760 4 2645 6581
7611 4 9497 6634 || 7°66r 4 7083 6339 || 7°711 4 4787 3380 || 7°761 4 2603 0337
7612 4 9448 1905 || 7°662 4 7036 5738 || 77712 4.4742 5730 || 7°762 4 2560 4520
7°613 4 9398 7670 || 7°663 4 6989 5607 || 7°713 4 4697 8528 || 7°763 4 2517 9128
7614 4 9349 3929 | 7°664 4 6942 5946 || 7°714 4 4653 1773 || 7°764 4 2475 4162
7615 4 9300 0682 || 7°665 4 6895 6755 | 7°715 4 4608 5464 || 7°765 4 2432 9620
7°616 4 9250 7928 || 7°666 4 6848 8033 || 7°716 4 4563 9602 || 7°766 4 2390 5502
7617 4 9201 5666 || 7°667 4 6801 9779 || 7°717 4 4519 4185 || 7°767 4 2348 1809
7618 4 9152 3896 || 7°668 4 6755 1993 || 7°718 4 4474 9213 | 7°768 4 2305 8538
7619 4 9103 2618 || 7°669 4 6708 4675 || 7°719 4 4430 4686 || 77769 4 2263 5691
7°620 4 9054 1831 || 7°670 4 6661 7824 || 7°720 4 4386 0604 || 7°770 4 2221 3267
7621 4 9005 1534 | 7°671 4 6615 1439 | 7°721 4 4341 6965 || 7°771 4 2179 1265
7622 4 8956 1728 || 7°672 4 6568 5521 || 7'722 4 4297 377° || 7°772 4 2136 9684
7623 4 8907 2411 || 7°673 4 6522 0068 || 7°723 4 4253 1017 |17°773 4 2094 8525
7624 4 8858 3583 || 7°674 4 6475 5080 | 7°724 4 4208 8708 || 7°774 4 2052 7787
7°625 4 8809 5243 || 7°675 4 6429 0557 || 7°725 4 4164 6840 | 7°775 4 2010 7469
7626 4 8760 7392 || 7°676 4 6382 6499 | 7°726 4 4120 5414 || 7°776 4 1968 7572
7627 4 8712 0028 || 7°677 4 6336 2904 || 7°727 4 4076 4429 | 7°777 4 1926 8094
7628 4 8663 3152 || 7°678 4 6289 9773 || 7°728 4 4032 3885 || 7°778 4 1884 9036
7629 4 8614 6762 || 7°679 4 6243 7105 || 7°729 4 3988 3781 || 7779 4 1843 0396
7630 4 8566 0858 || 7°680 4 6197 4899 || 7°73° 4 3944 4117 || 7'780 4 1801 2175
7631 4 8517 5440 || 7°681 4 O15r 3155 || 7°73! 4 3900 4893 || 7°781 4 1759 4371
7632 4 8469 0507 || 7°682 4 6105 1872 || 7°732 4 3856 6107 || 7°782 4 1717 6986
7633 4 8420 6059 || 7°683 4 6059 1051 || 7°733 4 3812 7760 | 7°783 4 1676 oor7
7634 4 8372 2095 | 7°684 46013 0690 117-734). 4 3768 9851 || 7784 4 1634 3466
7°635 4 8323 8614 | 7°685 4 5967 0789 || 7°735 4 3725 2380 | 7°785 4 1592 7330
7°636 4 8275 5617 || 7°686 4 5921 1348 || 7°736 4 3681 5347 || 7°786 4 1551 1611
7°637 4 8227 3103 || 7687 4 5875 2366 || 7°737 4 3637 8750 | 7°787 4 1509 6307 |
7°638 4 8179 1071 || 7°688 4 5829 3843 || 7°738 4 3594 2589 | 7°788 4 1468 1418 |
7°639 4 8130 9521 || 7°689 4 5783 5778 ||7°739 4 3550 6864 || 7°789 4 1426 6944 |
7640 4 8082 8452 || 7°690 4 5737 8172 || 7740 4 35°7 1575 || 7°79° 4 1385 2584
7641 4 8034 7864 | 7°691 4 5092 1022 |) 7°741 4 3463 6721 | 7°791 4 1343 9238
77642 4 7986 7756 || 7692 4 5646 4329 || 7°742 4 3420 2302 | 7°792 4 1302 6005
7643 4 7938 8128 | 7693 4 5600 8093 || 7°743 4 3376 8316 | 7°793 4 1261 3186 |
7644 4 7890 8980 | 794 4 5555 2313 || 7°744 4 3333 4765 | 7°794 4 1220 0779
7°645 4 7843 0310 || 7°695 4 5509 6988 || 7°745 4 329° 1647 | 7°795 4 1178 8784
7646 4 7795 2119 | 7°696 4 5464 2119 || 7°746 4 3246 8961 || 7°796 4 1137 7201
7647 4 7747 4406 | 7°697 4 5418 7704 || 7°747 4 3203 6709 | 7°797 4 1096 6030
7648 4 7699 717° || 7°698 4 5373 3743 | 7°748 4 3160 4888 | 7-798 4 1055 5269 |
7649 4 7652 o4it || 7°699 4 5328 0236 || 7°749 4 3117 3499 | 7°799 4 1014 4919

190 MR F. W. NEWMAN’S TABLE
im e-% x e-? | @ e-#
= : |

7800 4 9973 4979 || 7°850 3 8975 1968 | 7-900 3 7°74 3540 ||
7801 4 0932 5449 || 7°851 3 8936 2411 || 790% 3 7037 2982 ||
7802 4 0891 6328 || 7°852 3 8897 3243 || 7902 3 7000 2794
7°803 4 0850 7616 | 7-853 3 8858 4464 || 7°903 3 6963 2976
7804 4 0809 9312 | 7°854 3 8819 6074 || 7904 3 6926 3528 ||
7°805 4 0769 1417 | 7°855 3 8780 8072 | 7°905 3 6889 4449
7806 4 0728 3929 || 7°856 3 8742 0458 |, 7'906 3 6852 5739
7°807 4 0687 6849 || 7°857 3 8703 3231 || 7°907 3 6815 7398
7808 4 0647 0176 | 7°858 3 8664 6391 | 7°9908| - 3 6778 9424
7°809 4 0606 3909 | 7°859 3 8625 9938 || 7909 3 6742 1819
7810 4 0565 8048 | 7:860 3 8587 3871 | 7-910 3 6705 4580
7811 | 4 0525 2592 || 7°861 3 8548 8190 || 7-911 3 6668 7709
7812 4 0484 7542 || 7°862 3 8510 2895 || 7-912 3 6632 1205
| 7813 4 0444 2897 || 7°863 3 8471 7984 | 7913 3 6595 5067
7814 4 0403 8656 || 7°864 3 8433 3459 || 7914 3 9558 9295
7°815| 4 0363 4820 || 7°865 3 8394.9317 || 7°915 3 6522 3888
7816 4 0323 1386 || 7°866 3 8356 5560 || 7-916 3 6485 8847
7°817 4 0282 8357 || 7°867 3 8318 2186 | 7917 3 6449 4170
7818 4 0242 5730 || 7°868 3 8279 9195 | 7°918 3 6412 9858
7°819 | 4 0202 3505 || 7°869 3 8241 6588 || 7-919 3 6376 5910
| 7820! 4 0162 1683 || 7°870 3 8203 4362 || 7"920 3 6340 2326
7821 4 0122 0262 || 7°871 3 8165 2519 || 7921 3 6303 9106
7822 | 4 0081 9242 || 7°872 3 8127 1057 || 7922 3 6267 6248
7°823| 4 0041 8623 || 7°873 3 8088 9977 | 7°923 3 6231 3753
7°824 | 4 ooo 8405 |] 7°874 3 8050 9277 || 7°924 3 6195 1620
78251 3 9961 8586 || 7°875 3 8012 8958 | 7925 3 6158 9850
| 7°326 3 9921 9167 || 7°876 3 7974 9019 || 7°926 3 6122 8441
|7°827| 3: 9882 0148 | 7°877 3 7936 9460 | 7°927 3 6086 7393
| 7°828 3 9842 1527 || 7°378 3 7899 0280 || 7°928 3 6050 6706
7°829 3 9802 3304 || 7°879 3 7861 1479 | 7°929 3 6014 6379
| 7830 3 9762 5480 || 7°880 3 7823 3057 || 7°930 3 5978 6413
| 7°831 3 9722 8053 || 7°881 3 7785 50°13 || 7°93! 3 5942 6806
| 7°832 3 9683 1024 || 7°882 3 7747 7347 || 7°932 3.5906 7559
(7833 3 9643 4391 || 7°883 3 7710 0058 || 7°933 3 5870 8671
|7°834 3 9603 8155 || 7°884 3 7672 3146 || 7°934 3 5835 0142
| 7°835 3 9564 2315 || 7°885 3 7634 6612 || 7°935 3.5799 1971
| 7°836 3 9524 6870 | 7°386 3 7597 0453 || 7°936 3 5763 4158
| 7°837 3 9485 1821 || 7°887 3 7559 4671 || 7°937 3 5727 6702
| 7°838 3 9445 7166 || 7°888 3 7521 9264 || 7°938 3 5691 9604
7°839 3 9406 2906 || 7889 3 7484 4232 || 7°939 3 5656 2863
7°840 3 9366 go4o || 7°890 3 7446 9575 || 7°940 3 5620 6478
7°841 3 9327 5568 || 7801 3 7409 5293 || 7°941 3 5585 0450
7°342 3 9288 2489 | 7°892 3 7372 1384 |) 7°942 3 5549 4777
7°343 3 9248 9803 || 7°893 3 7334 785° || 7°943 3 5513 9460
7°844 3 9209 7509 || 7°894 3 7297 4688 || 7°944 3 5478 4498
7°845 3 9170 5608 || 7°895 3 7260 1900 ||-7°945 3 5442 9891
7846 3 9131 4098 || 7°896 3 7222 9484 || 77946 3 5407 5638
7°347 3 9092 2980 || 7°897 3 7185 7441 || 7°947 3 5372 1740
7848 3 9053 2252 || 7°898 3 7148 5769 || 7°948 3 5336 8195
7849 3 9014 1915 || 7°899 3 7111 4469 || 7°949 3 5301 5003

[7*800—7'999]

NNNN ONIN NI
Wwonowonwo wowwoworvo

8

nanan naNnou o

n
oI an £fWNHO

E959

5266 2165
5230 9679
5195 7545
5160 5763
5125 4333

599° 3255
SRI) L8)="//
5020 2149
4985 2122

495° 2445

4915 3117
4880 4139
4845 5509
4810 7228
aati) 3s

4741 1709
4706 4471
46071 7580
4637 1036
4602 4838
4567 8986
4533 3480
4498 8319
4464 3503
9032
4395 4905
4361 1122
4326 7682
4292 4586
4258 1833

4223 9422
4189 7354
4155 5628
4I2E 4243
4087 3199

4053 2496
SOLO RZ ae
3985 2112
3952 2420
3917 3087

3883 4083
3849 5418
3815 7092
3781 9104
3748 1454
3714 4141
3680 7166
3647 0527 |
3613 4224
3579 8258

BDWWWW WWWWW WWWWW WWWWW WHWWW WWWWW WWWWW WWWWW WWWwWWww WwWwWww
-
-
nN
Ko}

[S:ooo—S-199] OF THE DESCENDING EXPONENTIAL. 191

— = ~--— = +4
wv GX x Cu av (GH | @ Ca

1
8-000 3 3546 2628 || 8050 3 1910 1922 || 8100 3 0353 9138 || 8150 2 8873 5360
8001 3 3512 7333 || 8051 3 1878 20980 || 8-101 3 0323 5751 || S151 2 8844 6769
8-002 3 3479 2373 || 8052 3 1846 4356 || 8102 3 0293 2666 || 8152 2 8815 8466
8-003 3 3445 7748 || 8053 3 1814 6051 || 8103 3 0262 9885 || 8153 2 8787 0451
8004 3 3412 3457 | 8054 3 1782 8064 || 8104 3 0232 7407 | 8154 2 8758 2725
8005 3 3378 9501 || 8055 3 1751 0395 |) 8105 3 0202 5230 || 8155 2 8729 5286
8-006 3 3345 5878 || 8:056 3 1719 3043 || 8106 3 0172 3356 || 8156 2 8700 8134
8:007 3 3312 2589 || 8:057 3 1687 6009 || 8107 3 0142 1783 || 8157 2 8672 1270
8-008 3 3278 9633 || 8-058 3 1655 9291 || 8108 3 O112 0512 || 8158 2 8643 4692
8-009 3 3245 7010 || 8059 3 1624 2890 || 8109 3 0081 9542 || 8159 2 8614 8400 |
8-oro 3 3212 4719 || 8:060 3.1592 6805 | 8110 3 0051 8873 8160 2 8586 2395
| Sor1 3 3179 2760 || 8061 3 1561 1036 || 8111 3 0021 8505 || 8-161 2 8557 6675 |
8-o12 3 3146 1133 || 8062 3 1529 5583 || 8112 2 9991 8436 || 8-162 2 8529 1241 |
8-013 3 3112 9838 || 8:063 3 1498 0445 || 8113 2 9961 8668 | 8-163 2 8500 6093
S014 3 3079 8874 || 8:064 3 1466 5622 [oom 2 9931 9199 || 8°164 2 8472 1229
8015 3 3046 8240 || 8:065 3.1435 1114 || 8115 2 9902 0029 || 8165 2 8443 6650
8-016 3 3013 7937 || 8:066 3 1403 6920 || 8116 2 9872 1158 || 8-166 2 8415 2356 |
8-017 3 2980 7964 || 8:067 3 1372 3040 || 8117 2 9842 2587 || 8-167 2 8386 8345 |
8-018 3 2947 8321 || 8068 3 1340 9474 || 8118 2 9812 4313 || 8-168 2 8358 4619
8-019 3 2914 9007 || 8:069 3 1309 6221 fees) 2 9782 6338 || 8169 2 8330 1176
8-020 3 2882 0023 || 8070 3 1278 3281 || 8120 2 9752 8660 || 8170 2 8301 8016
8-021 3 2849 1367 || 8071 3.1247 0654 || 8121 2 9723 1280 || 8171 2 8273 5140
8-022 3 2816 3040 || 8:072 3 1215 8340 || 8-122 2 9693 4198 || 8-172 2 8245 2546
8-023 3 2783 5041 || 8:073 3 1184 6337 || 8-12 2 9663 7412 || 8-173 2 8217 0235
8:024 3 2750 7370 || 8074 3 1153 4647 || 8124 2 9634 0923 || 8174 2 8188 8205
8025 3 2718 0026 | 8:075 3 1122 3268 || 8125 2 9604 4730 || 8°175 2 8160 6458
8:026 3 2685 3010 || 8:076 3 1091 2260 || 8-126 2 9574 8833 || 8°176 2 8132 4992
8:027 3 2652 6320 || 8:077 3 1060 1443 || 8-127 2 9545 3232 || $177 2 8104 3808
8-028 3 2619 9957 || 8078 3 1029 0997 || 8128 2 9515 .7927 || 8178 2 8076 2905
8-029 3 2587 3920 || 8079 3 0998 0861 || 8129 2 9486 2916 || 8-179 2 8048 2282
8030 3 2554 8209 | 8:080 3 0967.1035 || 8°130 2 9456 8201 || 8:180 2 8020 1940
8-031 3 2522 2823 || 8-081 3 0936 1519 || 8131 2 9427 3780 || $181 2 7992 1878
8-032 3 2489 7763 || 8082 3 0905 2312 || 8132 2 9397 9653 || 8-182 2 7964 2096
8-033 3 2457 3028 || 8-083 3 0874 3414 || 8133 2 9368 5820 || 8183 2 7936 2594
8034 3 2424 8617 (8-084 3 0843 4825 || 8-134 2 9339 2281 || 8-184 2 7908 3371
8035 3 2392 4530 || 8085 3 0812 6545 || 8135 2 9309 9036 | 8185 2 7880 4427 |
8036 3 2360 0768 | 8:086 3 0781 8572 || $136 2 9280 6083 | 8°186 2 7852 5762 |
8°037 3 2327 7329 | 8087 3 0751 0907 || 8°137 2 9251 3423 || 8187 2 7824 7375
8-038 3 2295 4213 || 8088 3 0720 3550 || 8-138 2 9222 1056 || 8188 2 7796 9267
8'039 3 2263 1420 || 8089 3 0689 6500 || 8139 2 9192 8981 | 8-189 2 7769 1437
8-040 3 2230.8950 | 8090 3 0658 9757 || 8140 2 9163 7198 | 8*190 2 7741 3884
8-041 3 2198 6802 | 8'ogt 3 0628 3321 || 8141 2 9134 5707 || 8191 2 7713 6609
8-042 3 2166 4976 | 8092 3 0597 7190 || 8142 2 9105 4507 | 8192 2 7685 9611
8043 3 2134 3472 || 8093 3 0567 1366 || $143 2 9076 3598 | 8-193 2 7658 2890
8:044 3 2102 2289 | 8:094 3 0536 5848 || 8-144 2 9047 2979 || 8194 2 7630 6445
8045 3 2070 1427 | 8:095 3.0506 0634 || 8145 2 go18 2652 || 8-195 2 7603 0277
8-046 3 2038 0886 | 8096 3 0475 5726 || 8146 2 8989 2614 || 8196 2.7575 4384
8:047 3 2006 0666 | 8-097 3 0445 1123 || 8-147 2 8960 2866 | 8197 2 7547 8768
8-048 3 1974 0765 | 8:098 3 0414 6824 || 8148 2 8931 3408 || 8198 2 7520 3427
8-049 3 1942 1184 | 8:099 3 0384 2829 || 8-149 2 8902 4239 | 8199 2 7492 8361

MR F. W. NEWMAN'S TABLE

[8:200—8'399]

192
x e-= Hi e-%
8-200 2 7465 3570 || S:250 2 6125 8557
8-201 2 7437 9054 || 8251 2 6099 7429
8-202 2 7410 4812 || 8252 2 6073 6562
$:203 2 7383 0844 || 8-253 2 6047 5956
8-204 2 7355 7150 |] $254 2 6021 5610
8-205 2 7328 3729 || 8255 2 5995 5525
8-206 2 7301 0582 || 8-256 2 5969 5699
8-207 2 7273 7708 || 8'257 2 5943 6133
8-208. 2 7246 5107 || 8258 2 5917 6827
8-209 2 7219 2778 || 8259 2 5891 7780
8-210 2 7192 0721 || 8-260 2 5865 8991
8-211 2 7164 8936 || 8-261 2 5840 0461
8-212 2 7137 7423 || 8262 2 5814 2190
8-213 2 7110 6182 || 8-263 2 5788 4177
8-214 2 7083 5211 || 8-264 2.5762 6422
8-215 2 7056 4511 || 8-265 2 5736 8924
8-216 2 7029 4082 || 8-266 2 5711 1684
| 8-217 2 7002 3923 || 8°267 2 5685 4701
8-218 2 6975 4034 || 8-268 2 5659 7974
8-219 2 6948 4415 || 8269 2 5634 1505
8-220 2 6921 5065 || 8-270 2 5608 52091
8-221 2 6894 5984 || 8-271 2 5582 9334
8-222 2 6867 7173 || 8°272 2 5557 36032
8-223 2 6840 8630 || 8-273 2 5531 8187
| 8-224 2 6814 0356 || 8-274 2 5506 2996
| 8-225 2 6787 2349 || 8275 2 5480 8061
| 8-226 2 6760 4611 || 8-276 2 5455 3380
| 8-227 2 6733 7140 || 8-277 2 5429 8954
8-228 | 2 6706 9936 || 8:278 2 5404 4782
| 8-229 2 6680 3000 || 8:279 2 5379 0864
| 8-230 2 6653 6330 || 8-280 2 5353 7200
8-231 2 6626 9927 || 8281 2 5328 3790
8-232 2 6600 3790 || 8:282 2 5303 0632
| 8-233 2 6573 7919 || 8283 2 5277 7728
aa 2 6547 2314 || 8284 2 5252 5077
8235 2 6520 6975 || 8:285 2 5227 2678
| 8236 2 6494 1900 || 8-286 2 5202 0531
8°237 2 6467 7091 || 8287 2 5176 8637
8-238 | 2 6441 2546 || 8-288 2 5151 6994
| 8239 | 2 6414 8266 || 8-289 2 5126 5603
8-240 | 2 6388 4249 || 8-290 2 5101 4463
(S241) 2 6362 0497 || 8-291 2 5076 3574
| 8242 2 6335 7008 || 8-292 2 505% 2935
| 8-243 2 6309 3783 || 8°293 2 5026 2548
8244 2 6283 o821 Wage 2 5001 2410
8-245 | 2 6256 8r2r || 8-295 2 4976 2523
8-246 | 2 6230 5684 | 8-296 2 4951 2885
| 8°247 2 6204 3510 || 8297 2 4926 3497
8-248 2 6178 1597 || 8298 2 4901 4358
8-249 2 6151 9946 | 8-299 2 4876 5468

|

NNN N NN NNN N NNN NN NNN NN by Ny NN NNN NN NNN NN N NNN N NNNN ND

N

NNN NN

6827
8434
0290
2394
4745

7344
o1g0
3284
6624

O2I0 |;

4043
8122
2447
7Ol7
1833
6893
2199
7749

3543
9582

5864

2390 |

9159
6172

3427

0925.

8665
6647
4872
3337

2045

ey
0183

9613 |)
9283 |

9104.
9345
9735

0365 |

1235

2343 ||

3690
5275
7°99
g161
1461
3998
6773
9784

3033 ||8°

nN NN N NNN N NNN NN NNN NN Nn Nn NN NNN NN NNN NN N NNN ND NNNN WN

NNN NN

[8:400—8'599] OF THE DESCENDING EXPONENTIAL. 193

xv en * x“ CoE x“ Ge x“ Ore
8400 2 2486 7324 || 8450 2 1390 O4T5 || 8-500 2 0346 8369 || 8°550 I 9354 5099
8-401 2 2464 2569 || 8-451 2 1368 6622 || 8-sor 2 0326 5002 || 8-551 I 9335 1651
8402 2 2441 8039 || 8:452 2 1347 3042 || 8-502 2 0306 1839 || 8552 I 9315 8396
8°403 2 2419 3733 || 8453 2 1325 9676 || 8-503 2 0285 8879 | 8553 1 9296 5334
8-404 2 2396 9651 || 8454 2 1304 6523 || 8-504 2 0265 6121 || 8554 I 9277 2465
8°405 2 2374 5794 || 8°455 2 1283 3583 || 8-505 2 0245 3566 || 8555 I 9257 9789
8°406 2 2352 2160 || 8456 2 1262 0855 || 8-506 2 0225 1214 || 8556 I 9238 7306
8:407 2 2329 8749 | 8-457 2 1240 8341 || 8507 2 0204 9064 | 8557 I 9219 5015
8-408 2 2307 5562 || 8-458 2 1219 6039 || 8-508 2 0184 7116 || 8558 I 9200 2916
8409 2 2285 2598 || 8-459 2 1198 3949 || 8509 2 0164 5369 || 8559 I g18I 1009
8410 2 2262 9857 || 8460 2 1177 2071 || 8-510 2 0144 3825 || 8560 I 9161 9294
8-411 2 2240 7338 || 8-461 2 1156 0404 || 8-511 2 0124 2482 || 8561 I 9142 7770
8-412 2%22a8 5042 || 8462 2 1134 8950 || 8-512 2 O104 1340 || 8562 I 9123 6438
8413 2 2196 2968 || 8-463 2 1113 7706 || 8-513 2 0084 0399 || 8563 I 9104 5297
8-414 2 2174 1116 || 8°464 2 1092 6674 || 8514 2 0063 9659 || 8564 I 9085 4347
8415 2 2151 9486 || 8465 2 1071 5853 || 8-515 2 0043 9120 || 8565 I 9066 3588
8-416 2 2129 8077 || 8466 2 1050 5243 || 8516 2 0023 8781 || 8566 I 9047 3020
8-417 2 2107 6890 || 8:467 2 1029 4843 || 8517 2 0003 8642 || 8567 I 9028 2642
8-418 2 2085 5923 || 8-468 2 1008 4653 || 8-518 I 9983 8703 || 8568 I goog 2455
8°419 2 2063 5178 || 8-469 2 0987 4673 || 8519 1 9963 8965 || 8569 1 8990 2457
8:420 2 2041 4653 || 8470 2 0966 4903 || 8'520 I 9943 9425 || 8570 1 8971 2650
8-421 2 2019 4348 || 8°471 2 0945 5343 || 8521 I 9924 0086 || 8571 I 8952 3032
8:422 2 1997 4264 || 8-472 2 0924 5993 || 8522 I 9904 0945 || 8572 t 8933 3604
8423 2 1975 4400 || 8-473 2 0903 6851 || 3'523 1 9884 2004 || 8573 t 8914 4365
8424 2 1953 4755 || 8474 2 0882 7919 || 8524 1 9864 3261 || 8574 t 8895 5315
8-425 2 1931 5330 | 8-475 2 0861 9195 || 8525 1 9844 4717 || 8575 1 8876 6454
8-426 2 1909 6124 || 8:476 2 0841 0680 || 8-526 I 9824 6372 || 8576 1 8857 7782
8427 2 1887 7138 || 8:477 2 0820 2374 || 8527 1 9804 8224 || 8°577 1 8838 9298
8-428 2 1865 8370 || 8-478 2 0799 4276 || 8528 I 9785 0275 || 8578 I 8820 1003
8-429 2 1843 9821 || 8:479 2 0778 6385 || 8529 I 9765 2524 || 8579 I 8801 2896
8430 2 1822 1490 || 8:480 2 0757 8703 || 8530 I 9745 4970 || 8580 1 8782 4977
8-431 2 1800 3378 || 8-481 2 0737 1228 || 8531 I 9725 7614 || 8581 1 8763 7246
8-432 2 1778 5483 || 8°482 2 0716 3960 || 8532 I 9706 0455 || 8582 I 8744 9703
8°433 2.1756 7807 || 8-483 2 0695 6900 || 8°533 1 9686 3493 || 8583 1 8726 2347
8434 2 1735 0348 || 8-484 2 0675 0046 || 8534 1 9666 6728 || 8584 1 8707 5178
8435 2 1713 3106 || 8:485 2 0654 3400 || 8°535 I 9647 0159 || 8585 1 8688 8196
8-436 2 1691 6081 || 8-486 2 0633/6960 || 8-536 I 9627 3787 || 8586 1 8670 1402
8-437 2 1669 9274 || 8:487 2 0613 0726 || 8'537 I 9607 7612 || 8587 1 8651 4793
8°438 2 1648 2683 || 8-488 2 0592 4698 || 8:538 I 9588 1632 || 8583 1 8632 8372 |
8°439 2 1626 6308 || 8-489 2 0571 8876 || 8539 I 9568 5848 || 8589 I 8614 2137 |
8-440 2 1605 o150 || 8-490 2 0551 3260 || 8-540 I 9549 0260 || 8590 t 8595 6088
8-441 2 1583 4208 || 8-491 2 0530 7850 || 8541 I 9529 4868 || 8501 1 8577 0224
8442 2 1561 8482 || 8-492 2 0510 2644 || 8542 I 9509 9670 || 8592 1 8558 4547
8°443 2 1540 2971 || 8493 2 0489 7644 || 8'543 1 9490 4668 | 8593 1 8539 9055
8444 2 1518 7676 || 8-494 2 0469 2849 || 8544 I 9470 9861 || 8'°594 1 8521 3749
8°445 2 1497 2595 || 8°495 2 0448 8258 || 8:545 I 9451 5248 || 8595 1 8502 8628
8-446 2 1475 7730 || 8'496 2 0428 3872 || 8°546 I 9432 0830 || 8°596 1 8484 3692
8°447 2 1454 3080 || 8497 2 0407 9691 || 8547 I 9412 6607 || 8597 1 8465 8940
8-448 2 1432 8644 | 8-498 2 0387 5713 | 8548 1 9393 2577 || 8598 1 8447 4374
8°449 2 I4Il 4423 | 8-499 2 0367 1939 | 8°549 I 9373 8741 || 8°599 1 8428 9991

Vou. XIIT. Parr TIT, 26

194

| & C*
8-600 1 8410 5794
8-601 1 8392 1780
8-602 1 8373 7950
8-603 t 8355 4304
| 8°604 1 8337 0841
| 8-605 1 8318 7562
| 8-606 1 8300 4466
8-607 1 8282 1553
| 8-608 1 8263 8823
8-609 1 8245 6275
8-610 1 8227 3910
8-611 1 8209 1728
8-612 1 8190 9727
18-613 1 8172 7908
18-614 1 8154 6271
|$°615 1 8136 4815
8-616 1 8418 3541
| 8-617 1 8100 2448
8-618 1 8082 1536
|8*619 1 8064 0805
8°620 1 8046 0255
8-621 1 8027 9885
8622 1 8009 9695
8°623 I 7991 9685
8624 1 7973 9855
8-625 I 7956 0205
| 8-626 17938 0735
8°627 I 7920 1444
8628 I 7902 2332
| 8629 1 7884 3399
8630 1 7866 4645
8-631 1 7848 6070
8-632 1 7830 7673
8-633 1 7812 9454
8°634 17795 1414
8°635 17777 355%
8-636 1 7759 5867
8637 I 7741 8360
8-638 I 7724 1030
8639 I 7706 3878
8°640 1 7688 6902
8641 1 7671 O104
8-642 1 7653 3482
8°643 1 7635 7937
| 8644 1 7618 0768
8645 1 7600 4675
/8°646 | 1 7582 8758
8°647 1 7565 3017
8-648 1 7547 7452
8649 I 7530 2063

an
a
loa)
nN

oP

an

oO
~_

MR F. W. NEWMAN’S TABLE

Cam wv Cae
I 7512 6848 || 8°700 1 6658
I 7495 1809 || 8701 1 6641
I 7477 6944 || 8-702 I 6625
1 7460 2255 || 8703 I 6608
I 7442 7740 || 8704 I 6592
T 7425 3399 || 8705 1 6575
I 7407 9233 || 8-706 1 6558
I 7390 5241 || 8°707 I 6542
I 7373 1422 | 8708 1 6525
I 7355 7778 || 8709 1 6509
I 7338 4307 || 8-710 1 6492
I 7321 1009 || 8-711 1 6476
1 7303 7885 || 8712 1 6459
r 7286 4933 | 8713 I 6443
1 7269 2155 ||8°714 I 6426
I 7251 9549 || 8-715 I 6410
I 7234 7116 ||8'716 I 6394
r 7217 4855 | 8717 1 6377
I 7200 2766 || 8-718} I 6361
1 7183 0849 | 8719 1 6345
1 7165 gto4 || 8720 I 6328
I 7148 7531 || 8°72 I 6312
I 7131 6129 || 8-722 I 6296
I 7114 4898 || 8-723 I 6279
I 7097 3839 | 8724 I 6263
I 7080 2951 || 8725 1 6247
I 7063 2233 || 8°726 1 6231
t 7046 1686 || 8-727 I 6214
I 7029 1310 || 8-728 1 6198
I 7012 1104 || 8-729 I 6182
1 6995 1067 || 8730 1 6166
1 6978 1201 || 8-731 I 6150
1 6961 1505 || 8°732 I 6133
1 6944 1978 | 8-733 I 6117
I 6927 2621 || 8-734 I 6101
1 6910 3433 || 8°735 1 6085
1 68y3 4414 || 8°736 I 6069
1 6876 5564 || 8-737 1 6053
1 6859 6883 || 8°738 I 6037
1 6842 8370 || 8-739 1 6021
1 6826 0026 | 8740 I 6005
1 6809 1850 || 8°741 I 5989
1 6792 3842 | 8742 1 5973
1 6775 6002 | 8743 I 5957
1 6758 8330 | 8744 I 5941
1 6742 0826 | 8-745 I 5925
1 6725 3489 || 8°746 I 5909
1 6708 6319 | 8°747 t 5893
1 6691 9316 8-748 I 5877
1 6675 2480 |8°749 r 5861

[3600—8°799]

ae Ss eS eS Sa Se Se eS x Ss S&S x Ss See oe Le en en in en ee Be | Ss Ss SS Oe |

Lon I oe Bo oo

DESCENDING EXPONENTIAL.

OF THE
xv Cae

3075 || 8850 I 4338 1736
2417 || 8851 I 4323 8426
1910 || 8852 I 4309 5259
1553 || 8°853 I 4295 2236
1347 || 8°854 I 4280 9355
1291 || 8855 I 4266 6617
1384 || 8-856 I 4252 4021
1628 || 8-857 I 4238 1569
2021 || 8-858 I 4223 9258
2564 || 8859 I 4209 7090
3256 || 8860 I 4195 5064
4097 || 8-861 I 4181 3180
5088 || 8-862 I 4167 1437
6227 || 8-863 I 4152 9837
7515 || 8864 I 4138 8378
8952 || 8°865 I 4124 7060
0537 || 8-866 I 4110 5884
2271 || 8°867 I 4096 4848
4153 || 8868 I 4082 3954
6182 || 8-869 I 4068 3200
8360 || 8-870 I 4054 2588
0686 || 8-871 I 4040 2115
3159 || 8872 I 4026 1783
5779 || 8873 I 4012 1592
8547 || 8874 I 3998 1540
1462 || 8875 I 3984 1629
4524 || 8876 I 3970 1857
7733 || 8877 I 3956 2225
1089 || 8878 I 3942 2732
4591 || 8°879 I 3928 3379
8239 || 8-880 I 3914 4165
2034 || 8-881 I 3900 5091
5975 || 8882 I 3886 6155
0062 || 8883 I 3872 7359
4295 || 8884 I 3858 8700
8674 || 8:885 I 3845 o181
3198 || 8°886 I 3831 1800
7867 |) 8887 1 3817 3557
2682 || 8:888 I 3803 5453
7642 || 8°889 I 3789 7486
2747 || 8890 I 3775 9658
7996 || 8891 I 3762 1967
3391 || 8892 1 3748 4414
8929 || 8893 I 3734 6998
4613 || 8°894 I 3720 9720
0440 || 8895 I 3707 2579
6412 || 8°896 I 3693 5575
2527 || 8897 I 3679 8708
8787 || 8°898 I 3666 1977

8°899 T 3652 5384

5190

a“ Ca%
8:go00 I 3638
8*gor I 3625
8-902 I 3611
8-903 I 3598
8*904 1 3584
8'905 I 3570
8-906 I 3557
$907 I 3543
8*908 I 3530
8909 I 3516
8910 I 3503
8‘o1t I 3489
8-912 I 3476
8-913 I 3462
S914 1 3449
8-915 I 3435
8-916 I 3422
8917 I 3408
8-918 I 3395
8-919 I 3382
8-920 I 3368
8-921 I 3355
8-922 I 3342
8-923 1 3325
8-924 Tass 05
8925 I 3302
8-926 I 3288
8:927| 1 3275
8928 I 3262
8929 I 3249
8°930 T 3235
8°931 Eee 2
8°932 I 3209
8°933 I 3196
8-934 I 3182
8°935 1 3169
8-936 I 3156
$937 I 3143
8-938 I 3130
8°939 I 3117
8940 I 3104
8941 I 3091
8-942 I 3077
3943 I 3064
8°944 I 3051
8°945 T 3038
8-946 I 3025
8947 I 3012
8-948 I 2999
8949 I 2986

195
av 64

8-950 I 2973 7160
8'951 1 2960 7488
8952 I 2947 7945
8°953 1 2934 8532
8954 I 2921 9248
8955 I 2909 00903
8°956 I 2896 1068
8°957 I 2883 2171
8-958 I 2870 3403
8°959 1 2857 4764
8960 I 2844 6254
8-961 I 2831 7872
8962 1 2818 9618
8-963 1 2806 1492
8-964 1 2793 3495
8°965 I 2780 5625
8-966 I 2767 7884
8-967 I 2755 0270
8-968 I 2742 2783
8-969 I 2729 5424
8:970 1 2716 8192
8971 I 2704 1087
8'972 I 2691 4110
8°973 1 2678 7259
8-974 I 2666 0535
8°975 1 2653 3938
8-976 1 2640 7467
8977 —I 2628 1123
8°978 I 2615 4905
8979 I 2602 8813
8980 I 2590 2847
8-981 I 2577 7008
8-982 I 2565 1293
$983 I 2552 5795
8-984 I 2540 0242
8°985 I 2527 4904
8986 I 2514 9692
8-987 I 2502 4605
8-988 1 2489 9643
8989 I 2477 4806
8-990 I 2465 0093
8-991 T 2452 55°95
8-992 I 2440 1042
8'993 I 2427 6703
8°994 I 2415 2489
8995 I 2402 8398
8-996 as |
8°997 1 2378 0589
8-998 I 2365 6871
8-999 I 2353 3276

et oe i es A en ee A I Le ee oe on ee | ele | a ee | ei an il an in |

a |

NNN

by Ga Ga Oa Oo

NNN NN
NNN HN NN

9804
6456
3231
o129
7151
4295
1562
8952
6464
4299
1856
9735
7736
5860
4105
2471
0960
9569
8300
7153
6126
5220
4436

3772
3228

2805
2502
2320
2258
2315

2493
2790
3207
3744
4400
5175
6070
7083
8215
9466
0836
2325
3932
5657
750°

9462
1541
3739
6054
8487

MR F. W. NEWMAN'S TABLE

ee |

i a

ee ee Le ln ee | es ln en oe es en ee oe i

ne |

e7e

1739
1727
1715
1703
1692
1680
1668
1657
1645
1633

1622
1610
1599
1587
1575

1564
1552
1541
1529
1518
1506
1495
1483
1472
1460

1449
1437
1426
1414
1403
1392
1380
1369
1358
1346

1335
1324
1312
1301
1290
1278
1267
1256
1245
1233

1037
3704
6489
9391
2410
5546
8799
2169
5655
9258

2976

6812 ||

0763
4830
90x35

3312
7726
2256
6902
1663

6539
1529
6635
1856
7192
2642
8206
3885
9679
5586

1607
7743
3992

©355
6831

3421
O124

6941
3870
913
8068

5337

2718

o2II

7817

5536 |
3366 |

1309
9363
753°

HHH ee (iin tien tien Eien! Ds ln len le | (ion ies ill en Eilon! en ils len len! (esis tele ies Eilon! Ce a Oe see me

wee eee

[9°00c—9"199]

e-%

0621 9803
0611 3636
0600 7575
0590 1621
0579 5772
0569 0029
0558 4392
0547 8860

0537 3404
0526 8113

+ + Se et

Se HH Se et

0516 2808
0505 7788
0495 2782
0484 7882
0474 3087

0463 8396
0453 3810
0442 9328
0432 4951
0422 0678

Le oe I oe oe oe

Le oe oe oe Bo

O41I 6510
0401 2445
0390 8485
0380 4628
0370 0875

0359 7226
0349 3681
033959239
0328 6900
0318 3665

0308 0533
0297 7504
0287 4578
0277 1755
0266 9034

HH He ee Ln A oe oe oe

He OW

0256 6417
0246 3902
0236 1489
0225 9179
0215 6971

HH HH

0205 4865
o195 2861
0185 0959

O174 9159
0164 7461

0154 5864
0144 4369
0134 2975
0124 1683
OII4 0492

HH Se Se

bo oe oe oe

[9'200—9'399]

OF THE DESCENDING EXPONENTIAL.

9402
8413
7526
6738
6051

5466
3 4980 |

4596
4311
4127

4043
4059
4175
4390
4706

5121
5636
6250
6963
7776

8688
9698
0808
2017
3324
473°
6235
7838
9539
1338

3236
5232
7326
9517
1807

4194
6678
9260
1940
4716

759°
0561
3629
6794
0055
3414
6869
0420
4068
7812

WHWWWH WWWWO
NNN NN
[o)

Nyy NN
Oo On aM KW ND H

woo evovvrou YVvVvvurv~s
e}

G2 Go
KH
H

Nome)
Ge
W Ww
win

197

s
8

wn NN hl

Mmmm DnNMmMDMOH
Noo Oo Ul Os
-NH ONE NN

Se

=

MR F. W.

NEWMAN'S TABLE

9565
ogt5
2343
3850
5436
7°99
8841
0662
2560
4537

6591
8723
0933
3221
5587
8030
O551
3149
5825
8577

I407
4315
7300
0360
3498
6713
0005
3373
6818

2339

3937
7611

1362
5189
9091
3070
7125
1256
5463
9745
4103
8537
3046
7631
2291

7026

- 1837

6722
1683

6719

7485 1830
7477 7°15
747° 2276
7462 7611
7455 3020
7447 8505
7449 4063
7432 9697
7425 5404
7418 1186

7410 7042
1493 2972
7395 8976
7388 5054
7381 1206

7373 7431
7366 3731
7359 0104
7351 6551
7344 3071

7336 9664
7329 6331
7322 3072
7314 9885
7307 6772

7300 3732
7293 0764
7285 7870
7278 5049
7271 2300

7263 9624
7256 7021
7249 449°
7242 2032
7234 9646
7227 7332
7220 5091
7213 2922
7206 0825
7198 8800

7191 6848
7184 4967

7177 3158
7170 1420

7162 9755
7155 8161

7134 3808
7127 2500

[9'400—9'599]

e-%

7120 1263
7113 0097
7105, 9993
7098 7979
7°91 7027

7084 6145
7977 5334
7970 4595
7°63 3925
7056 3327

7949 2799
7042 2341
7935 1954
7028 1637
O21 1391
7OI4 1214
7007 1108
7000 1072
6993 1106
6986 1210

6979 1384
6972 1627
6965 1940
6958 2323
6951 2776

6944 3298
6937 3889
693° 4550
6923 5280
6916 6079

6909 6948
6902 7885
6895 8892
6888 9967
6882 1112

6875 2325
6868 3607
6861 4958
6854 6377
6847 7865

6840 9422
6834 1046
6827 2739
6820 4501
6813 6330
6806 8228
6800 0194
6793 2228
6786 4329
6779 6499

[9°600—9'799] OF THE DESCENDING EXPONENTIAL. 199

x Ce ae e-% zw. e-% x“ e-#

oe y..tk Evacey
9600 6772 8736 || 9°650 6442 5567 ||9°700 6128 3495 || 9°750 5829 4664
g°6o0r 6766 1042 || 9°651 6436 1173 || 9°7o1 6122 2242 || 9-751 5823 6398
9602 6759 3414 ||9°652 6429 6844 || 9702 6116 1050 || 9°752 5817 8191
9°603 6752 5855 || 9°653 6423 2580 9°703 6109 9920 |9°753| ~ 5812 0042
9°604 6745 8363 || 9°654 6416 8379 || 9-704 6103 8851 || 9°754 5806 1951
9°605 6739 0938 || 9°655 6410 4243 || 9°705 6097 7842 | 9°755 5800 3918
9°606 6732 3581 || 9°656 6404 0171 || 9-706 6091 6895 || 9°756 5794 5943
9°607 6725 6291 || 9°657 6397 6163 || 9°707 6085 6008 | 9°757 5788 8026
9°608 6718 9068 || 9°658 6391 2219 || 9-708 6079 5183 || 9°758 5783 0167
9°609 6712 1913 || 9°59 6384 8339 || 9°709 6073 4418 | 9°759 5777 2365
9°610 6705 4824 || 9660 6378 4522 || 9°710 6067 3714 || 9°760 5771 4622
g6rr 6698 7803 || 9°66r 6372 0770 || 9°711 6061 3071 || 9°761 5765 6936
g°612 6692 0849 || 9°662 6365 7081 || 9712 6055 2488 || 9°762 5759 9308
9°613 6685 3961 || 9°663 6359 3455 ||9°713 6049 1966 || 9'763 5754 1737
9614 6678 7141 | 9°664 6352 9894 || 9°714 6043 1504 || 9°764 5748 4225
9°615 6672 0387 || 9°665 6346 6396 || 9-715 6037 1103 || 9°765 5742 6769
9°616 6665 3700 || 9666 6340 2961 || 9716 6031 0762 || 9°766 5730 9371
9617 6658 7079 || 9°667 6333 9590 ||9°717 6025 0481 || 9°767 5731 2030
9618 6652 0526 || 9°668 6327 6282 || 9°718 6019 0261 || 9°768 5725 4747
9619 6645 4038 || 9°669 6321 3037 || 9°719 6013 o10r || 9°769 5719 7521
9620 6638 7618 | 9°670 6314 9855 ||9°720 6007 0000 || 9°770 5714 0352
9°621 6632 1263 | 9°671 6308 6737 || 9°721 6000 9960 || 9°771 5708 3240
9°622 6625 4975 || 9°672 6302 3682 || 9°722 5994 998r || 9°772 5702 6185
9623 6618 8753 || 9°673 6296 0690 || 9°723 5989 0060 || 9°773 5696 9188
9°624 6612 2597 || 9°674 6289 7760 || 9°724 5983 0200 || 9°74 5691 2247
9°625 6605 6508 || 9°675 6283 4894 || 9°725 5977 0400 || 9°775 5685 5363
9°626 6599 0484 | 9°676 6277 2091 || 9°726 5971 0660 || 9°776 5679 8536
9°627 6592 4527 || 9°677 6270 935° || 9°727 5965 9979 || 9°777 5674 1766
9°628 6585 8635 | 9°678 6264 6672 | 9°728 5959 1358 || 9°778 5668 5053
9°629 6579 2810 | 9°679 6258 4057 || 9°729 5953 1796 || 9°779 5662 8396
9°630 6572 7050 || 9°680 6252 1504 || 9°730 5947 2294 || 9°780 5657 1796
9°631 6566 1356 | 9°681 6245 9013 || 9°731 5941 2851 || 9°781 5651 5252
9°632 6559 5727 || 9682 6239 6586 || 9°732 5935 3468 || 9-782 5645 8765
9°633 6553 0164 || 9°6383 6233 4220 || 9°733 5929 4144 || 9°783 5640 2335
9°634 6546 4667 || 9684 6227 1917 || 9°734 5923 4880 || 9°784 5634 5961
9°635 6539 9235 | 9°685 6220 9676 || 9°735 5917 5675 || 9°785 5628 9643
9°636 6533 3868 | 9°686 6214 7498 || 9°736 5911 6529 || 9'786 5623 3351
9°637 6526 8567 || 9°687 6208 5381 || 9°737 5905 7442 || 9°787 5617 7176
97638 6520 3331 || 9°688 6202 3327 || 9°738 5899 8414 || 9°788 5612 1027
9°639 6513 8160 || 9°689 6196 1335 | 9°739 5893 9445 || 9°789 5606 4934
9°640 6507 3055 || 9°690 6189 9404 || 9°740 5888 0535 || 9°790 5600 8897
“9641 6500 8014 | 9691 6183 7536 || 9°741 5882 1684 || 9°791 5595 2916
9°642 6494 3039 || 9°692 6177 5729 || 9°742 5876 2891 || 9°792 5589 6991
9°643 6487 8128 | 9°693 6171 3984 || 9°743 5870 4158 || 9°793 5584 1122
9°644 6481 3282 | 9°694 6165 2301 ||9°744 5864 5483 | 9°794 5578 5309
9°645 6474 8501 || 9°695 6159 0680 || 9°745 5858 6867 || 9°795 5572 9552
9°646 6468 3785 || 9°696 6152 9120 || 9°746 5852 8309 || 9°796 5567 3850
9°647 6461 9134 || 9°697 6146 7621 || 9°747 5846 9810 || 9°797 5561 8204
9°648 6455 4547 || 9°698 6140 6185 || 9°748 5841 1370 || 9°798 5556 2614
9°649 6449 0025° || 9°699 6134 4899 || 9°749 5835 2987 || 9°799 555° 7079

200 MR F. W. NEWMAN’S TABLE [9'800—9'999]

——
x e-* ae e-@ x e-% | ae e-%
9'800 | 5545 1599 || 9°850 5274 7193 || 9°900 §017 4682 | 9°950 4772 7634
9°Sor 5539 6175 || 9°852 5269 4472 || 9901 5012 4532 || 9°951 4797 993°
9802 5534 0807 || 9°852 5264 1804 || 9:902 5097 4433 || 9°952 4763 2274
9803 | 5528 5494 ||9°853 5258 9188 | 9'903 5002 4383 || 9°953 4758 4666
9804 5523 0236 || 9°854 5253 6625 || 9°904 4997 4384 || 9°954 4753 7105
9805 5517 5033 || 9°855 5248 4115 || 9°905 4992 4435 || 9°955 4748 9591
9806 5511 9886 || 9°856 5243 1657 || 9°906 4987 4535 || 9°956 4744 2126
9807 5506 4794 || 9°857 5237 9252 || 9°907 4982 4686 || 9°957 4739 4797
| 9°808 5500 9756 || 9°858 5232 6899 || 9°908 4977 4886 || 9°958 4734 7336
| 9°809 5495 4774 || 9°859 5227 4598 || 9°909 4972 5136 || 9°959 4730 0012
9810 5489 9847 || 9°860 5222 2350 || 9°9I0 4967 5436 || 9°960 4725 2736
(9811 5484 4974 || 9°861 5217 O154 || 9911 4962 5785 || 9°961 4729 5507
| 9812 5479 0157 || 9°862 5211 8010 || 9912 4957 6184 || 9°962 4715 8325
| 9°S13 5473 5394 || 9°863 5206 5918 || 9°913 4952 6633 || 9°963 4711 I1g0
9814 5468 0686 || 9°S64 5201 3878 || 9°914 4947 7131 || 9°964 4706 4103
| 9°815 5462 6033 || 9°865 5196 1890 || 9°915 4942 7678 || 9°965 4701 7062
| 9816 5457 1434 || 9°866 5190 9954 || 9°916 4937 8275 || 9°966 4697 0068
9817 5451 6890 || 9°867 5185 8070 || 9°917 4932 8922 || 9°967 4692 3122
9818 5446 2400 9868 5180 6238 || 9°918 4927 9618 || 9°968 4687 6222
9819 5440 7965 || 9°869 5175 4457 ||9°919 4923 0363 || 9°969 4682 9369
9820 5435 3584 || 9°870 5170 2729 || 9°920 4918 1157 || 9°970 4678 2563
9821 5429 9258 || 9871 5165 1052 || 9921 4913 2000 |} 9°971 4673 5804
| 9°822 5424 4986 || 9°872 5159 9427 || 9°922 4908 2893 || 9°972 4668 gog2
982 5419 0768 || 9°873 5154 7853 || 97923 4903 3834 || 9°973 4664 2426
9824 5413 6604 || 9°874 5149 6331 || 9°924 4898 4825 || 9°974 4659 5807
9825 5408 2495 || 9°875 5144 4860 || 9°925 4893 5865 || 9°975 4654 9234
9°826 5402 8439 | 9°876 5139 3441 || 9°926 4888 6953 || 9°976 4650 2708
9827 5397 4438 || 9°877 5134 2073 ||9°927 4883 8091 || 9°977 4645 6229
9828 5392 0490 || 9°878 5129 0757 || 97928 4878 9277 ||9°978 4640 9796
9829 5386 6597 | 9°879 5123 9492 || 9°929 4874 0512 || 9°979 4636 3409
9°830 5381 2757 || 9°880 5118 8278 || 9°930 4869 1796 || 9°980 4631 7069
9°831 5375 8971 || 9°881 5113 7115 || 9°931 4864 3129 || 9981 4627 0775
9°832 537° 5239 || 9°882 5108 6004 || 9°932 4859 4510 || 9°982 4622 4528
9°833 5365 1560 || 9°883 5103 4943 || 9°933 4854 5940 || 9°983 4617 8326
9°834 5359 7936 || 9°884 5098 3934 || 9°934 4849 7418 || 9°984 4613 2171
9°835 5354 4365 || 9°885 5093 2975 || 9°935 4844 8945 || 9°985 4608 6062
9836 5349 0847 || 9°886 5088 2068 || 9°936 4840 0520 || 9°986 4603 9999
9°837 5343 7333 || 9°887 5083 1211 || 9°937 4835 2144 || 9°987 4599 3982
9838 5338 3972 || 9°888 5078 0405 || 9°938 4830 3816 || 9°988 4594 8011
9°839 5333 0615 || 9°889 5072 9650 || 9'939 4825 5536 || 9°989 4590 2086
9°840 5327 7311 || 9°890 5067 8946 || 9°940 4820 7305 || 9°990| 4585 6207
9841 5322 4060 || 9°891 5062 8292 || 9°941 4815 gi21 || 9991! 4581 0373
9°842 5317 0863 || 9°892 5057 7689 || 9°942 4811 0986 || 9'992 | 4576 4586
9°843 5311 7719 || 9°893 5052 7137 || 9°943 4806 2899 || 9°993 | 4571 8844
9°844 5306 4627 || 9894 5047 6635 || 9°944 4801 4861 | 9°994 4567 3148
9°845 5301 1589 || 9°895 5042 6184 || 9°945 4796 6870 || 9°995 4562 7498
9°846 5295 8604 || 9°896 5037 5783 || 9°946 4791 8927 || 9°996 4558 1893
9°847 5290 5672 || 9°897 5032 5432 || 9°947 4787 1032 || 9°997 4553 6334
9°848 5285 2793 || 9°898 5027 5132 || 9'948 4782 3185 || 9°998 4549 0820
9°849 5279 9966 || 9°899 5022 4882 | 9'949 4777 5385 || 9°999 4544 5352

[10:000-—r0"199] OF THE DESCENDING EXPONENTIAL.

xv (ge wv (OR xv | er | x
wteretele) 4539 9930 || 10°050 | ~~ 4318 5749 | 1O‘Ioo | 4107 9555 | ror
10°00I 4535 4552 || 19051 4314 2585 | 1o'ro1 4103 8496 IO‘I51
10002 4530 9221 || 10052 4309 9464 |) Io‘ro2 | 4099 7478 || 1o‘152
10°003 4526 3934 || 10°053 4305 6386 | ro'103 | 4095 6501 | 10°153
10°004 4521 8693 || 10°054 | 4301 3351 | 107104 | 4091 5565 || 10°154
10°005 4517 3497 || 10°055 4297 9359 | 1o'ros 4087 4670 | 10°
10°006 4512 8346 || 10°056 | 4292 7410 | 1o'r06 | 4083 3816 || 1o-
10°007 4508 3240 | 10°057 4288 4504 IO°IO7 | 4079 3002 |i 10°
107008 4503 8179 || 107058 4284 1641 | Lor08 | 4075 2230 || 10°
10°009 4499 3163 || 10°059 4279 8821 | Lo‘ro9g | 4071 1498 || Io
poke) fe) 4494 8193 || 10°060 4275 6044 || Io'1Io | 4067 0807 |! Io
IO‘OLI 4490 3267 || 10061 4271 3309 || Io'rrT | 4063 0156 || 10°
10012 4485 8386 || 10°062 4267 0617 | Ior12 | 4058 9546 || 10°
10013 4481 3550 || 10°063 4262 7968 || ro'173 | 4054 8977 || 10°
10014 4476 8759 || 10°064 4258 5361 | 1o1r4 | 4050 8448 | 10°
IO‘OI5 4472 4013 || 10°065 4254 2707 || to:rrs | 4046 7960 | 10°
10°016 4467 9311 || 10°066 4250 0275 || 10116 | 4042 7512 || Io"
10'017 4463 4654 || 10°067 4245 7796 || 10117 |} 4038 7105 | LOR
10°018 4459 0042 || 107068 4241 5360 || ror18 | 4034 6738 || 10°
CES, 4454 5474 || 10°069 4237 2966 || 1o"11g | 4030 6412 | 19°
10°020 4450 0951 || 10°070 4233 0614 | 10°I20 | 4026 6125 || 10°
10°02T 4445 6472 || 10°071 4228 8305 || ro'r2t | 4022 5879 || Io
10°022 4441 2038 || 10°072 4224 6037 IO°I22 4018 5674 || 10°
10°023 4436 7648 || 10°073 4220 3813 | LO°123 | 4914 5508 | Io |
10°024 4432 3303 || 10°074 4216 1630 || 1o‘124 4010 5383 | 10°

|
10°025 4427 goor || 10°075 4211 9489 IO°125 | 4006 5297 | 10°
10'026 4423 4745 || 10°076 4207 7391 || 10°126 4902 5252 || 10°176
10°027 4419 0532 || 10°077 AIT) Seisit) ||| MTU Z7] | 3998 5247 | 10°I77
10'028 4414 6363 || 10°075 4199 3320 || 10128 3994 5282 || 10'178
10'029 4410 2239 || 10°079 4195 1348 || 1o°r29 | 3999 5356 || 10179
10°030 4405 8159 || 10°080 4199 9417 || 10°130 3986 5471 || 197180
10°03f 4401 4123 | 107081 4186 7529 || 10°13 3982 5625 |; 19°181
10'032 4397 O131 || 10°082 4182 5682 || r0°132 | 3978 5820 | 10182
10°033 4392 6183 | 107083 4178 3877 | 10°33 | 3974 6054 | 10"
10°034 4388 2278 || 10°084 4174 2114 || 10°34 | 3979 6327 || Io° |
| | |

10°035 4383 8418 || 10°085 4170 0393 || 10°135 | 3966 6641 to"
10°036 4379 4601 | 107086 4165 8714 | 10136 | = 3962 6994 | 10°
10°037 4375 0829 | 10°037 4161 7076 | 10°137 | =. 3958 7387 || 10°
10°038 4370 7100 || 10°088 4157 5479 | 10°138 3954 7819 || 10°
10°039 4366 3415 | 10°089 4153 3925 || 10°139 | 3950 8291 || Io"
I0"040 4361 9773 || 10°090 4149 2412 || 10°r40 | 3946 8803 || Io°
10°04 4357 6175 || 10°091 4145 9940 || 1o°r41 3942 9354 | 10° |
10°042 4353 2621 | 10'092 4140 9510 | 10'142 | 3938 9944 | 10° |
10'043 4348 giro || 10°093 4136 8121 || r0°r43 | 3935 9574 || 10°
10'044 4344 5642 || 10°094 4132 6773 || 10°144 | 3931 1243 || 10°
T0'045 4340 2218 || 10°095 4128 5467 | ro'r45 3927 1951 10°
10°046 4335 8838 || 10°096 4124 4202 || 10'146 | 3923 2699 || 10°
10°047 4331 5501 || 10°097 * 41202979 || 10°147 | 3919 3486 | 10°197
10°048 4327 2207 | 10°098 4116 1796 || 10°r48 | 3915 4312 || 10'198 |
10'049 4322 8956 | 10°099 4112 0655 | 10149 | 391L 5177 | 107199 |

Wow, UU Irae IN

MR F. W. NEWMAN’S TABLE [10*200—10°399 |

x Gs" | &% aad wv e-#
To"250 SL ee eh ||) aheisis 3363 3095 || 10°350 3199 279°
|| ro"251 3532 2161 || 10°301 3359 9479 || 10°351 3196 0813
|| ro‘252 3528 6856 || 10°302 3356 5896 || 10°352 3192 8868
10°253 3525\ 7587 || 10:303 3353 2347 | 19°353 3189 6955
| T0254 3521 6353 | 10°304 3349 8831 || 10°354 3186 5074
ee) 3518 1154 | 10°305 3346 5349 || 10°355 3183 3225
| 10°256 3514 5991 | 10°306 3343 Igor || 10°356 3180 1408
|) 10°257 3511 0863 || 10°307 3339 8486 || 10°357 3176 9622
| 10258 3507 5769 | 107308 3330 5104 || 10°358 | . 3173 7868
| 10°259 3504 O7II | 10°309 3333' 2755 || 197359 3170 6146
| 10°260 3500 5688 | 10°310 3329 8440 || 10°360 3167 4456
10°261 3497 0700 || To°311 3326 5158 || 10°361 3164 2798
|| t0:262 3493 5746 || to"312 3323 1910 || 10°362 316r 1171
| 10°263 3490 0828 || 10°313 3319 8695 || 10°363 3157 9575
|| 10°264 3486 5944 || 10°314 3316 5512 || 10°364 3154 8011
10°265 3483 1096 || 10°315 3313 2364 || 10°365 3151 6479
10°266 3479 6282 || 107316 3309 9248 || 10°366 3148 4978
10°267 3476 1504 | TO°317 3306 6165 || 10°367 3145 3509
|| 10°268 3472 6759 || 10°318 3303 3115 || 10°368 3142 2071
10°269 3469 2050 || 10°319 3300 0099 || 10°369 3139 0665
| 10°270 3465 7375 || 10°320 3296 7115 || 10°370 3135 9290
10°271 3462 2735 || 10°321 3293 4165 || 10°371 3132 7946
10°272 3458 8130 || 10°322 3290 1247 || 19°372 3129 6634
10°273 3455 3559 || 10°323 3286 8362 || 10°373 3126 5353
10°274 3451 9023 || 19°324 3283 5510 || 19°374 3123 4103
10°275 3448 4521 || 10°325 3280 2691 || 10°375 3120 2885
|| 10°276 3445 0054 || 10°326 3276 ggos || 10°376 3117 1698
10°277 3441 5621 || 10°327 3273 7151 || 19°377 3114 0542
10°278 3438 1222 || 10°328 3279 4431 || 10°378 3110 9417
|| 10°279 3434 6858 || 10°329 3267 1742 || 10°379 3107 8323
|| 10°280 3431 2529 || 10°330 3263 9087 || 10°380 3104 7260
| 10°281 3427 8233 || 10°331 3260 6464 || 10°38 3101 6228
|| 10°282 3424 3972 || 10°332 3257 3874 || 10°382 3098 5227
| 10°283 3429 9745 |) 10°333 3254 1316 || 10°383 3095 4258
10°284 3417 5553 | 19°334 3250 8791 || 10°384 3992 3319
| 10'285 3414 1394 | 19°335 3247 6299 || 10°385 3089 2411
| 10'286 3410 7270 || 10'336 3244 3839 || 10°386 3086 1534
10'287 3407 3180 || 10°337 3241 1411 || 10°387 3083 0688
|, 10°288 3403 9124 || 107338 3237 9016 || 10°388 3079 9873
} 10°289 3400 5101 || 10°339 3234 6653 || 10°389 3076 9088
||) 0290 3397 II13 || 10°340 3231 4323 || 10°390 3°73 8334
10°291 3393 7159 || 10°341 3228 2024 || 10°391 3070 7611
10292 3399 3239 || 10°342 3224 9759 || 19°392 3067 6919
10°293 3386 9353 || 10°343 3221 7525 || 10°393 3064 6258
| 10°294 | 3383 5500 || 10°344 3218 5324 | 10°394 3061 5627
10°295 3380 1682 || 10°345 3215 3154 || 10°395 3058 5026
| 10°296 3376 7897 || 10°346 3212 1017 || 10°396 3°55 4457
10°297 3373 4146 || 10°347 3208 8912 || 10°397 3052 3917
10'298 3370 0429 || 10°348 3205 6839 || 10°398 3049 3409
10°299 3366 6745 || 10°349 3202 4799 || 10°399 3046 2931

[10°400—.10'599] OF THE DESCENDING EXPONENTIAL. 203
& Cae | @ e-% |. & Ore | 2 Dake
I0°400 3043 2483 | r07450 2894 8273 | 10°500 2753 6449 | I0"550 2619 3481
Io"4o1 3040 2066 | 10°451 2891 9339 | I0’5o1 2750 8927 || 10°551 2616 7301
10°402 3237 1679 || 10"452 2889 0434 I T0"502 2748 1431 || 10°552 2614 1147
10°403 3934 1322 | 10°453 2886 1558 | 10°503 2745 3964 || 10°553 2611 5019
10°404 3031 0996 | 10°454 2883 2711 || 10°504 2742 6523 || 10°554 2608 8917
10°405 3028 0700 |! 10°455 2880 3893 | 10'505 2739 QIII || 10°555 2606 2840.
10°406 3025 0435 || 10456 2877 5103 || 10°506 2737 1725 || 10°556 2603 6790
10°407 3222 0199 | 10°457 2874 6342 | 10°507 2734 4367 || 10°557 2601 0766
10°408 3218 9994 || 107458 2871 7611 || 10°508 2731 7036 || 10558 2598 4768
HOS9 3015 9819 || 10°459 2868 8907 | 10°509 2728 9733 | 10°559 2595 8797
Io"410 3012 9675 | 10°460 2866 0233 | I0°510 2726 2457 || 10°560 2593 2851
Io"41I 3909 9560 | 10°461 2863 1587 || 10511 2723 5208 || 10°561 2599 6931
10°412 3006 9476 || 10°462 2860 2969 |) 10°512 2720 7987 || 10°562 2588 1037
I0°413 3°03 9421 || 107463 2857 4381 || 107513 2718 0792 || 10°563 2585 5169
TO"414 3000 9397 || 10°464 2854 5821 | 10°514 2715 3625 | 10°564 2582 9327
1o"415 2997 9402 || 10°465 2851 7289 || 10°515 2712 6485 || 10°565 2589 3510
10°416 2994 9438 || 10°466 2848 8786 || 10°516 2709 9372 || 107566 2577 7720
10°417 2991 9503 || 10°467 2846 0312 || 10°517 2707 2286 || 10°567 2575 1955
10°418 2988 9599 || 10°468 2843 1865 || 10°518 2704 5227 || 10°568 2572 6216
Io"419 | 2985 9724 || 10°469 2840 3448 || 10°519 2701 8196 | 10°569 2570. 0503
10°420 | 2982 9879 | 10'470 2837 5059 || 10°520 2699 II9gI | 10°570 2567 4815
10"421 | 2989 0064 | 107471 2834 6698 || To'521 2696 4213 | 10°571 2564 9153
10°422 2977 0279 || 10°472 2831 8365 || 10°522 2693 7263 || 10°572 2562 3517
10°423 2974 0524 || 10°473 2829 oo061 || 10°523 2691 0339 || 10°573 2559 7900
10°424 2971 0798 | 107474 2826 1785 || 10°524 2688 3442 || 10°574 2557 2321
10°425 2968 r102 || 10°475 2823 3537 || 10°525 2685 6572 | 10°575 2554 6761
10°426 2965 1436 | 10°476 2820 5318 || 10°526 2682 9729 || 10°576 2552 1227
10°427 2962 1799 || 10°477 2817 7127 || 10°527 2680 2912 || 10°577 2549 5719
10°428 2059 2192 | 10°478 2814 8964 || 107528 2677 6123 , 10°578 2547 0236
10°429 2956 2615 || 107479 2812 0829 || 107529 2674 9360 || 10°579 2544 4778
|
10°430 2953 3067 | 10°480 2809 2722 || 10°530 2672 2624 || ro‘580 2541 9346
10°431 2950 3549 |) 10°481 2806 4643 || 10°531 2669 5915 || 10°581 2539 3940
10°432 2947 4060 || 107482 2803 6593 || 10°532 2666 9233 || 10°582 2536 8559
10°433 2944 4601 || 10°483 2800 8570 | 10°533 2064 2577 | 10°583 2534 3203
10°434 2941 5171 | 19°484 2798 0576 || 10°534 2661 5947 || 10°584 2531 7872
10°435 2938 577° || 10°485 2795 2609 || 10°535 2658 9345 | 10°585 2529 2567
10°436 2935 6399 || 10°486 2792 4670 || 10°536 2656 2769 | 10°586 2526 7287
10°437 2932 7058 | 10°487 2789 6760 || 10°537 2653 6219 || 10°587 2524 2032
10°438 2929 7745 | 10°488 2786 8877 || 10°538 2650 9696 || 10°588 2521 6803
10°439 2926 8462 | 10°489 2784 1022 10°539 2648 3200 | 10°589 2519 1599
10°440 2923 9208 | 10°490 2781 3195 | 10°540 2645 6730 | 10°590 2516 6420
10°44I 2920 9984 || 10°491 2778 5396 || 10°541 2643 0287 || 1o‘591 2514 1266
10°442 2918 0788 || r0"492 2775 7624 || 10542 2649 3870 | 10592 2511 6137
10°443 2915 1622 | 10°493 2772 9881 | 10°543 2037 7479 || 10°593 2509 1034
10°444 2912 2484 | 10°494 2770 2165 || 10°544 2635 I1I5 || 10°594 2506 5955
| i} |
10 445 2929 3377 | 10°495 2767 4476 | 10°545 2632 4777 | 10°595 2504 ogo2
19°446 2906 42098 || 10°496 2764 6816 || 10°546 2629 8465 || 10°596 2501 5873
10°447 2903 5248 || 10°497 2761 9183 || 10°547 2627 2180 || 10°597 2499 0870
10°448 2900 6227 | 10'498 2759 1577 || 10°548 2624 5921 || 107598 2496 5892
10°449 2897 7236 | 10°499 2756 3999 | 10'549 2621 9688 | 10°599 2494 0938

MR F. W. NEWMAN'S

|

non

NN we NN
Ne cl ool cella oe
NN bh ON Ge

oO

NNN NN NN NN NN

NNN NN

2384

i]
oP
~
de

f
+
No)
io)

oo
ty
ty

e-% an

tow
Gs Gs Go
Os Oo

Ny Go Ge Go
Ont SO

NNN NN

N

be bw NN ND
Cro WWW WW OW Go Oo Go Oo Go
Hw N LW

bw NN to

0841 || 10°700

7153 || 10°701
3487 | 10°702
9846 || 10°703
6227 || 10°704
2633 || 10°705
g062 || 10°706
GHG) |) SSA
1ggt || 10°708
8491 || 10°709
5014 || 10°710

156r || to"711

8131 || 10712
4725 || 19°713
1342 || ro"714
7982 || 10°715
4646 || 10°716
1333 | 19°717
8043 || 10°718
Witt |) ee Le)
1533 || t0"720
8314 || 10°72
5117 || 107722
TOAST pL OMS
8793 | 19°724
5666 || 10°725
2562 || 10°726
9481 | 10°727
6423 || 10°728
3388 || 10°729
0376 || 10°730
7387 |, 10°731
4421 || 10°732
1478 | 10°733

8558 10°734

5661 | 10°735

2787 || 10°736
9935 || 19°737
7107 || 10°738
4301 |) 10°739
1518 | 19°740
8758 || 10°741
6o2r || 10°742
3306 | 19°743
0614 | 10°744
7945 | 19°745
5298 || 10°746
2674 || 10°747
0073 | 10°748

7494 || 19°749

e-z

2254 4938
2252 2404
2249 9893
2247 7404
2245 4938 |

2243 2494
2241 0073
2238 7674
2236 52098
2234 2944

2232 o612 |
2229 8302

2227 6015
2225 3750
2223 1508
2220 9287
2218 7089.
2216 4913

2214 2759
2212 0628

2209 8518
2207 6431
2205 4365
2203 2322
2201 0301

2198 8301
2196 6324
2194 4369
2192 2435 |
2190 0524

2187 8634
2185 6767
2183 4921
2181 3097
2ASTfS) EEG) ||

2176 9514 |
2174 7756

2172 6019 |
2170 4303 |
2168 2610

2166 0938
2163 9288
2161 7660 |
2159 6053
2157 4468
2155 2904

2153 1362 |
2150 9841

2148 8342
2146 6864

[10°600—10°799]

+

eee

xv e-z
To"750 2144 5408 |
DOB 2142 3974 |
10°752 2140 2560
10°753 2138 1168
10°754 2135 9798
10°755 2133 8449
10°756 2131 7121
10°757 2129 5815
10°758 2127 4529
10°759 2125 3265
10°760 2123 2023
10°761 2121 o8o1
10°762 2118 g6or
10°763 2116 8422
10°764 2114 7264
10°765 2112 6128
10°766 2110 5012
10°767 2108 3918
10°768 | 2106 2844
10°769 2104 1792 |
19°770 2102 0761
PeTips 2099 975°
10°772 2097 8761
10°773 2095 7793
10°774 2093 6846
10°775 2091 5919
10°776 2089 5014
10°777 2087 4129
10°778 2085 3265
10°779 2083 2423
10°780 2081 1601
10°781 2079 0799 |
10°782 2077 0019
10°783 2074 9259
10°784 2072 8521
10°785 2070 7802
10'786 2068 7105
10°787 2066 64238
10°788 2064 5772
10°789 2062 5137
10°790 2060 4522
10°791 2058 3928
10°792 2056 3354
10°793 2054 2801
10°794 2052 2268
10°795 2050 1756
10°796 2048 1265
10°797 2046 0794
10°798 2044 0343
10°799 2041 9913

[10°800—r10'999]

x e-%
10°800 2039 9503
1o°801 2037 OII4
10°802 2035 8745
Bens 2033 8397
10°804 2031 8068
10°805 2029 7760
10°806 2027 7473
10°807 2025 7205
10°808 2023 6953
10809 2021 6732
1o'810 2019 6525
ro'811 2017 6338
10°812 2015 6172
10°813 2013 6026
10°814 2011 5900
poets 2009 5794
10°S16 2007 5709
eee 2005 5643
eoere 2003/5597
10°819 2001 5572
8 1999 5566
pore 1997 5581
ee 1995 5015
10°23 | *1993 5669
gaSe4 1991 5744
10°825 1989 5838
10°826 1987 5952
10°827 1985 6086
10°828 1983 6240
19°829 1981 6413
10°830 1979 6607
10°831 1977 6820
10°832 1975 7053
10°833 1973 7306
10°834 1971 7579
10'835 1969 7871
10'836 1967 8183
10°837 1965 8514
10°838 1963 83866
10'839 1961 9237
10°840 1959 9627
10'S41 1958 0037
10°842 1956 0467
10°843 1954 0917
10°844 1952 1385
10°845 1950 1874
10°846 1948 2382
10847 1946 2909
10°848 1944 3456
10°849 1942 4022

OF THE DESCENDING EXPONENTIAL.

4608
5213
5837
6481
7144
7827
8528
9252.

999°

0750
1525
2327

3144

3980

4836

5711

6604 |
USE 1

$449
9400

0371
1360

2368 || -
3395 ||

4441

5506 |
6590 ||

7693
8815

9955
III5
2293
3490
4706
5941

7194
8466

9757

1007
2395

3742
5108

6492

7895
9316
0756
2215
3692
5187
6701

205
Pe | @ e-*
1845 8234 | 10'950 1755 8015
1843 9785 | 10°95! 1754 0466
1842 1354 || 10°952 E7152 2934
1840 2942 | 10°953 1750 5420
1838 4548 || 10°954 1748 7923
1836 6173 || 10°955 1747 0444
1834 7816 | 10°956 1745 2983
1832 9477 |, 10°957 1743 5538
1831 1157 || 10°958 1741 8112
1829 2855 || 10°959 1740 0702
1827 4571 || 10960 1738 3310
1825 6306 || 10°961 1736 5935
1823 8059 || 10°962 1734 8578
1821 9830 || 10°963 1733 1238
1820 1619 || 10°964 1731 3916
1818 3427 || 10'965 1729 661a
1816 5252 || 10°966 1727 9323
1814 7096 || 10°967 1726 2052
1812 8958 |j 10°968 1724 4798 |
1811 0838 || 10°969 1722 7562
1809 2736 || 10°970 1721 0343
1807 4653 || 10°971 171g 3141
1805 6587 || 10°972 1717 5957
1803 8539 || 10°973 1715 8789
1802 o51o || 10°974 1714 1639
1800, 2498 | 10°975 1712 4506
1798 4505 || 10°976 1710 739°
1796 6529 || 10°977 1709 0291
1794 8572 || 10°978 1707 3210
1793 0632 || 10°979 1705 6145
1791 2711 || 10°980 1703 9097
1789 4807 || 10981 1702 2067
1787 6921 || 10'982 1700 5053
1785 9053 | 10°983 1698 8057
1784 1203 || 10°984 1697 1077
1782 3371 || 10°985 1695 4114
1780 5556 || 107986 1693 .7169
1778 7760 || 10°987 1692 0240
1776 9981 || 107988 1690 3328
1775 2220 || 10°989 1688 6433
1773 4476 || T0"990 1686 9556
1771 6751 || Io°991 1685 2694
1769 9043 || 10°992 1683 5850
1768 1353 || 10°993 1681 9023
1766 3680 ! 10°994 1680 2212
1764 6025 || 10°995 1678 5418
1762 8388 || 10°996 1676 8641
1761 0768 || 10°997 1675 1881
1759 3166 || 10°998 1673 5138 |
1757 5592 1671 8411 |

;

206 MR F. W. NEWMAN’S TABLE [11000—11'199]
!
r e= | 2@ ie xv oo x =
= |-— = eee _ ——
117000 1670 1701 || Ir‘050 | 1588 7149 || II‘I0o I51I 2324 || Ir‘I50 1437 5287
II‘Oor 1668 5007 | It‘o51 1587 1270 | II‘ror 1509 7219 || ITI51 1436 o919
II"002 1666 8331 | T1052 | 1585 5407 | II‘102 1508 2129 || I1'152 1434 6565
II-003 1665 1671 | 11053 1583 9559 | II‘103 1506 7055 | II‘I53 1433 2226
rr e04 1663 5027 | 11054 | 1582 3727 || TI'104 1505 1995 | IT'I54 1431 7901 |
II‘005 1661 8401 | 11055 1580 7912 | II105 1503 6951 || TI°I55 1430 3590
1r'006 1660 1791 | 11056 | 1579 2112 || 11°106 1502 1921 || 11156 1428 9293
II‘007 1658 5197 | 11057 | 1577 6327 || 11107 1500 6907 || II‘I57 1427 5011
11-008 1656 8620 | 11°058 | 1576 0559 | 11108 1499 1907 | 11158 1426 0743
II‘009 | 1655 2060 | 11°059 | 1574 4806 |, I1"109 | 1497 6923 || II"159 1424 6490
II‘o10 | 1653 5516 | 11060 | 1572 9069 || II‘II10 | 1496 1954 || 11160 1423 2250
IQ‘OIr | 1651 8989 | 11°061 1571 3348 || 11x11 | 1494 6999 || 11161 1421 8025
II‘ol2 | 1650 2478 | 117062 1569 7643 || Il‘112 1493 2060 || 11-162 1420 3814
Ir‘o13 1648 5984 | 117063 1568 1953 || II‘113 I49I 7135 || rr°163 1418 9618
II‘or4 | 1646 9506 | 11'064 1566 6279 || 11114 1490 2225 || 11°164 I417 5435
II‘or5 | 1645 3045 | 11:065 1565 0620 || 11115 1488 7331 Ir'165 1416 1267
Iro16 |. 1643 6600 || 11°066 1563 4977 || 11116 1487 2451 || 11°166 1414 7113
IVor7 | 1642 0172 | 11'067 1561 9350 || II°rx7 | 1485 7586 || 11°167 1413 2973
rro18 1640 3760 | 11068 1560 3739 || 11118 1484 2735 || 11°168 1411 8847
II‘org | 1638 7364 || 11°069 1558 8143 || 11-119 1482 7900 || rr‘169 I4IO 4735
TI‘020 | 1637 0985 || I1‘070 1557 2562 || 11120 1481 3080 || rr°r70 1409 0637
II‘o21 | 1635 4622 || 11°071 1555 6998 || 11121 1479 8274 || 11-171 1407 6554
11022 | 1633 8276 || 11°072 1554 1448 || 11-122 1478 3483 || 11°172 1406 2484
II'023 | 1632 1946 || 11°073 1552 5915 || 11123 1476 8707 || 1r°173 1404 8429
T1'024 1630 5632 | 11'074 1551 0397 | 11124 1475 3946 || 11174 1403 4387
Tr025 | 1628 9334 | II‘075 1549 4894 || II‘125 1473 9199 || 11175 1402 0360
11026 | 1627 3053 || 117076 1547 9407 || 11126 1472 4467 || 117176 1400 6347
II‘027 1625 6788 || 11°077 1546 3935 || T1127 1470 9750 || I1°177 1399 2347
11-028 1624 0539 || 11'078 1544 8479 || 117128 1469 5048 || 11°178 1397 8362
I1"029 1622 4307 || 11°079 1543 3038 || Ir129 1468 0360 | 11179 1396 439°
I1'030 1620 Sog1 || 11'080 1541 7613 || 11130 1466 5687 || 11180 1395 0433
1I'031 1619 1891 || 11-081 1540 2203 || I1°131 1465 1029 || 11181 1393 6490
I1‘032 1617 5707 || 117082 1538 6808 || r1°132 1463 6385 || 11°182 1392 2560
11033 1615 9539 || 11°083 1537 1429 || 11°33 | 1462 1756 || 11°183 1390 8644
11034 1614 3388 || 11°084 1535 6066 || 11°134 1460 7142 || 11°184 1389 4743
11035 1612 7253 | T1085 1534 0717 || I1°135 1459 2542 || rr°185 1388 0855
11036 1611 1133 || 11086 1532 5394 || 11°136 1457 7956 | 11°186 1386 6981
11°037 1609 5030 || 11°087 1531 0066 || 11°137 1456 3386 || 117187 1385 3121
11038 | 1607 8963 | 117088 1529 4764 || 11°138 | 1454 8830 || 11188 1383 9275
11'039 1606 2872 | 11089 1527 9477 || 11°139 1453 4288 | r1°189 1382 5442
II"040 1604 6818 || 11‘090 1526 4205 || 11140 | T451I 9761 || II‘1g0 1381 1624
Iro41 | 1603 0779 || II‘ogI 1524 8949 || II‘I4I 1450 5249 .|| I1°1g1 1379 7819
11042 | 1601 4756 || 11'092 1523 3707 || 11°142 1449 0751 | II‘1g2 1378 4028
11043 | 1599 8749 | I1'093 1521 8481 || 11°143 1447 6267 || 117193 1377 O251 |
11'044 1598 2759 | 11094 1520 3270 || 11144 1446 1798 | II'I194 1375 6488 |
11045 | 1596 6784 || r1'095 | 1518 8075 | 11°145 1444 7343 || I1°195 1374 2738
117046 1595 0825 | 11'096 1517 2894 || 11°146 1443 2903 | I1‘196 1372 goo2
11047 1593 4882 I1‘097 515 7729 || 11°147 1441 8478 || 11°197 1371 5280
11048 1591 8955 | 11098 1514 2579 || 11°148 1440 4066 || 11°198 1370 1572
11°049 159° 3044 | II’099 1512 7444 || 11°149 1438 9670 | II"199 1368 7877

[11:200—11'399] OF THE DESCENDING EXPONENTIAL. 207
!

2 Cee Ng e-# a er | @ e-*

|

| II‘200 1367 4196 || I1'250 1300 7298 || I1°300 1237 2924 || 11°350 1176 9490
II‘201 1366 0529 || Il'251 1299 4297 || I1‘301 1236 0557 || 11351 1175, 7726
I1'202 1364 6875 || 11-252 1298 1309 || I1°302 1234 8203 || 11°352 1174 5974
11'203 1363 3235 | 11'253 1296 8334 | 11°303 1233 5861 | 11°353 1173 4234
11°204 1361 9608 i II'254 1295 5372 || 11°304 1232 3531 || 11°354 1172 2506
II'205 1360 5996 | 11'255 1294 2423 || 11°305 1231 1214 |} 11°355 II7I 0789
11'206 1359 2396 || 11°256 1292 9487 || 11°306 1229 8909 || 11°356 1169 9084
II‘207 1357 S811 || 11°257 1291 6564 || 11°307 1228 6616 || 11°357 1168 7391
I1‘208 1356 5239 || 11°258 1290 3654 || 11°308 1227 4336 || 11°358 1167 5709
I1‘209 1355 1680 || 11°259 1289 0757 | I1I°309 1226 2067 || 11°259 1166 4039
II‘210 1353 8135 || 11°260 1287 7873 || 11°310 1224 9812 || 11°360 1165 2381
E20 1352 4604 || 11°261 1286 5001 || 11:311 1223 7568 || 11°361 1164 0735
Il’212 1351 1086 || 11°262 1285 2143 || 11312 1222 5336 || 11°362 1162 gtoo
I1‘213 1349 7582 || 11°263 1283 9297 || 11°313 I22I 3117 || 11°363 1161 7476
II'214 1348 4ogi || 11°264 1282 6464 || 11°314 1220 OgIO || 11°364 1160 5865
II‘215 1347 0614 || 11°265 1281 3644 | Bios 1218 8715 || 11°365 II59 4265
I1‘216 1345 7150 || 11°266 1280 0837 || 11°316 1217 6533 || 11°366 1158 2676
II‘217 1344 3699 || 11°267 1278 8042 || 11-317 1216 4362 || 11°367 II57 1099
11218 1343 0262 || 11°268 1277 5261 || 11°318 I215 2204 || 11°368 TI55 9534
II‘219 1341 6839 || 11°269 1276 2492 || I1°319 1214 0058 || 11°369 1154 7980
I1I‘'220 1340 3429 |] I1°270 1274 9736 || 117320 1212 7924 || 11°370 1153 6438
II‘221 1339 0932 || 11°271 1273 6993 || 11°321 1211 5802 || 11°37 1152 4907
I1°222 1337 6649 || 11272 1272. 4262 || 11°322 1210 3692 || 11°372 TI51 3388
Er223 Tego) g2i79) |l| Lxs273 I27I 1544 || 11°323 1209 1595.|| 11°373 1150 1881
I1‘224 1334 9922 || 11'274 1269 8839 || 11°324 1207 9509 || 11°374 1149 0384
I1‘225 1333 6579 || 11'275 1268 6146 || 11°325 1206 7436 || 11°375 1147 8900
11'226 1332 3249 || 11°276 1267 3466 || 11°326 1205 5374 || 11°376 1146 7427
I1'227 1330 9932 || 11°277 1266 0799 || 11°327 1204 3325 || 11°377 1145 5965
11°228 1329 6629 || 11'278 1264 8145 || 11°328 1203 1288 || 11°378 1144 4515
I1'229 1328 3339 || 11°279 1263 5503 || 117329 1201 9262 || 11°379 1143 3076
I1‘230 1327 0062 || 11°280 1262 2874 || 11°330 1200 7249 || 11°380 1142 1649
I1'231 1325 6799 || 11°281 1261 0257 || 11°331 1199 5248 || 11°381 II4Il 0233
I1‘'232 1324 3549 || 11°282 1259 7653 || 11°332 1198 3259 || 11°382 1139 8828
1I'233 1323 0312 || 11°283 1258 5062 || 11°333 1197 1281 || 11°383 1138 7435
11'234 132r 7088 || 11284 1257 2483 || 11°334 TI95 9316 || 11°384 1137 6053
TI'235 1320 3878 || 11°285 1255 9917 || 11°335 1194 7363 || 11°385 1136 4683
11'236 1319 0680 || 11°286 1254 7363 |) 11°336 1193 5421 | 11°386 1135 3324
11'237 1317 7496 || 11°287 1253 4822 || 11°337 IIg2 3492 || 11°387 1134 1976
11'238 1316 4325 || 11°288 1252 2294 || 11°338 IIQI 1574 || 11-388 1133 0640
II'239 1315 1168 || 11289 1250 9778 || 11°339 1189 9669 || 11°389 II3I 9315
II‘240 1313 8023 || 11°290 1249 7274 || 11°340 1188 7775 || 117390 1130 8001
II‘241 1312 4892 || 11'291 1248 4783 || 11°341 1187 5893 || 11°39! 1129 6699
II'242 I31I 1773 || 11‘292 1247 2305 || 11°342 1186 4023 || 117392 1128 5408
1I‘243 1309 8668 || 11°293 1245 9839 || 11°343 1185 2165 || 11°393 1127 4128
I1‘244 1308 5576 || 11-294 1244 7385 || 11°344 1184 0319 || 11°394 1126 2860
T1‘245 1307 2497 || I1°295 1243 4944 || 11°345 1182 8484 || 11°395 1125 1602
11246 1305 9431 || 11°296 1242 2515 || 11°346 1181 6662 || 11°396 1124 0356
I1°247 1304 6378 || 11°297 I24I 0099 || 11°347 1180 4851 || 11°397 I122 g122
11248 1303 3338 || 11°298 1239 7695 || 117348 I17Q 3052 || 11°398 1121 7898

| 11249 1302 O31T || 11°299 1238 5303 | 11349 1178 1265 || 11°399 1120 6686

208 MR F. W. NEWMAN'S TABLE [11'400—11'599]
> e-* x e-% x e= x e-%
I1*400 | TIIrg 5485 || 11°450 1064 9475 || II°500 IOI3 0093 |} T1550 963 6043
II‘4or | 11x18 4295 || 1F451 1063 8830 || II‘5o1r IOII gg68 || 11551 962 6412
II‘402 | III7Z 3116 || 11-452 1062 8197 || II"502 IOTO 9854 || 117552 961 6790
II*403 | 1116 1949 |} 11°453 1061 7574 || 117503 1009 9749 || 11553 960 7178
I1*404 | III5 0792 || 117454 1060 6962 || I1°504 1008 9654 || 117554 959 7576
II"4o5 | 1113 9647 || 11°455 1059 6360 || II*505 1007 9569 || 11°555 958 7983
11-406 | 1112 8513 || 11°456 1058 5769 || 117506 1006 9495 || 117556 957 8400
11°407 | IIIT 7390 || 11°457 1057 5189 || II‘507 1005 9430 || 11°557 956 8826
11-408 | 1110 6278 || r1°458 1056 4619 || I1°508 1004 9376 || 11°558 955 9262
EEOon 109 5177 | 11"459 1055 4059 || T2509 TQ9389S32" |) 22559 954 9708
II"410 1108 4088 || 11°460 1054 3511 || 11510 1002 9297 || 11°560 954 0163
II‘41I | 1107 3009 || 11°461 1053 2972 || II‘51 IOOI 9273 || 11°561 953 0627
II'412 I106 1942 | 117462 IO52 2445 || 11512 1000 9259 || 11°562 952 1102
T1413 t105 0885 | 117463 TO5I 1927 | I1'513 2999 9255 | 11°563 951 1585
IIl'414 1103 9840 || 11°464 IO50 142K || 11514 998 9260 | 117564 950 2078
|
II‘415 t102 8806 || 11°465 1049 0925 || 11515 997 9276 || 11°565 949 2581
Ir"416 10x 7782 || 11°466 1048 0439 | 11°516 996 9302 || 11°566 948 3093
II‘417 1100 6770 | 11°467 1046 9964 || 11°517 995 9337 || 11°567 947 3615
11418 1099 5769 | 11°468 1045 9499 | 117518 994 9383 || 11°568 946 4146
11-419 1098 4779 | 11°469 1044 yo45 | II‘519 993 9439 | 11°569 945 4687
11I"420 1097 3799 | I1°470 1043 8601 II"520 992 9504 |, II°570 944 5237
II‘421 1096 2831 | 11°471 1042 8168 || 11-521 991 9580 | 11°571 943 5796
II"422 1095 1874 | 11°472 IO4I 7745 || 117522 990 9665 | 117572 942 6365
11°423 1094 0927 | 11473 1040 7332 || 11523 989 9760 | 11°573 941 6943
11°424 10g2 9992 | 117474 1039 6930 | 117524 988 9866 | 117574 949 7531
11°425 Togit 9067 | 11°475 1038 6538 | II‘'525 987 998r || 117575 939 8128
11°426 Togo 8154 | 117476 1037 6157 || 117526 987 o106 | 117576 938 8735
11427 1089 7251 | 11°477 1036 5786 | 117527 986 o240 || 11°577 937 9351
11°428 | 1088 6359 | 11°478 1035 5425 || 117528 985 0385 || 117578 936 9976
11°429 1087 5478 | 11'479 1034 5075 | 11'529 984 0540 | 11°579 936 o611
11°430 1086 4608 | 11°480 1033 4735 || 11530 983 0704 | 11°580 935 1255
11431 1085 3749 | 11°481 1032 4406 || I11°531 982 0878 || 117581 934 1908
11432 1084 2901 | 11°482 1031 4086 || 117532 981 1062 | 117582 933 2571
11°433 1083 2063 | 11°483 1030 3777 || 117533 980 1256 | 117583 932 3243
11°434 1082 1237 | 11°484 1029 3479 | 11°534 979 1460 | 11°584 931 3925
11°435 to81 0421 | 11°485 1028 3190 11535 978 1673 || 11°585 930 4615
11°436 1079 9616 | 117486 1027 2912 | 11°536 977 1896 | 11°586 929 5315
11°437 1078 8821 | 117487 1026 2645 || 117537 976 2129 | 117587 928 6025
11°438 1077 8038 | 11°488 1025 2387 || 11°538 975 2372 | 117588 927 6743
11°439 1076 7265 | 11°489 1024 2140 || 11°539 974 2625 || 117589 926 7471
11°440 1075 6504 | 11°490 1023 1903 || 11°540 973 2887 || 11°590 925 8208
11°441 1074 5752 | I1°491 1022 1676 || 11541 972 3159 || 11'591 924 8955
11"442 1073 5012 | 11492 1021 1459 || 117542 971 3441 || 11°592 92379729
11°443 1072 4282 | 11°493 1020 1253 || 11°543 97° 3732 || 117593 923 0475
11°444 107I 3563 | 11°494 IOIg 1057 || 11°544 969 4033 || 117594 922 1250
1I°445 1070 2855 | 11°495 1018 0871 || 11°545 968 4344 || 11°595 921 2033
11°446 1069 2158 | 117496 Io17 0695 || 11°546 967 4664 || 11°596 g20 2825
11°447 1068 1471 | 11°497 1016 0529 || 11°547 966 4995 | 11°597 919 3627
11°448 1067 0795 | 117498 IOI5 0374 || 11°548 965 5334 || 117598 918 4438
11°449 1066 o129 | 11°499 IOI4 0229 || 11°549 964 5684 || 11°599 917 5258

[11°600—11'799]

OF THE DESCENDING EXPONENTIAL.

|

av Ge | @ Gr?
I1*600 916 6088 || 11-650 871 9052
II‘601 g15 6926 || 11°651 871 0338
I1‘602 914 7774 || 11°652 870 1632
11‘603 913 8631 || 11°653 869 2934
117604 92 9497 || 117654 868 4246 |
11605 gi2 0372 || 117655 867 5566 |
11°606 gII 1256 || 11°656 866 6895 |
11°607 gIO 2149 || 11°657 865 8232 |
11°608 909 3051 || 11658 864 9578
I1°609 908 3963 || 11°659 864 0933
Ir‘610 907 4884 || 11°660 863 2206
Ir‘611 go6 5813 || 11°661 862 3668
11612 905 6752 || 11°662 861 5049
11613 904 7700 || 11°663 860 6438 |
11614 903 8657 || 11°664 859 7836
Ir'615 go2 9622 || 11665 858 9242
11616 go2z 0597 || 117666 858 0657
11617 gor 1581 |) 11°667 857 2081
11618 goo 2574 || 117668 856 3513
11619 899 3576 || 11669 855 4954
11620 898 4587 || 11°670 854 6403
11621 897 5607 || 11°671 853 7861
11622 896 6636 || 11°672 852 9328
11'623 895 7674 || 11°673 852 0803
11624 894 8720 || 11°674 851 2286
11°625 893 9776 || 11°675 850 3778
11626 893 0841 || 11°676 849 5279
11627 892 1914 || 11°677 848 6788
11°628 891 2997 || 11°678 847 8305
r1'629 890 4088 || 11°679 846 9831
11'630 889 5189 || 11°680 846 1365
11°631 888 6298 || 11-681 845 2908
11632 887 7416 | 11682 844 4460
11°633 886 8543 | 11°683 843 6019
11°634 885 9679 || 117684 842 7587
11635 885 0824 || 11°685 841 9164
11636 884 1977 || 11°686 841 0749
11637 883 3140 || 11°687 840 2343
11'638 882 4311 || 11°688 839 3944
11°639 881 5491 || 11°689 838 5555
I1°640 880 6680 | 11°690 Ce fyi he}
11641 879 7878 || 11°69 836 8800
11°642 878 9084 || 11°692 836 0436
11°643 878 0300 || 11°693 835 2080
11°644 877 1524 || 11°694 834 3732
11°645 876 2757 || 11°695 833 5392
11°646 875 3998 || 11°696 832 7061
11°647 874 5249 || 11°697 831 8738
11648 873 6508 || 117698 831 0423
11649 872 7776 || 11°699 830 2117

VoL. Mile De IE

x“ (De xv
I1°700 829 3819 | I1°750
II‘7O1 828 5529 | 11°751
I1°702 827 7248 || 11°752
I1I'703 826 8975 || 11°753
I1I°704 826 0710 | 11°754
11795 825 2453 | 11°755
11°706 824 4205 | 11°756
11°707 823 5965 || 11°757
I1°708 822 7733 || 11°758
I1°709 821 9510 || 11°759
II‘710 82I 1294 | 11°760
LEAL 820 3087 || 11°761
II‘712 819 4888 I1°762
II‘713 818 6697 || 11°763
11-714 817 8515 | 117764
II‘715 817 0340 || 11°765
II‘716 816 2174 || 11°766
pasa, 815 4016 || 11°767
I1°718 814 5866 || 11°768
II‘719 813 7724 || 11°769
I1°720 812 9590 ||-11°770
II°721 812 1465 || 11°771
I1°722 811 3348 || 11°772
T7723 810 5238 || 11°773
11°724 809 7137 || 11°774
LE7Z5 808 9044 || 11°775
11°726 808 0959 || 11°776
Bie 27 807 2882 || 11°777
11°728 806 4813 || 11°778
I1‘729 805 6752 || 11°779
1I°730 804 8700 || 117780
11°731 804 0655 || 11°781
11732 803 2618 || 11°782
LIe733 802 4590 || 11°783
11734 801 6569 || 117784
11735 800 8557 || 11°785
11°736 800 0552 || 11°786
11°737 799 2556 || 117787
11°738 798 4567 || 11°788
11°739 797 6586 | 11°789
II'740 796 8614 || 117790
I1‘741 796 0649 || 11°791
II°742 795 2692 || 11°792
T1743 794 4744 || 11°793
T1744 793 0803 || 11°794
11°745 792 8870 || 11°795
11°746 792 0945 || 11°796
11°747 791 3028 || 11°797
11°748 79° 511g || 117798
11°749 789 7218 || 11°799

209

MR F. W. NEWMAN’S TABLE

210
| wv ae
| r1°S00 750 4558
| 11‘Sor 749 7°57
| 11°So2 748 9564
11°803 748 2078
11-804 |} 747 4600
| r1°So5 746 7129
| 11°806 745 9665
| 11807 | 745 2209 ||
| 11°808 744 4761
11‘809 743 732°
1r°8r0 742 9886 ||
| rx‘S1r 742 2460
rr-S12 741 5041
11813 740 7630
11814 740 0226
| rr°815 739 2829
| 11816 738 5440
| 11817 737 8059
} r1°818 737 0084
11°819 736 3317
| 11-820 | 735 5958
11821 734 8605
11°822 734 1260
11823 733 3923
11°824 732 6592
11°825 731 9270
11°826 731 1954
11°827 730 4646
11°828 729 7345
11°829 729 0051
| 117830 728 2765
11°831 727 5485
| 11°832 726 8214
| 11°833 726 0949
| 11°834 725 3692
11°835 724 6442
11°836 723 9199
| 11°837 723 1963
| 11°838 722 4735
| 11°839 721 7514
| 11°840 721 0300
| 11°841 720 3093
| 11°842 | 719 5894
11°843 718 8702
11°844 718 1516
| 11°845 717 4339
11°846 716 7168
11°847 | 716 0004
11°848 715 2848

| 11°849

5699

0405

3 3618

6837

0064 ||

3297

6537
9784
3038
6298
9565
2839
6120
9407
2701
6001

9309
2623
5944
9271
2605

5946
9293
2647
6008

9375

2749
6130
9517
2QI1
6311

9718
Shs
6552
9978
3412
6852
0298
3751
7211
0677

4149
7628
1114
4696
8105

1610
5121
8640

2164 |

5695

e-«

[11Soo—r1'999]

9233
2777
6327
9884
3447

7°17
0593
4176
7765
1360

4962
8571
2185
5896

9454

3067
6707
0354
4007
7666

1331
5993
8681
2306
6057

9754
3457
7167
0883
4605
8334
2068
5809
9557
3310
797°
0836
4608
8387
2172

5963
9760
3563
7373
1188

5010
8838
2673

6513
0360

[12‘000—12'199]

4212
8071
1936
5807
9684
3568
7457
1353
S235)
g162
3076
6996
0922

4854
8793

2737
6687
0643
4606

8574

2549
6529
0516
4508
8597
2511
6522
0538
4561
8589
2623
6664
0710
4762
8821

2885
6955
1031
5113
g201

Sas
7394
1499
5611

9728 |
3851 |

7981
2115
6256
0403

OF THE DESCENDING EXPONENTIAL. iat
i
v e-# WH e-# ay e-&
12°050 584 4555 || 12°100 555 9513 || 12150 528 8372
12°051 583 8714 || 12°101r 555 3950 || 12151 528 3087
12°052 583 2878 || 12°102 554 8405 || 12152 527 7806
12053 582 7048 || 12°103 554 2860 || 12:153 527 2531
12°054 582 1224 || 12°104 553 7320 || 12°154 526 7261
BOSS 581 5406 || 12°105 553 1785 || 12°155 526 1997
12°056 589 9593 || 12°106 552 6256 || 12°56 525 6737
12°057 580 3787 || 12°107 552 0732 || 12°57 525 1483
127058 579 7986 | 12°108 551 5214 || 12°158 524 6234
12°059 579 2191 || 12°r09 550 9702 || 12°59 524 oggt
12°060 | ° 578 6401 || 12‘I10 550 4195 || 12°160 523 5752
12°061 578 0618 || 12111 549 8694 || 12°161 523 0519
12'062 577 4840 || 12°112 549 3198 || 12°162 522 5291
12°063 576 9068 || 12113 548 7707 || 12°163 522 0069
12°064 576 3302 || 12°114 548 2222 || 12°164 52I 4851
12°065 575 7541 | 12°115 547 6743 || 12°165 | 520 9639
12°066 575 1787 2116 547 1269 || 12°166 520 4432
12°067 574 6038 || 12°117 546 5800 |} 12°167 519 9230
12'068 574 0295 || D208 546 0337 || 12°168 519 4033
12069 573 4557 |, 12°r19 545 4880 || 12°169 | 518 8842
12'070 572 8825 || 12°120 544 9427 || 12°170 518 3656
12071 572 3099 || 12°121 544 3981 || 12°171 517 8475
12°072 57 7379 || 12°122 543 8539 || 12°172 UL eek)
12'073 571 1665 || 12°123 543 3104 || 12°173 516 8128
I2°074 579 5956 || 12°124 542 7673 || 12°174 516 2962
12'075 579 0253 || 12°125 542 2248 || 12°175 515 7802
12°076 569 4555 || 12°126 541 6829 || 12°176 515 2647
12°077 568 8864 || 12°127 541 1415 || 12°277 514 7497
12°078 568 3178 || 12128 540 6006 || 12°178 514 2352
12°079 567 7497 || 12°129 540 0603 || 12°179 513 7212
12°080 567 1823 || 12°130 539 5205 || 12'180 513 2077
12‘081 566 6154 |) 12°13 538 9812 || 12°181 512 6948
12°082 566 0490 || 12°132 538 4425 || 12°182 512 1824
12°083 565 4833 || 12°133 537 9°43 || 12°183 511 6704
12°084 564 g18r || 12°134 537 3067 || 12°184 5Ir 1590
12°085 564 3534 || 12°535 536 8296 || 12°185 510 6481
12'086 563 7894 || 12°136 536 2930 || 12°186 510 1377
12°087 563 2259 || 12°137 535 757° || 12°87 509 6278.
12'088 562 6629 || 12°138 535 2215 || 12°188 509 1185
12'089 562 1005 |} 12°39 534 6866 | 12°189 508 6096
12090 561 5387 || 12°140 534 1522 || 12°190 508 1012
I2'09g1 | 560 9775 || 12°I41 533 6183 || 12°191 5°7 5934
12°092 | 560 4168 || 12°142 533 0849 | 12°192 507 0861
12°093 559 8566 || 12°143 532 5521 || 12°93 506 5792
12°094 559 2970 || 12°144 532 0198 | 12°194 506 0729
12°095 558 7380 || 12°145 531 4881 | 12°195 505 5671 |
12°096 558 1796 || 12°146 530 9568 | 12°196 505 0618
|| 12°097 557 6217 || 12°147 530 4261 || 12°197 504 5570
12°098 557 0643 || 12°48 529 8960 | 12°198 504 0526
12°099 556 5075 || 12°149 529 3664 | 12°199 503 5488 |

212 MR F. W. NEWMAN'S TABLE [12'200-—12"399]

e-* wv Ga% xv Cae | @& Cm
12°200 503 0456 || 12°250 478 5117 || 12°300 455 1744 || 12°350 432 9753
12°201 | 502 5428 || r2°251 478 0335 || 12°301 454 7195 || 12°351 432 5425
2°202 502 0405 || 12°252 477 5557 .|| 427302 454 2050 || 12°352 432 1102
2°203 501 5387 || 12°253 477 0783 || 12°303 453 8110 | 12°353 431 6783
12°204 SOL 0374 || 12°254 476 6015 | 12°304 453 3574 || 127354 431 2469
12°205 500 5366 || r2:255 476 1251 || 12°305 452 9042 || 12°355 430 8158
2°206 500 0363 || 12°256 475 6493 || 12°306 452 4516 || 12°356 43° 3852
12°207 499 5365 || 12°257 475 1738 || 12°307 451 9993 || 12°357 429 9551
12°208 499 0372 || 12°258 474 6989 |} 12°308 451 5476 || 12°358 429 5253
2°209 498 5385 || 12°259 474 2244 || 12°309 451 0962 || 12°359 29 0969
498 0402 || 12°260 473 7505 || 12°310 45° 6454 || 12°360 428 6671
II 497 5424 || 12°261 473 2769 || 12°311 452 1950 || 12°361 428 2387
I2°212 497 O451 || 12°262 472 8039 || 12°312 449 7450 || 12°362 427 8107
12°213 496 5483 || 12°263 472 3313 || 12°313 449 2955 || 12°363 427 3831
2°214 496 0520 || 12°264 471 8592 || 12314 448 8464 || 12°364 426 9559
12°215 495 5562 || 12°265 471 3876 ||-12°315 448 3978 || 12°365 426 5291
12°216 495 0609 || 12°266 470 9165 || 12°316 447 9496 || 12°366 426 1028
12°217 494 5661 || 12°267 470 4458 || 12°317 447 5019 || 12°367 425 6769
12°218 494 0717 |} 12°268 469 9756 || 12°318 447 0546 |} 12°368 425 2515
12°219 493 5779 || 12°269 469 5058 || 12°319 446 6078 || 12°369 424 8264 :
12°220 493 0846 || 12°270 469 0366 || 12°320 446 1614 |} 12°370 424 4018
12°221 492 5917 || 12°271 468 5678 || 12-321 445 7154 | 12°371 423 9776
12°222 492 0994 || 12°272 468 0994 | 12°322 445 2699 | 12°372 423 5539
12°223 491 6075 || 12°273 467 6316 || 12°323 444 8249 || 12°373 423 1305
12°22 491 1162 || 12°274 467 1642 || 12°324 444 3893 || 12°374 422 7076
12°225 492 6253 || 12°275 466 6972 || 12°325 443 9361 || 12°375 422 2851
12226 492 1349 | 12°276 466 2308 || 12°326 443 4924 || 12°376 421 8630
12°22 489 6450 | 12°277 465 7648 || 12°327 443 0492 || 12°377 421 4414
12°228 489 1556 || 12°278 465 2992 || 12°328 442 6063 || 12°378 421 0202
12°22 488 6667 | 12°279 464 8342 || 12°329 442 1639 || 12°379 420 5993
12°230 488 1783 || 12°280 464 3696 || 12°330 441 7220 || 12°380 420 1790
12°231 487 6904 || 12°281 463 9054 || 127331 441 2805 || 12°381 419 7590
12°232 487 2029 || 12°282 463° 4418 || 12°332 440 8394 || 12°382 419 3394
12°233 486 7160 || 12°283 462 9786 || 12°333 440 3988 || 12°383 418 9203
12°234 486 2295 || 12°284 462 5158 || 12°334 439 9586 || 12°384 418 5016
12°235 485 7435 || 12°285 462 0535 || 12°335 439 5189 | 12°385 418 0833
12°236 485 2580 || 12°286 461 5917 || 12°336 439 0796 | 12°386 417 6654
12°237 484 773° || 12°287 461 1303 || 12°337 438 6407 || 12°387 417 2480
12°238 484 2885 || 12°288 462 6694 || 12°338 438 2023 || 12°388 416 8309
12°239 483 8044 | 12°289 469 2090 || 12°339 437 7643 || 12°389 416 4143
12°240 483 3209 || 12°290 459 7499 || 12°340 437 3268 | 12°390 415 9981
12°241 482 8378 || 12°291 459 2895 || 12°341 | ~ 436 8897 || 12°391 415 5823
12°242 482 3552 || 12°292 458 8304 || 12°342 436 4530 || 12°392 415 1669
12°243 481 8731 || 12°293 458 3718 || 12°343 436 0168 || 12°393 414 7520
12°244 481 3914 || 12°294 457 9137 || 12°344 435 5810 | 12°394 414 3374
12°245 482 9103 || 12295 457 4560 || 12°345 435 1456 | 12°395 413 9233 |
| 12°246 480 4296 || 12°296 456 9988 || 12°346 434 7107 || 12°396 413 5096
12°247 479 9494 || 12°297 456 5420 || 12°347 434 2762 || 12°397 413 0963
12°248 479 4697 || 12°298 456 0857 || 12°348 433 8421 || 12°398 412 6834
12°249 478 9995 | 12°299 455 6298 || 12°349 433 4085 || 127399 412 2709

[12:400—12°599] OF THE DESCENDING EXPONENTIAL. 213

en. e* | ow e-2 x e-# x e-*
12°400 411 8589 | 12°450 391 7723 || 12°500 372 6653 || 12°550 354 4902
12°401 4IIl 4472 |} 12°451 391 3807 || I2°50r 372 2928 || 12°551 354 1359
12°402 411 0360 || 12°452 390 9895 || 12°502 371 9207 || 12°552 353 7819
T2°403 410 6251 || 12°453 39° 5987 || 12°503 372 5490 || 12°553 353 4283
12°404 410 2147 || 12°454 390 2083 |] 12°504 GH e eye) I aiedacicys! 353 0751
12°405 409 8047 || 12°455 389 8183 || 12°505 370 8066 || 12°555 352 7222
12°406 409 3951 | 12°456 389 4287 || 12°506 370 4360 || 12°556 352 3696
12°407 408 9859 || 12°457 389 0394 || 12°507 370 0658 || 12°557 352 O174
12408 408 5771 || 12°458 388 6506 || 12°508 369 6959 || 127558 351 6656
12°409 408 1688 || 12°459 388 2621 || 12°509 369 3264 || 12°559 351 3141
I2°410 407 7608 | 12°460 387 8741 || 12°510 368 9572 || 12°560 350 9630
12‘411 407 3532 || 12°46 387 4864 || 12°51 368 5885 || 12°56r 350 6122
12°412 406 9461 |) 12°462 387 oggt || 12°512 368 2200 || 12°562 350 2617
I2°413 406 5393 || 12°463 386 7122 || 12°513 367 8520 || 12°563 349 9117
12°414 406 1330 | 12°464 | | 386 3257 || 12°514 367 4843 || 12°564 349 5619
12°415 405 7271 | 12°465 385 9395" |) 12°515 367 1170 || 12°565 349 2125
12°416 405 3216 || 12°466 385 5538 || 12°516 366 7501 || 12°566 348 8635
12°417 404 9164 || 12°467 385 1684 || 12°517 366 3835 || 12°567 348 5148
12°418 404 5117 || 12°468 384 783 12°518 366 0173 || 12°568 348 1665
12°419 404 1074 || 12°469 384 3989 || 12°519 365 6515 || 12°569 347 8185
I2'420 403 7035 || 12°470 384 0147 || 12°520 365 2860 || 12°570 347 4708
12°42 403 3000 || 12°471 383 6308 || 12°521 364 9209 || 12°571 347 1235
12°422 402 8969 || 12°472 383 2474 || 12°522 364 5562 || 12°572 346 7766
12°423 402 4942 || 12°473 382 8643 || 12°523 364 1918 || 12°573 346 4300
12°424 402 ogIg || 12°474 382 4817 || 12°524 363-8278. || 12°574 346 0837
12°425 401 6900 || 12°475 382 0994 || 12°525 363 4642 || 12°575 345 7378
12°426 4or 2885 || 12°476 381 7175 || 12°526 363 1009 || 12°576 345 3922
12°427 400 8875 || 12°477 381 3359 || 12°527 362 7380 || 12°577 345 047°
12'428 400 4868 || 12°478 380 9548 || 12°528 362 3754 || 12°578 344 7022
12429 400 0865 || 12°479 380 5740 || 12°529 362 0132 || 12°579 344 3576
12°430 399 6866 || 12°480 | - 380 1936 || 12°530 361 6514 || 12°580 344 0134
12°431 399 2871 || 127481 379 8136 || 12°531 361 2899 || 12°581 _ 343 6696
12°432 398 8880 || 12°482 379 4342 || 12°532 360 9288 || 12°582 343 3261
12433 398 4893 || 12°483 379 0548 || 12°533 369 5681 |) 12°583 342 9829
12°434 398 ogto || 12°484 378 6759 || 12°534 360 2077 || 12°584 342 6401
12°435 397 6931 || 12°485 378 2974 || 12°535 359 8476 || 12°585 342 2977 !
12°436 397 2957 || 12°486 377 9193 || 12°536 359 4880 || 12°586 341 9555
12°437 396 8986 || 12°487 377 5416 || 12°537 359 1287 || 12°587 341 6138
12°438 396 5019 || 12°488 377 1642 || 12°538 358 7697 || 12°588 341 2723
12°439 396 1056 || 12°489 376 7873 || 12°539 358 4111 || 12°589 340 9312
12°440 395 7096 || 12°490 376 4107 || £2°540 358 0529 || 12°590 340 5904
12°441 395 3141 || 12-491 376 0344 || 12°541 357 6950 || 12°591 340 2500
12°442 394 9190 || 12492 | = 3375 6586 | 12°542 357 3375 || 12°592 339 9°99
12°443 394 5243 || 12°493 375 2831 || 12°543 356 9803 || 12°593 339 5702
12°444 394 1300 || 12°494 374 9080 || 12°544 356 6235 || 12°594 339 2308
12°445 393 7360 || 12°495 374 5333 | 12°545 356 2671 || 12°595 338 8917
12°446 393 3425 | 12°496 374 1590 | 127546 355 9IIO || 127596 338 5530
12°447 392 9494 || 12°497 373 7859 || 12°547 355 5553 || 12°597 338 2146
12°448 392 5506 || 12°498 373 4114 | 12°548 355 1999 | 12°598 337 8766

| 12°449 392 1642 || 12°499 373 0382 || 12°549 354 8449 | 12°599 337 5389

MR F. W. NEWMAN'S TABLE

es | @& Ca

337 2015 12°650 320 7560
336 8645 || 12°651 320 4354
336 5278 || 12°652 320 II51
336 I914 | 12°653 319 7952
335 8554 || 12°654 319 4755
335 5197 || 12°655 319 1562
335 1844 || 12°656 318 8372
334 8493 | 12°657 318 5185
334 5147 || 12°658 318 2002
334 1803 || 12°659 317 S821
333 8463 || 12°660 317 5644
333 5126 || 12°661 317 ‘2470
333 1793 || 12°662 316 9299
332 8463 || 12°663 316 6132
332 5136. || 12°664 316 2967
332 1812 | 12°665 315 9806
331 8492 12°666 315 6647
331 5175 | 12°667 315 3492
331 1862 || 12°668 315 0340
330 8552 || 12°669 314 7192
330 5245 || 12°670 314 4046
330 1941 || 12°671 314 0904
329 8641 || 12°672 313 7764
329 5344 | 12°673 313 4628
329 2050 | 12°674 313 1495
328 8760 | 12°675 312 8365
328 5473 || 12°676 312 5238
328 2189 || 12°677 312 2115
327 8908 | 12°678 311 8994
327 5631 || 12°679 311 5877
327 2357 || 12°680 311 2762
326 9086 || 12°681 310 9651
326 5819 || 12°682 310 6543
326 2555 || 12°683 310 3438
325 9294 | 12°684 310 0336
325 6036 | 12°6385 309 7237
325 2782 || 12°686 309 4142
324 9531 || 12°687 309 1049
324 6283 || 12°688 308 7960
324 3038 || 12°689 308 4873
323 9797 | 12°690 308 1790
323 6558 || 12°691 307 8710
323 3323 || 12°692 307 5632
323 0092 || 12°693 307 2558
322 6863 || 12°694 306 9487
322 3638 || 12°695 306 6419
322 0416 || 12°696 306 3354
321 7197 || 12°697 306 0293
321 3982 | 12°698 305 7234
321 0769 | 12°699 305 4178

[12°600—12°799]

277

277
277
276
276
276

2320
9419
6521
3626
0734

7845
4959
2075
9194
6317

3442
0570
7792
4834
1971
gI10
6253
3398
0546
7697
4851
2007
9167
6329
3494
0662
7833
5006
2183
9362

6544
3729
og16
8107
5300

2496
9695
6897
4IOI
1399
8519
5732
2947
0166
7387
4611
1838
9067
6300
3535

[12°800—r12"999] OF THE DESCENDING EXPONENTIAL. 215
| —— | =
x C= x e-% | @ Ce | @ | G-=
12°800 276 0772 || 12°850 262 6128 || 12°900 249 8050 | 12'950 237 6219
12°Sor 275 8013 || 12°851 262 3503 || 12°901 249 5553 || 12°95 237 3844
12°802 275 5250 || 12°852 262 o881 || I2*902 249 3059 || 12°952 237 IA71
12°803 275 2503 || 12°853 261 8261 |] 12°903 249 0567 | 127953 236 gror
12°S04 274 9751 || 12°854 261 5644 || 12°904 248 8078 | 127954 236 6733
12°805 274 7003 || 12°855 261. 3030 || 12°905 248 5591 || 12°955 236 4367
12°806 274 4257 || 12°856 261 0418 || 12°906 248 3106 || 12°956 236 2004
12807 274 1515 || 12°857 260 7809 || 12°907 248 0625 || 12°957 235 9643
12°808 273 8774 || 12°858 260 5203 || 12°908 247 8145 || 12°958 235 7285
12°809 273 6037 || 12°859 269 2599 || 12"909 247 5668 || 12°959 235 4929
12°810 273 3302 || 12°860 259 9998 || 12°910 247 3194 || 12°960 235 2575
12811 273 0570 || 12°861 259 7399 || 12'911 247 0722 || 12°961 235 0224
12‘812 272 7841 || 12°862 259 4803 || 12°912 246 8253 || 12°962 234 7875
12°813 272 5115 || 12°863 259 2209 || 12°913 246 5786 || 12°963 234 5528
12°814 272 2391 || 12°864 258 9618 || 12‘914 246 3321 || 12°964 234 3184
12°815 271 9670 || 12°865 258 7030 || 12°915 246 0859 || 12°965 234 0842
12‘816 271 6951 || 12°866 258 4444 || 12916 245 8399 || 12°966 233 8502
12°817 271 4236 || 12°867 258 1861 || 12917 245 5942 || 12°967 233 6165
12°818 27E 1523 || 12°868 257 9281 || 12°918 245 3488 || 12°968 233 3830
12°819 270 8813 || 12°869 257 6703 || 12°919 245 1035 || 12°969 233 1497
12°820 270 6105 || 12°870 257 4127 || 12°920 244 8586 || 12970 232 9167
12821 270 3401 || 12°871 257 1554 || 12°921 244 6138 || 12°971 232 6839
12°822 270 0699 |] 12°872 256 8984 || 12°922 244 3693 || 12°972 232 4513
12°823 269 7999 || 12°873 256 6416 || 12°923 244 1251 || 12°973 232 2190
12°824 269 5303 || 12°874 256 3851 || 12'924 243 8811 || 12°974 231 9869
12°825 269 2609 || 12°875 256 1289 || 12°925 243 6373 || 12°975 231 7550
12°826 268 ggt7 || 12°876 255 8729 || 12°926 243 3938 || 12°976 22 TS 2 a8
12°827 268 7229 || 12°877 255 6171 || 12°927 243 1505 || 12°977 231 2919
12°828 268 4543 || 12°878 255 3616 || 12°928 242 9075 || 12°978 231 0608
12°829 268 1860 || 12°879 255 1064 || 12°929 242 6647 || 12°979 230 8298
12'830 267 9179 || 12°880 254 8514 || 12°930 242 4222 || 12°980 230 5991
12°831 267 6501 || 12°881 254 5967 || 12°931 242 1799 || 12°981 230 3686
12°832 267 3826 || 12°882 254 3422 || 12°932 24I 9378 || 12°982 230 1384
12°833 267 1154 || 12°883 254 0880 || 12°933 241 6960 || 12°983 229 9083
12°834 266 8484 || 12°884 253 8340 || 12°934 241 4544 || 12°984 229 6785
12°835 266 5817 || 12°885 253 5803 || 12°935 241 2131 || 12°985 229 4490
12°836 266 3152 || 12°886 253 3269 || 12°936 240 9720 || 12°986 229 2196
12°837 266 0490 || 12°887 253 0737 || 12°937 240 7311 || 12°987 228 ggos
12°838 265 7831 || 12°888 252 8207 || 12°938 240 4905 || 12°988 228 7617
12°839 265 5175 || 12°889 252 5080 || 12'939 240 2502 || 12°989 228 5330
12'840 265 2521 || 12°890 252 3156 || 12°940 240 O100 || 12°990 228 3046
12°841 264 9870 || 12891 252 0634 || 12°94r 239 7701 || 12°99 228 0764
12'842 264 7221 || 12°892 251 8114 || 12°942 239 5305 || 12°992 227 8484
12°843 264 4575 || 12°893 251 5598 || 12°943 239 2911 || 127993 227 6207
12844 264 1932 || 12°894 251 3083 || 12°944 239 0519 | 12°994 227 3932
12°845 263 9291 || 12°895 251 0571 || 12°945 238 8130 || 12°995 227 1659
12°846 263 6653 || 12°896 250 8062 || 12°946 238 5743 || 12°996 226 9389
12°847 263 4018 || 12°897 250 5555 || 12°947 238 3358 || 12°997 226 7120
12°848 263 1385 || 12°898 250 3051 || 12°948 238 0976 || 12°998 226 4854
12°849 262 8755 || 12°899 250 0549 || 12°949 237 8596 || 12°999 226 2591

216 MR F. W. NEWMAN’S TABLE
vw | C=" wv Cae xv Cam xv
13"000 226 0329 || 13°050 215 0092 || 13*100 204 5231 || 13°150
13001 225 8070 || 13°051 214 7943 || I13°101 204 3186 || I3°I51
13002 225 5813 13'052 214 5796 || 13°102 204 1144 || 13°152
13°003 225 3558 || 13°053 214 3651 || 13°103 203 gI04 |) 13153
13004 225 1306 || 13'054 214 1508 || 13°104 203 7066 || 13°154
13°005 224 9956 || 13°055 213 9368 || 13°105 203 5030 || 13°I55
137006 224 6808 || 13°056 213 7230 || 13°106 203 2996 || 13°156
13°007 224 4562 | 13°057 213 5094 || 13-107 203 0964 || 13°157
13°008 224 2319 || 13°058 213 2960 || 13°108 202 8934 || 13158
13009 224 0078 || 13'059 213 0828 |} 13°109 202 6906 || 13°159
13010 | 223 7839 | 13060 212 8698 || 13110 202 4880 || 13160
13011 223 5602 || 13061 212 6570 || 13111 202 2856 || 13°161
13012 223 3367 || 13:062 212 4445 || 13°112 202 0834 || 13162
13013 223 1135 13'063 212 2321 || 13113 201 8815 || 13°163
13'014 222 8905 |] 13°064 212 0200 || 13°114 201 6797 || 137164
13°O15 222 6677 | 13065 211 8081 || 13°15 201 4781 || 13°165
137016 222 4452 || 13°066 21I 5964 || 13°116 201 2767 || 13°166
13°017 222 2228 || 13°067 211 3849 || 13°117 201 0755 || 13°167
13°018 222 0007 || 13068 211 1736 || 13°118 200 8746 || 13°168
13‘019 221 7788 || 13'069 210 9626 || 13119 200 6738 || 13°169
13020 221 5572 || 13°070 210 7517 || 13°r20 200 4732 || 13°70
13°02 22 3357 || 13°07 210 5411 || 13121 200 2729 || 13171
13°022 22I 1145 || 13°072 210 3306 || 13°122 200 0727 || 13°172
13023 220 8935 || 13°073 210 1204 || 13°123 199 8727 || 13°173
13024 220 6727 || 13°074 209 9104 || 13124 199 6729 || 13°174
13°025 220 4521 || 13°075 209 7006 || 13125 199 4734 || 13°I75
13026 220 2318 || 13°076 209 4910 || 13126 199 2740 || 13°176
13027 220 O117 || 13077 209 2816 || 13127 199 0748 || 13177
13'028 219 7918 || 13°078 209 0724 || 13°128 198 8758 || 13178
13'029 219 5721 || 13079 208 8634 || 13°129 198 6771 || 13°179
13030 219 3526 || 13°080 208 6547 || 13°130 198 4785 || 13°180
13031 219 1334 || 13°081 208 4461 || 13°131 198 2801 || 13181
13032 218 9144 || 13°082 208 2378 || 13°132 198 o819 || 13°182
13033 218 6956 || 13°083 208 0296 || 13°133 197 8839 || 13°183
13034 218 4770 || 13°084 207 8217 || 13°134 197 6861 || 13°184
13°035 218 2586 || 13°085 207 6140 || 13°135 197 4886 || 13°185
13'036 218 o405 || 13086 207 4065 || 13°136 197 2912 || 13°186
13'037 217 8225 || 13°087 207 1992 || 13°137 197 ©0940 || 13°187
13038 217 6048 |} 13°088 206 gg2t || 13°138 196 8970 || 13°188
13°039 217 3873 || 137089 206 7852 || 13°139 196 7002 || 13°189
13040 217 1700 || 13°090 206 5785 || 13°140 196 5036 || 13°190
13041 216 9530 || 13'091 206 3721 || 13°14! 196 3072 |} 13191
13°042 216 7361 || 13°092 206 1658 || 13°142 196 1110 || 13°192
13/043 216 5195 || 13°093 205 9597 || 13°%43 195 9150 || 13°193
13°044 216 3031 | 13°094 205 71539 || 13°244 195 7191 || 13°194
13045 216 0869 || 137095 205 5482 || 13°145 195 5235 || 13°195
13046 215 8709 || 13°096 205 3428 || 13°146 195 3281 || 13°196
13°047 215 6552 || 13°097 205 1375 || 13°147 195 1329 || 13°197
13°048 215 4396 || 137098 204 9325 || 13148 194 9378 | 13198
13°049 | 215 2243 | 13°099 204 7277 || 13149 194 743° || 13°199

[13'000—13'199]

188

188

186

[13'200—13'399] OF THE DESCENDING EXPONENTIAL. 217

wv Cae x (Ome a“ Ge wv Dat
| 13°200 185 o6o1 || 13°250 176 0346 || 13°300 167 4493 || 13°35° 159 2827
| 13°201 184 8751 || 137251 175 8587 ||.13°301 167 2819 || 13°351 159 1235
| 13°202 184 6904 || 13°252 175 6829 || 137302 167 1147 || 13°352 158 9645
| 13°203 184 5058 || 13°253 175 5073 || 13°303 166 9477 || 13°353 158 8056
|) #3'204 184 3213 || 13°254 175 3319 |) 13°304 166 7809 || 13°354 158 6469
| I3°205 184 1371 || 13°255 175 15066 || 13°305 166 6142 || 13°355 158 4883
13°206 183 9531 || 13°256 174 9816 || 137306 166 4476 || 13°356 158 3299
13207 183 7692 || 13°257 174 8067 || 13°307 166 2813 || 13°357 158 1716
13208 183 5855 || 13°258 174 6320 || 13°308 166 1151 || 13°358 158 0135
| 13°209 183 4020 |] 13°259 174 4574 || 13°309 165 9488 | 13°359 157 8556
| 13210 183 2187 || 13°260 174 2831 || 13°310 165. 7832 || 13°360 157 6978
| 13211 183 0356 || 13°261 174 1089 || 13°311 165 6175 || 13°36 157 5402
| 13212 182 8527 || 13°262 173 9349 || 13°312 165 4519, || 13°362 157 3827
13°213 182 6699 |) 13°263 173 7610 || 13°313 165 2866 || 13°363 157 2254
13214 182 4873 || 13°264 173 5873 || 13°314 165 1214 || 13°364 157 0682
13°215 182 3049 || 137265 173 4138 || 13°315 164 9563 |} 13°365 156 9113
13216 182 1227 || 13°266 173 2405 || 13°316 164 7914 || 13°366 156 7544
13217 181 9407 || 13°267 173 0673 || 13°317 164 6267 || 13°367 156 5977
181 7588 |} 13°268 172 8944 || 13°318 164. 4622 || 13°368 156 4412
18I 5771 || 13°269 172 7216 || 13°319 164 2978 || 13°369 156 2849
181 3957 || 13°270 172 5489 || 13°320 164 1336 13°370 156 1287
181 2144 || 13°271 172 3765 || 13°321 163 9695 || 13°37! 155 9726
181 0332 || 13°272 172 2042 || 13°322 163 8057 || 13°372 155 8167
180 8523 || 13°273 172 0321 || 13°323 163 6419 || 13°373 155 6610
180 6715 || 13°274 171 8601 |] 137324 163 4784 || 13°374 155 5054
180 4909 || 13°275 171 6883 || 13°325 163 3150 || 13°375 155 3500
180 3105 || 13°276 171 5167 || 13°326 163 1517 || 13°376 155 1947
180 1303 || 13°277 I7I 3453 || 13°327 162 9887 || 13°377 155 0396
179 9503 || 13°278 I7I 1740 || 13°328 162 8258 || 13°378 154 8846
179 7794 || 13279 171 0039 || 13°329 162 6630 || 13°379 154 7298
179 5907 || 13280 170 8320 || 13°330 162 5004 || 13°380 154 5752
: 179 4112 || 13°281 170 6613 || 13°331 162 3380 || 13°381 154 4207
179 2319 || 13°282 170 4907 || 13°332 162 1758 || 13°382 154 2663
179 0528 || 13°283 170 3203 |} 13°333 162 0137 || 13°383 154 1122
178 8738 || 13°284 170 I501 || 13°334 161 8517 || 13°384 153 9581
178 6950 || 13°285 169 9800 || 13°335 161 6900 || 13°385 153 8042
178 5164°|| 13°286 169 81o1 || 13°336 161 5284 || 137386 153 6505
178 3380 || 13'287 169 6404 || 13°337 161 3669 || 13°387 153 4969
178 1597 || 13°288 169 4708 || 13°338 161 2056 || 13°388 153 3435
177 9817 || 13289 169 3014 || 13°339 161 0445 || 13389 153 1903
! 177 8038 || 13'290 169 1322 || 13°340 160 8335 || 13°39° 153 0371
| 177 6261 || 13'291 168 9632 |) 13°34 160 7227 || 13°391 152 8842
177 4485 || 13°292 168 7943 |) 13°342 160 5621 || 13°392 152 7314
177 2712 || 13'293 168 6256 || 13°343 160 4016 || 13°393 152 5788
177 0940 |) 13'294 168 4570 || 13°344 160 2413 || 13°394 152 4262
176 9170 || 13'295 168 2887 || 13°345 160 0812 || 13°395 152 2739
176 7402 || 13°296 168 1205 || 13°346 159 9211 || 13°396 T5212
176 5635 || 13°297 | . 167 9524 || 13°347 159 7613 || 13°397 151 9696
176 3870 || 13°298 167 7845 || 13°348 159 6016 || 13°398 151 8177
176 2107 || 13°299 167 6168 || 13°349 159 4421 || 13°399 151 6660

oon -

218 MR F. W. NEWMAN’S TABLE [13'400—13'599]

v | C-* wv Cae wv Cn” xv Cae
| 13°400 I5I 5144 || 13°450 144 1250 || 137500 137 0959 || 13°550 130 4097
13°401 151 3630 | 13°451 143 9809 || 13°50 136 9589 || 13°551 130 2793
13°402 I5i 2117 || 13°452 143, 8370 |] 13°502 136 8220 || 137552 130 1491
13°403 I51r 0605 -| 13°453 143 6932 || 13°503 136 6852 || 13°553 130 O190
13°404 50 9096 | 13°454 143 5496 | 13°504 136 5486 | 13°554 129 8891
13°405 150 7587 | 13°455 143 4061 13°505 136 4121 || 13°555 129 7592
13°406 150 6080 | 13°456 143 2628 || 13°506 136 2758 || 13°556 129 6295
13°407 150 4575 | 13°457 143 1196 || 13°507 136 1396 || 13°557 129 5000
137408 150 3071 | 13°458 142 9766 | 13°508 136 0035 || 137558 129 3706
13°409 150 1569 | 13°459 142 8337 || 13509 135 8676 || 13°559 129 2413
13°410 150 0068 | 137460 142 6909 || 13°510 135 7318 |] 13°560 129 1121
13411 149 8569 | 137461 142 5483 || 13°511 135 5961 || 13°561 128 9831
13°412 149 7071 | 13°462 142 4058 || 13°512 135 4606 || 13°562 128 8542
13413 149 5575 || 137463 142 2635 || 13°513 135 3252 || 13°563 128 7254
13°414 149 4080 137464 142 1213 || 13°514 135 1899 || 137564 128 5967
13°415 149 2586 || 13°465 I4I 9792 || 13°515 135 0548 || 13°565 128 4682”
13°416 149 1095 || 137466 141 8373 || 137516 134 9198 || 137566 128 3398
13417 148 9604 || 13°467 I4I 6955 |! 13°517 134 7850 |} 13°567 128 2115
13°418 148 8115 || 13°468 I4I 5539 || 137518 134 6502 || 137568 128 0833
13419 148 6628 || 13°469 I4I 4124 |} 13°519 134 5157 || 13°569 I27 9553
13°420 148 5142 || 13°470 I4I 2711 || 137520 134 3812 || 13°570 127 8274
13421 148 3658 || 13°471 I4I 1299 || 13°521 134 2469 || 13°571 127 6996
13422 148 2175 || 13°472 140 9888 |] 137522 134 1127 || 13°572 127 5720
13°423 148 0693 13°473 140 8479 || 13°523 133 9787 || 13°573 127 4445
13°424 147 9213 || 13474 140 7071 || 13°524 133 8448 || 13°574 127 3172
| 13°425 147 7735 || 13°475 140 5665 || 13°525 133 7110 || 13°575 127 1899
13426 147 6258 || 13°476 140 4260 |] 13°526 133 5773 || 13°576 127 0627
13°427 147 4782 || 13°477 149 2856 || 13°527 133 4438 || 13°577 126 9357
13°428 147 3308 13°478 140 1454 |] 13°528 133 3105 || 13°578 126 8089
13429 147 1836 | 73479 149 0054 || 13°529 133 1772 || 13579 126 6821
13°430 147 0365 || 13°480 139 8654 || 137530 133 0441 || 13°580 126 5555
13°431 146 8895 || 137481 139 7256 || 13°531 132 QIII || 13°581 126 4290
13°432 146 7427 || 13°482 139 5860 || 13°532 132 7783 || 137582 126 3026
13°433 146 5960 | 13°483 139 4465 || 13°533 132 6456 || 137583 126 1764
13°434 146 4495 || 13°484 139 3°71 |) 13°534 132 5130 || 13°584 126 0503
13°435 146 3031 || 13°485 139 1678 || 13°535 132 3805 |} 13°585 125 9243
13°436 146 1569 || 13°486 139 0288 || 13'536 132 2482 || 13586 125 7984
13°437 146 o108 || 13°487 138 8898 || 13°537 132 1160 || 137587 125 6727
13°438 143 8649 || 13°488 138 7510 || 13°538 131 9840 |} 13°588 125.5471
13°439 145 7191 || 137489 138 6123 || 13°539 131 8521 || 13°589 125 4216
13°440 145 5734 || 13°49° 138 4737 || 13°540 131 7203 || 13°59° 125 2962
13441 145 4279 || 13°491 138 3353 || 13°541 131 5886 || 13°591 125 1710
13°442 | 145 2826 | 13°492 138 1971 || 137542 131 4572 || 13°592 125 0459
13°443 145 1374 || 13°493 138 0589 || 13°543 13I 3257 || 13°593 124 9209
13°444 144 9923 || 13494 137 9209 || 13°544 131 1945 || 13°594 124 7960
13°445 144 8474 || 13°495 137 7831 || 13°545 131 0633 || 13°595 124 6713
13°446 144 7026 || 13°496 137 6454 || 137546 139 9323 || 137596 124 5467
13°447 144 5580 |} 13°497 137 5078 || 13°547 _ 130 8015 || 13°597 124 4222
13°448 144 4135 || 13°498 137 3704 || 13°548 130 6707 || 13°598 124 2978
13°449 144 2692 | 13°499 137 2331 || 13°49 130 $402 || 13°599 124 1736

[13°600—13°799]

OF THE DESCENDING EXPONENTIAL.

bs e-# oh p e-% x e-#
13°600 124 0495 || 13°650 117 9995 || 13°700 112 2446 |
13601 123 9255 || 13°651 117 8816 || 13°701 II2 1324 |
13602 123 8016 || 13°652 E17 7638 || 137702 II2 0204
13603 123 6779 || 13°653 117 6461 || 13°703 III 9084 |
13°604 123 5543 | 13°654 117 5285 | 13°704 IIL 7965 |
13°605 123 4308 || 13°655 II7 4110 || 13°705 11r 6848
13°606 123 3074 || 13°656 II7 2937 || 13°706 III 5732
13°607 123 1842 | 13°657 117 1764 || 13°707 III 4617
13°608 123 0611 || 13°658 II7 0593 || 13°708 III 3503
13609 122 9381 || 13°659 II6 9423 || 13°709 III 2390 |
13610 122 8152 || 13°660 116 8254 || 13°710 IIL 1278
13611 122 6924 || 13°661 116 7087 || 13°71 III o167
13612 122 5698 || 13°662 116 5920 || 13°712 IIO 9057
13°613 122 4473 || 13°663 116 4755 || 13°713 IIo 7949
13614 122 3249 || 13°664 116 3590 || 13°714 I10 6842
13615 122 2026 || 13'665 116 2427 || 13°715 110 5735
13616 122 0805 || 13°666 116 1266 || 13°716 IIo 4630
13617 I2r 9585 || 13°667 116 o105 || 13°717 IIo 3526
13°618 121 8366 || 13°668 II5 8945 || 13°718 IIO 2423
13619 12x 7148 || 13°669 II5 7787 || 13°719 IIO 1321
13°620 I2I 5932 || 13°670 I1I5 6630 || 13°720 IIO 0220
13621 I2t 4716 || 13°671 TI5 5474 || 13°72 109 gi2I
13°622 121 3502 || 13'672 II5 4319 || 13°722 10g 8022
13623 121 2289 || 13°673 II5 3165 || 13°723 10g 6925
13°624 I2r 1078 || 13°674 II5 2013 || 13°724 109 5828
13625 120 9867 || 13°675 II5 086z || 13°725 109 4733
13626 120 8658 || 13°676 II4 g71i || 13°726 10g 3639
13°627 120 7450 || 13°677 114 8562 || 13°727 10g 2546
13628 120 6243 || 13°678 II4 7414 || 13°728 10g 1454
13629 I20 5037 || 13°679 114 6267 || 13°729 10g 0363
13°630 120 3833 || 13°680 II4 5121 || 13°730 108 9273
13631 120 2630 || 13°681 114 3977 || 13°73 108 8184
13'°632 I20 1428 || 13°682 I14 2833. || 13°732 108 7097
13633 120 0227 || 13°683 114 169t || 13°733 108 6010
13634 I1g 9027 || 13°684 II4 0550 |] 13°734 108 4925
13°635 I1g 7829 || 13°685 II3 9410 || 13°735 108 3840
13°636 I1g 6631 || 13°686 113 8271 || 13°736 108 2757
13°637 II9g 5435 || 13°687 113 7133 || 13°737 108 1675
13638 IIg 4241 || 13°688 113 5997 || 13°738 108 0594
13°639 I19 3047 || 137689 113 4861 || 13°739 TO7 9514
13°640 I1g 1854 || 13°690 113 3727 || 13°740 107 8435
13°641 IIg 0663 || 13°691 I13 2594 || 13°741 107 7357
13642 118 9473 || 13°692 113 1462 || 13°742 107 6280
13°643 118 8284 || 13°693 II3 0331 || 13°743 107 5204
13°644 118 7097 || 13°694 Il2 g2o1 || 13°744 107 4129
13°645 118 5910 || 13°695 112 8073 || 13°745 107 3056
13646 118 4725 || 13°696 II2 6945 || 13°746 107 1983
13°647 118 3541 || 13697 II2 5819 || 13°747 107 ogi2
13648 118 2358 || 13°698 II2 4693 || 13°748 106 9842
13649 118 1176 || 13°699 II2 3569 || 13°749 106 8772

106

106

106
106
106
105
TQ5

105
105
105
105
105

105
105
104
104
104

104
104
104
104
104

104
104
103
103
103

103
103
103
103
103

103
102
102
102
102

102
102
102
102
102

102
Ior
IoL
IoL
Ior

29—2

13°849 96 7065 || 13°899 9 ggor | 13°949 87 5037 || 13°999 83 2361

MR F. W. NEWMAN’S TABLE

e* x ee xv
96 6098 || 13°900 gt 8981 || 13°950
96 5133 || 13°901 gt 8063 || 13°951
96 4168 || 13°902 QI 7145 || 13°952
96 3204 || 13°903 gt 6228 || 13°953
96 2242 || 13°904 9I 5313 || 13°954
96 1280 || 13°905 91 4398 || 13°955
96 0319 || 137906 91 3484 || 13°956
95 9359 || 13°907 QI 2571 || 13°957
95 8400 || 13°908 gt 1659 || 137958
95 7443 || 13°909 gt 0748 || 13°959
95 6486 || 13’910 9° 9837 || 13'960
95 553° || 13911 9° 8928 || 13961
95 4575 || 13912 go S8org || 13°962
95 3620 || 13913 go 7112 || 13°963
95 2667 || 13°914 go 6205 || 13°964
95 1715 || 13°915 9° 5299 || 13°965
95 0764 || 137916 9° 4395 || 137966
94 9814 || 13°917 9° 3491 || 13°967
94 8864 || 13°918 go 2588 || 13°968
94 7916 || 13°919 go 1685 || 13°969
94 6968 || 13920 go 0784 || 13°970
94 6022 || 13°921 89 9884 || 13971
94 5076 || 13°922 89 8984 || 13°972
94 4132 || 13°923 89 8086 || 13°973
94 3188 || 13°924 89 7188 || 13°974
94 2245 |} 13°925 89 6292 || 13°975
94 1304 || 13°926 89 5396 || 13°976
94 0363 || 13°927 89 4501 || 13°977
93 9423 || 13°928 89 3607 || 13°978
93 8484 || 13°929 89 2714 || 13°979
93 7546 || 13°930 89 1821 || 13°980
93 6609 || 13°931 89 0930 || 13°981
93 5673 || 13°932 89 0039 || 137982
93 4737 || 13°933 88 9150 || 137983
93 3803 || 13°934 88 8261 || 13°984
93 2870 || 13°935 88 7373 || 13°985
93 1937 || 13°936 88 6486 || 13°986
93 1006 || 13°937 88 5600 || 13°987
93 0075 || 13°938 88 4715 || 13°988
92 9146 || 13°939 88 3831 || 13°989
92 8217 || 13°940 88 2947 || 13°990
92 7289 || 13°941 88 2065 || 13°991
g2 6362 |: 13°942 88 1183 || 13°992
92 5437 || 13°943 88 0303 || 13°993
92 4512 | 13°944 87 9423 || 13°994
92 3587 | x3:945 87 8544 || 13°995
92 2664 || 13°946 87 7666 || 13°996
g2 1742 13°947 87 6788 || 13°997
g2 0821 | 13'948 87 5912 || 13°998

[13 800—13"999]

e-%

[14°000—14'199]

OF THE DESCENDING EXPONENTIAL.

0974
0184

9394
8605
7817

70°29
6243
5457
4672
3888

3104
2321
1540
0758
9978
9198
8420
7642
6864
6088

5312
4537
3763
2990
2217

1445
0674
9904
9134
8366

7598
6830
6064
5298
4533
3769
3006
2243
1481
0720

9960
9200
8442
7683
6926
6170
5414
4659
3995
3151

|

2398
1646
0895
or44
9395
8646
7897
7150
6403
5657

4912
4167
3423
2680
1938

1197
0456
9716
8976
8235

7500
6763
6026
5291
4556
3821
3088
2355
1623
0892
o16r
9432
8703
7974
7247
6520
5794
5068
4343
3619
2896
2174
1452
0731
oo1o

geou
8572
7854
7136
6419

222 MR F. W. NEWMAN’S TABLE [14'200—14°399]

wee Se ee EE SS
| | |

xz e-z x e-% x“ e-z H £ e-#
| 14°200 68 0798 || 14°250 64 7595 || 14°300 61 6012 || 14°350 58 5968
I4°201 68 orr8 |} 14°251 64 6948 || 14°301 6r 5396 || 14°351 58 5383
14°202 67 9438 || 14'252 64 6301 || 14°302 61 4781 || 14°352 58 4798
14°203 67 8759 || 14°253 64 5655 || 14°303 61 4166 | 14°353 58 4213
14°204 67 8080 || 14°254 64 5010 || 14°304 61 3552 || 14°354 58 3629
14°205 67 7403 || 14255 | 464: 4365 || 14°305 61 2939 || 14°355 58 3046
14°206 67 6725 || 14°256 64 3721 || 14°306 61 2327 || 14°356 58 2463
14°207 67 6049 || 14°257 64 3078 || 14°307 61 1715 || 14°357 58 1881
14°208 67 5373 || 14°258 64 2435 || 147308 61 1103 || 14°358 58 1299
| I4°209 67 4698 || 14°259 64 1793 || 14°309 61 0492 || 14°359 58 0718
14°210 67 4024 || 14°260 64 II51 || 14°310 60 9882 || 14°360 58 0138
I4°21r 67 335° || 14°261 64 0511 || 14°31 60 9272 || 14°361 57 9558
I4°212 67 2677 }| 14°262 63 9870 || 14°312 60 8663 || 14°362 57 8979
14°213 67 2005 || 14°263 63 9231 || 14:313 60 8055 || 14°363 57 8400
14°214 67 1333 || 14°264 63 8592 || 14°314 60 7447 || 14°364 57 7822
14°215 |. 67 0662 || 14°265 63 7954 || 14°315 60 6840 || 14°365 57 7244
14°216 66 9992 || 14°266 63 7316 || 14°316 60 6233 || 14°366 57 6667
14°217 66 9322 || 14°267 63 6679 || 14°317 60 5627 || 14°367 57 6091
14°218 66 8653 || 14°268 63 6043 || 14°318 60 5022 || 14°368 57-5515
14°219 66 7985 || 14°269 - 63 5407 || 14°319 60 4418 || 14°369 57 4940
14°220 66 7317 || 14:270 63 4772 || 14°320 60 3814 || 14°370 57 4365
I4°221 66 6650 || 14°271 63 4137 || 14°321 - 60 3210 |} 14°371 57 3791
14°222 66 5984 || 14°272 63 3504 || 14°322 60 2607 || 14°372 57 3218
14°22 66 5318 || 14°273 63 2870 || 14°323 60 2005 || 14°373 57 2645
14°224 66 4653 || 14°274 63 2238 || 14°324 60 1403 || 14°374 57 2072
14°22 66 3989 || 14°275 63 1606 || 14°325 60 0802 || 14°375 57 1501
14°226 66 3325 | 14°276 63 0975 || 14°326 60 0202 || 14°376 57 0929
14°22 66 2662 | 14°277 63 0344 || 14°327 59 9602 || 14°377 57 9359
14°228 66 2000 || 14°278 62 9714 || 14°328 59 9002 || 14°378 56 9789
14°229 66 1338 | 14°279 62 9085 || 14°329 59 8404 || 14°379 56 9219
14°230 66 0677 || 14°280 62 8456 || 14°330 59 7806 || 147380 56 8650
14°231 66 oo17 || 14°281 62 7828 || 14°33 59 7208 || 14°381 56 8082
14°232 65 9357 || 14°282 62 7200 || 14°332 59 6611 ‘|| 147382 56 7514
14°233 65 8698 | 14°283 62 6573 || 14°333 59 6015 || 14°383 56 6947
14°234 65 8040 | 14°284 62 5947 || 14°334 59 5419 | 14°384 56 6380
14°235 65 7382 || 14°285 62 5321 || 14°335 59 4824 || 14°385 56 5814
14°236 65 6725 || 14°'286 62 4696 || 14°336 59 4230 || 14°386 56 5249
14°237 65 6069 || 14°287 62 4072 || 14°337 59 3636 || 14°387 56 4684
14°238 65 5413 || 14'288 62 3448 || 14°338 59 3042 || 14°388 56 4119
14°239 65 4758 | 14°289 62 2825 || 14°339 59 2450 || 14°389 56 3555
14°240 65 4104 || 14°290 62 2203 || 14°340 59 1857 || 14°390 56 2992
14°241 65 345° || 14°291 62 1581 || 14°341 59 1266 || 14°391 56 2429
14°242 65 2797 || 14°292 * 62 0959 || 14°342 59 0675 || 14°392 56 1867
14°243 65 2144 || 14°293 62 0339 || 14°343 59 0085 || 14°393 56 1306
14°244 65 1492 || 14°294 61 9719 || 14°344 58 9495 || 14°394 56 0745
14°245 65 0841 || 14°295 61 9099 || 14°345 58 8906 || 14°395 56 0184
14°246 65 o1gt | 14°296 61 8480 || 14°346 58. 8317 || 14°396 55 9624
14°247 64 9541 || 14297 61 7862 || 14°347 58 7729 |: 14°397 55 9065
14°248 64 8892 || 14°298 61 7245 || 14°348 58 7141 || 14°398 55 8506
| 14°249 64 8243 | 14°299 61 6628 || 14°349 58 6555 || 14°399 55 7948

[14'400—14'599]

OF THE DESCENDING EXPONENTIAL.

14°550
14°551

| 14°552

14'553
14°554

EaSao
14°556
ETO
14°558
14°559
14°560
14561
14'562
14563
14°564
14°565
14°566
14°567
14°568
14°569

14°570
14°571
14°572
14°573
14°574

14°575
14°576
14°577
14°578
14°579
14°580
14°58r
14'582
14583
14°54

| 14°585

14°586
14°587

| 147588

14°589

| 14°590
14°591

14°592

| 14°593

14°594
14°595

| 14°596

14°597
14°598
14°599

MR F. W. NEWMAN'S TABLE

14°650

| 14°651

14°652
14°053
14°654
14°655
14°656
14°657
14°658
14°659
14°660
14°661
14°662

14663

14°664

14°665
14°666
14°667
14'668
14°669

14°670
14°671
14°672
14°673
14674
14°675
14°676
14°677
14°678
14°679
14°680
14°681
14°682
14°683
14°684

14°685
14°686
14°687
14°688
14°689

14°690
14°691
14°692
14°693
14°694
14°695
14'696
14°697
14°698
14°699

4096
3662
3229
2796
2363

1931
1499
1068

0637
0207

9777

9347
8918

8489 ||

8061

7633
7206
6779
6352
5926

5500
5°75
4650
4226
3802

3378
2955
2532
2110
1688

1266
0845
0425
0004
9585
9165
8746
8328
7910
7492
7°75
6658
6241
5825
5410
4995
4580
4166
3752
3338

e-%

[14°600—14'799]

[14800 —14'999] OF THE DESCENDING EXPONENTIAL. 225

14800 37 3630 || 14°850 35 5408 || 14°900 33 8074 | 14'950 32 1586
14801 37 3256 |) 14°851 35 5052 || 14°901 33 7736 || 14°95! 2 1265
14°802 37 2883 || 14°852 35 4698 || 14°902 33 7399 || 14°952 32 0944
14°803 37 2511 | 14°853 35 4343 || 14°903 33 7062 || 14953 32 0623
14°804 37 2138 14°854 35 3989 || 14°904 33 6725 || 14°954 32 0302
14°805 37 1766 ||, 34°855 35 3635 || 14°905 33 6388 || 14°955 31 9982
14°806 37 1395 || 14°856 35 3282 || 14°906 33 6052 || 14°956 31 9662
14°807 37 1024 || 14°857 356-9298 BOO 33 5716 || 14°957 31 9343
14°808 37 0653 || 14°858 35 2576 || 14°908 33 5380 || 14°958 31 9024
14 809 37 0282 || 14°859 35 2223 || 14°909 33 5045 || 14°959 31 8705
14°810 36 9912 || 14°860 35 1871 || 14’910 33 4710 || 14°960 31 8386
14°81 36 9542 || 14°861 35 1520 || 14°911 33 4376 || 14°961 31 8068
14°812 36 9173 || 14°862 35 1168 || 14:912 33 4042 || 147962 SE TUS
14°813 36 8804 || 14°863 35 0817 || 14°913 33 3708 || 14°963 31 7433
14814 36 8435 || 14°864 35 0467 || 14914 33 3374 || 147964 3 7115
14°815 - 36 8067 |) 14°865 35 0116 || 14°915 333041 || 14°965 31 6798
14°816 36 7699 || 14°866 34 9766 || 14°916 33 2708 || 14°966 31 6482
14°817 36 7332 || 14°867 34 9417 || 14°917 33 2376 || 14°967 31 6165
14°818 36 6965 || 14°868 34 9068 || 14918 33 2043 || 147968 31 5849
14°819 36 6598 || 14°869 34 8719 || 14919 33 1712 || 14°969 31 5534
14°820 36 6231 || 14°870 34 8370 || 14°920 33 1380 || 14°970 31 5218
14°821 36 5865 || 14°871 34 8022 || 14'921 33 1049 || 14°971 31 4993
14°822 36 5500 || 14°872 34 7674 || 14°922 33 0718 || 14°972 31 4589
14°823 36 5134 || 14°873 34 7327 || 14°923 33 0387 || 14°973 Se Ceti
14824 36 4769 || 14°874 34 6979 || 14°924 33 0957 || 14°974 31 3960
14°825 36 4405 || 14°875 34 6633 || 14°925 32 9727 || 14°975 31 3646
14°826 36 4041 || 14°876 34 6286 || 147926 32 9398 || 14°976 31 3333
14827 36 3677 || 14°877 34 5949 || 14°927 32 9068 || 14°977 amy s029
14°828 36 3313 || 14°878 34 5594 || 14°928 32 8739 || 14°978 31 2707
14°829 36 2950 || 14°879 34 5249 || 14°929 32 8411 || 14°979 SE Bae
14'830 36 2587 || 14°880 34 4904 || 14°930 32 8083 || 14°980 31 2082
14°831 36 2225 || 14°881 34 4559 || 14°931 2 7755 || 14°981 She LH)
14°832 36 1863 || 14°882 34 4215 || 14°032 2 7427 || 14°982 3r 1458
14°833 36 1501 || 14°883 34 3871 || 14°933 2 7100 || 14°983 SRE TAY
14°834 36 1140 || 14°884 34 3527 || 14°934 32 6773 || 14°984 31 0836
14°835 36 0779 || 14°885 34 3184 |) 14°935 32 6446 |) 14°985 31 0525
14°836 36 0418 || 14°886 34 2841 || 14°936 32 6120 || 14°986 3I 0215
14°837 36 0058 || 14°887 34 2498 || 14°937 32 5794 || 14987 3° 99°95
14°838 35 9698 || 14°888 . 34 2156 || 14°938 32 5468 || 147988 3° 9595
14°839 35 9339 || 14°889 34 1814 || 14°939 32 5143 || 14°989 go 9286
14°840 35 8980 || 14°890 34 1472 || 14°940 2 4818 || 147990 30 8977
14°841 35 8621 || 14°81 34 1131 || 14°941 32 4493 || 14°991 30 8668
14°842 35 8262 || 14°892 34 0799 14942 2 4169 |} 14°992 8359
14'843 35 7904 || 14°893 34 0449 || 14°943 32 3845 |) 14°993
14°844 35 7547 || 14°894 34 0109 || 14°944 32 3521 || 14°994
14°845 35 7189 || 14°895 33 9769 || 14°945 32 3198 |) 14°995
14°846 35 6832 |) 14°S96 33 9429 || 14°946 32 2875 || 14°996
14°847 35 6476 || 14°897 33 9099 || 14°947 2 2552 || 14°997
14°848 35 6119 || 14°898 33 8751 || 14°948 32 2230 || 14°998
14°849 35 5763 || 14°899 33 8412 || 14°949 32 1907 || 14°999

Wem, UDR (eae JOR

226 MR F. W. NEWMAN'S TABLE [15:000—15'199]

x | e-* | & e* x“ Ce x er
| 15000 30 5902 || 15"050 29 0983 || I5"100 27 6792 || 15°150 26 3292
15001 30 5596 || 15°051 29 0692 || T5101 27 6515 || 15°15 26 3029
15"002 30 529% || 15'052 29 0402 || I5"102 27 6239 || 15152 26 2766
15°003 30 4986 || 157053 29 O112 | I5"103 27 5963 | 15153 26 2504
| 15°004 3° 4680 | 15°054 28 9822 || 15°104 27 5087 || 15°154 26 2241
15'005 30 4376 || 15°055 28 9532 || 15°105 27 5411 || 15°155 26 1979
15"006 30 4072 || 15°056 28 9243 || 15°106 27 5136 |} 15°156 26 1717,
| 15°007 30 3768 || 15°057 28 8954 || 15°107 27 4861 |} 15°157 26 1456
| 15°008 30 3465 || 15°058 28 8665 || 15°108 27 4586 || 15158 26 1194
15009 30 3161 || 15°059 28 8376 15109 27 4312 || 15°159 26 0933
| 15"0ro 30 2858 || 15°060 28 8088 || 15"110 27 4038 || 15'160 26 0673
I5‘Orr 30 2556 || 15°061 28 7800 || I5‘III 27 3764 || 15161 26 0412
15012 30 2253 || 15062 28 7512 || 15112 27 3490 || 15°162 26 o152
15°013 30 1951 || 15°063 28 7225 || 15°113 27 3217 || 15°163 25 9892
15014 30 1649 || 15°064 28 6938 || 15114 27 2944 || 15°164 25 9632
15"O1s 30 1348 || 15-065 28 6651 || 15115 27 2671 || 15°165 25 9373
15016 30 1047 || 15°066 28 6365 |} 15116 27 2398 || 15°166 25 QII3
15‘O17 30 0746 || 15°067 28 6078 |} 15117 27 2126 || 15°167 25 8854
|} 157018 30 0445 || 157068 28 5792 || 15118 27 1854 || 15°168 25 8596
15"019 30 0145 || 15°069 28 5507 || 15119 27 1582 || 15°169 25 8337
15"020 29 9845 || 15070 28 5221 || 15°120 27 Sts || eS en oO 25 8079
15021 29 9545 || 15'071 28 4936 || 15121 27 1040 |] 15°r71 25 7821
15°022 29 9246 || 15072 28 4652 || 15122 27 0769 || 15°172 25 7563
15°023 29 8947 |] 15°073 28 4367 || 15123 27 0498 || 15°173 25 7306
15°024 29 8648 || 15°074 28 4083 || 15°124 27 0228 || 15°174 25 7049
15'025 29 8349 || 15°075 28 3799 || 15°125 26 9958 || 15°175 25 6792
15026 29 8051 || 15°076 4 28 3515 || 157126 26 9688 || 15°176 25 6535
15'027 29 7753 || 15°°77 28 3232 || 15127 26 9418 |} 15°177 25 6279
| 15°028 29 7456 || 15°078 28 2949 |} 15°128 26 9149 || 15°178 25 6023
| 15°029 29 7158 || 15°079 28 2666 || 15129 26 8880 |} 15°179 25 5767
15030 29 6861 || 15080 28 2383 || 15°130 26 8611 || 15°180 25 5511
15031 29 6565 || 15°081 28 2101 || 15°131 26 8343 || 15181 25 5256
15032 29 6268 || 15°082 28 1819 || 15°132 26 8075 |} 15°182 25 5000
15°033 29 5972 || 15°083 28 1538 || 15°133 26 7807 || 15°183 25 4746
15034 29 5676 | 157084 28 1256 || 15°134 26 7539 || 15184 25 4491
157035 29 5381 || 157085 28 0975 || 15°135 26 7272 || 15°185 25 4237
15036 29 5086 | 15°086 28 0694 || 15°136 26 7004 || 15°186 25 3983
15037 29 4791 || 15°087 28 0414 |} 15137 26 6738 || 15°187 25 3729
15°038 29 4496 | 157088 28 0133 || 15°138 _ 26 6471 || 15°188 25 3475
15°039 29 4202 | 15°089 27 9853 15139 26 6205 || 15"189 25 3222
15040 29 3908 || 15"090 27 9574 || 15"140 26 5939 || 15"190 25 2969
15041 29 3614 | 15°091 27 9294 || 15°141 26 5673 || 15‘19r 25 2716
15'042 29 3320 || 15°092 27 goIs || 15°142 26 5407 || 15"192 25 2463
15°043 29 3027 || 15°093 27 8736 || 15°143 26 5142 || 15193 | ° 25 2211
15°044 29 2734 || 15°094 27 8458 || 15°144 26 4877 || 15°194 25 1959
15°045 2y 2442 || 15°095 27 8179 || 15°145 26 4612 || 15°195 25 1707
15'046 29 2150 || 15'096 27 790% || 15°146 26 4348 || 15°196 25 1455
15'047 29 1858 || 15°097 27 7624 || 15°147 26 4083 || 15°197 25 1204
15048 29 1566 || 15098 27 7346 || 15°148 26 3820 || 15198 25 0955
| 15°049 29 1274 15099 27 7069 || 15°149 26 3556 || 15°199 25 0704

| i
Vance. h a SS ee et

[15'200—15'349 | OF THE DESCENDING EXPONENTIAL. 227

xv Cae av C-% av Cat x Cm
15200 25 0452 || 15'240 24 0631 || 15°280 23 1196 || 15°320 22 2131
15°201 25 0201 || 15°241 24 0391 |] 15281 23 0965 || 15°321 22 1909
15°202 24 9951 || 15242 24 0150 || 15°282 23 0734 || 15°322 22 1687
15°203 24 9701 || 15°243 23 9910 || 15°283 23 0503 || 15°323 22 1465
15204 24 9452 || 15°244 23 9671 || 15°284 23 0273 || 15°324 22 1244
15°205 24 9202 || 15°245 23 9431 || 15°285 23 0043 || 15°325 22 1023
15206 24 8953 || 15°246 23 g1g2 || 15°286 22 9813 || 157326 22 0802
I5'207 24 8704 || 15°247 23 8953 || 15°287 22 9583 |) 15°327 22 0581
15208 24 8456 || 15°248 23 8714 || 15°288 22 9354 || 15°328 22 0361
15209 24 8208 || 15°249 23 8475 || 15°289 22 9124 || 15°329 22 O140
15‘210 24 7960 || 15°250 23 8237 || 15°290 22 8895 || 15°330 2I 9920
15211 24 7712 || 15°25 23 8000 || 15°291 22 8667 || 15°331 2I g701
15212 24 7464 || 15°252 23 7761 || 15°292 22 8438 || 15°332 21 9481
15'213 24 7217 || 15°253 23 7523 || 15°293 22 8210 || 15°333 “21 g262
15°214 24 6970 || 15°254 23 7286 || 15°294 22 7982 || 15°334 2I go42
15215 24 6723 || 15°255 23 7049 || 15°295 22 7754 || 15°335 21 8823
15216 24 6476 || 15°256 23 6812 || 15:296 22 7526 |) 15°336 21 8605
15°217 24 6229 || 15°257 23 6575 || 15°207 22 7299 || 15°337 21 8386
15218 24 5984 || 15°258 23 6339 || 15°298 22 7072 || 15°338 21 8168
T5219 24 5738 || 15°259 23 6102 || 15°299 22 6845 || 15°339 207,959
15'220 24 5492 || 15°260 23 5866 || 15°300 22 6618 || 15°340 PS ay
15°221 24 5247 || 15°26 23 5631 || 15°301 22 6391 || 15°341 2I 7514
15°222 24 5002 ||) 15°262 23,5395) ||) L502 22 6165 || 15°342 21 7297
15°223 24 4757 || 15°203 23 5160 || 15°303 22 5939 || 15343 21 7080
15°224 24 4512 || 15°264 23 4925 || 15304 22 5713 || 15°344 21 6863
15°225 24 4268 || 15°265 23° 4690 || 15°305 22 5488 || 15°345 21 6646
15°226 24 4024 || 15°266 23 4455 || 15°300 22 5262 || 15°346 21 6430
15°227 24 3780 || 15°267 23 4221 || 15°307 22 5037 || 15°347 21 6213
15°228 24 3536 || 15°268 23 3987 || 15°308 22 4812 || 15°348 21 5997
15°229 24 3293 || 15°269 23 3753 || 15309 22 4587 | 15°349 21 5781
15°230 24 3050 || 15°270 23 3519 || 15°310 22 4363
15231 24 2807 || 15'271 23 3286 || 15°311 22 4139
15'232 24 2564 || 15°272 23 3053 || 15°312 22 3915
15'233 24 2322 || 15°273 23) 2620) ||| Lh-3r3 22 3691
15'234 24 2079 || 15°274 23 2587 || 15°314 22 3467
15°235 24 1837 || 15°275 23 2355 || 15°315 22 3244
15'2306 24 1596 || 15°276 23 2123 || 15°316 22 3021
15°237 24 1354 || 15°27 23 1891 || 15°317 22 2798
15°238 24 1113 || 15°278 23 1659 || 15°318 22 2575
15°239 24 0872 || 15°279 23 1427 || 15°319 22 2353 |

30—2

228 MR F. W. NEWMAN’S TABLE
[Second Part. Fourteen decimal places.)
:
x | e-= se | e-* | 2 e-x
ne |
| |
| 15°35° 2155 6572 || 15°45° 1950 5193 || 15°55° 1764
157352 2151 3502 || 157452 1946 6222 || 15°552 1761
15°354 2147 0518 || 15°454 1942 7328 || 15°554 1757
15°356 2142 7620 || 15°456 1938 8512 |] 15°556 1754
15358 2138 4808 | 15°458 1934 9774 || 15°558 1750
15°360 2134 2081 || 15°460 1931 1113 |} 15°560 1747
15°362 2129 9439 | 157462 1927 2530 |] 157562 1743
| 157364 | 2125 6882 || 15-464 1923 4023 |} 15°564 1740
15°366 2121 4411 || 15°466 T9QIg 5593 || 15'°566 1736
2117 2025 || 15°468 I9I5 7241 |} 15°568 1733

9028
3765
8573
3451
8399

3416
8504
3661
8890
4187

[5°350—15'748]

v
15650
15°652
15°654
15°656
15°658
15°660 1581 0602
15662 1577 gor2
15°664 1574 7485
15'666 1571 6021
15°668 1568 4620

[15°750—16'148] OF THE DESCENDING EXPONENTIAL. 929

[Second Part. Fourteen decimal places.|

Cae | @ e-%

1183 0498 | 16050 1070 4677
1180 6861 || 16°052 1068 3289
1178 3270 || 16°054 1066 1944
1175 9727 || 16°056 1064 0641
1173 6231 |] 16:058 1061 9381
1171 2781 || 16°060 1059 8164
1168 9379 || 16:062 1057 6989
1166 6023 || 16°064 1055 5855
1164 2714 || 16°066 1053 4765
I161 9452 || 16°068 IO5I 3716
T159 6234 || 16°070 1049 2710
1157 3065 || 16:072 1047 1746
1154 9942 || 16°074 1045 0823
1152 6865 || 16:076 1042 9943
I150 3835 || 16:078 1040 QI03
1148 o8sr || 16°080 1038 8306
1145 7913 || 16082 1036 7550
1143 5019 || 16°084 1034 6836
II4r 2172 || 16°086 1032 6164
1138 9370 || 167088 "1030 5532
1136 6615 || 16090 1028 4941
1134 3905 || 16092 1026 4391
1132 1240 |) 16°094 1024 3883
1129 8620 || 16°096 1022 3417
1127 6045 || 16°098 1020 2990
1125 3517 || 16°100 1018 2604
1123 1032 || 16"102 Io16 2259
1120 8593 || 16104 IOI4 1955
1118 6198 || 16°106 IoI2 1691
1116 3848 |} 16°108 1010 1468
III4 1543 || 16110 1008 1285
IIII 9282 |} 16112 1006 1143
I10g 7066 || 16°114 I004 IO41
1107 4894 || 16°116 I002 0979
1105 2766 || 16118 I000 0957
1103 0683 || 16°r20 998 0975
1100 8644 || 16°122 996 1033
1098 6647 || 16124 994 I131
1096 4696 || 16°126 992 1268
1094 2789 || 16°128 990 1446
1092 0926 || 16°130 988 1662
1089 g106 || 16132 986 1919
1087. 7330 || 16°134 984 2215
1085 5596 || 16136 982 2551
1083 3907 || 16°138 g80 2925
re8r 2261 || 16°140 978 3338
1079 0658 || 16°142 976 3791
1076 9097 || 16144 974 4283
1074 7581 || 16146 972 4814
1072 6107 || 16148 97° 5383

230 MR F. W. NEWMAN’S TABLE [16:150—16'548]

[Second Part. Fourteen decimal places.|

| |
Sel see | ie ls nese x e-% x cn ye
| |

| 16°150 968 5992 || 16°250 876 4248 || 16°350 793 0220 || 167450 717 5559
16°152 966 6639 | 16°252 874 6737 || 16°352 “791 4376 || 16°452 716 1223

| 16°154 964 7325 16°254 872 9261 || 16°354 789 8563 || 16°454 714 6915
| 16°156 962 8050 || 16°256 871 1820 || 16°356 788 2781 || 16°456 713 2635
| 16-158 960 8813 || 16258 869 4414 | 16°358 786 7032 || 16°458 711 8384
| 16°160 958 9615 | 16-260 867 7042 || 16°360 785 1313 || 16°460 710 4161
16°162 957 0455 || 16°262 865 9706 || 16°362 783 5626 || 16°462 708 9968
| 16164 | 955 1333 || 16-264 864 2403 | 16364 781 9971 || 16°464 707 5802
| 16°166 953 2249 || 16°266 862 5136 | 16°366 780 4346 | 16°466 706 1665
16168 951 3204 || 16°268 860 7903 | 16°368 778 8753 || 16468 704 7555
16°170 949 4197 | 16°270 859 o704 | 16°370 777 319% || 16°470 (5) ht);
16172 947 5228 || 16272 857 3540 || 16°372 775 7660 |, 16°472 7OI 9422
16°174 945 6296 || 16°274 855 6410 || 16°374 774 2160 || 16°474 700 5397
16°176 943 7403 || 16°276 853 9314 || 16-376 772 6692 || 167476 699 1400
16178 941 8546 | 16°278 852 2253 16°378 771 1253 || 16°478 697 7431
16°180 939 9728 | 16°280 850 5225 || 167380 769 5846 || 16°480 696 3490
16182 938 0947 || 16°282 848 8232 || 167382 768 0470 || 16°482 694 9577

| 16°184 936 2204 || 16°284 847 1273 || 16°384 766 5124 || 167484 693 5692
| 16°186 934 3498 || 16°286 845 4347 || 16°386 764 980g || 16°486 692 1834
| 16°188 | 932 4830 || 16°288 843 7455 || 16°388 763 4525 || 16°488 690 8004
| 16"1g0 930 6199 | 16°290 B42 0597 || 16°390 761 9271 || 16°490 689 4202
16°1g92 928 7605 || 16-292 840 3773 || 16°392 760 4048 || 16°492 688 0427
16°194 926 9049 || 16°294 838 6982 || 16°394 758 8855 || 16°494 686 6680
16°196 925 0529 || 16°296 837 0225 || 16°396 757 3693 || 16°496 685 2960
16°198 923 2046 || 16°298 835 3501 || 16°398 755 8560 || 16°498 683 9268
16°200 g21 3601 || 16-300 833 6811 || 16°400 754 3458 || 16°500 682 5603
16°202 919 5192 || 16°302 832 0154 || 16°402 752 8386 || 16°502 681 1966
16°204 917 6820 || 16°304 830 353° || 16°404 751 3345 |) 16°504 679 8355
16°206 915 8485 || 16°306 828 6940 || 16°406 749 8233 || 16°506 678 4772
16°208 914 0186 || 16-308 827 0382 || 16°408 748 3352 || 16°508 677 1216
16°210 gt2 1924 || 16-310 825 3858 || 16:410 746 8400 || 16°51r0 675 7687
| 16°212 gio 3698 || 16-312 823 7367 || 16-412 745 3478 || 167512 674 4186
| 16°214 g08 5509 || 16°314 822 0909 || 16°414 743 8586 || 16°514 673 O711
16°216 9°26 7356 || 16°316 820 4484 || 16°416 742 3724 || 16°516 671 7263
| 16°218 904 9240 || 16°318 818 8091 || 16°418 740 8891 || 167518 670 3842
16°220 903 1159 || 16°320 817 1731 || 16°420 739 4088 || 16°520 669 0447

| 16°222 QOI 3115 || 167322 815 5404 || 16°422 737 9315 || 167522 667 7080
16°224 899 5107 || 16°324 813 gIIo || 16°424 736 4571 || 16°524 666 3739

| 16°226 897 7135 16°326 812 2848 || 16°426 734 9857 || 16°526 665 0425
16°228 895 9199 || 16°328 810 6618 || 16°428 733 5172 || 16528 663 7137
16°230 894 1298 | 16°330 809 0421 || 16°430 732 0516 || 16°530 662 3876
16°232 892 3433 || 16°332 807 4257 | 16°432 730 5890 || 16°532 661 0642
16°234 890 5604 || 16°334 805 8124 | 16°434 729 1292 || 16°534 659 7434
16°236 888 7811 || 16°336 804 2024 || 16°436 727 6724 || 167536 658 4252
16°238 887 0053 || 16°338 802 5956 || 167438 726 2185 || 167538 657 1097
16°240 885 2330 || 16°340 800 9920 || 16°440 724 7675 || 16°540 655 7968

| 16-242 883 4644 || 16°342 799 3917 || 16°442 723 3195 || 16°542 654 4865
| 16°244 881 6992 || 16°344 797 7945 || 16°444 721 8743 || 16°544 653 1788
16°246 879 9376 || 16°346 796 2005 || 16°446 720 4320 || 16°546 651 8738
| 16°248 | 878 1795 || 16°348 794 6097 || 16°448 718 9925 || 167548 650 5713

[16°550—16'948]

«

OF THE DESCENDING EXPONENTIAL.

[Second Part.

Fourteen decimal places.|

v Gate “x (Dae a“ Oe x CR”
16°550 649 2715 || 16°650 587 4851 || 16°750 531 5785 || 16850 480 9921
16°552 647 9742 || 16°652 586 3113 || 16°752 530 5164 || 16°852 480 0311
16°554 646 6796 | 16°654 585 1399 || 16°754 529 4564 || 16°854 479 0720
16°556 645 3875 || 16°656 583 9707 || 16°756 528 3986 || 16°856 478 1149
16°558 644 0980 || 16°658 582 8040 || 16°758 527 3429 || 16°858 477 1596
16'560 642 8111 || 16°660 581 6395 || 16°760 526 2892 || 16°860 476 2062
16°562 641 5268 || 16°662 580 4774 || 16°762 525 2377 || 16°862 475 2547
16'564 640 2450 || 16°664 579 3176 || 16°764 524 1883 || 16°864 474 3052
16566 638 9658 |, 16°666 578 1601 || 16°766 523 1410 || 16°366 473 3575
16°568 637 6892 || 16°668 577 0050 || 16°768 522 0957 || 16°368 472 4118
16'570 636 4150 || 16°670 575 8521 || 16°770 521 0526 || 16°870 471 4679
16°572 635 1435 || 16°672 574 7016 || 16°772 520 C115 || 16°872 470 5259
16°574 633 8745 || 16°674 573 5533 || 16°774 518 9726 || 16°874 469 5858
16'576 632 6080 || 16°676 572 4074 || 16°776 517 9356 || 16°876 468 6476
16°578 631 3440 || 16°678 571 2637 || 16°778 516 go008 || 16°878 467 7112
16°580 630 0826 || 16°680 570 1223 || 16°780 515 8680 || 16°880 466 7767
16°582 628 8237 || 16°682 568 9833 || 16°782 514 8373 || 16°882 465 8441
16°54 627 5673 || 16°684 567 8464 || 16°784 513 8087 || 16°884 464 9133
16°586 626 3134 || 16°686 566 7118 || 16°786 512 7821 || 16°886 463 9844
16'588 625 o621 || 16°688 565 5795 || 16°788 5IL 7575 || 16°888 463 0574
16'590 623 8132 || 16°690 564 4495 || 16°790 510 7350 || 16°890 462 1322
16592 622 5668 || 16692 563 3227 || 16°792 509 7146 || 16°892 461 2088
16°594 621 3229 || 16°694 562 1962 || 16°794 508 6962 || 16°894 460 2873
16°596 620 0815 || 16°696 561 0729 || 16°796 507 6798 || 16°896 459 3677
16°598 618 8426 || 16-698 559 9519 |) 16°798 506 6654 || 16°898 458 4499
16°600 617 6061 || 16°700 558 8331 .|| 16°800 505 6531 || 16900 457 5339
16602 616 3722 || 16°702 557 7106 || 16°802 504 6428 || 16°902 456 6197
16604 615 1406 || 16°704 556 6023 || 16°804 503 6346 || 16°904 455 7°74
16°606 613 9116 || 16°706 555 4902 || 16°806 502 6283 || 16:906 454 7969
16°608 612 6850 || 16°708 554 3803 || 16°08 501 6241 || 16°908 453 8882
16'610 611 4609 || 16°710 553 2726 || 16°810 500 6218 || 16910 452 9813
16°612 610 2392 || 167712 552 1672 || 16°812 499 6215 || 16:912 452 0763
16°614 609 o199 || 16°714 551 0640 || 16°814 498 6233 || 16:914 451 1730
16°616 607 8031 || 16°716 549 9630 || 16°816 497 6271 || 16°916 450 2716
16°618 606 5887 || 16°718 548 8641 || 16°818 496 6328 || 16:918 449 3719
16°620 605 3767 || 16°720 547 7675 || 16°820 495 6405 || 16°920 448 4741
16°622 604 1672 || 16°722 546 6731 || 16°822 494 6502 || 16°922 447 5780
16°624 602 g6or || 16°724 545 5808 || 16°824 493 6619 || 16:924 446 6838
16°626 601 7553 || 16°726 544 4907 || 16°826 492 6756 || 16°926 445 7913
16°628 600 5530 || 16°728 543 4028 || 16°828 491 6912 || 167928 444 9006
16°630 599 3531 || 16°730 542 3171 || 16°830 49° 7088 || 16°930 444 O117
16°632 598 1556 || 16°732 541 2336 || 16°832 489 7284 || 167932 443 1246
16°634 596 9605 || 16°734 540 1522 || 16°834 488 7499 || 16°934 442 2392
16°636 595 7678 || 16°736 539 0730 || 16°836 487 7734 || 16°936 441 3556
16°638 594 5775 || 16738 537 9959 |) 16°838 486 7988 |, 16°938 440 4738
16°640 593 3895 || 16°740 536 9210 |) 16-840 485 8262 || 16-940 439 5937
16°642 592 2039 || 16°742 535 8482 || 16842 484 8555 || 16942 438 7154
16°644 591 0207 || 16°744 534 7776 || 16°844 483 8868 || 16°944 437 5389
16°646 589 8398 || 16°746 533 7091 || 16°846 482 9200 || 16°946 436 9641
16°648 588 6613 || 16°748 532 6428 || 16°848 481 9551 || 16°948 436 ogto

232 MR F. W. NEWMAN’S TABLE [16:950—17'420]

[Second Part. Fourteen decimal places.]

xv e-* xv Ca wv (Dats xv Cae
16°950 435 2197 | 17°050 393 8030 || 17°150 356 3277 || 17°25° 322 4187
16°952 434 3501 | 17°052 393 0162 |] 17°152 355 6158 || 17°252 321 7745

| 16954 433 4823 | 177054 392 2320 || 17154 354 9053 || 17°254 S20 eS r5
| 167956 432 6162 | 177056 391 4473 || 17°156 354 1962 || 17:256 320 4899
| 16°958 431 7518 | 17°058 399 6652 | 17°158 353 4885 |) 17°258 319 8496
16"960 430 8892 || 17°060 389 8846 || 17°160 352 7822 || 17°260 319 2105
16°962 430 0283 || 17°062 389 1056 || 17°162 352 0774 || 17°262 318 5728
16'964 429 1691 | 17°064 388 3282 || 17164 351 3739 || 17°264 317 9363
16°966 28 3116 |, 17°066 387 5523 || 17°166 350 6719 || 17°266 317 3010
| 167968 427 4558 || 17°068 386 7780 || 17°168 349 9712 || 17°268 316 6670
| 16°970 426 6018 || 17°070 386 0052 || 17°170 349 2720 || 17:270 316 0343
| 16°972 425 7494 || 17°072 385 2340 || 17°172 348 5741 || 17°272 315 4029
| 16°974 424 8988 | 17°074 384 4643 || 17°174 347 8777 || 17°274 314 7727
16°976 424 0498 || 177076 383 6961 || 17°176 347 1826 || 17°276 344 1438
16°978 423 2026 | 17°078 382 9295 || 17°178 346 4889 || 17°278 313 5162
16°980 422 3570 || 17°080 382 1644 || 177180 345 7966 || 17°280 312 8898
16"982 421 5132 || 177082 381 4008 || 17'182 345 1057 || 17°282 312 2646
16°984 420 6710 || 177084 380 6388 || 17°184 344 4162 || 17°284 311 6407
16°986 419 8305 || 177086 379 8783 || 17°186 343 7281 || 17°286 311 o180
167988 418 9917 || 177088 379 1193 || 17°188 343 0413 || 17°288 310 3966
16"990 418 1545 || 17‘090 378 3618 || 17°190 342 3559 || 17°290 309 7764
16"992 417 3191 || 17°092 377 6058 || 17°192 341 6719 || 17°292 309 1575
16°994 416 4852 || 17°094 376 8514 || 17°194 340 9892 || 17°294 308 5398
16°996 415 6531 || 177096 376 0984 || 17°196 340 3079 || 17°296 307 9234
16°998 414 8226 || 17°098 375 347° || 17°198 339 6280 || 17°298 307 3081
17000 413 9938 || 17°100 374 5979 || 17°200 338 9494 || 17°300 306 6941
| 177002 413 1666 || 17°102 373 8486 || 17°202 338 2722 || 17°305 305 1645
| 177004 412 3411 || 17°104 373 1016 || 17°204 337 5964 || 17°310 303 6425
| 17°006 “AIL 5173 || 17°106 372 3562 || 17°206 336 9218 || 17°315 302 1280
177008 410 6951 || 17°108 371 6122 || 17°208 336 2487 || 17°320 300 6212
17010 409 8745 || 17110 370 8697 || 17°210 335 5768 || 17°325 299 1218
17012 409 0556 || 17°112 370 1287 || 17°212 334 9064 || 177330 297 6299
17014 408 2383 || 177114 369 3892 || 177214 334 2372 || 17°335 296 1455
17016 407 4226 || 177116 368 6512 || 17°216 333 5694 || 17°340 294 6685
17018 406 6086 || 17118 367 9146 || 17°218 332 9029 || 17°345 293 1988
17'020 405 7961 || 17°120 367 1795 || 17°220 332 2378 |) 17°350 291 7365
| 177022 404 9854 || 17°122 366 4459 || 17°222 331 5740 || 17°355 290 2814 |-
17°024 404 1762 || 17°124 365 7137 || 17°224 330 QII5 || 17°360 288 8336
17°026 403 3687 || 17°126 364 9830 || 17°226 330 2504 || 17°365 287 3931
| 17°028 | 402 5628 || 17°128 364 2538 || 17°228 329 5905 || 17°370 285 9597
|
| 7°030 | 401 7584 || 17°130 363 5269 | 17'230 328 9320 || 17°375 284 5335
| 177032 400 9557 || 17°132 362 7997 || 17°232 328 2748 || 17°380 283 1143
17034 400 1546 || 17°134 362 0748 || 17°234 327 6189 || 17°385 281 7023
1770361 399 355% || 17°136 361 3514 || 17'236 326 9643 || 17°390 280 2973
17°038 | 398 5572 || 177138 360 6294 || 17°238 326 3110 || 17°395 278 8993
17°040 397 7608 || 17°140 359 9089 || 17°240 325 6599 || 17°400 277 5083
| 177042 | 396 9661 || 17°142 359 1898 || 17°242 325 0084 || 17°405 276 1242
17044 | 396 1730 || 17°144 358 4721 || 17°244 324 3590 || 17°410 274 747°
| 17°046 | 395 3814 || 17°146 357 7559 || 17°246 323 7109 || 17°415 273 3768
| 177048 | 394 5914 || 177148 357 0411 || 17°248 323 0642 || 17°420 272 0133

[17°425—18'420] OF THE DESCENDING EXPONENTIAL. 233

[Second Part. Fourteen decimal places.|

wv Gnu xv (Gat: av Cae x Cae
17°425 270 6566 || 17°675 210 7876 || 177925 164 1615 |) 18°175 127 8491
17°430 269 3067 || 17°689 209 7363 || 17°939 163 3428 || 18180 127 2115
17°435 267 9635 || 177685 208 6902 || 17°935 162 5281 || 18°185 126 5770
17°440 266 6271 || 17°690 207 6494 || 17°940 161 7175 || 18:190 125 9457
17°445 265 2973 || 17°695 206 6137 || 17°945 160 grog || 18°195 125 3175
17°450 263 9741 || 17°700 205 5832 | 17°950 160 1084 || 18:200 124 6925
U5) 262 6575 || 17°705 204 5579 | 17°955 159 3098 || 18-205 124 0706
17°460 261 3475 || 17°710 203 5370 || 17°960 158 5153 || 18:210 123 4518
17°465 260 ©440° || 17°715 202 5225 || 17°965 157 7247 || 18°215 122 8361
17479 258 7470 || 17°720 201 5124 || 17°970 156 9380 || 18°220 122 2234
17°475 257 4565 || 17°725 200 5074 || 17°975 156 1553 || 18°225 T2t 6139
17°489 256 1725 || 17°73° 199 5°73 17'980 155 3765 || 18°230 I2I 0073
17°485 254 8948 || 17°735 198 5123 || 177985 154 6015 || 18°235 120 4038
17°490- 253 6235 || 17°740 197 5222 || 17°990 153 8304 || 18°240 119 8033
| 17°495 252 3586 || 17°745 196 537° || 17°995 153 0632 || 18°245 Ig 2057
17°509 25I 0999 || 17°759 195 5568 || 18000 152 2998 || 18:250 118 6112
177595 249 8475 || 17°755 194 5815 || 18005 I51 5402 || 18-255 118 o196
; 17°510 248 6014 || 17°760 193 6110 || 18-010 150 7844 || 18°260 II7 4310
| 17515 247 3615 || 17°765 192 6454 || 18015 150 0323 || 18:265 116 8453
17°520 246 1278 || 17°770 191 6845 || 18020 149 2841 || 18°270 I16 2625
17525 244 9002 || 17°775 190 7285 || 18:025 148 5395 || 18°275 115 6827
177530 243 6788 || 17°789 189 7772 || 18-030 147 7987 || 18°280 II5 1057
17°535 242 4634 || 17°785 188 8307 |} 18:035 147 0615 || 18°285 II4 5316
17°540 241 2541 || 17°790 187 888g || 18:040 146 3280 || 18-290 113 9604
17545 240 0509 || 17°795 186 9518 || 18°045 145 5982 || 18°295 II3 3920
177550 238 8536 || 17°800 1&6 0194 || 18:050 144 8720 || 18°300 112 8264
17555 237 6623 || 17°805 185 og16 || 18055 144 1495 || 18°305 II2 2637
17°560 236 4770 || 17°810 184 1685 || 18-060 143 4305 || 18°310 III 7038
17°565 235 2975 || 17°815 183 2499 || 18065 142 7152 || 18°315 III 1467
17°57° 234 1240 || 17°820 182 3360 18'070 142 0034 || 18°320 IIO 5923
17°575 232 9563 || 17°825 181 4266 || 18°075 I4I 2951 || 18°325 IIO 0407
17°580 231 7944 || 17°830 180 5217 || 18-080 140 5904 || 18°330 10g 4919
17°585 230 6383 || 17°835 179 6213 || 18°085 139 8892 || 18°335 108 94538
17°590 229 4880 || 17°840 178 7255 || 18-090 139 1915 || 18-340 108 4025
17595 228 3435 || 17°845 177 8341 || 18:095 138 4973 || 18°345 * 107 8618
17600 227 2046 || 17°850 176 9471 || 18100 137 8065 || 18-350 107 3238
| 17°605 226 07134 || 17°855 176 0646 || 18-105 137 1192 || 18°355 106 7886
17'610 224 9439 || 17°860 175 1865 || 18110 136 4353 || 18°360 106 2559
17°615 223 8220 || 17°865 174 3127 || 18115 135 7549 | 18°365 105 7260
17°620 222 7057 || 17°870 173 4433 || 18120 135 0778 || 18°370 105 1987
17°625 221 5949 || 17°875 172 5783 || 18125 134 4041 || 18°375 104 6740
17°630 220 4897 || 17°880 I7I 7175 || “18-130 133 7338 || 18°380 104 1519
17°635 21g 3900 || 17°885 170 8611 || 18-135 133 0667 || 18°385 103 6325
| 17°640 218 2958 || 17°890 170 0089 || 18-140 132 4031 || 18-390 103 1156
| 17°645 217 2070 || 17°895 169 1610 | 18145 131 7427 || 18-395 102 6013
17°650 216 1237 || 17°909 168 3173 | 18150 131 0856 || 18-400 102 0896
17°655 215 0458 || 177905 167 4778 || 18°155 130 4318 || 18-405 IOI 5804
17°660 213 9732 || 17°910 166 6425 || 18'160 129 7813 || 18-410 IOI 0738
17°665 212 9060 || 17°915 165 8114 || 18°165 I29 1340 || 18°415 100 5697
17°670 211 8442 || 17°920 164 9844 | 18170 128 4900 || 18-420 I00 0681

Worn, SUN (ee JUL 31

Fourteen decimal places.)

234 MR F. W. NEWMAN’S TABLE
[Second Part.

x e-% x ere x Ess
18°425 | 99 5690 || 18675 | 77 5444 || 18'925 | 60 3917
18439 | 99 0724 || 18680 | 77 1576 || 18°930 | 60 ogo5
18-435 | 98 5783 || 18°685 | 76 7728 || 18°935 | 59 7908
18440 | 98 0866 || 18690 | 76 3899 || 18940 | 59 4925
18445 | 97 5974 || 18695 | 76 0089 || 18945 | 59 1958
18450 | 97 1106 |} 18:700 | 75 6298 || 18950 | 58 g006
18°455 | 96 6263 || 18°705 | 75 2526 |} 18955 | 58 6068
18462 | 96 1444 || 18'710 | 74 8773 || 18960 | 58 3145
T8°465 | 95 6648 || 18°715 | 74 5039 |) 18965 | 58 0237

p22)" 95 S877 P28 72: | 742393 | S97 | G7 aes
18°475 | 94 7130 || 18725 | 73 7625 || 18'°975 | 57 4463
18-485 | 94 2406 || 18:730 | 73 3946 || 18980 | 57 1598
18°485 | 93 7725 || 18°735 | 73 0286 |] 18°985 | 56 8747
18-499 | 93 3029 || 18740 | 72 6643 || 18°99 | 56 5910
18°495 | 92 8375 || 18°745 | 72 3019 || 18995 | 56 3088
18*500 2 3745 || 18°750 |] 71 9413 || 19°000 | 56 0280
18°505 | gr 9138 || 18°755 | 71 5825 || 19°005 | 55 7485
18510 | gt 4554 || 18°760 | 7x1 2255 || 1g°0IO | 55 4705
18515 | 90 9992 || 18°765 | 70 8703 || T9'°0T5 | 55 1938
18520 | 90 5453 || 18°770 | 70 5168 || 19°020 | 54 9185
18°525 | 90 0938 || 18°775 | 76 1651 || 19°025 | 54 6446
18°532 | 89 6444 || 18°780 | 69 8152 || 19°030 | 54 3721
18°535 | 89 1973 || 18°785 | 69 4670 |] 19°035 | 54 1009
15°540 | 88 7524 || 18°790 | 69 1205 || 19040 | 53 83rt
15°545 | 88 3098 || 18°795 | 68 7757 || 19°045 | 53 5626
18°550 | 87 8693 || 18°800 | 68 4327 || 19°050 | 53 2954
| 18°555 | 87 4311 || 18°805 | 68 o914 || 19°055 | 53 0296
18560 | 86 9950 || 18°810 | 67 7518 || 19°:060 | 52 7651
18°565 | 86 5611 || 18°815 | 67 4139 || 19°065 | 52 5020
18570 | 86 1294 || 18°820 | 67 0776 || 19':070 | 52 2401
18°575 | 85 6998 || 18825 | 66 7431 |] 19°075 | 51 9796
18589 | 85 2724 || 18°830 | 66 4ro2 || 19°080 | 51 7203
| 18°585 | 84 8471 || 18°835 | 66 0790 || 19'°085 | 51 4624
18590 | 84 4239 || 18°340 | 65 7494 || r9°090 | 51 2057
18°595 | 84 0029 || 18845 | 65 4215 || 19°095 | 50 9593
18690 | 83 5839 || 18°85c | 65 og52 || 19°109 | 50 6962
18605 | 83 1670 || 18°355 | 64 7706 || 19°105 | 50 4433
18610 | 82 7522 || 18°860 | 64 4475 || 19‘t110 | 59 1917
18615 | 82 3395 || 18°865 | 64 1261 || 19115 | 49 9414
18°625 | 81 9288 || 18870 | 63 8962 || 19°t20 | 49 6923
18°625 | 81 5202 || 18°875 | 63 4880 || 19'125 | 49 4445
18°639 | 81 1136 || 18°889 | 63 1713 |] 191130 | 49 1979
18°635 | 80 7og1 || 18°885 | 62 8563 |] 19°135 | 48 9525
18640 | 80 3365 || 18°895 | 62 5428 || 19°:140 | 48 7084
18°645 | 79 9269 || 18°895 | 62 2309 || 19'145 | 48 4654
18650 | 79 5075 || 18:900 | 61 9205 || 19°150 | 48 2237
18°655 | 79 1109 || 18905 | 61 6116 || 19°155 | 47 9832
18°660 | 78 7164 ||.18:910 | 61 3244 || 19°169 | 47 7439
18665 | 78 3238 || 18°915 | 60 9986 || 19°165 | 47 5057
18.670 18920 | 60 6944 || 19170 | 47 2688

oh wee
T9°I75 | 47 0331
19180 | 46 7985
19185 | 46 5651
Ig‘'I90 | 46 3328
19195 | 46 ro18
1g'200 | 45 8718
19'205 | 45 6430
T9210 | 45 4154
19'215 | 45 1889
19'220 | 44 9635
19°225 | 44 7392
19°230 | 44 5161
19°235 | 44 2941
19°240 | 44 0732
19°245 | 43 8534
19'250 | 43 6346
19°255 | 43 4170
19'260 | 43 2004
19265 | 42 9850
I19'270 | 42 7706
EOi275 | 42° 5573
19'280 | 42 3450
19285 | 42 1338
19°290 | 41 9237
19'295 | 41 7146
19°300 | 41 5065
T9395 | 41 2995
T93IO | 41 0935
19°315 | 40 8886
19°320 | 40 6846
19°325 | 40 4817
19°330 | 40 2798
19°335 | 40 0789
19°340 | 39 8799
19°345 | 39 0801
19°350.| 39 4822
19355 | 39 2853
19°360 | 39 0894
19°365 | 38 8944
19°37 | 38 7004
19°375 | 38 5074
19°380 | 38 3154
19°385 | 38 1243
T9399 | 37 934
19°395 | 37 7449
19°400 | 37 5567
19°405 | 37 3693
19410 | 37 1830
19°415 | 36 9975
19'420 | 36 8130

[18-425—19°670]

[19°675—20'920] OF THE DESCENDING EXPONENTIAL. 235
[Second Part. Fourteen decimal places.|
By ee x Ce le er i & ee x Ca,
| | 1
19°675 | 28 5270 || 19°925 | 22 2168 || 207175 | 17 3025 | 20°425 | 13 4752 20'675 | 10 4945
19680 | 28 3847 || 19°939 | 22 ro60 || 20189 | 17 2162 |) 20°439 | 13 4080 20°680 | IO 4421
19685 | 28 2431 || 19°935 | 21 9958 || 20°85 | 17 1303 || 20435 | 13 3411 || 20°685 | Io 3901
19690 | 28 1023 || 19°940 | 21 8861 || 20°190 | 17 0449 || 20°440 | 13 2746 || 20°690 | 10 3382
19695 | 27 9621 || 19°945 | 21 7769 || 207195 | 16 9599 || 20°445 | 13 2084 || 20°695 | 10 2867
19°700 | 27 8227 || 19°950 | 21 6683 || 20:200 | 16 8753 || 20°450 | 13 1425 || 20°700 | 10 2354
19°705 | 27 6839 || 19°955 | 21 5602 || 20°205 | 16 7g1x || 20°455 | 13 0769 || 20°705 | IO 1843
19710 | 27 5458 || 19°960 | 21 4527 || 20°210 | 16 7074 20°469 13 O117 || 20°710 | TO 1335
19715 | 27 4084 || 19°965 | 21 3457 || 20°215 | 16 6240 |) 207465 | 12 9468 || 20°715 | 10 0830
19°720 | 27 2717 || 19°979 | 21 2392 || 20°220 | 16 5411 || 20°470 | 12 8822 || 20°720 | 10 0327
19°725 | 27 1357 || 19°975 | 21 1333 || 20°225 | 16 4586 || 20°475 | 12 8180 || 20°725 9 9827
19°730 | 27 0904 || 19°989 | 21 0279 || 20°230 | 16 3765 || 20480] 12 7541 || 20°730 | 9 9329
19°735| 26 8657 || 19°985 | 20 9230 || 20°235 | 16 2949 || 20°485 | 12 6905 | 20°735 | 9 8833
19740 | 26 7317 || 19°99° | 20 8187 || 20°240 | 16 2136 || 20°499 | 12 6272 || 20°740 | 9 8340
19°745 | 26 5984 || 19°995 | 20 7148 || 20°245 | 16 1327 || 20°495 | 12 5642 || 20°745 | 9 7850
19°750 | 26 4657 || 20°000 | 20 6115 || 20°250 | 16 0523 || 20°500 | 12 sors || 20°750 | 9 7362
19°755 | 26 3337 || 20°005 | 20 5087 || 20°255 | 15 9722 | 20°505 | 12 4392 || 20°755 | 9 6876 |
19°769 | 26 2924 || 20°010 | 20 4064 || 20°260 | 15 8925 || 20519 | 12 3771 || 20°760 9 6393 |
19°765 | 26 0717 || 20°015 | -20 3047 || 20°265 | 15 8133 || 20°515 | 12 3154 || 207765 | 9 5912 |
19°77° 25 9417 || 207020 20 2034 || 20°270 15 7344 || 20°520 I2 2540 || 20°770 9 5434
19775 | 25 8123 || 20°025 | 20 1026 || 20°275 | 15 6559 || 20°525 | 12 1929 || 20°775 9 4958
19°780 | 25 6835 || 20°030 | 20 0024 || 20280 | 15 5779 || 20°530 | 12 1320 || 20780 | 9 4484
19°785 | 25 5555 || 20°035 | 19 go26 || 20°285 | 15 5002 || 20°535 | 12 0715 || 20°785 9 4013
19°790 | 25 4280 |) 20°040 | 19 8033 || 20°290 | 15 4229 |/ 20°540 | 12 o113 |] 20°799 | 9 3544
19°795 | 25 3012 || 20°045 | 19 7046 || 20°295 | 15 3459 || 20°545 | 12 9514 |] 20°795 | 9 3078
19820 | 25 1750 || 20°050 | 19 6063 || 20:°300 | 15 2694 || 20°550 | 11 8918 || 20°800 | 9 2614
19805 | 25 0494 || 20°055 | 19 5085 || 20°305 | 15 1932 || 20°555 | 11 8325 || 20°85 Q 2152
19810 | 24 9245 || 20°060 | 19 4112 || 20-310 | 15 1175 || 20°560 | 11 7735 || 20°810 | 9 1692
19815 | 24 8502 || 20°065 | 19 3144 || 20°315 | 15 o4en || 20°565 | rr 7148 || 20°815 Q 1235
19820 | 24 6765 || 20°070 | 19 2181 || 20°320 | 14 9670 || 20°570 | 11 6563 || 20°820 | 9 0780
19°825 | 24 5534 || 20°075 | 19 1222 || 20°325 | 14 8924 || 20°575 | 11 5982 |] 20°825 9 0327
19°830 | 24 4309 || 20°080 | 19 0268 || 20°330 | 14 8181 |) 20°580 | x11 5404 || 20°830 | 8 9876
19°835 | 24 3091 || 20°085 | 18 9319 || 20°335 | 14 7442 || 20°585 | 11 4828 || 20°835 | 8 9428
19°840 | 24 1879 || 20°099 | 18 8375 || 20°340 | 14 6707 || 20°590 | 11 4255 || 20°840 8 8982
19'845 | 24 0672 || 20095 | 18 7436 || 20°345'] 14 5975 || 20°595 | 14 3685 || 20°845 | 8 8538
19°850 | 23 9472 || 20'100 | 18 6sor || 20'350 | 14 5247 || 20°600 | 11 3118 || 20°850 8 8097
19855 | 23 8277 || 20105 | 18 5571 || 20°355 | 14 4523 || 20%605 | rx 2554 || 20°855 | 8 7657
19°860 | 23 7089 || 207110 | 18 4645 || 20°360 | 14 3802 || 20°610 | rr 1993 || 20°860 | 8 7220
19865 | 23 5907 || 207115 | 18 3724 || 20°365 | 14 3085 || 20°615 | x1 1434 |] 20°865 | 8 6785
19870 | 23 4730 | 20°120 | 18 2808 || 20°370 | 14 2371 || 20°620 | rr 0879 || 20°870 | 8 6352
19°875 | 23 3559 || 20°125 | 18 1896 || 20°375 | 14 1661 || 20°625 | 11 0326 || 20°875 8 5922
19°880 | 23 2394 || 20:130 | 18 0989 || 20°380 | 14 0954 |) 20°630 | 10 9775 || 20°880 8 5493
19°885 | 23 1235 || 20°135 | 18 0086 || 20°385 | 14 0251 || 20635 | 10 9228 20°885 8 5067
19°890 | 23 0082 || 20°140°| 17 9188 || 20°390 | 13 9552 || 20°640 | 10 8683 20°890 | 8 4642
19°895 | 22 8934 || 207145 | 17 8294 || 20°395 | 13 8856 || 20°645 | 10 8141 20°895 8 4220
19°900 | 22 7793 || 20°150 | 17 7405 || 20°409 | 13 8163 |) 20°650 | 10 7602 || 2079900 | 8 3800
19905 | 22 6656 || 20155 | 17 6520 || 20°405 | 13 7474 || 20°655 | 10 7065 || 20°:905 8 3382
19910 | 22 5526 || 20°160 | 17 5640 || 20'410 | 13 6788 || 20660 | ro 6531 || 20°910 8 2966
19915 | 22 4401 || 20°165 | 17 4764 || 20°415 | 13 6106 || 20665 | Io 6000 || 20°915 8 2553
19°920 | 22 3282 || 20170 | 17 3892 || 20°420 | 13 5427 || 20°70 | 10 5471 | 207920 | 8 2141 |
3 1—2

236 MR F. W. NEWMAN'S TABLE [20°925—22'170]
[Second Part. Fourteen decimal places.)
x C3 | v e-2 x em x Cw v I
20°925 8 1731 || 21175 6 3652 || 21°425 4 9572 || 21°675 3 8607 || 21°925 3 0067
| 20°930 8 132 217180 6 3335 || 217430 4 9325 || 217680 3 8414 || 21°930 | 2 9917
20°935 8 og18 || 21185 6 3019 |] 21°435 4 9079 || 21°685 3 8223 || 21°935 | 2 9768
20°940 8 0514 || 217190 6 2705 || 21°440 4 8834 || 21°690 3 8032 || 21°940 2 9620
20°945 8 or13 |} 21-195 6 2392 || 21°445 4 8591 || 21°695 3 7843 || 21°945 | 2 9472
| 20°950 7 9713 || 21°200 6 2081 |} 21°450 4 8348 || 21°700 3 7654 || 21950 | 2 9325
20°955 7 9316 | 205 6 1771 || 21°455 4 8107 |} 21°705 3 7466 || 21°955 2 9179
209°969 7 8920 I'2I0 6 1463 || 21°460 4 7867 || 21-710 3 7279 || 21969 | 2 9033
20°965 7 8526 215 6 1156 || 21-465 4 7629 || 21-715 3 7093 || 21°965 | 2 8888
20°970 7 8135 || 21-220 6 0851 || 21°470 4 7391 || 21°720 3 6908 || 21°970 | -2 8744
| 20°975 | 7 7745 || 21225 | 6 0548 | 21°475 | 4 7155 || 21°725 | 3 6724 |] 21°975 | 2 860%
20°989 77257 || 217230 6 0246 || 21-480 4 6920 || 21°730 3 6541 || 219980 | 2 8458
20°985 | 7 6971 || 21°235 | 5 9945 || 21485 | 4 6686 || 21°735 | 3 6359 || 21-985 | 2 8316
20°99) 7 6588 || 21-240 5 9646 | 21°490 4 6453 |} 21°740 3 6177 || 21990 | 2 8175
20°995 | 7 6206 || 21245 | 5 9349 || 21495 | 4 6221 | 21°745 | 3 5997 || 21995 | 2 8034
21°000 7 5826 || 21-250 5 9953 || 21500 4 5990 || 21°750 3 5817 || 22°000 2 7895
21005 | 7 5447 || 21255 | 5§ 8758 |] 21505 | 4 5761 || 22°755 | 3 5639 || 22°005 | 2 7756
21°010 7 5071 || 217269 5 8465 || 21-510 4 5533 || 21°769 3 5461 || 22°010 | 2 7617
2T015 7 4697 || 217265 5 8174 || 217515 4 5306 || 21°765 3 5284 || 22°015 2 7479
21029 7 4s2 21°270 5 7884 || 217520 4 5080 || 21°770 3 5108 || 22°029 2 7342
21°025 7 3953 || 20275 | 5 7595 || 21°525 | 4 4855 || 21°775 | 3 4933 || 22.025 | 2 7206
21°030 7 3585 || 21°28) 5 7308 || 21°530 4 4631 || 217780 3 4759 || 22°030.| 2 7070
217035 7 3218 || 21°285 5 7022 || 21°535 4 4409 || 21°785 3 4585 |} 22°035 2 6935
21°040 7 2852 || 21°290 5 6737 || 21°540 4 4187 || 21°799 3 4413 || 22°040 2 6801
21'045 7 2489 || 217295 5 6454 || 217545 4 3967 || 21°795 3 4241 || 227045 | 2 6667
21050 7 212 217300 5 6173 || 21°550 4 3747 ||. 21°800 3 4071 || 22°050 | 2 6534
21°055 7 1768 || 21°305 5 5893 || 21°555 4 3529 || 21°805 3 3901 || 22°055 | 2 6402
21°069 7 I4I0 |} 21°310 5 5614 || 21°560 4 3312 || 21°810 3 3732 || 22°060 2 6270
21065 7 1054 || 21°315 5 5337 || 217565 4 3096 || 21°815 3 3563 || 22°065 2 6139
21°070 7 0699 || 21°320 5 5061 || 21°570 4 2881 || 21°820 3 3396 || 22°070 | 2 6009
21075 | 7 0347 || 21°325 | 5 4786 | 21°575 | 4 2667 || 21825 | 3 3229 || 22°075 | 2 5879
21°089 6 9996 || 21°330 5 4513 || 21°580 4 2455 || 21°830 3 3064 || 22°080 | 2 5750
21085 6 9647 || 21°335 5 4241 |} 21'585 4 2243 || 21°835 3 2899 || 227085 2 5622
21'095 6 9299 || 21°340 5 397° || 21°599 4 2032 || 21°840 3 2735 || 22°090 2 5494
217095 6 8954 || 21°345 5 3701 || 217595 4 1823 || 21°845 3 2571 || 22°095 2 5367
21I°100 6 8610 || 21°350 5 3433 || 21°600 4 1614 || 21°850 3 2409 || 22°100 2 5240 |
21°I105 6 8268 |} 21°355 5 3167 || 217605 4 1406 || 21°855 3 2247 || 22°105 2 5114 |
21°10 6 7927 || 217369 5 2902 || 21°610 4 1200 || 21°860 3 2086 || 22110 | 2 4989 |
21115 6 7588 || 21°365 5 2638 || 21615 4 0994 || 21°865 3 1926 || 22°115 | 2 4864
21°120 6 7251 || 217370 5 2376 || 21620 4 0790 || 21°870 3 1767 || 22°%20 2 4740
21°12 6 6916 || 21°375 5 2114 || 21°625 4 0586 || 21°875 3 1609 || 22°125 2 4617
21°130 6 6582 || 21°380 5 1854 || 21°639 4 0384 || 21°880 3 1451 || 22°30 2 4494
21°135 6 6250 || 21°385 5 1595 || 21°635 4 0183 |] 21°885 |. 3 1294 || 22°135 | 2 4372
21'140 | 6 5920 || 21°390 5 1338 || 21°640 3 9982 || 21°890 3 1138 || 22°140 | 2 4250
21°145 6 5591 || 21°395 5 1082 || 21°645 3 9783 | 21°895 3 0983 || 22°145 2 4129
21°15) 6 5264 || 21°400 5 0827 || 21°650 3 9584 || 21'900 3 0828 || 22°150 2 4009
21155 | 6 4938 | 21°405 | § 0574 || 21°655 | 3 9387 || 219905 | 3 0674 || 22°155 | 2 3889
21°160 6 4614 | 21°410 5 0322 || 21'660 3 9190 || 21'910 Brogan ||| 22060.) |252770
21°165 6 4292 | 21°415 5 0071 || 21665 3 8995 | 21°915 3 0369 || 22°165 2 3652
21'170 6 3971 || 21°420 4 9821 || 21670 3 8801 || 21'920 30218 |] 22°170 | 2 3534

[22°175—23°420] OF THE DESCENDING EXPONENTIAL. 237

[Second Part. Kourteen decimal places.]

|
xv Oe vw Cae wv Cm av Cm av E-%
22°175 2 3416 || 22°425 t 8237 || 22°675 I 4203 || 22°925 I I06r || 23°175 8614
22°180 2 3299 || 22°430 I 8146 || 22°680 I 4132 || 22°930 I 1006 || 23°180 8571
22°185 2 3183 || 22°435 1 8055 || 22°685 I 4061 || 22°935 I Og5I || 23°185 8529
22°190 2 3068 || 22°440 I 7965 || 22*699 I 3991 || 22°940 I 0896 || 23°190 8486
22°195 2 2953 || 22°445 1 7876 || 22°695 I 3921 || 22°945 I 9842 || 23195 8444
22°20) 2 2838 || 22°450 I 7786 || 22°700 I 3852 || 22°950 t 0788 || 23°200 8402
22°205 2 2724 || 22°455 I 7698 || 22°705 I 3783 || 22°955 1 0734 || 23°205 8360
22°210 2 2611 || 22°460 I 7699 || 22°710 I 3714 || 22°960 I 0681 || 23:210 8318
22°215 2 2408 || 22°465 I 7522 || 22°715 I 3646 || 22°965 I 0627 || 23°215 8277
22°220 2 2386 | 22°470 I 7434 || 22°720 I 3578 || 22°970 I 0574 || 23220 8235
22°225 2 2274 || 22°475 I 7347 || 22°725 I 3510 || 22°975 I 0522 || 23°225 8194
22°230 2 2163 || 22°480 17200 |(227720 I 3442 || 22°980 I 0469 || 23°230 8153
22°235 2 2053 || 22°485 Te 775) | 227735 I 3376 || 22°985 I 0417 || 23°235 8113
22°240 2 1943 || 22°499 I 7089 || 22°740 I 3309 || 22990 I 0365 || 23°240 8072
22°245 2 1833 | ANS T 7004 || 22°745 13243) || 122199 Sa lpe LROS HSN 2312455 8032
22°250 2 1724 || 22°500 I 6919 || 22°750 I 3176 || 23°000 I 0262 || 23°250 7992
22°255 2 1616 || 22°505 1 6835 || 22°755 I 3111 || 23°005 I 0211 || 23°255 7952
22°269 2 1508 || 22°510 I 6751 || 22°760 I 3045 || 23,010 I 0160 || 23°260 7912
22°265 2 14or |} 22°515 1 6667 || 22°765 I 2989 || 23°015 I O10g || 23°265 7873
22°270 2 1294 || 22°520 I 6584 || 22°770 I 2915 || 23020 I 0059 || 23°270 7834
Be5275 2 1188 || 22°525 I 6501 || 22°775 1 2851 || 23°025 I 0009 || 23°275 7795 |
22°280 2 1082 || 22°539 I 6419 || 22°780 I 2787 || 23°030 © 9959 || 23°280 7756
22°285 | 2 0977 || 22°535 | 1 6337 || 22°785 | 1 2723 || 23°035 9909 || 23°285 7717
22°292 | 2 0873 || 22°540 1 6256 || 22°790 I 2660 || 23'040 9860 || 23°290 7679
22°295 2 0768 || 22°545 I 6174 || 22°795 I 2597 || 23°045 g8ro || 23°295 7640
22°300 2 0665 || 22°550 I 6094 || 22°800 I 2534 || 23°050 9761 || 23°300 7602
22°395 2 0562 | 22°555 I 6014 | 22°805 I 247% || 23°055 9713 || 23°3°5 7564
22°310 2 0459 || 22°560 I 5934 || 22°810 I 2409 || 23'060 9664 || 23°310 7526
22°315 2 0357 || 22°565 I 5854 || 22°815 I 2347 || 237065 9616 || 23°315 7489
22°320 2 0256 || 22°570 I 5775 || 22°820 I 2286 || 23°070 9568 || 23°320 7452
22°325 Sporn 22575 m5697 || 22°825 I 2224 || 23°075 9520 || 23°325 7415
22°330 2 0054 || 227580 | 5618 || 22°830 I 2163 || 23'080 9473 || 237339 7378
22°335 | 1 9954 || 22°585 | 1 5540 || 22°83 I 2103 || 23°085 9426 | 23°335 734°
22°340 | 1 9855 || 22°599 | 1 5463 || 22°840 | 1 2042 || 23°090 9379 || 237340 7304
22°345 I 9756 | 22°595 I 5386 || 22°845 I 1982 || 23°095 9332 || 23°345 7267
22°350 I 9657 || 22°600 I 5309 || 22°850 I 1922 || 23°100 9285 || 23°350 7231
22°355 I 9559 || 22°605 I 5233 || 22°855 I 1863 || 23°105 9239 || 23°355 7195
22°360 I 9461 || 22°610 I 5157 || 22°860 I 1804 || 23°110 9193 || 23°360 7159
22°365 I 9364 || 22°615 I 5081 || 22°865 I 1745 || 23°115 9147 || 23°365 7123
22°370 | I g268 || 2z°620 I 5006 || 22°870 I 1686 || 23120 gio2 || 23°37° 7088
22°375 I gt72 || 22°625 I 4931 || 22°875 I 1628 || 23125 9256 || 23°375 71°53
22°380 I 9076 | 22°630 I 4856 || 22°889 I 1570 || 23130 gorr || 23°380 7018
22°385 1 8981 | 22°635 I 4782 || 22°885 Te Dow |e252035 8966 || 23°385 6983
22°390 1 8886 | 22°640 I 4709 || 22°890 |} I 1455 || 237140 8921 || 237399 6948
22°395 1 8792 || 22°645 I 4635 || 22°895 | 1 1398 || 23145 | 8377 || 23°395 6913
22°400 1 8698 || 22°650 I 4562 || 22'909 | x 1341 || 23°150 8832 || 23°400 6879
22°405 1 8605 || 22°655 I 4490 || 22:905 | 1 1284 || 23°155 8788 || 237405 6844
22°410 1 8512 || 22°660 I 4417 || 22°910 I 1228 || 23169 8745 || 237410 6810
22°415 I 8420 || 22°665 I 4345 || 22°915 I 1172 || 23°165 8701 || 23°415 6776
22'420 1 8328 || 22°670 I 4274 || 22°920 I 1116 || 23°170 | 8658 || 237420 6742
( | !

238 MR F. W. NEWMAN’S TABLE [23°425 —24°670]

[Second Part. Fourteen decimal places.

i ] |

| a2 |) ex xv e-* x e-% x en xv e-*
— ————— See ——__—— | | OOo |
23°425 6709 || 23°675 5225 || 23°925 4069 || 24 175 3169 || 24°425 2468
23°430 6675 || 23°680 5199 || 237930 4049 || 24°180 3153 || 24°430 2455

| 23°435 6642 || 23°685 5173 || 23°93 4029 || 247185 3137 || 24°435 2443
23440 6609 || 23°690 5147 || 23°940 4009 || 24°199 3122 || 24°440 2431
23°445 6576 || 23°695 5122 || 23°945 3989 || 24°195 3106 || 24°445 2419
237450 6543 || 23°700 5096 || 23°95° 3969 || 24°200 3091 || 24°450 2407
23°455 6510 || 23°795 5070 || 23°955 3949 || 24°205 3076 || 24°455 2395
23°460 6478 || 23°710 5045 || 23°960 3929 || 24°210 3061 |} 24°460 2383
237465 6446 || 23°715 5020 || 23°965 3910 || 24°215 3045 || 24°465 2371
23°470 6414 || 23°720 4995 || 23°97° 3890 || 24°220 3030 || 24°470 2359
23°475 6382 || 23°725 4970 || 23°975 3871 || 24°225 3015 || 24°475 2348
237480 6350 || 23°730 4945 || 23°980 3852 || 24°230 3000 || 247489 2336
23°485 6318 | 23°735 4920 || 23°985 3833 || 24°235 2985 || 24°485 2324
23°490 6286 || 23°740 4895 || 23°99° 3814 || 24°240 2970 || 24°499 2313
23°495 6255 | 23°745 4871 || 23°995 3794 | 24°245 2955 || 24°495 2301
23°500 622 23°750 4847 || 247000 3775 || 24°250 2940 || 247500 2290
23°505 6193 || 23°755 4823 || 24°005 3756 || 24°255 2925 || 24°505 2278
23°510 6162 || 23°760 4799 || 24010 3737 || 24°260 2910 || 24°510 2267
23°515 6131 || 23°765 4775 || 24°015 3718 || 24°265 2895 || 24°515 2256
237520 6100 || 23°770 4751 || 24°020 3700 || 24°270 2881 || 24°520 2244
23°525 6070 || 23°775 4728 || 24°025 3682 || 24°275 2867 || 24°525 2233
23°530 6040 || 23°780 4704 || 24°030 3663 || 24°280 2853 || 247530 2222
23°535 6010 || 23°785 4680 || 24°035 3645 || 247285 2839 || 24°535 2211
23°540 5980 || 23°790 4657 || 247040 3627 || 24°290 2825 || 247540 2200
23°545 595° || 23°795 4634 || 24°045 3609 |) 24°295 2810 || 24°545 2189
23°550 5921 || 23°800 4611 || 24°050 3591 || 24°300 2797 || 24°550 2178
23°555 5891 || 23°805 4588 || 24055 [| = 3573 || 24°305 2783 || 24°555 2167
237560 5861 || 23°810 4565 || 24°060 3555 || 24°310 2769 || 24°560 21506
23°565 5832 || 23°815 4542 || 24°065 3536 || 24°315 2755 || 24°565 2146
23370 5803 | 23°820 4519 || 24°070 3519 || 24°320 2741 || 24°57° 2135
23°575 5774 || 23°82 4497 || 24°075 3502 || 24°325 2728 || 24°575 2124
23°580 5745 || 23830 4474 || 24°080 3485 || 24°330 2714 || 24°5S0 2113
23°585 5716 | 23°835 4452 || 247085 3467 || 24°335 2700 || 24°585 2103
23°59 5687 || 23-840 4430 || 24°090 3450 || 24°340 2687 || 24°599 2092
23°595 5660 | 23°845 4408 || 24°095 3433 || 24°345 2674 || 24°595 2082
23'600 5632 || 23°850 4386 || 24°100 3416 || 24°350 2660 || 24°690 2072
23°605 5604 || 23°855 4364 || 24°105 3398 | 24°355 2647 || 24°605 2061
23610 5576 || 23°860 4342 || 24°110 3381 || 24°360 2634 || 24°610 2051
23°615 5548 || 237865 4320 || 24°115 3364 || 24°365 2621 || 24°615 2041
23620 5521 || 23°870 4299 || 24°120 3347 || 24°370 2608 || 24°620 2031
23°62 5493 || 23°875 4278 || 24°125 3331 || 24°375 2595 || 24°625 2021
23°630 5465 || 23°880 4256 || 24°130 3314 || 24°380 2582 || 24°630 2010
23°635 5438 || 23°885 4235 || 24°135 3297 || 24°385 2569 || 24°635 2000
23640 5411 || 237890 4214 || 24°140 3281 || 24°390 2556 || 24°640 1990

| 23°645 5384 || 23°895 4193 | 24°145 3265 || 24°395 2543 || 24°645 1981
23°650 5357 || 23°900 4172 | 24°150 3249 || 24400 2530 || 24°650 1971
23°655 533° || 23°905 4151 | 24°155 3233 || 24°405 2518 || 24°655 1961
23'660 5303 || 23°910 4130 || 24°160 3217 || 24°410 2505 || 24°660 Ig51
23°665 5277 | 237915 4109 || 24°165 3201 || 24°415 2493 || 24°665 1941
23°670 5251 || 23°920 4089 || 24°170 3185 || 24°420 2480 || 24°670 1932

i"
SS ee

[24°675—25'920]

OF THE DESCENDING EXPONENTIAL.

[Second Part.

Fourteen decimal places.|

e-#

1497
1489
1482
1475
1467
1460
1453
1445
1438
1431

1424
1417
1410
1403
1396
1389
1382
1375
1368
1361

1354
1348
1341
1334
1328
1321
1314
1308
1301
1295

1288
1282
1276
1269
1263

1257
1250
1244
1238
1232

1226
1219
1213
1207
1201

1195
1189
1183

1177
II7I

wv E=%
25°175 1165
25°180 I160
25°185 TI54
25°190 1148
25°195 1142
25200 1137
25°205 I131
25°210 1126
25°215 1120
25°220 III5
25°22 1109
25°230 1103
25°235 1098
25°240 1092
25°245 1087
25°250 1082
25°255 1076
25°260 1071
25°265 1065
25°270 1060
25°275 To55
25°280 1050
25°285 1044
25°290 1039
25) A9)5) 1034
25°300 1029
25°305 1024
25°310 101g
25°315 1013
25°320 1008
25°325 1003
25°330 998
25) S35) 993
25°340 988
25345 984
257309 979
23 5)5) 974
25°360 969
25°35 964
25°379 959
25°375 955
25°380 950
25°385 945
25/399 940
25°395 936
25°400 931
25°405 926
25°410 g22
25°415 OL]
25°420 gi2

239
HY Ome AG Ee-%

25°425 908 || 25°675 797
25°430 903 || 25°680 704
25°435 899 || 25°685 700
25°440 894 || 25°690 697
25°445 890 || 25°695 693
25°450 885 || 25°700 690
25°455 88r || 25°705 686
25°460 877 || 25°710 683
25°465 872 || 25°715 679
25°470 868 || 25°720 676
25°475 864 || 25°725 673
25°480 859 || 25°730 669
25485 855 || 25°735 666
25°490 851 |] 25°740 663
25°495 847 || 25°745 659
25°500 842 || 25°750 656
25°55 838 || 25°755 653
25°510 834 || 25°760 649
25°515 830 || 25°765 646
25°520 826 || 25°770 643
25°525 822 || 25°775 640
25°530 817 || 25°780 636
25°535 813 || 25°785 633
25°540 899 || 25°790 630
25°545 805 || 25°795 627
25°550 851 || 25:°800 62

25°555 797 |) 25°805 621
25°560 793 || 25°810 618
25°565 789 || 25815 615
25°570 785 || 25°820 612
25°575 781 || 25°825 609
25°580 777 || 25°830 606
25°585 774 || 25°835 603
25°590 770 || 25:840 600
25°595 766 |) 257845 597
25°600 762 || 25°850 594
25605 758 || 25°855 591
25610 755 || 25°60 585
25°O15 751 || 25°865 585
25°620 747 || 25°870 582
25°625 743 || 25°875 579
25°630 740 || 25°880 576
25°635 736 || 25°885 573
25°64 732 || 25°890 570
25°645 729 || 25°895 567
25°650 725 || 25°900 565
25°655 721 25°905 562
25°660 718 |, 25'910 559
25°665 714 | 25°925 556
25°670 711 25°920 553

is)

SD eS ee) NNN Wh
mon Minoo Nooo

COPonwKns Vueuwws
om Ss WOU 4 4 Os Oo

omomonmonon

ton
So
Om

25°985
25°99°
25°995
26°000
26°005
26°010
26°015
26°020

| 26°025

26°030
26°035
26°040
26°045

26050
26°055
26°069
26°065
26'070

26'075
26080
26°085
26°090
26°095
26°109
26°105
26110
26°115
26°120

26°125
26°130
26°135
26°140
26°145
26°150
26°155
26°160
26°165
26°1790

474

465 |
| 26°350
|| 26°355

453
451

440

431

| 26-175
26°180

26°185
26°190
26°195
26°200
26°205

|| 26°210

26°215
26°220
26°225
26°230

| 26°235

26°240
26°245
26°250
26°255
26°260
26°265
26°270

| 26°275

267289
26°285
267299
26°295
26°300
26°305
26°310
26°315
26°320
26°325
26°330
26°335
26°340
26°345

26°360
26°365
26°370
26°375
2673890
26°385
26°390
26°395

| 26°400

267405
26°410
26°415

26°420

MR F. W. NEWMAN’S TABLE

Fourteen decimal places.)

259
2571

255
254

251
250
249

247

245

241
240
239

235

230
228
227
226
225

224
223
222
221
219
218
217
216
215
214

213
212
211
210
209

208
207
206
204
203

[25°925—27°170]

[27°175—27°635] OF THE DESCENDING EXPONENTIAL. 241

[Second Part. Fourteen decimal places.|

| xv Cut wv Ge x“ (G2 wv Ca
27-175 158 || 27°275 143 || 277375 129 || 27°475 117
| 27°180 15 27°280 142 || 27°380 128 || 27°480 116
| 27°185 156 || 27°285 141 |] 27°385 128 || 277485 116
27°190 155 || 27°290 140 || 27°390 127 || 27°490 II5
27195 BOS 275295 FAS | 277395 127 || 27°495 114
27°200 154 || 27°300 139 || 27°400 126 || 277500 114
27°205 153 || 277395 138 || 27°405 125 |) 277505 113
27°20 152 || 27°310 138 || 27°410 125 || 27°510 113
27°215 152 || 27°315 137 || 27°415 124 |} 27°515 112
27°220 I51 || 27°320 136 || 27°420 124 || 27°520 112
27°225 150 | 27°325 136 || 27°425 123 || 27°525 II || 27°625 IOI
27°230 149 || 277330 135 || 277430 122 || 27°530 III || 27°630 Too
27°235 148 || 27°335 134 || 27°435 122 | 27°535 IIo || 27°635 =O)
27°240 148 || 27°340 134 || 27°440 121 |} 27°540 IIO
27°245 147 || 27°345 133 || 27°445 E20 || 277545 WeNg)
27°250 146 || 27°350 132 | 27°450 120 || 27°550 108
27°255 145 || 27°355 132 || 27°455 119 || 27°555 108
27°269 145 || 277360 131 || 27°460 118 || 277560 107
27°265 144 || 27°365 130 || 277465 118 || 27°565 107
27°270 | 143 || 27°370 130 || 27°470 117 || 277570 106

Wert XtiL, Pane Lit) - 32

ly

ae
7 tr i. ogee
- _

a a

V. Tables of the Exponential Function. By J. W. L. Guatsuer, M.A., F.R.S.,
Fellow of Trinity College, Cambridge.

[Read May 21, 1877.]

THE present paper contains four tables, in each of which the functions tabulated are

e, e”, loge” and log,,e*. The ranges of the four tables are as follows:

Table I. From «=0:001 to «=0:100 at intervals of 0-001.
Table I]. From x=0:01 to x=2°'00 at intervals of 0°01.
Table III]. From x=01 to «=10°0 at intervals of 0°1.

Table IV. From w=1 to «=500 at intervals of unity.

In all the tables the first nine figures of e”, and the first nine significant figures

of e* are given. The logarithms are in all cases given to ten places of decimals.

Since log,,(e’) and log,,(e”) are equal respectively to wlog,,e and wlog,,(e*) it is
evident that the logarithmic results in the tables are merely multiples of log,,e and
log,,(e"). They were readily calculated in this manner and the values of e* and e~
were derived from them by means of ten-figure logarithms, the tenth figure being
rejected. The last figure is therefore in general correctly given to the nearest unit, but

it may be in error by a unit where the tenth figure is a 4, 5 or 6.

Mr Newman in the table which precedes this paper gives the values of e” from —
2=0°001 to 2=15'350 to twelve places of decimals at intervals of 0:001, from #=15350
to #=17'300 at intervals of 0-002, and thence to x=27°635 at intervals of 0:005 to
fourteen places. The introduction contains (pp. 148, 149) a table of e” from #=0'1 to
x=87'0 at intervals of 01 to eighteen places. The only other tables of exponential functions
that I know of are the following:

(i) On p. 188 of the first volume of Schulze’s Sammlung logarithmischer trigono-
metrischer...Tafeln (Berlin, 1778) there is a table giving the values of e*, for x=1, 2, 3,...24
to 28 or 29 figures, and for 2=25, 30 and 60 to 32 or 33 places.

32 --2

244 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

(ii) A table of log,,(e*) to seven decimal places and of e* to seven figures from r=0'01
to =1000 at intervals of 0°01 was given in Vega’s Tabule logarithmico-trigonometrice
(1797), and has been retained in the later editions of this work. Kdéhler’s Logarithmisch-

trigonometrisches Handbuch also contains the table of e’.

(iii) In the eighth and ninth volumes* of Crelle’s Journal Gudermann has given a
table of log,,sinha, log, cosha and log,,tanha from 2=2 to e=5 at intervals of 0-001
to nine decimal places, and from x=5 to x=12 at intervals of 001 to ten places.
Gudermann’s papers on the hyperbolic trigonometrical functions were afterwards collected
together and published as a separate work under the title Theorie der Potenzial- oder

cyklisch-hyperbolischen Functionen (Berlin, 1833), and this table occupies pp. 263—336.

The tables in the present paper are portions of some that I calculated as long ago
as 1872. It was then my intention to calculate extensive tables of e” and e~ for
publication in a separate form; but the scheme was not carried out and the tables
were left in an incomplete state. In 1876 Mr Newman communicated: the first part of
his table to the Society; and I thought it would be desirable to supplement it by the
tables here given. Owing to various causes Mr Newman's table has been a long time
passing through the press, and of course this paper has been kept back so that the two
might appear together.

The tables were verified as follows:

Tables I. and II. The values of e and e” were verified by differences. All the values
of e~ given in the two tables are included in Mr Newman’s great table, and the values
were compared. No error was found, but there were of course occasional differences of
a unit in the last figure: the figure was in these cases changed so as to agree with
Mr Newman’s more extended value.

During the time that his large table was being printed Mr Newman sent me a
table of & from x=0 to r=1 at intervals of 0:001 to twelve decimal places. The
values of e* in Table I. were compared with this table and, as in the case of e”%, the
last figure was changed so as to agree with it.

Table III. The values of &* were compared with Mr Newman’s table and the last
figure changed in cases where a discrepancy occurred as in Tables I. and II. The values
of & were recalculated.

Table IV. The values of & and e” were recalculated.

The tables were also compared with Schulze’s and Vega’s tables (i) and (ii), described
above, as far as the extent of the different tables permitted.

The columns giving e” and e~ are placed side by side, as the two functions are
often required in combination as in the case of the hyperbolic sine, cosine and tangent.

* Vol. viu1. pp. 195—212, 301—316; Vol. 1x. pp. 81—96, 193—208, 297—304.

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 245
Gudermann’s table (iii) mentioned above begins at w=2, and it is for this reason
that 2=2 was taken as the limit of Table II.

With respect to the use of the tables, it may be remarked that they may be
conveniently combined in interpolations: thus, for example,

ef 37 — ef? x 69 5T — §66°6863310 x 1-03769302,
and log e*57 = log (e**) + log (e%") = 1°8240368240 + 0°0160688958.

For the sake of completeness I reproduce here Schulze’s table referred tu in (1).

v &

1 | 2.718 281 828 459 045 235 360 287 471
2 | 7. 389 056 098 930 650 227 230 427 460
3 20.085 536 923 187 667 740 928 529 652
4 54.598 150 033 144 239 078 110 261 19
5 148 . 413 159 102 576 603 421 115 580 O1
6 403 . 428 793 492 735 122 608 387 180 5
U 1096 . 633 158 428 458 599 263 720 238 1

8 2980 . 957 987 041 728 274 743 592 099

) 8103 . 083 927 575 384 007 709 996 688

10 22026 . 465 794 806 716 516 957 900 641

11 59874 .141 715 197 818 455 326 485 75

12 162754. 791 419 003 920 808 005 204 77

13 442413 . 392 008 920 503 326 102 777 5

14 1202604 . 284 164 776 777 749 236 769 7

15 3269017 . 372 472 110 639 301 855 040

16 8886110 . 520 507 872 636 763 023 722

17 | 24154952 .'753 575 298 214 775 435 130

18 65659969 . 137 330 511 138 786 503 12

19 178482300 . 963 187 260 844 910 003 4

20 485165195 . 409 790 277 969 106 829 3

21 1318815734 . 483 214 697 209 998 880 2

22 | 3584912846 .131 591 561 681 159 934
23 | 9744803446 . 248 902 600 034 632 654
24 | 26489122129 . 843 472 294 139 162 068

C= 72004899337 . 385 872 524 161 351 466 126
= 10686474581524 . 462 146 990 468 650 741 2

e® = 114200738981568428366295718 . 314 472

This table was partially verified in the following manner. Since

a1
a

Hse apSeOF cog FEO =

246 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

it follows that unity added to the sum of the first twelve powers of e is equal to
eS —

1 : :
<A , and that unity added to the sum of the first twenty-four powers of e is equal
ae N

Retaining 23 places of decimals, we find by addition from Schulze’s table

ltete’...+e7= 257473 . 706 979 5383 059 990 318 032 45,
and by division, taking Schulze’s value of e”,

oil
e-1

which verifies the values of the first thirteen powers of e to 22 places of decimals.

= 257473 . 706 979 533 059 990 318 032 37,

Similarly by addition we find
lteteé...+e%= 41905174194 . 247 197 714 849 662 8,
and by division, taking Schulze’s value of e”,

< =f = 41905174194 247 197 714 849 663 0.

The values from e“ to e* are thus verified to 15 places of decimals.

2. In connexion with the exponential function I may here give the following
1 1 1 1

21° 31? 4p" 500?
the values of series having factorials in their denominators.

wp

values of which I worked out on account of their use in calculating

The figures enclosed in brackets denote the numbers of ciphers occurring between the

decimal point and the first significant figure. From a to a the number of significant

figures given is twenty-eight.

i 1 .

5 = 05, = 0001 38,

1 016 1 0,000 198 412 ¢

4-016, 7 = 01000 198 412 6,

1 ‘ 1 : :

P= 00416, == 0'000 024 S01 5873,

+ = 0-0088, 510000 002 755 731 922 398 589 065 8,
517000 000 275 573 192 239 858 906 53,
1; =0:000 000 025 052 108 385 441 718 7,
= 0:000 000 002 O87 675 698 786 809 897 921 009 032 120 143 231 254 342 365

453 476 564 5,

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 247

13 | 0: (9) 160 590 438 368 216 145 993 923 771
14 | 0-(10) 114 707 455 977 297 247 188 516 979
15 | 0:(12) 764 716373 181 981 647 590 113 198
16 | 0-13) 477 947 733 238 738 529 743 820 749
17 | 0:(14) 281 145 725 434 552 076 319 894 558
18 } 0-(15) 156 192 069 685 862 264 622 163 643
19 | 0-(17) 822 063 524 662 432 971 695 598 123
20 | 0-(18) 411 031 762 331 216 485 847 799 061
21 | 0-(19) 195 729 410 633 912 612 308 475 743
22 | 0:(21) 889 679 139 245 057 328 674 889 744.
23 | 0:(22) 386 817 017 063 068 403 771 691 193
24 | 0-(23) 161 173 757 109 611 834 904 871 330
25 | 0:(25) 644 695 028 438 447 339 619 485 321
26 | 0:(26) 247 959 626 322 479 746 007 494 354
27 | 0:(28) 918 368 986 379 554 614 842 571 683
28 | 0:(29) 327 988 923 706 983 791 015 204 172
29 | 0(30) 113 099 628 864 477 169 315 587 645
30 | 0:(32) 376 998 762 881 590 564 385 292 152
31 | 0:(33) 121 612 504 155 351 794 962 997 468
32 | 0(35) 380 039 075 485 474 359 259 367 089
33 | 0:(36) 115 163 356 207 719 502 805 868 814
34 | 0(38) 338 715 753 552 116 184 723 143 573
35 | 0:(40) 967 759 295 863 189 099 208 981 638
36 | O-(41) 268 822 026 628 663 638 669 161 566
37 | 0(43) 726 546 017 915 307 181 588 274 503
38 | 0-(44) 191 196 320 504 028 192 510 072 237
39 | 0:(46) 490 246 975 651 354 339 769 415 994
40 | 0-(47) 122 561 743 912 838 584 942 353 998
41 | 0:(49) 298 931 082 714 240 451 078 912 191
42 | 0-(51) 711 740 673 129 143 981 140 267 122
43 | 0-(52) 165 521 086 774 219 518 869 829 563
44 | 0-(54) 376 184 288 123 226 179 249 612 644
45 | 0(56) 885 965 084 718 280 398 332 472 542
46 | 0:(57) 181 731 540 156 147 912 680 972 291 8
47 | 0:(59) 386 662 851 396 059 388 682 919 769 8
48 | 0:(61) 805 547 607 075 123 726 422 749 520 4
49 | 0:62) 164 397 470 831 657 903 351 581 534 8
50 | 0:(64) 328 794 941 663 315 806 703 163 069 6

oononte rr weowoaaawonrt=ainairav aun wonton auwdrioaoorn

oo ff

248 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

These values were obtained by actual division, each result being deduced from the
preceding one; that is to say, the value of + was derived from that of CEST by
dividing it by n.

The results should be in all cases correct to the last figure, as several figures
were rejected.

By addition we find

p= 1548 080 634 815 243 778 477 905 620 757 061 682 6,

*
e-

lt Stata

oh +3 +geh 175 201 193 643 801 456 882 381 850 595 600 815 2,

giving
: e =2°718 281 828 459 045 235 360 287 471 352 662 497 8,

é€'=0°367 879 441 171 442 321 595 523 770 161 460 867 4,

which are correct to the last figure.

We do not thus obtain, es a good verification of the values of the reciprocals

of the factorials even as far as ae for, of those beyond = (in which only 28 significant

figures are given in the table on the preceding page) the first only, =e is verified to

its full extent, the next, te is verified to only 27 figures and so on, the first figure

alone being verified"in the case of sar

It seems worth while to give in detail the calculation of e” and e” by means of the
preceding values of the reciprocals of the factorials. The values of the different terms of
the 7 are shown on the opposite page, and it will be seen that ten figures (3287949417)

of o are thus verified. In order to complete the calculation it was necessary to find the values
of = and the subsequent terms to a few places of decimals. The last term included
‘3 that savoli 1

is that mvolving 62!"

The first column contains the values of the terms involving factorials of even numbers
and the second column those involving factorials of uneven numbers in the series

as 10s" _

1
1+10+ Pee i Sane

Gi + &e.

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION, 249

Calculation of e° and e from the series.

(Even terms.) (Uneven terms.)

50 10
416°666 666 666 666 666 666 666 667 166°666 666 666 666 666 666 666 667
1388°888 888 888 888 888 888 888 889 833 °333 333 333 333 333 333 333 333
2480°158 730 158 730 158 730 158 730 1984°126 984 126 984 126 984 126 984
2755 °731 922 398 589 065 255 731 922 2755°731 922 398 589 065 255 731 922
2087°675 698 786 809 897 921 009 032 2505°210 838 544 171 877 505 210 839
1147°074 559 772 972 471 385 169 798 1605°904 383 682 161 459 939 237 717
477 °947 733 238 738 529 743 820 749 764°716 373 181 981 647 590 113 199
156°192 069 685 862 264 622 163 643 281°145 725 434 552 076 319 894 558
41°103 176 233 121 648 584 779 906 82°206 352 466 243 297 169 559 812
8°896 791 392 450 573 286 748 897 19°572 941 063 391 261 230 847 574
1°611 737 571 096 118 349 048 713 3°868 170 170 630 684 037 716 912
247 959 626 322 479 746 007 494 644 695 028 438 447 339 619 485
32 798 892 370 698 379 101 520 91 836 898 637 955 461 484 257
3 769 987 628 815 905 643 853 11 309 962 886 447 716 931 559
380 039 075 485 474 359 259 1 216 125 041 553 517 949 630
338 871 575 355 211 618 472 115 163 356 207 719 502 806
2 688 220 266 286 636 387 9 677 592 958 631 890 992
191 196 320 504 028 193 726 546 017 915 307 132
12 256 174 391 283 858 49 024 697 565 135 434
711 740 673 129 144 2 989 310 827 142 405
37 618 428 812 323 165 521 086 774 220
1 817 315 401 561 8 359 650 847 183
80 554 760 708 386 662 851 396
3 287 949 417 16 439 747 083
123 979 993 644 695 964
4 331 935 23 392 452
140 647 787 625
4.254 24 675
120 : 721
3 20
11013232 920 103 323 139 721 376 087 11013 °232 874 703 393 377 236 524 556

By adding and subtracting the sums of the two columns we find
e° = 22026465 794 806 716 516 957 900 648,
Gv = 0:000 045 399 929 762 484 851 531.
The value of e° to 24 decimal places is given by Schulze in the table reprinted

on p. 245. It differs from that given by this calculation only in the last figure, the last
three figures in Schulze’s table being 641.

Won, 00 Jee Ue 33

250 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.
As an additional verification I have made the following calculation of e™:

e jv» _ 4539992976 +h

Loves
then — 10 = log’ (4539992976 +h) — 14 log, 10,
and therefore log, (4539992976 + h) = 14 log, 10 — 10.

The object is to determine from this equation, the value of log, 4539992976 being
known, for
4539992976 = 1296 x 1763 x 1987,

and, taking the logarithms of 1296, 1763 and 1987 from Wolfram’s table*, we find
log, 4539992976 = 22°236 191 301 861 907 O78 9.
Also 14 log, 10 —10 = 22°236 191 501 916 639 576 3.

Putting therefore x = 4539992976, we have
log, (w+ h) — log, x= 07000 000 000 054 732 497 355 4.

h h? he
Now log, (@ + h)—log,e =" — 3 a t+3 73— &e.,
Hie
and therefore h=z2 {log, (x ae h) = log, x} an 3 tn 4 = 4+ &e,

By multiplication we find
a flog, (a +h) — log, «} =0°248 485 153 552 351 449 550.

Taking log, (#+h)—log,e as an approximate value of E , we have

* = 0-000 000 000 054 732 497 355,
2
and therefore +~ = 0-000 000 000 006 800 106 505,
whence h=0248 485 153 559 151 556 055;

and, except for last-figure errors, this value should be correct as far as it extends. We
thus find
€”=0:000 045 399 929 762 484 851 535 591 515 560 6,

which differs from the value found above for &” by 4 in the twenty-fourth place.

* This table gives the hyperbolic logarithms of all on the first page of this paper, and was reprinted with
numbers up to 2,200 and of primes, and also ofagreat many _ the addition of six logarithms that were omitted through
composite numbers, up to 10,009 to 48 places of decimals. Wolfram’s death, in Vega’s Thesaurus logarithmorum com-
It was first published in Schulze’s Sammlung, referred to _pletus (1794).

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 251

To verify the accuracy of this value of e® I found by division the reciprocal of

22026465 794 806 716 516 957 900 643,

the result being
0:000 045 399 929 762 484 851 535 591 515 565,

10

which agrees to 32 places with the value of e” just found.

I have thought it worth while to give this calculation of ¢* at some length, as the
method affords perhaps the most convenient means of calculating e for an isolated value
of z when a considerable number of figures are required. The principle of the method is
as follows. The first nine or ten figures of the value being obtained from the Tables,
or calculated by logarithms independently, we seek for a number near to it which can be
resolved into factors, none of which exceed 10,009, the limit of Wolfram’s table. Denoting
this number by z, we then obtain log,v, and it only remains to calculate h from the formula

( hV h?
h =a {log, (e + h) —log,2} +4 —— $=, + &e.

by repeated approximation.

In calculating a table of ¢ for successive integral values of z, such as Schulze’s,
it might be well to form the table by actual multiplication, and to verify the final
value by an independent calculation in this manner.

In order to verify absolutely the accuracy of the values of the reciprocals of fac-
torials given above, to the full extent of the 28 figures, I formed the value of log, (50!)
by adding up the logarithms of 2, 3,...50, and I also calculated the logarithm of the

and thence deduced the value of

twenty-eight figure number given as the value. of =

log, (50 !).

Adding together the values of the logarithms of the first 50 numbers given in ©
Wolfram’s table and retaining 28 decimals, we find

log, (50 !) = 148-477 766 951 773 032 067 537 193 850 9.

Now taking the value of aT given on p. 247, we have
1 _ 328794941:663 315 806 703 163 069 6
50! — 10% ?

whence we ought to find

log, (50 !) = 78 log, 10 — log, 328794941°663 315 806 703 163 069 6.

To calculate the logarithm of 328794941663 315 806 703 163 069 6 we notice
that 328794943 = 17 x 19 x 569 x 1789, whence, taking the logarithms from Wolfram’s table,

log, 328794943 = 19610 944 840 857 706 621 131 134 384 5,
33—2

252 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

and putting 328794943 =, and denoting the number whose logarithm is required by a—h

we have
2 8
log, (w—h) =log, 2" 45-3 1 be.
where h=1°336 684 193 296 836 930 4.
Thus log, (50 !) = 73 log, 10 2 ae + es er 3 = + &e. — log, a,

and the final steps of the calculation are as follows:

73 log, 10 =168°088 711 788 565 334 933 313 376 191 9
4 065 403 747 091 198 207 4

8 263 753 813 4

22 4

168088 711 792 630 738 688 668 328 235 1
log,x= 19610 944 840 857 706 621 131 134 384 5

148:477 766 951 773 032 U67 5387 193 850 6

which differs by 3 in the last place from the value of log, (50!) found by addition.

The value of = is therefore verified, and as each value was derived from the pre-

ceding one by division, this affords a verification also of the values of the other reciprocals
of factorials.

A table of log,,(x!) from #=1 to 2=1200 to 18 places of decimals was given
by C. F, Degen in his “Tabularum Enneas” (Copenhagen, 1824). This table was re-
printed by De Morgan, in the Article “Probabilities” in the Encyclopedia Metropolitana,
the number of decimal places, however, being reduced to six.

The value of e was calculated by Mr Shanks to 205 places, and published by him to
this extent in Vol. vi, p. 397 (1854) of the Proceedings of the Royal Society. In his
“Rectification of the Circle” (1853) Mr Shanks had given the value to 137 places, and
this result I verified in 1871*, Mr Shanks calculated his value by means of the series

e=1l+7, £ ity z taty L + &e.,
and I used the continued-fraction formula

e—l. 1 A 1 1

Fe ois 6s 0a. ane ee
The value thus verified is
e=2°718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699
959 574 966 967 627 724 076 630 353 547 594 571 382 178 525 166
427 427 466 391 932 003 059 921 817 413 596 629 043 57.

* Report of the British Association for 1871 (pp. 16—18), (Sectional Proceedings).

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 253

The value of the reciprocal of e, given by Schulze (Sammlung, 1778, p. 188), is
e* = 0367 879 441 171 442 321 595 523 770 161 460 867 445 811 12.

T may mention that I have calculated the value of e from the series, retaining nine
decimal places. The sums of the terms involving even and uneven powers were found
to be respectively 4443055°260 253 99 and 4443055-260 258 88, giving

e” = 8886110520 507 87, ¢%=0:000 000 11.

This value of ¢™ is correct. The value of e® given in Schulze’s table (ante, p. 245)
is 8886110520 507 872..., which is thus directly verified to fifteen figures.

The modulus M of the common or Briggian logarithms is equal to log,, e, and its value
has been given by Professor J. C. Adams to 282 places of decimals in Vol. xxvmL., p. 93
(1878) of the Proceedings of the Royal Society. This value is reprinted in the article
“Logarithms” in the Encyclopedia Britannica (1882).

It may be here remarked that Gauss’s posthumous memoir “De curva lemniscata”
9
(Werke Vol. 111, pp. 413—432) contains the values of e-*, e-4*, and e~4” to 50 or more

25 49
places of decimals, and the value of e#* to 34 places. The values of e-®*, e~ 4" and e74”
are also given to sixteen, twenty-four, and twelve significant figures respectively.

The values of “Ve for integral values of n may be readily calculated from the series
by means of the values of the reciprocals given above, but they may also be very
conveniently obtained from the formula expressing %/e as a continued fraction, viz.

1
eA, Sa Sila Big sag Ne
2 ~ W—1+ 6n+ 10n + 14 + &e. *

254 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE I.
Values of e*, e~*, log,,(e), log, (e7*) from «=0'001 to #=0'T00 at intervals of o’oor.
x log,.(e*) é
o*001 0°00043 42945 I'0O I00 050
07002 0'00086 85890 I'0O0 200 200
0'003 0'00130 28834 I‘00 300 450
0°004 0°00173 71779 1°00 400 801
0°005 0°002T7 14724 I'0O0 501 252
07006 0°00260 57669 1°00 601 804
0'007 0700304 00614 T°00 702 456

0°008 0°00347 43559 I'00 803 209
07009 0°00390 86503 1°00 904 062

o'oIo 0'00434 29448 I‘OI 005 O17
o’oll 0'00477 72393 I’OI 106 072
o'ol2 0°00521 15338 I‘OI 207 229

O'013 0°00564 58283 T'or 308 487
o'o14 060608 01227 ror 409 846
O'OI5 0'00651 44172 I‘ol 511 306
o'016 0'00694 87117 Tor 612 869
O'O17 0'00738 30062 | I'or 714 532
o'018 0'0078I 73007 rot 816 298
oop de) 0'00825 15952 Tor 918 165
07020 0°00868 588096 I'02 020 134
0'021 o'00g912 o184r I'02 122 205
0'022 0'00955 44786 I'02 224 378
0°023 0°00998 87731 T'02 326 654
0'024 O'01042 30676 I'O2 429 032
0'025 0'01085 73620 T'O2 531-512
0°026 o'O1I29 16565 1°02 634 095
0'027 O°0I172 59510 1°02 736 780
0'028 O'01216 02455 t'02 839 568
0'029 0°0I1259 45400 1°02 942 459
0'030 0'01302 88345 I°03 045 453
0°031 0°01346 31289 1°03 148 550
0'032 0°01389 74234 T03 251 75.
BUCSs 0°01433 17179 T°03 355 954:
0°034 c'01476 60124 1°03 458 461
0°035 0701520 03069 1°03 561 971
0°036 0701563 46013 1°03 665 585
0'037 0'01606 88958 1°03 769 302
0'038 0°01650 31903 1'03 873 123
0039 0'01693 74848 1°03 977 048
0'040 0°01737 17793 1°04 O81 077
O'O4I 0'01780 60738 1°04 185 211
0°042 0'01824 03682 1704 289 448
0'043 0'01867 46627 104 393 789
0'044 o'or1gto 89572 104 498 235
0°045 001954 32517 1°04 602 786
0'046 001997 75462 104 707 441

0'047 0702041 18406 1°04 812 201
0°048 0°02084 61351 1'04 917 066
0'049 0°02128 04296 I'05 022 035
0'050 O'O217I 47241 TOs 1275110

i

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNOTION.

TABLE I. (continued).

255

Values of e”, e~*, log,,(e”), log,,(e~*) from «=o'001 to «=0'100 at intervals of o’oor.

x log,,(e) é Oe
O'051 0'02214 go186 105 232 289 0'950 278 671
0052 0°02258 33131 105 337 574 | 07949 328 867
0'053 0°02301 76075 1°05 442 964 0'948 380 o12
0'054 0°02345 19020 1°05 548 460 0°947 432 107
07055 0702388 61965 1°05 654 o61 0946 485 148
0056 | 0'02432 04910 T'05 759 768 | 07945 539 136
0°057 0°02475 47855 t'05 865 581 0'°944 594 069
0°058 0°02518 go800 1°05 97I 500 0°943 649 947
0'059 002562 33744 1°06 077 52 0°942 706 769
07060 0°02605 76689 1°06 183 655 o'941 764 534
o'061 002649 19634 1°06 289 891 0940 823 240
0'062 0'02692 62579 1°06, 396 234 0°939 882 887
0'063 0°02736 05524 1706 502 684 0°938 943 474
07064 0°02779 48468 1°06 609 240 0°938 005 000
07065 0°02822 gI413 1°06 715 go2 0'937 067 463
0'066 0'02866 34358 1'06 822 672 0936 130 864
0°067 002909 77303 1°06 929 548 0°935 195 201
0°068 0702953 20248 I'07 036 531 0°934 260 474
07069 0°02996 63193 I'°07 143 621 0'933 326 680
0'070 003040 06137 1°07 250 818 0°932 393 820
o7071 0°03083 49082 T'07 358 123 0°931 461 892
0'072 0°03126 92027 1°07 465 534 0'930 530 896
0'073 0°03179 34972 107 573 054 "929 600 830
0'074 0°03213 77917 1°07 680 681 07928 671 694
0'075 0°03257 20861 ro7 788 415 0927 743 486
0'076 0°03300 63806 107 896 257 0926 816 207
0'077 0'03344 06751 1°08 004 208 07925 889 854
0'078 0'03387 49696 1'08 112 266 0'924 964 427
0079 0°03430 92041 1°08 220 432 0°924 039 924
07080 0°03474 35586 108 328 707 0'923 116 346
o'08r 0°03517 78530 1°08 437 ogo 0°922 193 691
0082 0'03561 21475 108 545 581 O'921 271 959
0'083 0°03604 64420 108 654 181 0°920 351 147
0'084 0'03648 07365 108 762 889 O'919 431 256
0°085 0703691 50310 108 871 707 o'918 512 284
0'086 0°03734 93254 1°08 980 633 O°917 594 231
0°087 0'03778 36199 1°09 089 668 0°916 677 096
0'088 0'°03821 79144 1°09 198 812 o'915 760 877
07089 0°03865 22089 Tog 308 066 0914 845 574
o'090 0'03908 65034 I‘0g 417 428 0°913 931 185
O'0gr 0°03952 07979 1°09 526 gor O°913 O17 7II
0'092 0703995 50923 10g 636 482 O'912 105 150
0'093 0°04038 93868 1:09 746 174 O'9II 193 500
0'094 0°04082 36813 Tog 855 975 o'910 282 762
07095 0704125 79758 Tog 965 886 0°909 372 934
07096 0704169 22703 I'IO 075 go06 07908 464 016
0'097 0°04212 65647 I'Io 186 037 0°907 556 006
07098 0°04256 08592 I'Io 296 279 0'906 648 904
0°099 0704299 51537 I'1o 406 630 0905 742 708
o*100 0°04342 94482 I‘IO 517 092 0904 837 418

HIRT RT RTL RRR RR RR RT RT RR RRR RR RR RR]

SOUS SSUNUUSE
~_
on
O°
~

7220

682
6786

6699
6655

6482
6438
6395
6351
6308

6221

"95874
"95830

256 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE II.

Values of e*, e~*, log,,(e*), log,.(e~*) from «=o'o1 to x= 2°00 at intervals of oor. |

ae *) Botlog: (2) e e* log,,(e-*)

oror 0700434 29448 Tor 005 O©7 0990 049 834 199565 70552
0°02 0700868 58896 I'02 020 134 o'980 198 673 I°Q9I13I 41104
0°03 001302 88345 1°03 045 454 0°979 445 534 | 1°98697 11655
0°04 0'01737 17793 1°04 O81 077 0960 789 439 1°98262 82207
0°05 O°0217I 47241 | 1°05 127 IIo O°951 229 425 1797828 52759
0°06 0'02605 76689 1°06 183 655 0941 764 534 1°97394 23311

0°07 0°03040 06137 | I'07 250 818 0°932 393 820 1796959 93863
0°08 0°03474 35586 1°08 328 707 07923 116 346 196525 64414
o'09 0°03908 65034 I'0g 417 428 0'°913 931 185 196091 34966
o'Io 0°04342 94482 I‘IO 517 092 0°904 837 418 195657 05518
oll 0°04777 23930 I'Ir 627 807 0°895 834 135 1'95222 76070
o'12 0°0521I 53378 I'r2 749 685 0°886 920 437 194788 46622
O13 0705645 82826 113 882 838 0°878 095 431 194354 17174
o'r4 0'06080 12275 I°I5 027 380 0869 358 235 193919 87725
O'1s5 0°06514 41723 1°16 183 424 0'860 707 976 193485 58277
o'16 0'06948 7II7I I'l] 351 087 0°852 143 789 193051 28829
O'l7 0'07383 00619 1°18 530 485 0°843 664 817 792616 99381
o'18 0'07817 30067 | I°Ig 721 736 0°835 270 211 "92182 69933
o'1g 0708251 59516 | 1°20 924 960 0°826 959 134 "91748 40484

I

I

I

I

I

I

I

9
0 20 0°08685 88964 "22 140 276 0°818 730 753 91314 11036
o'2I o'09120 18412 "23 367 806 o'810 584 246 "90879 81588
0°22 0°09554 47860 "24 607 673 0802 518 798 "99445 52140
0°23 0'09988 77308 *25 860 oor 0'794 533 603 gQooIl 22692
0°24 0710423 06757 "27 124 915 0786 627 861 89576 93243
0°25 010857 36205 °28 402 542 0°778 800 783 "89142 63795
0'26 O'II291 65653 "29 693 009 0°77I 051 586 8
0°27 O'11725 95104 1°39 996 445 0°763 379 494 8
0°28 o'12160 24549 1°32 312 981 0°755 783 741 87839 75451
o'29 012594 53998 | 1°33 642 749 0°748 263 568 87405 46002
0°30 0°13028 83446 1°34 985 881 0°740 818 221 186971 16554
o'31 0°13463 12894 1°36 342 511 0°733 446 956 1'86536 87106
0°32 0°13897 42342 1°37 712 777 0°726 149 037 186102 57658
0°33 0°1433I 71790 1°39 096 813 0°718 923 733 185668 28210
0°34 0°14766 01238 1°40 494 759 O'711 770 323 | 1°85233 98762
0°35 0°15200 30687 I°4I 906 755 0°704 688 ogo 184799 69313
| 0°15634 60135 1°43 332 942 0°697 676 326 184365 39865
0°37. | 0'16068 89583 1°44 773 462 0°690 734 331 183931 10417
0°38 | 016503 19031. 1°46 228 459 0°683 861 409 1783496 80969
0°39 | 0°16937 48479 | 1°47 698 079 0°677 056 874 183062 51521
0°17371 77928 | 1°49 182 470 0670 320 046 1'82628 22072
041 | 017806 07376 1°50 681 779 0°663 650 250 182193 92624
0°42 0°18240 36824 1°52 196 156 0°657 046 820 181759 63176
0°43. | 018674 66272 P53 725752 0°650 509 095 181325 33728
0°44 | o'1g108 95720 I'55 270 722 0°644 036 421 180891 04280
0°19543 25169 | 1°56 831 219 0637 628 152 1'80456 74831
0°46 | 019977 54617 1°58 407 399 0°6314 283 646 180022 45383
0°47 0720411 84065 1°59 999 419 0'625 002 268 1°79588 15935
0°48 0°20846 13513 1°61 607 440 0°618 783 392 1'79153 86487
0°49 0°21280 42961 1°63 231 622 0612 626 394 1°78719 57039
0°50 0°21714 72410 1°64 872 127 0°606 530 660 1'78285 27590

88708 34347
88274 04899

to
on
~“I

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE II. (continued).

Values of e”, e~”, log,,(e”), log, (e*) from «=o'o1 to #=2°00 at intervals of ovor.

o'5I 0°22149 01858 | 1°66 529 120 0600 495 579
0°52 0°22583 31306 | 1°68 202 765 07594 520 548 |
0°53 0°23017 60754 1°69 893 231 0588 604 970 |
0°54 0°23451 goz02 | 1°71 600 686 0'582 748 252
0°55 0°23886 19650 | 1°73 325 302 0576 949 810
0°56 0°24320 49099 | 1°75 067 250 | 0571 209 064
0°57 0'24754 78547 | 1°76 826 705 | 0°565 525 439
0°58 0'25189 o7995 | 1°78 603 843 | 07559 898 367

7

yi

7

9)

17

Te

77

ey,
0°59 0°25623 37443 | 1°80 398 842 0°554 327 285 174376 62557
o'60 0°26057 66891 1°82 211 880 0°548 S11 636 1°73942 33109
061 0°26491 96340 1°84 043 140 | 0°543 350 869 1°73508 03660
0°62 0'26926 25788 | 1°85 892 804 0°537 944 438 1°73073 74212 |
0°63 0°27360 55236 | 1°87 761 058 07532 591 Sor 1°72639 44764
0°64 0'27794 84684 1°89 648 088 0°527 292 424 1°72205 15316
0°65 0°28229 14132 | I'91 554 083 0°522 O45 777 171779 85868
066 | 028663 43581 | 1°93 479 233 | 0°516 851 334 | 1°71336 56419 |
0°67 0°29097 73029 1°95 423 732 o'511 708 578 1'70902 26971
0°68 0°29532 02477 1-0}7 387) 773 0'506 616 992 1°70467 97523
0°69 0°29966 31925 | 1°99 371 553 o501 576 069 1°70033 68075
0°70 0°30400 61373 2200 ea 7/527 0496 585 304 169599 38627
o'71 0°30834 90822 2°03 399 126 o'49I 644 197 169165 09178
0°72 0°31269 20270 2°05 443 321 0°486 752 256 168739 79730
073 0°31703 49718 | 2°07 508 o61 0481 908 gg90 1°68296 50282
0°74 0°32137 79166 | 2°09 593 551 0477 113 916 167862 20834
0'75 0°32572 08614 | 2°11 700 002 0°472 366 553 167427 91386 |
0°76 0°33006 38062 | 2°13 827 622 0467 666 427 166993 61938 |
o'77 0°33440 67511 2°15 976 625 0°463 013 068 166559 32489 |
0°78 0°33874 96959 | 2°18 147 227 07458 406 ol! 166125 03041 |
079 0°34309 26407 2°20 339 643 07453 844 795 165699 73593
080 0°34743 55855 2°22 554 093 0449 328 964 165256 44145
o'81 0°35177 85303 2°24 790 799 0°444 858 066 164822 14697 |
o'82 0°35612 14752 2°27 049 984 0°440 431 654 164387 85248 |
0°83 0°36046 44200 | 2°29 331 874 0°436 049 286 163953 55800 |
0°84 0°36480 73648 | 2°31 636 698 0°431 710 523 163519 26352 |
0°85 0736915 03096 | 2°33 964 685 0°427 414 932 | 163084 96904 |
0°86 0°37349 32544 | 2°36 316 069 0'423 162 082 | 1:62650 67456 |
0°87 0°37783 61993 | 2°38 691 085 o°418 951 549 162216 38007
0°88 038217 91441 | 2°41 089 971 07414 782 gI2 161782 08559
0°89 0°38652 20889 | 2743 512 965 O'410 655 753 161347 79111
0'90 0°39086 50337 2°45 960 311 0°406 569 660 160913 49663
o'91 039520 79785 | 2°48 432 253 0402 524 224 160479 20215
092 0°39955 09234 2°50 929 039 0°398 519 O41 160044 90766
0'93 0740389 38682 2°53 450 918 0°394 553 710 159610 61318
0'94 0740823 68130 _| 2°55 998 142 0°390 627 835 159176 31870
0°95 0°41257 97578 2°58 570 966 0°386 741 023 1°58742 02422 |
0'96 0°41692 27026 2°61 169 647 0°382 892 886 1°58307 72974 |
0°97 | 0°42126 56474 2°63 794 446 0°379 083 038 1°57873 43520
0°98 0°42560 85923 2°66 445 62 0°375 311 099 157439 14077
0°99 0°42995 15371 2°69 123 447 0°371 576 691 1757004 84629 |
1°00 0°43429 44819 271 828 183 0°367 879 441 156570 55181

Vou. XIIL Parr Il. ants oe 34

258 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE II. (continued).

Values of e*, e~*, log,,(e), aa *) from w=o'o1 to «= 2°00 at intervals of ovor.

ole

oe 4 log, (e) | & ed | log, (e~”)

lor | 043863 74267 | 2°74 560 ror | 0°364 218 980 | eens 25733
To2 | 0°44298 03715 | 2°77 319 476 0°360 594 940 155701 96285
1:03 | 044732 33164 | 2°80 106 584 0°357 006 961 155267 66836
1°04 | 0°45166 62612 2°82 g2I 701 0°353 454 682 1°54833 37388
1°05 | 0°45600 92060 | 2°85 765 112 °°349 937 749 154399 07940
106 0°46035 21508 2°88 637 099 | 0°346 455 810 1°53964 78492
107 0°46469 50956 | 2°91 537 959 | 0343 008 517 1753539 49044
1°08 | 0746903 80405 | 2°94 467 955 | 0°339 595 526 | 1°53096 19595
1°09 | = 0°47338 09853 2°97 427 407 | 0336 216 494 | 1°52661 90147
ib fe) 0°47772 39301 3°00 416 602 0'332 871 084: | 1°52227 60700
Itt 0°48206 68749 3°03 435 839 |. 0329 558 961 151793 31251
112 048640 98197 3°06 485 420 0°326 279 795 | 1°51359 01803
1-33 0°49075 27646 3°09 565 650 0°323 033 256 1'50924 72354
I'l4 0°49509 57094 3°12 676 837 0319 819 022 150490 42906
Ils 0749943 86542 315 819 291 | 0°316 636 769 | 1750056 13458
1°16 0750378 15990 3°18 993 328 | 0-313 486 181 | 1:49621 840I10
liz 0750812 45438 3°22 199 264 0°310 366 g41 1°49187 54562
118 0751246 74886 3°25 437 420 | 0307 278 739 1°48753 25114
IIg | 0°51681 04335 3°28 708 121 0°304 221 264 1°48318 95665
1°20 | 052115 33783 3°32 O11 692 O'30I 194 212 1°47884 66217
r2r 0°52549 63231 3°35 348 465 0298 197 279 | 1°47450 36769
1:22 0°52983 92679 3°38 718 773 0°295 230 167 1°47016 07321
1°23 0°53418 22127 3°42 122 954 0°292 292 578 1°46581 77873
r24 | 0753852 51576 3°45 561 347 0289 384 218 146147 48424
r25 | 054286 81024 3°49 034 206 0'286 504 797 1°45713 18976
1°26 0'54721 10472 3°52 542 149 0'283 654 027 1°45278 89528
I'27 0°55155 39920 | 3756 085 256 0°280 831 622 1°44844 60080
1°28 0755589 69368 | 3°59 663 973 0'278 037 301 1°44410 30632
1°29 0°56023 98817 | 3°63 278 656 0°275 270 783 1°43976 01183
130 | 0°56458 28265 | 3°66 929 667 | 0:272 531 793 143541 71735
1°31 0°56892 57713 3°70 617 371 0°269 820 056 1°43107 42287
132 0°57326 87161 | 3°74 342 138 0'267 135 302 1°42673 12839
1°33 0°57761 16609 3°78 104 339 0°264 477 261 1'42238 83391
1°34 0°58195 46058 3°81 904 351 o'261 845 669 | 1741804 53942
1°35 0°58629 75506 | 385 742 553 0'259 240 261 141379 24494
1°36 | 0759064 04954 3°89 619 330 0°259 660 777 1°40935 95046
1°37 059498 34402, | 3°93 535 0709 | 0°254 106 g60 | 40501 65598
1°38 0°59932 63850 | 3°97 490 163 o'251 578 553 1'40067 36150
1°39 060366 93298 | 401 485 005 0'249 075 305 1°39633 06701
149 | o60801r 22747 | 4°05 519 997 | 0'246 596 964 | 1°39198 77253
141 | 061235 52195 4°09 595 541 0°244 143 283 138764 47805
142 | 061669 $1643 4°13 712 044 0'241 714 O17 1°38330 18357
143 | 062104 11091 417 869 919 0'239 308 g22 1°37895 88909
1°44 | 0'62538 40539 4°22 069 582 0°236 927 759 1°37461 59461
1°45 062972 69988 4°26 311 452 0°234 570 288 1°37027 30012
1°46 0763406 99436 | 4°30 595 953 0°232 236 275 136593 00564
1°47. | 063841 28884 4°34 923 514 0°229 925 485 1°36158 71116
1°48 0°64275 58332 4°39 294 568 0'227 637 688 1°35724 41668
1°49 0°64709 87780 4°43 799 552 0°225 372 656 r °35290 12220
1°50 065144 17229 | 4°48 168 907 0°223 130 160 1°34855 82771

= ee

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION,

Values of ¢”, e~*, log,,

TABLE ILI. (continued).

Dp)

a

(e*), log,,(e”) from xw=o'or to «=2'00 at intervals of oor.

x | log.(¢) e o log. (e~*)

151 0°65578 46677 4°52 673 079 0°220 gog 978 134421 53323
1°52 066012 76125 4°57 222 520 o218 711 887 133987 23875
53 0°66447 05573 461 817 682 0216 535 667 133552 94427
1'54 0'6688r 35021 4°06 459 027 O'214 381 IOI 1°33118 64979
1°55 0°67315 64470 4°71 147 018 O-212 247 973 132684 35531
1°56 © 67749 93918 4°75 882 125 0'210 136 O71 1°32250 06082
1°57 068184 23366 4°50 664 819 0'208 045 182 T'31815 76634
1°58 068618 52814 4°85 495 581 0°205 975 098 1°31381 47186
1°59 069052 82262 4°92 374 893 0'203 925 612 1°30947 17738
1°60 0°69487 11710 4°95 303 242 o'201 896 518 1°30512 88290
161 069921 41159 5°00 281 123 o'199 887 614 1°30078 58841
162 0°70355 70607 5°05 309 032 | o'197 898 699 | 1:29644 29393
1°63 LTS) Cielo) 5°10 387 472 195 929 574 129209 99945
1°64 0°71224 29503 Bit5 510 95 2'193 980 042 128775 79497
1°65 071658 58951 5°20 697 983 O'192 049 909 128341 41049
1°66 0°72092 88400 5°25 931 084 0190 138 980 127907 I1600
1°67 072527 17848 5°31 216 780 0188 247 066 127472 82152
1°68 0°72961 47296 5°36 555 597 0186 373 976 127038 52704
169 0°73395 70744 541 948 o71 0184 519 524 1°26604 23256
1°70 0°73830 06192 5°47 394 739 0182 683 524 126169 93808
ie 0°74264 35641 5°52 896 148 0180 865 793 1'25735 64359
172 0°74698 65089 5°58 452 846 0'179 066 148 125301 34911
173 0°75132 94537 5°64 065 391 o'r77 284 410 1°24867 05463
1°74 075567 23985 5°69 734 342 O°175 520 4o1 1°24432 76014
175 076001 53433 5°75 460 268 °'173 773 944 123998 46567
1°76 0°76435 82881 Frou) 243) 739 o'172 044 864 1'23564 17119
1'77 0°76870 12330 5°87 085 336 O°170 332 989 1°23129 87670
1°78 0°77304 41778 5°92 985 642 0168 638 147 1°22695 58222
1°79 0°77738 71226 5°98 945 247 0°166 g60 170 122261 28774
1°80 0°78173 00674 6°04 964 746 0°165 298 889 1'21826 99326
1°81 0°78607 30122 GII 044 743 0°163 654 137 1‘21392 69878
1°82 0°79041 59571 6°17 185 845 0'162 025 751 1'20958 40429
1°83 0°79475 89019 6°23 388 666 0°160 413 568 1°20524 10981
1°84 079910 18467 6:29 653 826 0'158 817 426 120089 81533
1°85 0°80344 47915 6°35 981 952 O'157 237 166 119655 52085
1°86 080778 77363 6°42 373 677 0155 672 630 T19221 22637
1°87 9°81213 06812 6°48 829 640 O'154 123 662 118786 93188
1°88 081647 36260 6°55 350 486 O°r52 590 106 1'18352 63740
1°89 082081 65708 6°61 936 868 O'151 o71 809 T'17918 34292
I‘g0 082515 95156 6°68 589 444 0'149 568 619 117484 04844
I°gt 0'82950 24604 6°75 308 880 0148 080 387 117049 75396
192 0°83384 54053 6°82 095 847 0°146 606 962 T16615 45947
I'93 083818 83501 6°88 951 024 07145 148 199 T'16r8r 16499
T-94 0°84253 12949 6°95 875 097 97143 703 950 115746 87051
1°95 0°84687 42397 7°02 868 758 O'142 274 072 I°I5312 57603
1°96 o'85121 71845 7°09 932 707 o140 858 421 114878 28155
1°97 085556 01293 7°17 067 649 0'139 456 856 114443 98707
1°98 0'85990 30742 7°24 274 299 0°138 069 237 T'14009 69258
1°99 0°86424 60190 73% 553 376 0°136 695 426 113575 39810
2°00 086858 89638 7°38 905 610 0135 335 283 I‘IZI4I 10362

260 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE III.
Values of e", e°*, log,,(e*), log, (e-*) from #=o'r to w=10'0 at intervals of o'1.

x log, (e*) e er log,,(e~*)
Or | 0°04342 94482 | I'lIO 517 092 0°904 837 418 1°95657 05518
O72 | 008685 88964 1°22 140 276 0818 730 753 I'91314 11036
o°3 | 0'13028 83446 1°34 985 881 0740 818 221. 1°86971 16554
GAS) (Or7e7T 7770251. | 1°49 182 470 0'670 320 046 1°82628 22072
O°5 0°21714 72410 1°64 872 127 0°606 530 660 1'78285 27590
06 0'26057 66891 1°82 211 880 | 0548 811 636 1°73942 33109

|) be ez 0°30400 61373. = 20 375 271 =| «= 0496 585 304 | 1'69599 38627
oS 0°34743 55855 2°22 554 993 | 0°449 328 964 1765256 44145
o'9 0°39086 50337 2°45 960 311 0°406 569 660 160913 49663
bie) 0°43429 44819 2°71 828 183 0°367 879 441 1°56570 55181
I'l 0°47772 39301 3°00 416 602 | 0°332 871 084 1°52227 60699
2 O'52115 33783 3°32 O11 692 | o-30r% 194 212 1°47884 66217
T'3 | 0756458 28265 3°66 929 667 =| 0-272 531 793 | 143541 71735
14 060801 22747 405 519 997 = -0'246 596 964 139198 77253
15 0°65144 17229 448 168 907. | 0°223 130 160 1°34855 82771
16 0 69487 11710 | 4°95 303 242 o'201 896 518 1°30512 88790
137 0°73830 06192 | 5°47 394 739 | 0182 683 52 1'26169 93808
18 0°78173 00674 6°04 964 746 | o'165 298 888 1'21826 99326
19 082515 95156 | 6°68 589 444 | o0'149 568 619 1'17484 04843
2°0 0°86858 89638 7°38 905 610 0°135 335 283 I'I3141 10362
251 o'g1201 84120 | 8:16 616 got O'122 456 428 1:08798 15880
2°2 0°95544 78602 | 9°02 501 350 o'110 803 158 1104455 21398
2°3 099887 73084 9°97 418 246 o'100 258 844 TOOI12 26916
24 | 1704230 67566 Ir'0 231 764 (1)907 179 533 | 2°95769 32434 |
2°5 108573 62048 12°r 824 940 (1)820 849 986 2°91426 37952 |
2°6 I'I2916 56529 13°4 637 380 (1)742 735 782 2°87083 43471 |
2°7 I°17259 51011 14°8 797 317 (1)672 055 127 2°82740 48989 |
2°8 1'21602 45493 16°4 446 468 (1)608 100 626 2°78397 54507
3 125945 39975 181 741 454 | (1)550 232 201 | 274054 60025 |
30 1°30288 34457 | 20°0 855 369 (1)497 870 684 | 269711 65543
sar 1°34631 28939 22° 979 513 (1)450 492 02 2°65368 71061
ie 1°38974 23421 | 24°5 325 302 | (1)407 622 040 2°61025 76579
oye) 1°43317 17903 | 271 126 389 | (1)368 831 674 | 256682 82096.
3°4 1°47660 12385 | 29°9 641 COL (1)333 732 700 2°52339 87615
Sy 1°52003 06867 | 33°71 154 520 | (1)301 973 834 2°47996 93133
3°6 156346 01349 36°5 982 344 | (1)273 237 224 | 2°43653 98651
a7 160688 95830 | 40°4 473 044 (1)247 235 265 2°39311 04170
38 1°65031 90312 44°7 o11 845 (1)223 707 719 2°34968 09688
3°9 | 1°69374 84794 49°4 024 491 (1)202 419 114 | 2°30625 15206
4°0 1°73717 79276 54°5 981 500 (1) 183 156 389 226282 20724 |
41 | 1°78060 73758 60°3 402 876 (1)165 726 754 2°21939 26242 |
4°2 182403 68240 | 66°6 863 310 (1)149 955 768 2°17596 31760
4°3 186746 62722 | 73°6 997 937 (1)135 685 590 | 2°13253 37278
44 | 1I'91089 57204 81°4 508 687, (1)122 773 399 208910 42796
4'5 1'95432 51686 goo 171 313 (1)111 089 965 2°04567 48314
4°6 199775 46168 99°4 843 157 (1)100 518 357 2'00224 53832
47 2°04118 40649 109° 947 173 (2)909 527 710 | 3°95881 59351
4°8 2°08461 35131 | I2I° 510 418 (2)822 974 705 3°91538 64869
4°9 2°12804 29613 | 134° 289 780 (2)744 658 307 387195 70387
5‘0 2°17147 24095 148° 413 159 (2)673 794 700 3°82852 75905

The numbers in parentheses denote the numbers of ciphers between the decimal point and the
first significant figure ; for example, e~* = 0:00673794700.

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE III. (continued).

Values of e”, e7*, log,(e7), log,(e*) from x=o'r to e=10'0 at intervals of
bs log, (e”) e Ae log, (e7*)
| | ——— = =
sor 221490 18577 164° 021 907 (2) 609 674 657 ae 81423
Ge 2°25833 13059 I8I° 272 242 (2) 551 656 442 3°74166 86941
553 2°30176 07541 200° 336 810 (2) 499 159 391 369823 92459
| 5°4 2°34519 02023 221° 406 416 (2) 451 658 094 | 3°65480 97977
) Rs 2°38861r 96505 244° 691 932 (2) 408 677 144 | 3°61138 03495
pews.0 2°43204 90987 | 279° 426 407 (2) 369 786 372 356795 09013
ee 2°47547 85468 | 298° 867 4or (2) 334 596 546 | 3°52452 14532
sce 2°51890 79950 | 330° 299 560 (2) 302 755 475 | 3748109 20050
Sack 2°56233 74432 365° 037 468 (2) 273 944 482 | 343766 25568
| 670 2°60576 68914 403° 428 794 (2) 247 875 218 3°39423 31086
6:5 2°64919 63396 445° 857 770 (2) 224 286 772 3°35080 36604
0:2 2°69262 57878 492° 749 O41 (2) 202 943 064 | 3730737 42122
6°3 2°73605 52360 544° 571 910 (2) 183 630 478 3°26394 47649
6"4 2°77948 46842 | 601° 845 038 (2) 166 155 727 3°22051 53158
| 675 2°82291 41324 | 665° 141 633 (2) 150 343 919 317708 58676
66 | 2°86634 35806 | 735° 095 189 (2) 136 036 804 313365 64194
| 6°7 | 299977 30288 812° 405 82 (2) 123 ogt 190 3709022 69712
| 68 | 295320 24769 | 897° 847 292 (2) 111 377 515 304679 75231
| 6°9 2799663 19251 992° 274 716 (2) 100 778 543 300336 80749
(RS 304006 13733 109 6°63 316 (3) 911 881 966 4°95993 86267
7a 308349 08215 I2I 1°96 708 (3) 825 104 923 491650 91785
72 3°12692 02697 133. 9°43 977 (3) 746 585 808 487307 97303
73 317934 97179 148 0°29 993 (3) 675 538 775 482965 02821
74 3°21377 91661 163 5°98 443 (3) 611 252 761 4°78622 08339
75 325720 86143 180 8°04 242 (3) 553 084 370 4°74279 13857
76 330063 80625 199 819 590 (3) 500 451 433 4+:69936 19375
Rah 3°34406 75107 220 8°34 799 (3) 452 827 183 465593 24893
78 3°38749 69588 244 0°60 198 (3) 499 734 979 4°61250 30412
79 343092 64070 269 7°28 233 (3) 379 743 54° 4°56907 35930
8:0 347435 58552 298 0°95 799 (3) 335 462 628 452564 41448
81 3°51778 53034 329 446 808 (3) 393 539 138 448221 46966
8:2 356121 47516 364 0°95 031 (3) 274 653 570 4°43878 52484
8°3 360464 41998 402 3°87 239 (3) 248 516 827 4°39535 58002
84 3°64807 36480 444 7°06 675 (3) 224 867 324 4°35192 63520
8'5 369150 30962 491 4°76 884 (3) 203 468 369 4°30849 69038
| 86 | 3°73493 25444 543 165 959 | (3) 184 105 794 4'26506 74556
8-7 3°77836 19926 600 2°91 222 (3) 166 585 811 | 4°22163 80074
8°8 382179 14407 663 4°24 401 (3) 150° 733 °75 417820 85593
89 | 3°86522 08889 733 197 354 3) 136 388 926 413477 QIIII
| 9 | 3°90865 03371 810 3°08 393 (3) 123 409 804 409134 96629
ot | 3795207 97853 895 5°29 270 (3) 111 665 808 4°04792 02147
| 92 | 3°99955° 92335 989 7°12 906 (3) 101 039 402 400449 07665
ie 953 4°03893 86817 10g 38°0 192 (4) 914 242 315 596106 13183
| oe 4°08236 81299 120 88°3 807 (4) 827 240 656 5°91763 18701
9°5 4°12579 75781 133 59°7 268 (4) 748 518 299 587420 24219
9°6 416922 70263 147 64°7 816 (4) 677 287 365 5°83077 29737
| 97 421265 64744 163 17°6 072 (4) 612 834 951 5°78734 35255
| 98 425608 59227 180 33°7 449 (4) 554 515 994 5°74391 40774
9°9 4°29951 53708 199 39°3 704 (4) 501 746 821 5°70048 46292
10'0 4°34294 48190 220 2674 658 (4) 453 999 298 5°65705

51810

261

Ol.

The numbers in parentheses denote the numbers of ciphers between the decimal point and the

first significant figure ;

for example, e

—10

= 01000045 3999298.

262 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

Values of e, e7",
x log,

I “43429
| 2 86858
3 1°30288

A ie Se ov

5 2°17 147

6 2°60576

7 3°04006
8 3°47435
9 390865

10 4°34294

Ir 477723

12 521153

13 5°64582

I4 6:08012

15 OST 441

16 6°94871

17 7°38300

18 7°81730

19 825159

20 8°68588

21 g'12018

22 9°55447

23 9°98877

24 10742306

25 10.85736

26 II‘°29165

27 11°72595

28 1216024

= 12°59453

30 13°02883

31 13°46312

2 | 13°89742
33 14°33171

34 14°76601

He oie hl Pseizdelei se)
36 | 15°63460

37 16'06889

| 38 16°50319
| 39 | 1693748
iv eesC 17°37177
1 A vax 17°80607
2 18°24036

43 18°67466

44 19'10895

45 | 19°54325

46 | 19°97754

47 | 20°41184

48 | 20°84613
| 21'28042
50 | 21°71472

>
©

log,

()

44819
89638
34457
79276
24095
68914
13733
58552
03371
48199
93009
37828
82647
27466
72285
17105
61924
06743
51562
96381
412090
86019
30838
75957
20476
65295
IOII4
54933
99752
44571
89390
34209
79028
23847
68666
13485
58304
03123
47942
92761
37580
82399
27218
72937
16856
61675
06495
51314
96133
42952

(e’

, log, (e") from 1 to 500 at intervals of unity.

TABLE IV.

342
611
131
673
734
479°

494
947
oor
pfs
942
289
591
657
553

es

Ne Ae

mmow TIA AniMNnNrf HWW DN

Ke)

Sn. SSS Semen SS SSSA fos Se a Sn SSN Sn Se ee ne A

Oo

13

|
|
|

|

e

(19) 105
(20) 387
(20) 142
(21) 524
(21) 192

—-z

879
335
870
156
794
875
881
462
4°09
999
O17
421
032
528
go2
535
993
299
279
I15
256
946
618
513
879
908
952
449
306
762
247
641
888
39°
511
952
304
913
482

4 835

288
952
513
113
251
306
399
516
288
874

[4s [4s [02 102] 2/09] a] Ht

10°87981
10°44552
TO'OI122
1157693
1114263
12°70834

14°10257
1566828
15°23398
16°79969
16°36539
1793110
1749680
17'06251
18°62822

20°89104
20°45674
20°02245
21°58815
21°15386
22°71957
22'28527

55181
10362
65543
20724
15995
31086
86267
41448
96629
51810
06991
62172
17353

M2534

27715
82896
38076
93257
48438
03619
58800
13981
69162
24343
79524
34795
89886
45067
00248
55429
10610
65791
20972
76153
31334
86515
41696
96877
52058
07239
62420
17601
72782
27963
83144
38325
935°5
48686
03867
59048

The numbers in square brackets denote the numbers of figures between the last figure given
and the decimal point; for example, the first nine figures of e” are 518470553, and there are

13 additional figures before the decimal point is reached.
numbers of ciphers between the decimal point and the first significant figure ; for example, in e
there are 21 ciphers between the decimal point and the figures 192874985.

The numbers in parentheses denote the

—50

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 263

TABLE IV.. (continued).

Values of e*, e~*, log,,(e"), log,(e) from 1 to 500 at intervals of unity.

180 485 139 | 35°25644 14477
663 967 720 | 36°82214 69658
244 260 074 | 36°38785 24839
898 582 594 | 37°95355 80020
339 57° 663 | 37751926 35201
121 609 930 | 37'08496 90382
447 377 931 | 38765067 45563

80 | 34°74355 85523 | 554 062 238
81 3517785 30342 159 609 731
82 35°61214 75161 | 409 399 6096
83 36°04644 19980 111 286 376
84 | 36°48073 64799 | 302 507 732

85 36'91503 09618 | 822 301 271
86 | 37°34932 54437 | 223 524 660

510

A log, (e*) e& en” log,,(e-*)
51 2214901 85771 149 934 908 ta] (22) 709 547 416 | 23°850908 1422
52 22°5833I 30590 383 100 800 [14] | (22) 261 027 g07 | 23°41668 69410
53 23°01760 75409 104 137 594 [15] | (23) 960 268 005 24°98239 24591
54 | 23'45190 20228 | 283 075 330 [15] | (23) 353 262 857 | 24°54809 79772
55 | 2388619 65047 | 769 478 527 15] (23) 129 958 143 | 2411380 34953
56 24°32049 09866 209 165 950 16| (24) 478 089 288 | 25°67950 g0134 |
57 | 24°75478 54685 | 568 572 000 [16] | (24) 175 879 220 | 25724521 45315
58 | 25°18997 99504 | 154 553 894 [17] | (25) 647 023 493 | 2681092 00496
"59 2562337 44323 420 I2I o40 [x7] (25) 238 026 641 26°37662 55677
60 26'05766 89142 II4 200 739 [18] | (26) 875 651 076 | 27794233 10858
61 26°49196 33961 310 429 794 Ls (26) 322 134 029 | 27°50803 66039
62 26°92625 78780 843 835 667 [18] | (26) 118 506 487 | 27°07374 21220
63 | 27736055 23599 | 229 378 316 tI (27) 435 961 000 | 28°63944 76404
64 27°79484 68418 623 514 908 [19 (27) 160 381 089 | 2820515 31582
65 28°22914 13237 169 488 924 [20] | (28) 590 009 054 | 29°77085 86763
66 28°66343 58056 460 718 663 [20] | (28) 217 052 201 | 29°33656 41944
67 | 29°09773 02875 | 125 236 317 [2 1 (29) 798 499 425 | 30°90226 97125
68 | 29'°53202 47694 | 340 427 605 [21] | (29) 293 748 211 | 30°46797 52306
69 29°96631 92513 925 378 172 [21] | (29) 108 063 928 | 30°03368 07487
79 | 30°40061 37332 | 251 543 867 [22] | (30) 397 544 974 | 31'59938 62668
WE 30°83490 82151 683 767 123 [22] | (30) 146 248 62 3116509 17849
72 31°26920 26970 185. 867 175 [23] | (31) 538 018 616 | 32°73079 73030
73 | 3170349 71789 | 505 239 363 [23] | (31) 197 925 988 | 3229650 28211
74 32°13779 16608 137 338 298 [24] | (32) 728 129 018 | 33°86220 83392
75 | 32°57208 61427 | 373 324 200 [24] | (32) 267 863 696 | 3342791 38573
76 | 33°00638 06246 | ror 480 039 [25] | (33) 985 415 469 | 34°99361 93754
77 | 33°44067 51066 | 275 851 346 [25] | (33) 362 514 092 | 34'55932 48934
78 | 33°87496 95885 | 749 841 700 [25] | (33) 133 361 482 | 34712503 04115
79 | 34°30926 40704 | 203 828 107 490 609 473 | 35°69073 59296

| (35)

(35)

1| Go

|| G6)

] | (37)

] | (37)

(38)

OI I ADAMNL fWWO NN HH

[
|
[
[
88 38°21791 44075 165 163 626 |
[
|
|

87 37°7836I 99256 | 607 603 023 [2 164 581 143 | 38°21638 00744

30| 605 460 189 | 39778208 55925
89 38°65220 88894 | 448 961 282 [30] | (38) 222 736 356 | 39°34779 11106
go 39°08650 33713 122 040 329 [3r| | (39) 819 40% 262 | 40'91349 66287
9t | 39°52079 78532 | 331 740 O10 [31] | (39) 30r 440 879 | 40°47920 21468
"92 | 39°95509 23351 | gor 762 841 [3r] | (39) 110 893 go2 | 40°04490 76649
93 40°38938 68170 245 124 554 |32] | (40) 407 955 867 | 4161061 31830
94 40°82368 12989 | 666 317 622 [32] | (40) 150 078 576 | 41°17631 87011
95 4125797 57808 | 181 123 908 [33] | (41) 552 108 228 | 42°74202 42192
96 41°69227 02627 492 345 829 [33] | (41) 203 109 266 | 42°30772 97373
97 | 4212656 47446 | 133 833 472 [34] | (42) 747 197 234 | 43°37343 52554
98 | 42756085 92265 | 363 797 095 [34] | (42) 274 878 501 | 4343914 07735
99 42°99515 37084 | 988 903 032 34 (42) Tor 122 149 | 43°00484 62916
100 | 43°42944 81903 | 268 81x 714 [35] | (43) 372 007 598 | 44°57055 18097

The numbers in square brackets denote the numbers of figures between the last figure given
and the decimal point; for example, the first nine figures of e” are 140934908, and there are
14 additional figures before the decimal point is reached. The numbers in parentheses denote the
numbers of ciphers between the decimal point and the first significant figure; for example, in ¢”
there are 22 ciphers between the decimal point and the figures 709547416. :

264 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE IV. (continued).

Values of e*, e~*, log,(e*), log,(e~) from 1 to 500 at intervals of unity.

i log, (e") e e* log, (e-*)

TOI | 43°86374 26722 | 730 795 998 Ee (43) 136 853 947 | 4413625 73278
102 44°29803 71541 198 626 484 [36 (44) 503 457 536 | 45°70196 28459
103. | 44°73233 16360 539 922 761 [36 (44) 155 211 676 | 45°26766 83640
104 45°16662 61179 146 766 223 |37 (45) 681 355 682 | 46°83337 38821
105 | 4560092 05998 | 398 951 957 [37] | (45) 25° 656 748 | 46°39907 94002
106 46°03521 50817 | 108 446 386 [38 (46) 922 114 642 | 47°96478 49183

°
i)

46°90380 40456 | Sor 316 426 {38 (46) 124 794 646 | 47°09619 59544
47°33809 85275 | 217 820 388 [39] | (47) 459 093 847 | 4866190 14725
| 110 47°77239 30094 592 097 203 |39 (47) 168 891 188 | 48:22760 69906
| ir | 48°20668 74913 160 948 707 te (48) 621 315 959 | 49°79331 25087

112 4864098 19732 45 |40 (48) 228 569 368 | 49°35901 80268
| 49°07527 64551 | 118 925 go2 [41] | (49) 840 859 712 | 50°92472 35449
| 49°50957 09379 | 323 274 119 fs (49) 399 335 001 | 5049042 90630

107 | 46°46950 95636 | 294 787 839 é (46) 339 227 019 | 47753049 04364

WM
wo

_

we

=

un

fo)

ios)

Ko}

_

On

115 | 49°94386 54189 | 878 750 164 [41 (49) 113 797 987 | 50°05613 45811
116 50°37815 99008 | 238 869 060 5162184 00992
117 50°81245 43827 649 313 426 [42 (50) 154 008 829 | 51°18754 56173
118 51°24674 88646 176 501 689 | 4; 52°75325 11354
119 | 5168104 33465 | 479 781 333 [43] | (51) 208 428 284 | 52°31895 66535
52°11533 78284 130 418 088 | 44 (52) 766 764 807 | 53°88466 21716
2m | 52°54963 23103 354 513 118 [44 (52) 282 077 009 | 53°45036 76897
122 52°98392 67922 963 666 567 |44 (52) 103 770 332 | 53°01607 32078
12 53°41822 12741 | 261 951 732 : (53) 381 749 719 | 5458177 87259

8
fo)

12 53°85251 57560 | 712 058 633 [45] | (53) 140 437 873 | 5414748 42440
| 54°28681 02379 | 193 557 604 [46 (54) 516 642 063 | 55°71318 97621
126 54°72110 47198 526 144 118 |46 (54) 190 061 994 | 55°27889 52802
127 55°15539 92017 143 020 800 |47 (55) 699 199 000 | 56°84460 07983
128 55758969 36836 | 388 770 841 |47
12 56°02398 81655 105 678 871 |48
139 | 56°45828 26474 | 287 264 955 [48
131 56°89257 71293 | 780 867 107

~
nN
wn

(55) 257 220 937 | 56°41030 63164
(56) 946 262 947 | 57°97601 18345
(56) 348 110 684 | 57°54171 73526
(56) 128 062 764 | 57710742 28707
(57) 471 116 580 | 58767312 83888
(57) 173 314 104 | 58°23883 39069

132 57°32687 16112 212 261 687 |
(58) 637 586 958 | 59°80453 94250

133. | 57°76116 60931 576 987 086
58°19546 05750 | 156 841 351
135 | 5862975 50569 | 426 338 995
| 136 | 59°06404 95388 | 115 890 954
| 137 | 59°49834 40207 | 315 024 275
138 | 59°93263 85026 | 856 324 762
139 | 60°36693 29846 | 232 773 204
140 60°80122 74665 632 743 171 [52
141 | 61°23552 19484 | 171 997 426 [53 (61) 581 404 049 | 62°76447 80516
142 | 6166981 64303 | 467 537 479 [53] | (61) 213 886 597 | 6233018 35697
1143 | 62°10411 Og122 127 089 863 [54] | (62) 786 844 816 | 63°89588 90878
| 144 | 62°53840 53941 | 345 466 066 |54] | (62) 289 464 031 | 63°46159 46059
| 145 62°97269 98760 939 °74 129 |54 (62) 106 487 866 | 63:02730 01240
146 | 63°40699 43579 | 255 266 814 [55] | (63) 391 746 966 | 64°59300 56421
147 | 63°84128 88398 | 693 887 142 [55 (63) 144 115 655 | 64:15871 11602
148 64°27558 33217 188 618 o81 |56 (64) 530 171 867 | 65°72441 66783
149 | 64°70987 78036 | 512 717 102 |56 (64) 195 039 330 | 65:29012 21964
| 150 | 65"x4417 22855 139.370 958,157] || (65) 727 509 597.) COBaRS2 7 7n45
| ;

w
P=

(58) 234 555 134 | 59°37924 49431
(59) 862 880 116 | 60°93595 04612
(59) 317 435 855 | 60°50165 59793
(59) 116 778 125 | 60°06736 14974
(60) 429 602 713 | 61763306 70154
(60) 158 042 006 | 61'19877 25335

The numbers in square brackets denote the numbers of figures between the last figure given
and the decimal point; for example, the first nine figures of e'” are 730705998, and there are
35 additional figures before the decimal point is reached. The numbers in parentheses denote the
numbers of ciphers between the decimal point and the first significant figure; for example, in e'”
there are 43 ciphers between the decimal point and the figures 136853947.

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE IV. (continued).

265

Values of ¢”, e~*, log,,(e”), log,(e*) from 1 to 500 at intervals of unity.
x log, (e”) e Ge log,,(e~*)
151 | 65°57846 67674 | 378 849 543 33) (65) 263 957 03> | 6642153 32326
152 6601276 12493 102 981 983 [58 (66) 971 043 646 | 67°98723 87507
153 | 6644705 57312 | 279 934 052 [58] | (66) 357 226 994 | 67'55294 42688
154 66°88135 02131 760 939 648 [58 (66) 131 416 467 | 67711864 97869
™55 | 67°31564 46950 | 206 844 842 2 (67) 483 454 164 | 68°68435 53050
156 | 67774993 91769 | 562 262 575 [59] | (67) 177 852 848 | 68:25006 08231
157 68:18423 36588 | 152 838 814 [60 (68) 654 284 062 | 69°81576 63412
158 | 6861852 81407 | 415 458 971 [60 (68) 240 697 655 | 6938147 18593
159 | 69'05282 26226 | 112 933 457 Pe (69) 885 477 188 | 79°94717 73774
160 69°4871I 71045 306 984 964 |6r (69) 325 748 853 | 70751288 28955
16 69°9214r 15864 834 471 649 te (69) 119 836 306 | 70707858 84136
162 70°35579 69683 226 832 gr2 |62 (7°) 440 853 133 | 71°64429 39317
163 70°79090 05502 616 595 783 |62 (70) 162 185 804 | 71:20999 94498
164 | 71°22429 50321 | 167 608 111 [63] | (71) 596 629 837 | 72°77579 49679
165 71°65858 95140 | 455 606 083 [63 (71) 219 487 851 | 72°34141 04860
166 72°09288 39959 123 846 574 [64 (72) 807 459 679 | 73°90711 60041
167 | 72°52717 84778 | 336 649 891 [64] | (72) 207 044 505 | 73°47282 15222
168 72°90147 29597 | 915 tog 280 |64 (72) ft09 276 566 | 73°03852 70403
169 | 73°39576 74416 | 248 752 493 [65] | (73) 492 006 022 | 7469423 25584
179 | 7383906 19236 | 676 179 381 [65 (73) 147 889 751 | 74°16993 80764
I7I_ | 74°26435 64055 | 183 804 612 [66] | (74) 544 055 988 | 75°73564 35945
172 74°69865 08874 499 632 738 |66 (74) 200 147 0F3 | 75°30134 91126
173 | 75°13294 53693 | 135 814 259 [67] | (75) 736 299 712 | 76°86705 46307
174 75°56723 98512 | 369 18% 433 [67] | (75) 270 869 527 | 76°43276 01488
175 | 76°00153 43331 | 100 353 918 [68] | (76) 996 473 301 | 77°99846 56669
176 76°43582 88150 242 790 232 [68] (76) 366 582 o41 | 77:5641r7 11850
177 | 7687012 32969 | 741 520 730 [638] | (76) 134 857 996 | 77712987 67031
178 7730441 77788 | 201 566 233 163 (77) 496 114 844 | 7869558 22212
179 | 77°73871 22607 | 547 913 828 [69] | (77) 182 510 451 | 78°26128 77393

use) 78°17300 67426 148 938 420 |70 (78) 671 418 429 | 79°82699 32574
181 78°60730 12245 404 856 6o1 |70] -| (78) 247 00% 036 | 79°39269 87755
182 | 79°04159 57064 | 110 051 434 [71] | (79) 998 666 032 | 80°95840 42936
183 | 79°47589 01883 | 299 150 814 [71 (79) 334 279 552 | 80°52410 98117
184 7991018 46702 | 813 176 221 [71 (79) 122 974 575 | 8008981 53298
185 80°34447 91521 221 044 214 |72 (80) 452 398 179 | 81°65552 08479
186 80°77877 36340 | 609 869 471 172] (89) 166 427 989 | 81°22122 63665
187 8121306 81159 163 336 8to [73] | (81) 612 254 357 | 82°78693 18841
183 | 8164736 25978 | 443 979 173 Hs! (Si) 225 235 791 | 82°35263 74022
189 82°08165 70797 126 686 052 |74 (83) 828 596 168 | 83'91834 29203
190 82°51595 15616 | 328 058 702 [74] (82) 304 823 495 | 83°48404 84384
19 82°95024 60435 | 891 756 007 [74] | (82) 112 138 297 | 83°04975 39565
192 | 83°38454 05254 | 242 404 415 [75| | (83) 412 533 741 | 84°61545 94746
193 83°81883 50073 658 923 516 [75| | (83) 151 762 682 | 84°18116 49927
194 | 84°25312 94892 | 179 113 982 He (84) 558 303 706 | 85°74687 05108
195 84°68742 39711 486 882 283 [76 (84) 265 388 455 | 85°31257 60289
196 8512171 84530 132 348 326 77\ (85) 755 581 go2 | 86°87828 15470
197 | 85755601 29349 | 359 760 050 [77] | (85) 277 963 048 | 8644398 70651
198 85°99030 74168 977 929 206 |77| (85) 102 256 891% | 8600969 25832
199 86°42460 18987 265 828 719 [73] (86) 376 182 078 | 87'57539 81013
200 | 86°85889 63807 | 722 597 377 [78] | (86) 138 389 653 | 87:141to 36194

The numbers in square brackets denote the numbers of figures between the last figure given

and the decimal point; for example, the first nine figures of e'”
57 additional figures before the decimal point is reached.
numbers of ciphers between the decimal point and the first significant figure; for example, in ¢~
there are 65 ciphers between the decimal point and the figures 263957030.

Vou. XIII. Parr II.

30

are 378849543, and there are
The numbers in parentheses denote the

151

266 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION,
TABLE IV. (continued).

Values of e*, e~*, log, (e*), log,(e7*) from 1 to 500 at intervals of unity.

| 96°41337 49825 | 259 044 862 (96) 386 033 520 | 97'58662 5or75

96°84766 94644 | 704 156 941

tse: log, (e*) e Re log,,(e~*)

201 87°29319 08626°| 196 422 332 2 (87) 509 107 o8t | 8870680 91374
202 87°72748 53445 | 533 931 255 [79] | (87) 187 290 028 | 8827251 46555
203 8816177 98264 | 145 137 563 tes (88) 689 cor 510 | 8983822 01736
204 88+59607 43083 | 394 524 800 [80 (88) 253 469 490 | 89°40392 56917
205 89'03036 87902 | 107 242 960 Isr (89) 932 462 145 90°96963 12098
206 89°46466 32721 | 291 516 588 [81 (89) 343 033 653 99°53533 97279
207 89°89895 77540 | 792 424 244 |8r (89) 126 195 029 | go*10104 22460
208 99°33325 22359 | 215 403 242 [82 (90) 464 245 566 | 91°66674 77641
209 9°°76754 67178 | 585 526 719 [82] | (90) 170 786 399 | 9123245 32822
210 gt20184 11997 | 159 162 664 [83 ee 628 288 o51 92°79815 88003
211 91°63613 56816 | 432 648 977 [83 Om) 22r meds 92°36386 43184
212. | 9g2°07043 01635 | 117 606 185 [84 (92) 850 295 414 | 93792956 98365
213 | 92750472 46454 | 319 686 757 [84 (92) 312 806 202 93°49527 53546
214 92°93901 91273 | 868 998 yor [84 (92). 115 074 971 93°06098 08727
215 93°37331 36092 | 236 218 338 [85] | (93) 423 337 159 | 94°62668 63908
216 93°80760 80911 | 642 108 o15 [85 (93) 155 737 937 | 94°19239 19089
21 9424190 25739 | 174 543 055 [86] | (94) 572.924 543 | 95°75809 74270
94°67019 70549 | 474 457 215 [86] | (94) 210 767 161 | 95°32380 29451
95"11049 15368 | 128 g70 843 [87] | (95) 775 369 053 | 96°88950 84632
95°54478 60187 | 350 579 098 |87] | (95) 285 242 334 | 96°45521 39813
95°97908 05006 | 952 972 790 [87] | (95) 104 934 791 | 96'02091 94994

| 97°28196 39463 | 191 409 702 (97) 522 439 558 | 98°71803 60537
97°71625 84282 | 520 305 514 [89 (97) 192 194 773 | 9828374 15718
9815055 29101 | 141 433 702 [go] | (98) 707 045 056 | 99°84944 70899°

WN NNN HNN ND Se

Ne ODO CONIA MNUFWNHOUKO MO NIAMNHLWNH OO GO

98°58484 73920 | 384 456 663 [90

(98) 260 107 340 | 99°41515 26080
99°01914 18739 | 104 506 156 [ox 100

(99) 956 881 429 | 10098085 81261

: (96) 142 0f3 796 | 97°15233 05356

22 99°45343 63558 | 284 077 185 [91] | (99) 352 017 006 | 100°54656 36442
23 99°88773 08377 | 772 201 850 [gx (99) 129 499 819 | 100711226 91623
23 100°32202 53197 | 209 906 226 [g2| | (100) 476 403 211 | 101°67797 46803
23 100°75631 98016 | 570 584 279 [92] | (100) 175 258 947 | 101'24368 01984
23 TOr'1g061 42835 | 155 roo 888 [93] | (101) 644 741 635 | 102°80938 57165
23 10162490 87654 | 421 607 925 [93] | (101) 237 187 193 | 102°37509 12346
23 102°05920 32473 | 114 604 916 [94] | (102) 872 562 919 | 103°94079 67527
23 102°49349 77292 | 311 528 461 [94| | (102) 320 997 959 | 10350650 22708
23 | 102°92779 22111 | 846 822 154 [94] | (102) 118 088 550 | 103'07220 77889
23 103°36208 66930 | 230 190 127 [95] | (103) 434 423 497 | 104°63791 33070
23 10379638 11749 | 625 721 640 [95] | (103) 159 815 473 | 104'20361 88251
| 24 104°23067 56568 | 170 088 776 [96] | (104) 587 928 270 | 105°76932 43432
|) 24 104°66497 01387 | 462 349 230 [96] | (104) 216 286 723 | 105°33502 98613
| 24 105'09926 46206 | 125 679 551 [97] | (105) 795 674 389 | 106°90073 53794
| 243 | 105°53355 91025 | 341 632 440 [97] | (105) 292 712 250 | 106°46644 08975
| 244 10596785 35844 | 928 653 253 [97] | (105) 107 682 819 | 106:03214 64156
245 | 106740214 80663 | 252 434 126 [98] | (106) 396 142 952 | 107°59785 19337
246 106°83644 25482 | 686 187 098 [98] | (106) 145 732 848 | 107°16355 74518
247 107'27073 70301 | 186 524 992 [99] | (107) 536 121 186 | 108°72926 29699
| 248 107°70503 15120 | 507 027 496 [99] | (107) 197 227 962 | 108°29496 84880
| 249 108°13932 59939 | 137 824 363 [100| | (108) 725 561 126 | 109°86067 40061
| 250 108'57362 04758 | 374 645 461 ie (108) 266 g19 022 | 109°42637 95242

!

The numbers in square brackets denote the numbers of figures betiveen the last figure given

and the decimal point; and the numbers in parentheses denote the numbers of ciphers between
the decimal point and the first significant figure.

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 267

TABLE IV. (continued).

Values of e*, e7*, log,(e"), log,(e7*) from 1 to 500 at intervals of unity.

x log,,(€”) & Can log,,(e7”)

251 109'0079g1 49577 | 101 839 195 [ror] | (109) 981 940 205 | 110799208 50423
252 10944220 94396 | 276 827 633 [101|'| (109) 361 235 614 | 110°55779 05694
253 | 109°87650 39215 | 752 495 525 [101] | (109) 132 891 156 | 110712349 60785
254 | 110°31079 84034 | 204 549 491 [102] | (110) 488 879 241 11168920 15966
25 11074509 28853 | 556 023 165 |1o02| | (110) 179 848 622 | 111°25490 71147
256 IIL'L7938 73672 | 151 142 767 [103] | (411) 661 626 106 | 112°82061 26328
257 11161368 18491 | 410 848 636 | 103] | (111) 243 398 642 | 112°38631 81509
258 112'04797 63310 | 111 689 238 [104] | (112) 895 413 564 | 113°95202 36690
259 | 112°48227 08129 | 303 578 362 [104] | (112) 329 494 242 | 113°51772 91871
11291656 52948 | 825 211 544 [104] | (112) 121 181 048 | 113°08343 47052
261 | 113°35085 97767 | 224 315 755 [105] | (113) 445 890 163 | 114°64914 02233

| 113°78515 42587 | 629 753 439 [105] | (r13) 164 ooo 715 | 114°21484 57413
263 11421944 87406 | 165 748 169 a (114) 693 324 914 | 115°78055 12594
264 114°65374 3222 450 550 237 [106] | (114) 221 950 832 | 115°34625 67775
265 11508893 77044 | 122 472 252 [107| | (115) 816 511 481 | 116°91196 22956
266 115°52233 21863 | 332 914 098 |107]| | (125) 309 377 787 | 11647766 78137
267 11595662 66682 | 904 954 342 |107| | (115) t10 502 813 | 116°04337 33318 |
268 116°39092 11501 | 245 992 094 [108] | (116) 406 517 129 | 117°69997 88499
269 116°82521 56322 | 668 675 840 |108] | (116) 149 549 294 11717478 43680
270 I17'25951 01139 | 181 764 939 [109] | (117) 550 161 108 | F18°74048 98861
271 | 117769389 45958 | 494 088 330 [109
272 11812899 99777 | 134 307 133 Be

N
a
N

(117) 202 392 961 | 118°30619 54042
(r18) 744 562 094 | 119°87199 09223
(118) 273 999 087 | 119°43760 64404
(118) 100 765 522 | 119°09331 19585
(119) 379 695 639 | 120°56991 74766
(119) 436 37% 394 | 120°13472 29947
(120) 501 681 993 | 121°70042 85128

273 | 118°56239 35596 | 365 084 638
274 118"99668 80415 | 992 402 938 [110
275 I19°43098 25234 | 269 763 087 |111
276 119°86527 70053 | 733 292 098 [rx
277 | 120°29957 14872 | 199 329 459 |112

|
|
278 120°73386 59691 | 541 833 645 an (120

) 184.558 491 | 12126613 40309
279 12116816 o4510 | 147 285 655 [113] | (121) 678 952 746 | 122°83183 95499
285 12160245 49329 | 400 363 920 [123 (121) 249 772 757 | 122°39754 50671
281 122°03674 94148 | 108 830 197 [114] | (122) 918 862 622 | 123°96325 05852
282 122°47104 38967 | 295 831 147 [114] | (122) 338 039 668 | 123°52895 61033
283 122°90533 83786 | 804 152 430 [114] | (122) 124 354 533 | 123°09466 16214
284 | 123°33963 28605 | 218 591 294 [115] | (123) 457 474 762 | 12466036 71395
285 123°77392 73424 | 594 192 742 [115] | (123) 168 295 560 | 124°22607 265706
286 124°20822 18243 | 161 518 333 Bee (124) 619 124 764 | 125°79177 81757
287 124°64251 63062 | 439 052 350 fae (124) 227 763 272 125°35748 36938
288 125°07681 07881 | 119 346 803 [117] | (125) 837 894 253 | 126°92318 g2119
289 125°51110 52700 | 324 418 245 [117] | (125) 308 244 070 | ¥26°48889 47300
290 125°94539 97519 | 881 860 219 117| (125) 113 396 656 | 126'05460 02481
29i _| 126°37969 42338 | 239 714 461 [118] | (126) 417 162 985 | 127°62030 57662
292 126°81398 87157 | 651 611 463 [118| | (126) 153 465 686 | 127718601 12843
2093 127°24828 31977 | 177 126 360 |rr9j | (127) 564 568 707 | 128°75171 68923
294 12768257 76796 | 481 479 366 [119] | (127) 207 693 220 | 12831742 23204
295 128°11687 21615 | 130 879 661 [r120| | (128) 764 069 659 | 129°88312 78385
2096 12855116 66434 | 355 767 804 [120] | (128) 281 082 298 | 129°44883 33566
297 128°98546 11253 | 967 077 157 ze (128) 103 404 366 | 129°01453 88747
298 129'41975 56072 | 262 878 826 [121] | (129) 380 403 493 | 130758024 43928
299 129°85405 00891 | 714 578 737 [121] | (129) 139 942 591 | 13014594 99109
300 130°28834 45710 | 194 242 640 bras (130) 514 820 022 | 131°71165 54299"

The numbers in square brackets denote the numbers of figures between the last figure’ given
and the decimal point; and the numbers in parentheses denote the numbers of ciphers between
the decimal point and the first significant figure.

30—2

268 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE IV. (continued).

Values of ¢*, e~*, log,(e"), log,(e~*) from 1 to 500 at intervals of unity.

Gi) adon, (2) & ee log,,(e-*)
301 130°72263 99529 | 528 006 237 [122] | (130) 189 391 702 | 431°27736 09471
302 | 13115693 35348 | 143 526 976 [123] | (131) 696 733 135 | 132°84306 64652
303 131°59122 80167 | 390 146 771 [123] | (131) 256 313 796 | 132°40877 19833
304 13202552 24986 | 106 o52 888 [124] | (132) 942 925 762 | 133°97447 75014
305 132°45981 69805 | 288 281 638 [124] | (132) 346 883 002 | 133754018 30195
306 132°S941r 14624 | 783 630 737 |124| | (132) 127 611 125 | 133°10588 85376
3°27 | 133°32840 59443 | 213 O12 919 [125] | (133) 469 455 094 | 134°67159 40557
328 | 133°76270 04262 | 579 029 148 [125] | (133) 172 702 878 | 13423729 95738

| 309 134°19699 49981 | 157 396 441 [126] | (134) 635 338 381 13580300 50919
310 134°63128 93990 | 427 847 886 [126

31 135°06558 38719 | 116 301 113 [127] | (135) 859 836 997 | 136°93441 61281

iene 135°49987 83538 | 316 139 203 [127] | (135) 316 316 354 | 136°50012 16462
313 | 135793417 28357 | 859 355 459 [127] | (135) 116 366 284 | 136°06582 71643
314 136°36846 73176 | 233 597 031 [128] | (136) 428 087 634 137763153 26524
315 136°80276 17995 | 634 982 563 [128] | (136) 157 484 640 13719723 82005

(137) 579 353 612 | 138°76294 37186
(137) 213 132 283 | 138°32864 92367

317. | 137°67135 07633 | 469 192 178
| 318 | 138:10564 52452 | 127 539 657
| 319 | 138°53993 97271 | 346 688 732
| 32 138°97423 42090 | 942 397 682
139°40852 86909 | 256 170 249
139°84282 31728 | 696 342 934
140°27711 76548 | 189 285 634
I40°7II4I 21367 | 514 531 700
141°14570 66186 | 139 864 217
141°58000 11005 | 380 199 360
142°01429 55824 | 103 346 455
142°44859 00643 | 280 924 790
142°88288 45462 | 763 632 751
143°31717 90281 | 207 576 903

(138) 784 069 851 | 139°89435 47548
(138) 288 443 179 | 139°46006 02729
(138) 106 112 315 | 13902576 57910
(139) 399 365 393 | 149°59147 13091
(139) 143 607 403 | 140715717 68272
(140) 528 302 110 | 141°72288 23452
(140) 194 351 485 |, 14128858 78633
(141) 714 979 157 | 142°85429 33814
133] | (141) 263 026 133 | 142°41999 88995
134] | (142) 967 619 067 | 143°98570 44176
134] | (142) 355 967 162 | 143°55140 99357
134] | (142) 130 953 Cor | 14311711 54538
135] | (143) 481 749 167 | 144°68282 09719
143°75147 35100 | 564 252 524 |135| | (143) 177 225 614 | 144°24852 64900
144°18576 79919 | 153 379 738 [136] | (144) 651 976 599 | 145°81423 20081
144°62006 24738 | 416 929 355 (144) 239 848 787 | 145°37993 75262

(134) 233 727 929 | 135°36871 06100
L
| 316 137°23705 62814 | 172 696 156 f

NH OM ON AUN LW NH

WwWwWWNNN HNN NN DN

Co Co Ge Gs G2 Go On Cn Ga Go We Ws Go ©
Los)

334 | 145°05435 69557 | 113 333 149 [137] | (145) 882 354 377 | 146°94564 30443
335 | 145°48865 14376 | 308 o71 439 [137] | (145) 324 600 035 | 146°51134 85624
336 | 145°92294 59195 | 837 424 995 [137] | (145) 119 413 680 | 146°07705 40805
337 | 146°35724 04014 | 227 635 715 [138] | (146) 439 298 377 | 147°64275 95986
338 146°79153 48833 | 618 778 027 [138] | (146) 161 608 841 | ¥47°20846 51167
339 | 147°22582 93652 | 168 201 307 [139] | (147) 594 525 703 | 148°77417 06348
349 | 147°66012 38471 | 457 218 555 [139] | (147) 218 713 783 | 148°33987 61529
341 148'09441 83290 | 124 284 889 [140] | (148) 804 603 044 | 149°90558 16710
342 148°52871 28109 | 337 841 356 [140] | (148) 295 996 918 | 149°47128 71891
343 148°96309 72928 | 918 348 018 [140] | (148) 108 891 181 | 149'°03699 27072
344 | 149°39730 17747 | 249 632 873 [141] | (149) 400 588 267 | 150°60269 82253
345 | 14983159 62566 | 678 572 502 [141] | (149) 147 368 188 | 15016840 37434
346 150°26589 07385 | 184 455 130 [142] | (150) 542 137 266 | 151°73410 92615
347 | 150°70018 52204 | 501 4or 028 |r42] | (150) 199 441 155 | 151°29981 47796
348 | 15113447 97023 | 136 294 93 [143] | (151) 733 703 005 | 152°86552 02977
349 151°56877 41842 | 370 488 033 [143] | (151) 269 914 251 152°43122 58158
359 | 152°00306 86661 | 100 709 089 [144] | (152) 992 959 040 | 153°99693 13339

The numbers in square brackets denote the numbers of figures between the last figure given
and the decimal point; and the numbers in parentheses denote the numbers of ciphers between
the decimal point and the first significant figure.

—_—

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION,

TABLE IV. (continued),
(7), log

og.o(e

269

e~*) from 1 to 500 at interyals of unity.

e en?

Values of e%, e~*, log,
x log, (e”)
351 152°43736 31480
352 152°87165 76299
353 | 153°30595 21118
354 | 153°74024 65938
355 154°17454 10757
356 | 154°60883 55576
357 155°04313 99395
358 | 155°47742 45214
359 T55‘91I7I 90933
360 15634601 34852
361 156°78030 79671
362 157°21469 24490
363 157°64889 69309
364 158°08319 14128
365 | 15851748 58947
366 158°95178 03766
367 | 159°38607 48585
368 | 159°82036 93404
369 160°25466 3822
370 160°68895 83042
371 161°12325 27861
372 161°55754 72680
373 | 16199184 17499
374 162°42613 62318
375 162°86043 07137
376 | 163°29472 51956
377 | 163°72901 96775
378 164°16331 41594
379 | 164°59760 86413
380 165°03190 31232
381 165°46619 76051
382 165°90049 20870
383 166°33478 65689
384 166°76908 10508
385 | 167°20337 55328
386 167°63767 oo147
387 168:07196 44966
388 168°50625 89785
389 | 168°94055 34604
399 169°37484 79423
391 16980914 24242
392 179°24343 69061
393 | 170°67773 13880
394 17111202 58699
395 | 17154632 03518
396 17198061 48337
397 | 172°41499 93156
398 | 17284920 37975
399 | 173°28349 82794
400 | 173°71779 27613

273
744

202 2

549
149
406
IIo
300
816
221
602
163
445
T21
329
894
243
661
179
488
132
361
g8r
266
725
197
535
145
395
107
292
795
216
587
159
434
118

320
872
237
644
175
476
129
351
956
259
706
192
522

380
160
136
427
819
345
961
649
087
146

686 [144] | (152) 365 289
106 [144] | (152) 134 382

612 [145] | (153) 494 365
994 [145] | (153) 181 866
540 [146] | (154) 669 o50
462 [146] | (154) 246 12

926 [147] | (155) 905 461
562 [147] | (155) 333 100
198 [747] | (155) 122 540
530 [148] | (156) 450 802
025 [148] | (156) 165 841

867 [149] | (157) 610 095
495 [149] | (157) 224 441
182 = (158) 825 673
761 [150] | (158) 303 748
093 [150] | (158) 111 742
328 [151] | (159) 411 078
566 [151 (159) 151 227
899 [152] | (160) 556 334
447 [152] | (160) 204 664
731 = (161) 752 917
306 [153] | (161) 276 982
275 [153] | (161) tor 896
535 SoH (162) 374 855
779 [154] | (262) 137 gor
506 PS (163) 507 311
935 i (163) 186 629
231 [156] | (164) 686 571
094 ts (164) 252 575
512 [157] | (165) 929 173
318 [357 (165) 341 82

776 |157] | (165) 125 749
593 [158] | (166) 462 608

004 [158] | (166) 170 184

5 97° [159] | (167) 626 o72 2

200 |159| | (167) 230 319

2 415 ee (168) 847 296

16c| ees 311 703
168) 114 669
161] | (169) 421 844
552 oa (169) 155 187
(170) 570 904

404 [162] | (170) 210 02
294 [163] | (171) 772 634
860 tres! (171) 284 236
163] | (171) 104 564

668 [164] | (172) 384 672
a (172) 141 512
135 5] | (473) 520 597
969 [165] | (173) 191 516

ac

log, (e

153'56263
153°12834
15469404
154°25975
15582545
15539110
156795686
156°52257
156°08828
157°65398
157°21969
158°78539
158°35110
159°91680
159°48251

15904821

160°61392

160°17963
161'74533

16131104

162°87674

162°44245
162‘00815

163°57386
163°13956

167°23091
168°79662
168°36232
169'92803
169°49374
169°05944
170°62515
170°19085
171°75656
17132226
172°88797
172°45367
172°01938
173°58509
ETB; 25°79
174°71650
174'28220

The numbers in square. brackets denote the numbers of figures between the last

and the decimal point;

and the numbers in

the decimal point and the first significant figure,

-2)

68520
23701
78882
34062
89243
44424
99605
54786
09967
65148
20329
75510
30691
85872
GNSS
96234
52415
06596
61777
16958
72139
27320
82501
37682
92863
48044
03225
58406
13587
68768
23949
79130
34311
89492
44672
99853
55°34
10215
65396
20501
75758
32939
86120
41301
96482
51663
06844
62025
17206
72387

figure given

parentheses denote the numbers of ciphers between

270 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

TABLE IV. (continued).

Values of e*, e~*, log, (e"), log, (e~") from 1 to 500 at intervals of unity.

l l
| @ log, (e”) & e* log, (e7*)

| 40x | 174"15208 72432 | 141 934
402 174°58638 17251

26 Hee (x74) 704 551 521 | 175°84791 27568 |
32

403 17502067 62070 | 104 876 02
58
81

5 [166] | (174) 259 190 020 | 175°41361 82749
2 [167] | (175) 953 506 796 | 176°97932 37930
404 17545497 06889 | 285 082 586 [167] | (175) 350 775 547 | 176°54502 93111
405 175°88926 51708 | 774 934 S812 |167] | (175) 129 043 112 | 17611073 48292
406 | 176°32355 96527 | 210 649 122 [168] | (176) 474 723 O81 | 177°67644 03473
407 176°75785 41346 | 572 603 680 |168] | (176) 174 640 862 | 177°24214 58654-
408 17719214 86165 | 155 649 818 [169] | (177) 642 467 826 | 178°80785 13835
499 | 17762644 30984 | 423 I00 071 [169] | (177) 236 350 705 | 178°37355 69016

410 178'06973 75803 | II5 O10 524 [170] | (178) 869 485 652 | 179°93926 24197

4II 178°49503 20622 | 312 631 016 |170| | (178) 319 865 896 | 179°50496 79378

412 | 17892932 65441 (178) 117 672 087 | 179°07067 34559

413 179°36362 10260 | 231 004 812 |171]| | (179) 432 891 416 | 180°63637 89740

414 179°7979I 55079 | 627 936 182 {171} | (179) 159 251 852 | 180°20208 44921

415 180°23220 99898 | 170 690 751 |172| | (180) 585 854 824 | 181°76779 OO102
| 416 180°66650 44718 | 463 985 567 [172] | (180) 215 523 945 | 18133349 55282
| 417 181°10079 89537 | 126 124 354 [173] | (181) 792 868 285 | 18289920 10463
418 181°53509 34356 | 342 841 539 [173] | (181) 291 679 942 | 182°46490 65644
419 181°96938 79175 | 931 939 925 [173] | (181) 107 303 054 | 182°03061 20825
420 | 182°40368 23994 | 253 327 536 [174] | (182) 394 745 875 | 183759631 76006
421 182°83797 68813 | 688 615 639 [174] | (182) 145 218 892 | 183°16202 31187
422 183°27227 13632 ] 187 185 138 |175] | (483) 534 230 448 | 184°72772 86368
423 | 183°70656 58451 | 508 821 958 [175] | (183) 196 532 399 | 18429343 41549
424 184°14086 03270 | 138 312 148 [176] | (184) 723 oc2 290 | 185°85913 96730
425 | 184°57515 48089 | 375 971 399 [176] | (184) 265 977 679 | 185°42484 519I1
426 185°00944 92908 | 102 199 622 [177] | (185) 978 477 197 | 186799055 07092
427 | 185°44374 37727 | 277 807 376 [177] | (185) 359 961 645 | 186°55625 62273
428 18587803 82546 | 755 158 743 |177] | (185) 132 422 489 | 186°12196 17454
429 186°31233 27365 | 205 273 429 [178] | (186) 487 155 xix | 187°68766 72635
430 186°74662 72184 | 557 991 031 |178] | (186) 179 214 350 | 187°25337 27816
431 187°18092 17003 | 151 677 688 [179| | (187) 659 292 750 | 188°81907 82997
432 187°61521 61822 | 412 302 703 |179| | (187) 242 540 248 | 188°38478 38178
| 433 188'04951 06641 | 112 075 495 [180] | (188) 892 255 710 | 189°95048 93359
| 434 18848380 51460 | 304 652 780 [180] | (188) 328 242 532 | 18951619 48540

435 188°91809 96279 | 828 132 117 |180]| | (188) 120 753 679 | 18908190 03721

436 189°35239 41098 | 225 109 648 [181] | (189) 444 227 960 | 190°64760 58902
| 437 189°78668 85917 | 611 gtr 467 [181] | (189) 163 422 334 | 190°21331 14083
| 438 190°22098 30736 | 166 334 782 [182] | (190) 601 197 168 | 191°77901 69264
439 190°65527 75555 | 452 144 816 [182] | (19d) 221 168 078 | 191°34472 24445

ms
nS
No)
oo
Lal
Oo
rs)
=
°
i
~
°

440 1g91‘08957 20374 | 122 905 704 [183] | (191) 813 631% 8g1 | 192'91042 79626
441 | 191°52386 65193 | 334 092 341 [183] | (r9I) 299 318 445 | 192747613 34807
442 19195816 roo12 | go8 157 139 [183] | (191) 110 113 102 | 192°04183 89988
443 192°39245 54831 | 246 862 705 |184] | (192) 405 083 466 | 19360754 45169
444 192°82674 99650 | 671 042 405 |184] | (192) 149 021 879 | 193717325 00350
445 | 193°26104 44469 | 182 408 237 [185] | (193) 548 220 856 | 194°73895 55531
446 | 193°69533 89289 | 495 836 997 [185] | (193) 201 679 182 | 194°30466 IO7I1
447 | 19412963 34108 | 134 782 470 [186] | (194) 741 936 248 | 19537036 65892
448 194°56392 78927 | 366 376 739 [186] | (194) 272 943 092 | 195°43607 21073
449 | 194°99822 23746 | 995 915 232 [186] | (194) 100 411 052 | 195°00177 76254
450 195°43251 68565 | 270 717 828 [187] | (195) 369 388 307 | 196756748 31435

The numbers in square brackets denote the numbers of figures between the last figure given
and the decimal point; and the numbers in parentheses denote the numbers of ciphers between
the decimal point and the first significant figure.

Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

Values of e*, e~*, log,,(e”), log,,(e-”) from 1 to 500 at intervals of unity.
x log, (€) e e* log,,(e~7)
19586681 13384 | 735 887 352 [187] | (195) 135 890 364 | 196713318 86616
196°30110 58203 | 200 934 g22 [188] | (196) 499 912 711 | 197°69889 41797
196°73540 03022 | 543 751 292 [188] | (196) 183 907 609 | 197°26459 96978
197°16969 47841 | 147 806 926 [189] | (197) 676 558 284 | 198°83030 52159
197°60398 92660 | 401 780 880 [189 (197) 248 891 883 | 198°39601 07340
198°03828 37479 | 109 215 367 [190] | (198) 915 622 070 | r99'96171 62521
198°47257 82298 | 296 878 146 [190] | (198) 336 838 535 199'°52742 17702
198°90687 27117 | 806 998 471 [190] | (198) 123 915 972 | 199°09312 72883
199°34116 71936 | 219 364 928 [191] | (199) 455 861 386 200°65883 28064
199°77546 16755 | 596 295 697 [191] | (499) 167 702 032 | 200'22453 83245
200'20975 61574 | 162 089 976 |192| | (200) 616 g4r 298 | 201°79024 38426
200°64405 06393 | 440 606 236 fee (200) 226 960 020 | 201735594 93607
20107834 51212 | 119 769 192 [193 (2cr) 834 939 253 20292165 48788
201°51263 g603r | 325 566 419 [193] | (201) 307 156 986 | 202°48736 03969
201'94693 40850 | 884 981 282 [193] | (201) 112 996 740 | 202°05306 59150
| 202°38122 85669 | 240 562 854 [194] | (202) 415 691 777 | 203°61877 14331
202°81552 30488 | 653 917 634 | 194] | (202) 152 924 459 | 203°18447 69512
20324981 75307 | 177 753 242 [195] | (203) 562 577 643 | 204°75018 24693
203°68411 20126 | 483 183 408 [195] | (203) 206 960 749 | 204°31588 79874
20411840 64945 | 131 342 868 [196] | (204) 761 366 047 | 205°88159 35055
204°55270 09764 | 357 026 931 [496] | (204) 289 090 916 | 205°44729 90236
20498699 54583 | 97° 499 818 [196] | (204) 193 039 690 | 205°01300 45417
205°42128 gg402 | 263 809 202 Be (205) 379 061 834 | 206757871 00598
205°85558 44221 | 717 107 760 [197] | (205) 139 449 056 | 206'14441 55779
206'28987 89040 | 194 930 099 25 (206) 513 004 407 | 207771012 10960
206°72417 33859 | 529 874 947 [198] | (206) 188 723 775 | 207°27582 66141
207°15846 78679 | 144 034 944 [199] | (207) 694 275 967 | 20884153 21321
207°59276 23498 | 391 527 571 [199] | (207) 255 409 855 | 208°40723 76502
208'02705 68317 | 106 428 228 [200] | (208) 939 600 347 209°97294 31683
208°46135 13136 | 289 301 919 [200] | (208) 345 659 650 | 209°53864 86864
208'89564 57955 | 786 404 148 |200] | (208) 127 161 079 | 209'10435 42045
209°32994 02774 | 213 766 811 201| (209) 467 799 467 | 210°67005 97226
209°76423 47593 | 581 078 436 [201] | (209) 172 093 807 | 210°23576 52407
210°19852 92412 | 157 953 496 [202] | (210) 633 097 734 | 211°80147 07588
210°63282 37231 | 429 362 117 [20a] | (210) 232 903 640 | 211°36717 62769
21106711 82050 | ¥16 712 724 [203] | (211) 856 804 611 | 212°93288 17950
2IT'50141t 26869 | 317 258 077 [203] | (21¥) 315 200 Sor | 212°49858 73131
211°93570 71688 | 862 396 865 |203] | (211) 115 955 $95 | 212°06429 28312
212°37000 16507 | 234 423 773 [204] | (212) 426 577 897 | 213°62999 83493
212°80429 61326 | 637 229 881 24 (212) 156 929 239 | 213°19570 38674
213°23859 06145 | 173 217 041 [205] | (213) 577 310 406 | 214'76140 93855
21367288 50964 | 470 852 734 [205] | (213) 212 380 629 | 214°32711 49036
21410717 95783 | 127 991 043 see (214) 781 304 673 | 215°89282 04217
214°54147 40602 | 347 915 727 [206] | (214) 287 425 926 | 215°45852 59398
214°97576 85421 | 945 732 997 [206] | (214) 105 738 089 | 215°02423 14579
215°41006 30240 | 257 076 882 207] (215) 388 988 692 | 216°58903 69760
215°84435 75059 | 698 807 417 [207] | (215) 143 100 942 | 216°15564 24941
216°27865 19878 | 189 955 550 |208] | (216) 526 438 947 | 217'72134 So122
216°71294 64697 | 516 352 721 |208]| | (216) 193 666 066 | 217°28705 35303
217°14724 09516 | 140 359 222 [209] (217) 712 457 641 | 218°85275 90484

TABLE IY. (continued).

271

The numbers in square brackets denote the numbers of figures between the last figure given
and the decimal point; and the numbers in parentheses denote the numbers of ciphers between
the decimal point and the first significant figure.

272 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION.

Postscript. The statement on p. 244 that the tables in this paper were compared
with Schulze’s and Vega’s tables, as far as the extent of the different tables permitted,
may perhaps convey the impression that no errors were detected by the comparison.
It seems therefore desirable to state that no errors were found in Schulze’s table, but
that the following errors (in which the discrepancy amounts to at least 3 in the last
figure) were found in Vega’s Tabule logarithmico-trigonometrice :

x = 0°46, loge” is given as 071991755 instead of —0°1997755

» 127, @& a 3:560860 7 3560853
, 146, , i 4305950 4305960
eyiela ‘ 5528964 " 5528961
ae : 9974185 9974182
, 3°30, , Fi 2003371 ‘ 200:3368

There were also a great many cases in which the discrepancy was a unit in the
last figure, and several in which it amounted to 2.

VI. On Functions of more than two variables analogous to Tesseral Harmonics.

By MJ: Me, Elms MA,

[Read January 29, 1883.]
THE objects of this paper are

(1) To develope a series of functions*, here called Normal Functions, of (7—1)
variables analogous to Tesseral Harmonics, and

(II) To show how to expand any function of (‘—1) variables in terms of them,
the values of the variables being restricted within certain limits.

The result marked (A) corresponds to the conjugate property of Tesseral Harmonics,
the expansion (B) to the expansion of an arbitrary function of two variables in a series
of Laplace’s Functions, and the expansion (C) to the expansion of an arbitrary function
of two variables in a series of Tesseral Harmonics.

The following is an abstract.

Part I. 1. By means of the formulae of transformation

27 SIMO SIN Osc. cisnaiitas eas sin 6, , sin 0, ,
Fe. =r tsinn (2) Shin). oaboosacobee sin 0., cos @,_,
Po— TSIM SIN) On x m-segesoee SIMO, COSO)-

7_,.=rsin@ sin @, cos 8
i—2 1 2 3
v

£,, =rsin @, cos 0,

x, =r cos 0,

the equation = + oe SE isin a+ oa =0 is transformed into the equation
Bet) aie tls leg neat dt
r” sin’® 0, <0 connog Sun Ce {30 ga = NCEE, a as
+ sin? @, sin* : apopen un (oh. Ga BOs al + sin? 6, ca ee esinsom (aa) =a

* These functions were obtained by Green in his paper statement. Articles 1 and 2 of Part I. of this paper corre-
‘On the d¢-ermination of the attractions of ellipsoids of | spond with the commencement of Article 14 of Green’s
variable densities.” He states in the preface that it would paper. They have been inserted in order to make the
be easy to show that the functions of Laplace are particular paper complete in itself.
cases of them. The present communication proves this

Von. XIII. Parr IIL 36

27 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

2. A solution w=? (sin 6)", ...... (cin 0." avs (sin 6.) oh 0... (an? in2 6.) is

sin
found, where ©, is a function of @, only, satisfying the equation

sin’ 6, 96 ?+(2p,,+t—m—1)sin@, cos 6 de sat (Piney — Pin) (Piney + Pm +t — m —1) sin’, O,, = 0.

m dé,

The indices p, p,, Py: +++» P-. are all integers such that no one is greater than any
one preceding it in the series.

The function (sin 6,)"O, ....-. (sin 0,)°" @,. vss... (sin 0,,)”°0,., eae) will be re-
2
ferred to in this paper as a Normal Function of the equation iat = ah aera +74 =0,
.

n
3. Putting cos 6,=y, ti-m—1=n, and denoting the expansion of (1—2uh+h’) 2
in powers of h by

Qt Qh + we ies hs Gent ae :

Pm
it is shown that ( x) Qpn+ 1S a possible value of ©.

Pn-1

4. If n be an even integer it is shown that

1 (a ) cos {( mit 3) a
m7

Crea (n—2) (n—4) ...... 4.2

n
Pm-1 + 5)

If n be an odd integer, and P, the 1" Legendre’s coefficient,

1 Gia
Qom-a = (n—2) (n—4) ......3.1 (x3) Greats) .

. If °-s. when Pant =P av but = @,,” when Pm-1 =p a3 and if @;,; “
, @®@,,” site
te Te are all finite when @,=0 or 7; then

fe ;
| (sin 6,,)""""@, @,”d0,, = 0.
“0

6. It is shown that

-/ s \2P_tn n an Pm=1 ahr Pm — 1 ee ae .
s m e 2 a) —— =» aa n 2 :
i (sin Ce) ( ny d6,, Pn. — P Pm in + Py, = rf (sin 0) (Qe 7) dé,

ANALOGOUS TO TESSERAL HARMONICS. 279

7. It is shown that

ae 2pmtn |n ste Dest ts sia 1 1 Tv 1
| (sin 0,)"""*" (@,)*d0, == ed
" Pas — Pm iG
0 Pama — P Pua +5 { (n)}

where [ is the symbol of the second Eulerian integral.

8. If 1,=(in6)"®, ...... (sin ,)?"O,, ...... (sin 8,,)"* ©, , ie Pen 6.)
u, = (sin 8,)"*@, ...... (sin 8,)""@",, ...... (sin 8,,)""@"_, Nee eS 6.)
and aS =(sin @:)** (sin 0,)*- ...... (sin 6, ,) d0,d0, ...... dé_,d0..,
then puss OS Oso di carinw oAteena vas ce cscaeet case ewnaeeees (A),

unless w,=w,' identically, the limits of integration for @_, being 0 and 27, but 0 and 7 for
the other variables.

9. It is shown that

/ u2dS = . - = ; - : Pateee DO ira Ps oe : : P aE — : multiplied by
1 1 ei,
ee wae eS pee FSG) = jj—5 Renwas [2 |1
unless p,, vanish, when this value must be doubled.
10. It is shown that ;
” (sin 0)” ©, ...... (sin 8,,)"" , ...... (sin 8," @, aes po 6.1)
is a rational integral homogeneous function of the variables LiD iosons x, of degree p,

11. It is shown that the different functions of degree p of the form in the
preceding article are all the independent rational integral homogeneous functions of

(pee #, of the same degree which satisfy the equation
dui du au
ditt das to + 7,270

Part II. 1. Preliminary propositions.

i
(2) If &, be the coefficient of h” in the expansion of (1—2h cos 0 +h?) 2" then
R, is greatest when cos@=1, 7 being a positive integer not less than 3.

36—-2

276 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

(8) If h<1, and 7 a positive integer not less than 2,

[Ss — h’) (sin 0)*d0

=A (sin 6)'* dé.
°(1—2h cos 6+ he)

2. In general any arbitrary function of the variables 0/, 0, ... 0;
s y y 1» Us

i-))
F(0/0,... 0) = 5 a 3 oe ae a ie F(6,0,...0,,) Rd0,d0, ... d0, 440, ,d0,. ...(B),
where R={S (2p +i—2) R,} (sin 8) (sin 0,)*...... (sin 0, ,)? (sin 8,.),
Os
and where 2, is the coefficient of h? im the expansion of (1— 2h cosO +h?) 2, and

’ /
U2, -s.. +2,%,
cos 0 = —-4+4—_—__ + ,

ae

The relation between the variables a/a,'...a; and 7’@,’...0’_, is the same as that

between z.z,...z, and 7r@....0,,. 6',, lies between 0 and 27; @,’...6'., between 0 and 7.
1 Ua) é 1 i-1 i-l ? i-2

3. In general any arbitrary function of the variables 0,0,’ ... 0,
Ge. s..8 gs) 1
= > (sin 0,)2@, ... (sm 6,,)?"@,, ... (sin 6’, .)?0',_, {C cos p, , 0.5 + D sin p,.7,,},

where C=

ie ee (sin YO, (Sim B,)P*""* On (Sin 8)", 00878 FO,0y 8) 40d

i *... | [in 6%"... (sin 8)" ©,”.., (sin 8.) 8%, (C05 p., 0,,)°46,48,...d8., + C.
0 “0-0

and D=

ie AN if (sind, ji*7@,... (sind, )?="7""@_...(sind,_,)"=" ©) (sin p,_,0,_,)\ 2 (G0,...0,_,)a 0d... 46..

eae I "["(sin 8,702... (sin 8,)”**"40,2... (sind,,)”-" 6", (sin p,_,0.,)°d0,d8,... d8,,
/0

Vina
“0

The summation extends to all possible positive integral values of the indices p,...p,_,,
such that in any one term no index is greater than the index p with which that term
is connected nor any index greater than any one which precedes it. The limits between
which 6,/0,...0_, lie are the same as in the last article.

bo
~I
> T

ANALOGOUS TO TESSERAL HARMONICS.

PAE I

1, The formulae of transformation for #,2,...#, given in the abstract are equivalent
to the following :—

= (af +af+..t a2)

= (@etar+ ...+ 2, a3
9, = tant | = a,

Ce i i ied

» 2 2 2\3
—— tan? {@ +, Hines a) it

ee re ii)

But du _ du eee du d0, a du_ d0.,
Ms dx, dr‘ dx, * a0, dx,*'"* d0,, de,’
1 d*u_drjd*’u dr, du dé, fe du Sa @r
da? dz, (ua dz, drdé, da, ye redeem ait) adr aes
d0,j d’u dr | du do, du = du a0,
* de, (ard dv," dO? de,*""* d0,d0,, dv,)* dO," da}
ca aeenetelntel etal cletatotetotetefeleyefaraieratelareaslersinvetetsreveyeisiel steve etavefoiereleleialeretelstcistelalaicteleverelsietteiatersts
Oe ff Crt he d*u = dé, we d*u d0,.\ du d’0.,
+ de (eae de, * d6., dé, da, dOL dee) dO, dete
du du d*u
Therefore da,’ + da? + oie
Hu l(aey 4 (YL de (ae 4. 4 a9
6dr & cary ze +o (aa cot dee
es d*u G dr d0, sy > du /dé., Glee 4 Ou dr
dr d0,\dx, dx,’ du, de, 2 add ( des det © ap ma

+ aa (laa) t+ (Ge) ft aac (Cae) t+ (Ge)

du (d0, d0,, , dd, dO,
+4 2 apae- (Ge Fei ee Ta) +
du (a0 dé du (a0. d*6__,
+96 (ae: af + oat) a (Gi tet a

Mr HILL,

And from the formulae of transformation it may be shown that

ON FUNCTIONS OF MORE THAN TWO VARIABLES

dr \* dr\*
(i) cas (an) =
Bie) ges
Geo a8 ede We ane
dé, dr d0,, dr _ 0
dx, dz dx. da :
dé.\* d6,,\" 1
or) ares ea 7° sin’, sin’6,...sin®6,,_,’
dé,, dé, dé,, do, _ 0
dz, dx, a ee
dl’, d*0,, _ _—— (t—m—1) cotd,,
dx,* 3 dz? r* sin*6, sin®6,...sin?@,_,
Fee: au eu d*u
Therefore det das des
au i—i du 1 (dau du
— = = if z) at aa se (2 — 2) cot 6, es |
1 d*u du |
r sin*@, ae He 4) cot 8g + ; | )
: — 5, aenaas :
+ Patel, ia Gul! <5) eae a aaa |
du il d*u

1
TR sin*@,...sin*0, , (aa

+ cot.

oom =|t r’ sin’@..

esi oe (ca,

2) =9

2. Put in the equation (I) w=r?.u,, where u, is a function of 6,0,...0,, only.
After dividing out by r?”*; it becomes
p(p+t—2)u,+ (git @- 2) cot 8, Taf ed lapel 3) cot 8 ght a
aes {ra +(¢—m—1) coté, al +... soba (UID),
a: See Ge + cot6,, oa) + Sin, i le (Sas) =i

Now put in the equation (II) u,=(sin6,)"'.©
and u, of A Pe: Ae

{ (cin 6.)

|

u

only.

F ¢ + (2p,+1—2) (sin8,)”—! cosé ee

(-
1° Uys

After dividing out by (sin@,)?-?

1d0
(sin@, 1-2),

ihe ©, is

a function of @, only,
, it becomes

8, +(p—p,)(p+p,+t—2) (sin 8),

+P, (p,+1—8)/

ANALOGOUS TO TESSERAL HARMONICS. 279

1 Che een du,)
3 2810", lie + (¢ Te 1) cot, do,) sr ..
1 d*u, )
sin? Che aa

GRA oe du,
+ eee (@ = 3) an at +...4 ad
du,

26. :) + sin®6,.. Ao

il du,
+ Sin 6,...sin? 0, , ae atieabe,

If ©, be chosen so that the coefficient of u, is p,(p,+7—3), then ©, must satisfy

the equation

dO, : dO : qos
sin’, — do + (2p, +7 — 2) 1 dg, + (P—P,) (p+p,+%—2) sin?@,.@,=0...... (1),
and wu, satisfies

; d*u, du 1 du du, ) .
DP, (ptt —3)u,+ age 3) cot, a et To ae {rg it (-m—1) coté,, do. +... |
1 du, 1 d*u, i
+ 5in?,...sin°0,, Gas prac al errs =) a)

Reticianactheeaeiameste ren srates (IIT)

The equation (III) is strictly similar to (II).

If therefore in it uv, be put =(sin6,)”.@,.u,, where @, is a function of 6, only
and w, is a function of @,...0,, only, then proceeding as before but employing the quantity
Ps — instead of p,(p,+%—8), ©, and wu, satisfy the equations

sin ag = 2+ (2p, +7 —3) sin 8, cos oe = *+ (p, —p,) (p, +p, +t 38) sin’ 6, . O, = 0...(2),
ChU Paar du 1 du du, \ }
P2(Patt—4) Uy Seo we acct a. he. tin? 6,...sin’@, oo za aie, 0.75} | (IV)
1 dt, ee) we) * 1 o,) =0 (oF
ae + Sin? OS Sino (ia. amy COR eae Sina eee sin’ 6. (aa |

Proceeding in this way the solution of the transformed equation will be

u=r (sin 0)” @, ...... (qin-6: "Owe sac (sin 0.) O, 5 » Uea3

where the quantities ©,...0,, satisfy the equations (3) ... (i — 2), similar to the equations

(1) and (2) :

).

sin’6, T° + (2p, + 1 — 4) sin @, cos @ 22: 78, Shae Ps) (Py + Ps +% — 4) sin’O, . O, =O .... cece (3)

TTT rrr errr rrr rrr rere ree eee eee eee eee eee eee ee eee ee eee ee ry

d® 4 :
m+ (2p +i—m—1)sin@, ,0088,,76- ™ 4 (Dm 1— Pm) (Pm + Pm+tt—m —1)sin’d,,.O,,=0.....(m).

Seon oe ee ee ee ee a ee ar

2
sin? 6. es + (2p_, + 1) sin 6, cos@,_, =

280 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

The quantities similar to w,, u,, Us, VIZ. Uy... Um s+ Ug U_, Will satisfy equations similar

to the equations II, IIL, IV. the last two of which will be

Cu ee
Dea (Pea + 1) thot (ae a. a. + cot 8, so) tara Weta
3 i
Pio Vis ar dé.” =0
Hence w=?" (sin 6,)" ©, «0... (sin:8..)P 7 OL dees: (sin 6,,)"'* ©, ee Din 6...) ;
Any function of the form (sin 6,)"®,...... (sin'@,,)" Ou s (sin 6.) @,_ (ee Di 6.)
will be referred to in this. paper as a Normal Function of the equation
du du du
da? ar da, SF soccor ae da? = 0,

or more briefly as a Normal Function.

3. Certain properties of the functions © will now be proved. The equations (1),
(2) cee. (¢—2) which they satisfy are all of the same form, and may be typified by the

equation (7).
Putting cos @,,= and i—m—1=n for brevity, this becomes
FO,

(1 — p’)
dy

d®
— (2p, +2 +1) pw aie + (Dmg — Pm) (Pm. + Pm + 2) Om = 0.

It is now possible to obtain further information of the nature of O,,
It will be shown that if
dl

— = Q)+ Qht+ Qh? +... EA es | ik Na as sacs
a — 2uh +h

pm
then =) Qom+ Satisfies the equation in @,,.

i.e. it is required to show that

pm+2 3 pmt+1 Dm

d\ 1 1
(1 — p’) (= 7D Qos Fe (2p,, +n+ 1) lad a OP == (Pies are Dm) (ae aio ate n) aa) Dyn =0.

Pm
But this is (Z) {a —p a ana *—(n+1) p rac + Pines (Pua +”) Qpm- f= 0.

It will be sufficient therefore to show that the term in brackets vanishes. Multi-

plying it by 1", it may be arranged thus:—

2 dq? m= 1 d Im—1 d d m— d pm—
(LH) 5 (Ooms) = (041) HT Ores BP) + rae (Qpas )f a ah Qo Wer),

ANALOGOUS TO TESSERAL HARMONICS. 281:

This will vanish if

i d d f(, a d 1
{0-H ga- @ +a + (bay (tay) mb =U)
du du dh = dh (1 —2uh +2

because the expression which must be demonstrated to vanish is the term containing

h”* on the left-hand side of the last equation, It is necessary therefore to show

that

(1=p')n(n +2) h8(1— Yuh +h)? n(n +1) hye (L— Qph + hey 2
+ {n (1 — 2uh + nyt (hy — 2h?) +n (n +2) 1h? (wu —h)? (1 — Qph + ny 2) =0,
+h (po — h) (l- 2Quh +R)?”

Le Qn yes

,

{n(n + 2) (h?— hp?) +n (n+ 2) (Wp? — 2h? + h')}

+ (1 — 2uh+ hy? {—n (n+1) hu+n (hp — 2h’) + n*h (w —h)}

n

Le, (l—2uh+h’ 2" n (n+ 2) h? (1 — Quh + h®) + le Onsen — 2nh’ — n7h?) =0,
be

which is identically true.

4, The form of Q,,, must now be more closely examined.

m-1

Firstly, let x be an even integer,
: F WV’ h®
Since log (1 — 2uh +h’) =— 2 @ cos 6, + 3g 098 26,,+ 3 00S Bidet Speco "

after differentiating both sides 5 times with regard to mw, and comparing the coefficients

with the coefficients in the equation

1

pm-1

A. ee a Qo + Q, h Tiseeees + Qrm-s h Pivevees
(1 —2ph + h’)?
it appears that
f n
a ais
*Pat (nn — 2) (n—4) 0000 4,2\d n
(n= 2) (n= 4) are
Secondly, let n be an odd integer.
1 :
Since Pit Ph +... + PR + cece

(1 — 2uh + n°)? Be

oP ane : (ON ae
where P. is the 7” Legendre’s coefficient, after differentiating both sides —z_ times

r

Vou. XIIL Parr III. | 37

282 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

n—l

with regard-to yz, and dividing by 2*, this becomes

nt
TLE. Deke (n—2)}. = .= (4) {Pat eth +P 2) ) acme ees \ ,
(1 — 2uh + h’y 2 (rma
Comparing with the series
1 Ym—
$<, = @)+ Qik +...... 8 Os a aaa Se
(1—2ph +h’)?
n—1
it appears that 0 e d )'P
is ar Pm-1 (nm — 2) (n—4) woe Deore (aa (pma+?5") c

In both cases Q,,,, is a rational integral function of mw of the degree p,, con-
taining only even or only odd powers of mw; and therefore ©,, is also a rational integral

function of pw of the degree p,,—Pm. This is not the most general form of ©,,, but
it is the only one which will be considered in this paper.

5. Suppose now that if p,,.=p'n.. Qn=9,,’ satisfies the equation (m) in article
2; but if p, ,=p",» 9,=,,"" satisfies it; it is required to determine the value of

[7 Gin 8,27" 0/0," Uy
0

Since the equations
2 ,

sin?@ =

F dO,’
m dé 2 a5 (2p, ats n) sin On Cos On Fe a (pee — Dn) (Dees + Pm oF n) sin”6,, 2 er a 0,

2a)

sin® An as Ar (2p,, + n) sin 6,, cos 6, ae at (4 is) (pies aml ate n) sin*6,, ‘2 @,,” = 0

’

hold good; it appears that, multiplyig the first of them by (sin@,)”»t""-2@”, and the
second by (sin 6,,)?"*"-?@,’ and subtracting, the result may be arranged as follows:

ae | sin 0.) ies {@.” oe az sg o- }] at (Glee 3 pes) (Dees ar Dee ar n) (sin 6,,) poate uy 6,9, = 0.

Integrating with regard to @,, between the limits 0 and 7,

T

; - ” e ‘ ' () € / ” 7, ” uw a, , a”
(sin 6,,)22™*” 1, d ey di Dm F(a) may) (Det? oe +n) (sin 6,,)22™+"@_'@,,d6,,=0.
dé, d6,, 0 ; : : . 0

z dO,’ d@,,” ; : :
Now ®©,’, qe.’ @,.’; 16 are finite when 6,=0 or aw since ©,’0,,” are rational

nm
integral functions of cos@, by the last article; therefore the term in brackets is zero
at both limits: p’,,,—p",,., 18 not zero, since p’

wt of , wm"
im, 20d p,,, are different: p,,+) m y+”
is not zero, since p’,,p’

m are positive integers or zero, and n is a positive integer
the least value of which is 1.

Therefore the equation shows that i (sin 6,,)2?=*" @,,@,, 0, = 0.
0

ANALOGOUS TO TESSERAL HARMONICS. 283
6. The value of the integral i “(sin 6,,)°P™*n@ *d@, can now be determined.
0

The working divides itself into two parts. The first part applies’ whether n be an
even or odd integer; the second part breaks up into two cases, the first corresponding
to m even, the second to n odd,

It has been shown that

3)
sin’, she 2 + (2pm + 2) sin 8, cos oe Ss ao (Prt — Pm) (Pmt + Pn + 2)8in"O,, . O,, = 0.

Multiplying by (sin@,,)?7"*""@, and fear between the limits 0 and 7, the result
may be arranged thus

(Pas Pn) (Pas + Pn +2) [ (sin 8)?" O%, 8,
0

aN (ie LO
=— i : {(in 6, 2pmtn @ de. = + (2p,, + 2) (sin @,,)??™+"-1 eos 8, a do

H Fa? Be i onen (Zo
—| Gino, +2@ al + (ine, ar (Se TB) as

(Pani~ Pa) (Paci + Pu +) | (sind,)**O%,d0,= |" (sin 8,)*m'2 (—"O Fy ) a8,
But @, = (ae) Owes
Dyas Pa) Paci + Pat») | (in 8)? (ae) Ooms} do
= i “(in Ga) Pmnes (a Qt dé.

Fe dn ere G3
Put P (Pm) =| (sin Cp yer \(=sc5-aa.) Qones} d6,,.

Then the last equation may be written

(pos =i) (Pina + Pm ae n) ® (Pm) = ® (Pr ate 1).

This is an equation which is true for all positive integral values of p,, less than p

m-1*

Hence changing p,, successively into p,—1, p,—2, ...... 2, 1 a series of equations is
obtained, whence it can be shown that

<< \n + Png + Pn — Pn m1
P(P.) i [M+ Py = it i ” [po a (9),
* 9 | + Dm it Pm ell ee we .
ae fe (in d,)m10,849, = — Sete | ' [ (sin 8)" (Qn,_,) Oy.
m—| m1 m

37—2

284 Mr HILL, ON FUNCTIONS OF MORE THAN -TWO VARIABLES

7. Let m be an even integer.

Then since Q,,,, satisfies the equation satisfied by ©,, when p,,=0, it appears that

m1

: tre n 2 iyo nt2 ago
Paci(Paca®) | (si @,)" (Qnes)® Wn = | (sin 8,)"* (S29) ds

Pe Part” [in 8)" (s cos {(Paus* 5) ent | dé,

= I “(in a abi lier cos { (Past 5) @nt | dB.

This is an equation which is true whatever positive number p,,, may be and what-
into p,,,+1, the equation

ever even integer nm may be. Changing » into n—2, and p

m-1
becomes

(PoitD (pPaat2—- 1) [Gin Ee ee (er ane {(Paus 5) ant | dé,

= i “(sin g2)* (2) cos {Paar 5) ant | d6,,.

Proceeding in this way there arise finally (by putting in the last equation n= 4,

n

2

sob Gardonare oe nde

= [si 6, | (aa) cos {(Po- + 5) ant | d6,.,

nD) onl me fsee [oeeDelan
[in 6." @ cos {(Pms+5) 0nt | 0,

= n\ |Puatn—1 f= 5) iy
as (px1+5) eel Ga Das 0 | cos {(Paat ) 6, ita

and for p4. Pmat

m-1

2; and n=2, and for p,,., Pat3-1) the two equations

nN Pra ti wes 1 7 s

= (Px = 5) fon 2 2 >
of jot 1 1 [Poa tT ogre
[cin Pn)” (pas) Cm = Ty (yA) as EDP” A ve? pee ol

Pnat 5 =
if n be an even integer

et ae 2+ Ppa t+ Pm—1 1 1 7
s 2pmtn (=)? SS —_—_—_— "9°
[(cin 6% 'n46,, iS ee nm’ (n—2) (7-4)... 4.2) 2

Pnat 5

bo
oo
ou

ANALOGOUS TO TESSERAL HARMONICS.

But if m be an odd integer

a a8 : d pm 2
f (sin 6,,)°?m*" \(Ganacaz.) Orn —

fe Peat Pa [Pea 1 ame WAG a a2
eee ee DG ea) Pome | &

[etmek Pinel pe iT
~ n+ Pmi-l [Pma—Pm {(n—2)(n—4)......5.3. 17

multiplied by

[ fn — n—-1 7}
Sercraaes pry + Tila |
ate <2 1 {
n—1 =) Bio
5) es io 2 (=5 me +P ns) ar il |
ie J
by a known property of Legendre’s coefficients. :
+, if m be an odd integer
7 | Pia + Pn — 1 1
| Gin @,)~in@, 199, =a Pe ; ee ee
0 (Da =O nm {(n—2)(n—4)...... HS ly
Pmnr-1 stg 2 *
Both forms are included in
7 a Deena Dred! il
[in 0, )7em*2@7d6,, = RF Pat Pa? ee ce
= z (ee Te Zeal Gin):
Pn-1 ts 5) A

where I’ is the symbol of the second Eulerian integral.

8. Suppose now that there are two solutions of the equation in wu, of Art. 2, viz.

u,, U, where
é é ca cos

u, = (sin 0,) "0, ...... (sin 6, ,)-?, ie Fae 0S)

' . 1? : ney Soy
and Ua (SING) 21 O)eraetenre| (SUELO ey!) ota? © eee ( p 0)
and if dS be a differential element having the same relation to the variables
i, Uy ovens £2, as an element of area of a sphere has to the three variables , #, #7, so
that | fdedS = | da dan,..... de, = | TardOede. 0,
where J is the Jacobian of w,2,...... xz, With regard to 76,0, sates @_,, so that

dS = = SAG. dO) cose dé,_, = (sin 6,)' 7d, (sin 0,)'°d, ...... (sin 6,,)° d6,., (sin 0.) d0_,d0_,.

3 7
Then [uu,ds=o unless u,w, are identical; the limits of inte- |

gration for 6,, being 0 and 27, and for the rest of the variables 0 ae

and 7.

236 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

The integral in question is the product of a number of definite integrals which can
be separately evaluated. It is necessary to commence with the last of these, viz.

es Py ee ; \ d..5

> |sin? PeaaVena \sin? 204

2 be different positive integers; and if p,, p’_, be the same,
the integral will still vanish unless the terms in brackets be both sines or both cosines,

and this vanishes if p,,p

It is therefore only necessary to consider the integral when p,,p’,, are the same.

Supposing this to be the case, but p,,p’,, different; then by Art. 5,

i-2

ia (sin Pee ones dé, =

It is therefore only necessary to consider the integral in the cases in which p,,p,,
are respectively equal to p’.,p'_, Hf this be so, but p,, different from p’_,; then
again by Art. 5,

ie (sin Ch ysis ©, 0’... dé,_. —

tt)

And so on, it appears that the integral in (A) will vanish unless the quantities p,,p,,---P,
be respectively the same as p’,,p',,---P, This corresponds to the conjugate property of
Tesseral Harmonics.

9. It is now required to determine the value of |[u,’ dS, ie.

ig es P. a, da [" (ine_)%21 0,2 d0,,...[ (sin 8,"*-2 0, dB, |" (sin 8,)”4-* 8,08,
0 10 0

sin

Supposing p,, not to vanish, this is

n).( Pe Satis 1 T ) [p+ p,ti-4 1 . BS

Dea Peo Pst Pe i [Ps— Ps nee Qi-s ir el.
siialeipliede By. eRe ee * rae
IP—Ps t= 2° us fr (’ =|

2 2
_ Pest Por [Pot Pst1 = [PtP ti-4* [ptmti-3 1
7. ae —Prs Pi. Pues seeeee Ps — =a P—P, DP, Pat seeee
1 1 7 ia = “ 2

ANALOGOUS TO TESSERAL HARMONICS. 287

Bot oF AOR FTO PEP pipe
: PST PE)
= PSs Sa
ia r(5*) {r( zr} eS tes)e AVQ)TA)HEMPa Wea)
7? 1 1

rif ) 5 ‘mgt tag vat} 7° gg Bah Ja Va

i—-1 OpSearnsct iC)

7 1

a oe EE EL sap poapes..b

u, dS
|p. LA Ne ies ccf D+p,+t—4 |p+p,4+%1-38 .
sues +Pis Pot fee lid ae ae PtP, multiplied by
~ |p -3 — Pi-2 |p i Pi-s |p 1 Pe IP — Py Ps
1 1 1 aritt
Hb oe Bg J=3) = “QT Gi). i—4fi—5...... 21’
Pi-s + 9 Pr: a: 2 P ar

where [' is the symbol of the second Eulerian Integral. If p,,=0, this value must be

doubled,

i-2

10. To show that

7? (sin 0,)”: @, (sin O,)®, ...... (sin 8,,)%-2 ©, Ie ae 6.)
is a rational integral homogeneous function of z,#,...... x, of degree p.

It is homogeneous and of degree p because r? is of degree p in these variables, and
all the quantities sin 6, © are of degree zero,

Moreover it was shown in Art. 4 that ©,, contains (cos 6,,)?=-1-?= and lower powers
of cos @,, whose indices are positive integers differing from p,,—p, by some even number
which may be denoted by 2q,,, so that the general term in ©, is a numerical multiple
SE (COSO, \Pa— Pa — 2am,

In like manner the general term in cos p,,0,, is a numerical multiple of
(cos 0, ,)Pi2-24, and that in sinp,,0@,, is a numerical multiple of

He Y4-1

(sin 6,_,) (cos 0,_,)?i-2- 241-1,
It is sufficient therefore to show that the quantities included in the form
r? (sin. 8,)?: (cos 6,)?-?:— 2% (sin 8,)?2 (cos O,)?~P2— 2% 2... Ginnie \7=2!(cosi On.) Pe aes seat
multiplied by (sin 6, ,)?-? (cos 6,_,)?i-8- Pi-2- "4-2 (cos O,_,)Pi-2- 24-3 ite: 6...)

sin
are rational integral functions of w,7,...... 2;

ie

Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES
Now r? (cos 6,)?~P:- 2a = (7 cos @,)P~ Pim 2%, pri t2h = (7 cos O,)P~P1- 2a, 72M, Pr ;
*, 7? (sin 0,)" (cos @,)P~21-24 = (r cos O,)P~Pi- 2a, 72h | (7 sin ,)?;
*, rP (sin 6,)?: (cos 0,)?-P:-2% (cos O,)P1— P2242
= (r cos 0,)P—P1- 2% 7° (r sin 8, cos O,)P1~P2- 2% (7 sin O,)P2 2a
= (r cos 0,)?-P1-24 72h (7 sin O, cos O,)?1-P2— 2% (r sin O,)74 (7 sin 8,)?* ;
* 7” (sin @,)?! (cos @,)P—?1- 2 (sin @,)?2 (cos O,)P1- P27 2%
1 1 2 2
= (7 cos 0,)P-M—2n 7% (7 sin A, cos O,)?1-P2- 242 (7 sin 8.) (7 sin , sin @,)?s,

Proceeding in this manner it appears that

r? (sin @,)? (cos 8,)?-P:-2% (sin O,)?2 (cos ,)P1~Pa- 22

(sin Chae (cos 0, ,)P-4- Pts 2at-s (sin 0,_,)Pi-2 (cos 0, ,)Pt-2-Pi-a—-2at-s

= (r cos 8,)P~P- 2%: p24 (r sin 8, cos O,)P-P2-24 (77 sin O,) 22

(r'sin 0; sin.@, ...... SinO,, COS 0,4)" *i ses
xi(rsin@ sind, 2... sin 6, ,)#-8 (r sin 6, sin 0, ...... sin 0, Cos 0,_,)Pi-8- Pi-2-2ai-2
multiplied by (r sin 6, sin 6,....... sin 6, ,) 4-2 (7 sin 6, sin 6, ...... sin 6,_, sin 0,)"'>
Every term except the last is a rational integral function of the variables. It
remains therefore to show that the expressions included in the form

: : (cos
: . 6 j-2—244-1—
(r sin @, sin 6, ...... sin 6, sin 8, ,)?'-2 (cos 0, ,)Pi-2- 241-1 ean 6.)

are rational integral functions of «,7,......”,. This form may be arranged thus:

(r sin @, sin 6, ...... sin 6,, cos 6,_,)P!-2729%-1-1 (r sin O, sin ( cooann sruau(ed 5) A

multiplied by G sin 0, sin 0, ...2.. sin 0,, See 6...)
and every term in this is a rational integral function of 2,2,

The result is that

r? (sin 0)? (cos 8,)P-”:-2% (sin 8,)P2 (cos 0,)P:~ P2~ 242

Pid Ce

(sin 8,,)”-* (cos 8,,)P--P'-s-24-8 (sin B, .)Pi-2 (cos 8_,)P-8-Pi-2-241-2 (cog Pe eager

= (x)? (0? + a +... + 07)n (av,_,)?1-P:- 2a (77+ a,° + ) oa

2

5 see

(a,)Pi-3— Pi-2— 2qi-2 a? + as se 5) 4-2 (a,)®!-2- 240-11 (a? + 7) 26-1 ©) ;
&.

1
11. To determine the number of different functions included in the form

r? (sin 8,)" ©, (sin 0), ...... (sin 6,,)""-* ©, , te Pra 0.)
which can be obtained by giving all possible integral values to the exponents p,, p,...+.. Pi»

which are such that no one is ever greater than p, and no one is ever greater than
any one which precedes it in the series.

ANALOGOUS TO TESSERAL HARMONICS. 289

The possible values of p, are
on pee ater dee astcb et ere naate obs clas Gelslainetotrorane (a).

When p,=0, the only possible value of p, is 0;
p,=1, the possible values of p, are 0, 1;
D2, ~ - 55 oa (Oe ben Bi
[RSeh oe i A y We) Bis
and so on.

Thus the values of p, giving rise to different functions may be arranged thus:
ORS ORs a Oe Dee On U2, ates he stercte siete oc 3. XD) dlse2 G) one'/Dioendcaorebagdea (B).
(it should be noticed that though p, may have the same value twice over, it gives

rise to different functions. Thus there is a function corresponding to p,=0, p,=0; whilst
a distinct function corresponds to p,=0, p,=1; and so on.)

Again the values of p, giving rise to different functions may be arranged thus:
DOM Owls OO nlesOrt e210; OF 1, 0} 1250) 1, 2) Sie .ce. cane ;
OD, Os thy O, Us 2, Oy 1h 2 BY ccoocuaee OTS Aap essa ocles (7).
The law of formation is seen to be that the numbers which stand in the n™ place
of the row (y) include all the numbers which stand in the first n places of the row (8).

The relation of the numbers in the row (8) to the numbers in the row (a) is the same
as the relation of the numbers in the row (y) to those in the row (@).

It is necessary therefore to find the number of the numbers standing in the row
corresponding to p,,, the number of zeros being counted separately, because if p,_,=0,
sin p,,0,,=0, and this does not give rise to a function. Every other value of p,, gives
rise to two functions.

The number of zeros in the row corresponding to
fa 08 Le
p, is 1+1+14...41 (p+1 terms),
P, is 1424+3+4...4(p+1) (p+1 terms),
and so on.

The number of other figures in the row corresponding to

p, is 1414+1+...4+1 (p terms),
Pp, 8 14+24+3+4...+p (p terms),

Pp, 18 ae ay eee (p terms),

and so on.

Vou. XIII. Parr III. 38

290 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

Now the series of figurate numbers of the first order is 1+1+1+...;
= » » » second , 1+4+24+3+4...;
» 5 >» » » third , 143+46+...;

and so on, the law of formation being the same as in the above.

Therefore the number of zeros in the row corresponding to p,, is the sum of the
first (p+1) terms of the figurate numbers of order 7-3; and the number of other figures
in the same row is the sum of the first p terms of the figurate numbers of order i—2,

Therefore the number of zeros is the (p+1)™ figurate number of order 7—2, and
the number of other figures is the p™ figurate number of order 7-1.

‘ n+r—2
But the x™ figurate number of r order is ae
Therefore the number of functions arising from the numbers in the row corresponding
[prta3. of |p+i—3

to p,, 1s I. p73 5° ips es2

It is required to show that this number is equal to the number of independent
rational integral homogeneous functions of 2,7,...7; of degree p, which satisfy the equation
du. du d*u
dat * datt cg 4
The most general form of a rational integral homogeneous function of 7 variables of degree
ptti—

rites

_

p contains - constants. If it satisfy

d*u ase au
det age 0 age
2 2 2
then since oat a a is a rational integral homogeneous function of degree

ea

p—2, there must exist ——

p+i-—3 . :
virtue of these, ns of the constants may be expressed as linear functions of the
Eee =
remainder, Therefore the most general form of a rational integral homogeneous function

1? au A
dn “beet aa a= =0 contains

linear relations amongst the arbitrary constants. In

of 7 variables of degree p which satisfies the equation 7 ae igi t

—=}= — ————. arbitrary constants involved in a linear manner,

|pti-3 \jp+i-3 |p+ saa |jp+i-3
ip \t-3 a [p—1|i-2 rz |p |e 1” |p- 2|¢—1°

ANALOGOUS TO TESSERAL HARMONICS. 291

Therefore the number of functions of degree p of the form considered in this article,
each of which is known to satisfy the equation

du du au
GES EI IE a: Ee

0,

is equal to the number of the independent rational integral homogeneous functions of
x,@,...0, of degree p which satisfy it. But the functions considered in this article are all
different, and are all rational integral homogeneous functions of «,,...7, of degree p.

Therefore the functions included in the form
3 ; 2 cos
r? (sin @,)”: ©, (sin#,)” ©,...(sin@,_,)'2O,_, . Den 6.)

are all the independent rational integral homogeneous functions of z,7,...2, of degree p
which satisfy the equation

du du au - 0
dais da, eae Stee
IPANRIE IDL

The expansion of an arbitrary function of the variables 6,0,...0,, will now be considered.

The proof of this expansion for any rational integral function of the quantities

cos6,, sin 8, cos6,, sin8, sin@, cos6,, ...... , sin, sin8,...sin8,,, cos8,,, sin8, sin#,...sinO,_, sin8,_,,
and the proof that the expansion for any arbitrary function if possible is unique, both
proceed in exactly the same manner as in the case of a function of two variables. They
will therefore be omitted, and the general proposition only given. The proof here set
forth is similar to one of those given of Laplace’s expansion of a function of two variables.

It is subject to similar criticisms.

1. The following preliminary propositions are necessary.
(2) To show that if R, be the coefficient of h? in the expansion of
1
(1 — 2h cos@ + i)?

then R, is greatest when cos@=1, 7 being a positive integer not less than 3.

Suppose that ee =Q,+Qh+ OM +...+ Oh? +...,

(1 — 2h cosO + h?) ?
1

and
(1 — 2h cos + h®)?

=P,+Ph+ Ph’ +...+ Ph? +....

38—2

292 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

1 1 1

tts. = |
(1—2h cosO +h) | (1—2heosO-+A%)* > (1—2hcosd-+h')*
(Ro+Rh+ BY +...4+ Bh? +...)

Then since

=(Q,+ Qh+ OJ? +...4+ Qh? +...) (Pot Pht PV +...4 Bh? t...).
Therefore R, = P,Q, + P,Q, + P29, +--+ P,Q.

Now it is known that each of the quantities P,, P,, P,...P, is greatest when cos? =1;
if therefore this property hold also for Q,, Q,, Q,-.-Q, it will hold for R,; i.e. if this property

hold for the coefficients in the expansion of t 7,» then it holds for the
(1 — 2h cos@ +h’) *
coefficients in the expansion of ee but it does hold for the coefficients in

(1—2h cos6 +h?)?

the expansion of i 33; therefore it holds for the coefficients in the expansion

(1 —2h cos0 +3)?
of ———— and so on universally.
(1 — 2h cos@ + h’)?
The proof holds good only when 7 is a positive integer not less than 3.

(8) To show that
(1 — hk’) (sin 6) de = [Gin 6)" d8,
0 (1—2h cosO+h*)? °°

where h <1, and 7 is a positive integer not less than 2.

Let SM ER we
(1 — 2h cos @ + h*)?
apg ee Re pee
(1—2h cos 6 + h’)*

“. (1-2)

UUW _.G +242, 0)R4G—242.1)4R, +... += 2 eee
(1 — 2h cos 6 + h*)*

iG) (1 — h’) (sin 6)* ae
9 (1—2h cos 6 + h’)’

eget 0) {Gin 6)? Rodd +(i—2 +21) i [Goin 6) R, dé
0 0

ra Ce ER) | “(sin 0)'*R,dO+......
0

But putting n=1-2, p,1=P, Pn=0, m=1, O,=8, in equation (m) of Art. 2, ©,

becomes f,.

ANALOGOUS TO TESSERAL HARMONICS. 293

Therefore sin’6 — +(i— 2) sin 6 cos 0 7 +p(p+i—2)sin’é. kh, =0.

Multiplying by (sin 6)** and integrating between the limits 0 and mw with regard
to the variable @

in gy-2t nak +p(p+t%—2) i (sin 0)" R, dd = 0,
i "(sin 0)? R,d0 =0 if p be not zero,
0
If p be zero, then since Rk, =1
i “(sin 0)" R,d0= ih “(sin 6)"*d8 ;

aa —h*) (sin 6)~ do _ (ci 6)2d0.

(1— 2h cos @+ 12? ,

The restriction 7 not less than 2 arises from the fact that throughout the paper, it has
been supposed that there are not less than two variables included in ,2,...... fi

2. Suppose now that

Goi Se ee papa, oP aA occ. 5 +a32=r77,
da=r (sin @.)' (sin @,)5° 24. (sin 6, ,)’ (sin 6,,) dO, dO, ...... dé_,dé,_,d0_,
DP = (a,—a + (a, — 2,2 + ee +(x, —2',)? =7* — 2rr’ cos 8 +r”
BEOF Geren asta 6,,) any function of @,6,...... 0_, whatever,
ea ge
= i Gwe. es =——= 6.1) do , the limits of integration for @_, being 0 and

Base tor 6. 0, 1.40. @,. the limits being 0 and 7.

Suppose that r’ is infinitely nearly equal to r but greater than it, then every
element of the integral w is very small except those which make JD infinitely small;
therefore for all elements which do not make D infinitely small, anything may be sub-

stituted for F'(0,0,...... @_,) in calculating the value of the integral. But when D is
infinitely small this cannot be done, for then both numerator and denominator of the
expression to be integrated are very small; so that in this case 9'(6,0,...... @_,) which is
very nearly equal to F'(0,0@,...... @'_,) cannot be replaced by anything else. It is possible
then to replace F'(0,0,...... 0_,) jis LO epace @’_,) im all the elements of the integral.
(Since it has been assumed that 6,8,...... 0, can be put equal to 06',6,...... Can
respectively, the limits between which 6@,6,...... 0, lie must be the same as_ those
between which 0,0,...... @_, are’ contained.)
Therefore | its a Ge ss #1) da = F(8',0', «+... 0’,.) | = do
2
=F (0.08 on s028',-5) fr es _ 8! d0,d9, ......d0,,d0_, 10.,
(r? — 2rr’ cos 8+ 7 yt

where ==N(eImO)) =" (Si ON)" See -: (sin 6)? (sin 6,,),.

294 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

Let the variables a,,......@, be transformed by an orthogonal transformation to new

variables ¥, Yo «+++ y,, and let Y, Yo ss y, be transformed into a set of variables
rh, py vveee $_, by a transformation similar to that given in the abstract of Art. 1 of
Part I. Moreover let ¢,=9. Then
7 (sin 0,)* (sin 8,)°° ....2 (sin 6,_,)* (sin 6,_,) dO, dO, ...... dé,_,d0,_,d0,_,

=r" (sin 6)" (sin G,)™....-- (sin $,.,)° sin d, .dOdq, ...... dh, _,dp,.db;43

and the limits for the new variables are the same as those for the old.

Therefore

2_ 9°) F(0.0, ali} Bah ; a ee 2 : i-2 é
[ Cee = do = F (6,,'... 8,1) Rants re) ue multiplied by
: 9 (r?—2rr’ cos 6+ 7°)’

| 5 fannie) Fug) <5 | jf eae { ie

Hence by Preliminary Proposition (8)
27?) F (0,0, ... 9.) do 1g! fi
ies ‘) os ST dC eae

i "(sin @)"?d0 multiplied by
0

ie
I ; (sin $,)** dd, ....+. i ‘ sin $,_, dg,_» “ae
0
=F (0/6; 8.3) 7 aa alll * (sin 0.) (sin 8.) «..+.. (sin 0...) 0, dO, ...0.. €,-.0,-1

and therefore as r’ approaches indefinitely near to r
7, 7 mw f2r
| emt Es ae +++ 9.) dg =F (6/0)... 8.) | “ I ip S'd0,d0,... d,.,d0,..

where S’ has the same value as on previous page.

A similar result would have been obtained if 7 had been infinitely nearly equal
to r but less than it.

Both results may be expressed thus :—

F (6/0; ...0,.::) | "ae [7 [7 (sin 0)" (sin 8)" «....» (Sin 8...) (sin 8.) 40,08, «.. 48,48... 18,
0 Jo 0 : 2 i-3 i-2 i-1

“(ff P0000."

(jen
where it is supposed that z 7 is expanded, and 7’ put =7 after the expansion has

.r'? §/d0,d0, ... d0,,d0,,,

been performed.

re — 7 il d
But er td ar (pa) + G- 2) pal -

ANALOGOUS TO TESSERAL HARMONICS. 295

Also je= bs . :
(r? — 2rr’ cos 8+ 7°)? i i —25 cos O+ (=) ie
1 Ue (*) '
‘eat G) +f if r'>r.
ee il r ,
Similarly pe = {r, +R, note +... 17 <9
1 _2 COS a: =") oh
pce 4, Ga G-8) 2 »(=sn eae sin aa mee Peeve
+
2
1 d e Bar
pt == G26)5 1 ==) gis BE

therefore if 7 > 1,

i ee !
es pea G-2 42.0) Rt 6-242. 5K, 4... 4-242p)(5) B+ ae he

dD 41-2
but if 7’ <7,

r—r il
Dp 1—2° 7

a {(-242.0) R+G-242.)°R 4... + (¢—2 + 2p) (=) R+ 7s f.

The series are convergent* except when r=r'; and in this case the unexpanded
value of uw having been shown to be finite and equal to

FOO,.....0,,)7 ii [ [in O° ...... (sin. O,,) dO s..+-. 0,449, 4,

2
the above

ie
it follows that the sum of either series, obtained by substituting for : 7

values in uw, approaches more and more nearly to this value as 7’ approaches to equality
with 7.

FOG, 0. 0.) | I : | TG (ein 0. =... (Gn, (Gin 6) db.d0,...... 0. d0,_d0,.
0 0/0
€ [lr
et. I oe i | F (6,0,...... 6,,) Rd0,d0, ...... d0,,d0,,40,.,
t—2J oo
where i= {"S"@p +7¢—2) R,} (sin 0,)*? (sin 8,)°° ...46. (sin 6,_,)’ (sin 0,_,).
p=0

* Because by Preliminary Proposition (a) R, is greatest the ratio of which to the preceding term can be made less
when cos 6=1, and then the general term is than unity by taking p sufficiently great.

p|t+p-3 3
(i- 2+2) (5) P ip p32’

296 Ms HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

This may be written
F(O.B'g esses: ..)=sqaaygR PO [-[[Pee, oh 0.) Ra0,dO, ... d0,_,40,,40,, ... (B)
where R has the same value as on previous page.

The above reasoning holds good except when 6’, is either 0 or 2r, or one of the
other variables say 6’,, is 0 or 7.

In the first of these exceptional cases, it may be shown that for F'(0/0,’... 6',)
on the left-hand side of the last equation there must be substituted

4 (F(0,9, ...... O30) so (OY OF csccse O,_, 217).
In the second if poe for F'(6,'0,' ...... 6’_,) must be substituted
px = (2m Fnu , 0 0 i—m-2 }
‘les | | eee eben 6.1) (sin 8j,,,) M7? weve (sin ,,) dO, «.+--. 40,40, ,
Be | 5 | | (sin Ee (sin 0.) dO yyy os0+ee d0_,d0,,

3. It remains to show that the general term of the expansion is a linear function
of the Normal Functions considered in this paper satisfying the equation in u, marked
(II) in Part I. Art. 2, and to determine the coefficients of the several terms.

Firstly, it satisfies the equation in u, for

( a ae d* . &. he ) 1 0
da dept da ; - an;
: . ; [(z, —%, : ar (x, — Hy YP veeeee af (x, = x;')"] F
th a? a 1
. (septate. SS =0
(ae 2 Ghee d = ; Se
: % 7 (2 — Orr’ cos 6+ 7°)?
PL, AU + ovceee + 1, ‘
where cos@ = Te ee , and the relation between the symbols z,’...... an
and (96) oos.+. @’_, is the same as that between 2,...... 6, Bnd 90) eae. ys
d* d’ iN 7 r'\? ;
‘ (aataat sete +a) (BtR + aren +f, (“) + os) =0, assuming 7’ <7.

hd & d* ;
(ae +a Fivveeee rE i") (Rr?) =0;
hed, 32a ee _d
“i Gar ae) + a4 (aga + C= 2) cot a7) + siete

i &
ay sin’ 0, we} | (R, 7?) =0;

ANALOGOUS TO TESSERAL HARMONICS. 297

2 E aes x Te) Ose ap7 a) teeta a, a a”, | see
. E (p+i-2)+ (gat (i 2) cot 8, a) * me — se | V=0
where V= less i [vF (0,0, «+++ 9.,) « Ry (sin 8,)** ...... (sin 8,,) d8, ...... dO, d8,.,;
. the general term in the expansion of £'(@/6,’...... @'_,) satisfies the equation in w,.

Secondly, it is a linear function of the Normal Functions considered in this paper.

f, is a rational integral function of cos@ of degree p, containing only even or only
odd powers of cos 6;

. R,r? is a rational integral function of 2,7,’ ...... wv; of degree p satisfying the
2 aa du ih au 0:
q ee do qe

.. by Art. 11 of Part IL, it is a linear function of functions of the form

r? (sin 6,’) @/ ...... (sin 6,,)-20",, see PC “

Let one of these functions be denoted by r?U', then R,==2A.U'; where A is a

function of Ze a OOrE oi only, and therefore of 6,0, ...... 6,_, only.
r r

i-1

Substituting this in the general term of the expansion, it becomes

Qn =
su" |". ai [= SPATE? A. F (0,0, 8.) S000, ...... dB, 10, . 48,
= 2U'A', where A’ is a constant.

But U’ is a normal function of the form considered in this paper; .. F'(0,0,...... Gem)
may be expanded in a series of functions each one of which is a linear function of the
normal functions considered in this paper.

Thirdly, to determine the coefficients of the several terms.

The following method is adopted because it will lead to the development cf A, in
Normal Functions.

R, is by the foregoing argument the sum of such terms as

(sin 6," @,’...... (sin ,,)P-#',, (es Pea ey

sin

multiplied by some quantity which is independent of the variables 0,6,’ ...... d=. But
since R, is a symmetrical function of the accented and unaccented variables, this quantity

Vout. XIII. Parr III. 39

298 Mr HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES

must contain (sin 6,)?: ©, ...... (sin 6,_,)?-? © mee un 0.) as a factor. Thus &, is the sum
of such functions as
B (sin 0) ©, ...... (sin)? 8", ae @’..) (sin 8)"@, ...... (sin 0,,,)"-*©,_ sete Ae
where B is a constant,
: RB OES Fils ‘le J, (in 6.) (sin 8,)"*...... (sin 8,_,)? (sin ,,) d0,d0, ......d0,,d0,, d0,.,
1 Pir Pr ple pH=n

a | ‘el | | F@bies Ou\S | @p+i—2) R, | W dew? d0..,

t—-=J0 o/0 5 p=

where #’,==B(sin0,):,’...(sin6_,)?'-0',_ fe DunG, a) (sin6,)"©, ... (sind,_,)?-20, alex pe 6...)

aw 00. crac @,_, are independent of 0,0,...... 0,_,; therefore in each term represented
in the summation of the second member of the last equation, they may be taken out-
side the signs of integration. After this has been done, multiply both sides of this
equation by

cos

(sin 6/)+#-2@"...... (sin 6_,)-2+1 @’,_, ie Drs Fe)

and integrate with regard to 6’_, from 0 to 27, and with regard to the other variables
from 0 to 7;

= Pie. sl ren pr. 6’_,)(sin 8,))>+i-2 @’......(sin 6)? 0’, Ce 1, .,) d0, Pe d6’..,
#0 /~0/0 sin
multiplied by | vel He (BNO i i aces (sin 0,.,) d@, ..... dé._,
fi | F : “Ane cos
= yf || FC8.---..) 2p+i-2) BGsin 8,)”++-*6,...(sin 8, .)>-=#48,, AS men a8... dO,
me S. Foi all ae gti aa eee ; ms COs . ; A
multiplied by |”... | (sin 6/2412 0... (sin 6-H", (9.8% a db... Ae,
J0 0/0 sin!

All the remaining terms disappear in virtue of the conjugate property of Art. 8;

and the first integral of the second member of the last equation is B(2p+i-— 2)
times the first integral of the first member ;

; me [ al, [ (eiiG.) 28 4.2. (sin 6,_,) dO, ...... dé,
Cy ean eer w (20 -
i eaha | | a _ (sin 8a? O...... (sin 8.) OF, ia Pix 0.1) BD. veins
™ w (lr hi
Observing that i ee [” [in 8). (Sin ,,,) 40, «0... d8,, =2 ns it follows by
0 ~0/0 i- I (41)

Part I. Art. 9, that

ANALOGOUS TO TESSERAL HARMONICS. 299

B=2|t—4 |t—-5...... B |2 (2p, + 1) (2p... +2) ---- (2p, +%—38) multiplied by
[Pis—P af Pi-2 : [Pi Pi-s ell (dee Tae

| Dig + Pin | Dig t+ Pine zp il

except when p,,=0, when only half this value is to be taken.

This form is free from integrals, but the other form is more convenient for the present
purpose,

The value of B is the same whether the term cos’p,,0,, or sin’p,,.0,, be employed to
calculate the integral in the denominator.

Tale N ie (sin 6,)'? ... (sin 0,.) dO, ... d0,_,

a “i [| Gin ayes} ... (in oy. ae ) a

where T, = (sin 6,)9@/ ... (sin 6',.,)*-20',_, (sin 6,)":O, ... (sin 0,..)"@,_, cos p,-2 (A... — 1).

ia

Dp

The summation extends to all possible positive integral values of the indices p, ... p,.
not greater than p, and such that in any one term no index is greater than one which

precedes it. If p,, be zero, the factor (eae is to be replaced by 1

Substituting this value of R,, it appears that

F (0/0, ... 0.) = % (sin ,)?O/’ ... (sin 0’,_,)"-0",_, {C cos p,_.0’,.. + D sin p,_.0’..} |
where

i ve ik [Gin G ett 20), S\ele (sin 0,._)"2"18; (cos eae) F (6,0, cae 6...) dé, <e dé,_,

ie [ [in Ort 2@ ? ... (sin 6,_,)??t-2+1@,_,” (cos p,_.9,.,) dO, ... d0,_,

["... [7 [Gin ayn, ... (sin 8,,)9-20,., (sin p,.0,.) F 0,0, --- 6.) a, -.. d
| J70/0

7 w (20
i ea | | (sin 8,)21+'-2 ? ... (sin O,.)°%-210, 2 (sin p, .0,,)?d8, ... dO... |
0 0/0

The summation in this case extends to all possible positive integral values of the

indices p,.-. p,., such that in any one term no index is greater than the index p with
which that term is connected, nor any index greater than any one which precedes it.

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INDEX TO THE TRANSACTIONS

OF THE

Cambridge Philosophical Society.

VOLS. I—XII.

I. NAMES OF AUTHORS.

Apams, Prof. J. C., Note on Sir G. B. Airy’s memoir
on the resolution of a certain Trinomial; Nov.
23, 1868; x1. 444—445,

Atry, Sir G. B., On the use of silvered glass for the
Mirrors of Reflecting Telescopes: Noy. 25,
1822; 11. 105—118.

— On the Figure assumed by a Fluid Homogeneous
Mass, &c., &c.: March 15, 1824; 11. 203—216.

—— On the Achromatism of the Eye-pieces of Tele-
scopes, and of Microscopes: May 17, 1824;
Il. 227—252,

—— On the defect in the Eye, and a mode of cor-

3 recting it: Feb. 21, 1825; 11. 267—271.

—— On the form of the teeth of Wheels: May 2, 1825;
11. 277—286.

—— On Laplace’s investigation of the Attraction of
Spheroids differing little from a Sphere ; May 8,
1826 ; 11. 379—390.

—— On the Spherical Aberration in the Eye-pieces of
Telescopes: May 14, 21, 1827; 111. 1—63.

— On Pendulums and Balances, and the Theory of
Escapements; Noy. 26, 1826; 11. 105—128.,

— On the Longitude of Cambridge Observatory :
Noy. 24, 1828; 111. 155—170.

— On a means of correcting the length of a Pen-
dulum by a Ball suspended by Wire: Nov, 16,
1829 ; 111. 355—360.

— On the conditions under which Perpetual Motion
is possible; Dec. 14, 1829; 111. 369—372.

Vou, XII.

Atry, Sir G. B., On the Nature of the Light in the two
Rays produced by the Double Refraction of
Quartz: Feb. 21, 1831; Iv. 79—123.

—— Addition to this memoir: Apr, 18, 1831; Iv. 199
— 208.

—— Ona remarkable modification of Newton’s Rings:
Noy. 14, 1831; Iv. 279—288.

—— On a new Analyzer, and its use in Polarization :
March 5, 1832; Iv. 313—322.

—— On the phenomena of Newton’s Rings with Sub-
stances of different refractive Powers: March
19, 1832; Iv: 409—424.

—— On a calculation of Newton’s Experiments on
Diffraction: May 7, 1833; v. 101—111.

— On the Latitude of Cambridge Observatory :
Apr. 14, 1834; v. 271—281.

—— On the Diffraction of an Object-Glass with cir-
cular aperture: Nov. 24, 1834; v. 283—291.
See EARNSHAW.

—— On the Intensity of Light in the neighbourhood

* of a Caustic; May 2, 1836: March 26, 1838;

vi. 879—402.

— On Triple Algebra: Oct. 28,1844; vir. 241—254,

—— Supplement to this memoir: May 8, 1848; vit.
595—599,

—— On a new construction of the Going-Fusee:
March 2, 1840; vit. 217—225.

— On an Eye affected by a mal-formation: May 25,
1846; vir. 361—362.

73

i INDEX TO TRANSACTIONS I—XII.

Arry, Sir G. B., Further observations on the same:
Feb. 12, 1872 ; x11. 392—393.

On the substitution of Ordinary Geometry for
the general Doctrine of Proportions: Dee. 7,
1857 ; x. 166—172.

—— Suggestion of a Proof that every Equation has

a Root: Dec. 6, 1858; x. 283—289.

— Supplement to this memoir: Dec. 12, 1859; x.
327—330.

On the factorial Resolution of the Trinomial

a" — 2 cos na+ = : Noy. 9, 1868; x1. 426—443.

Ak, C. K., On the origin of Electricity: Dec. 7, 1863;
xr. 6—20.

ALDERSON, Jas., On a Spermaceti Whale, stranded in
Yorkshire: May 16, 1825; 1. 253—266.

— On an Artificial formation of Plumbago: Feb. 21,
1825; 1. 441—443.

Awnstep, D. T., On some Fossil Multilocular Shells
found in Cornwall: Feb. 26, 1838; v1. 415—
422.

— On a portion of the Tertiary Formations of
Switzerland: May 20, 1839; vu. 141—152.

—— Onsome Phenomena of the Weathering of Rocks:
March 2, 1868; x1. 387—395.

BaBBaGE, C., On the notation employed in the Calculus
of Functions: May 1, 1820; 1. 63—76.

—— On the General Term of a New Class of Infinite
Series: May 3, 1824; m1. 218—225.

— On the influence of Signs in Mathematical Reason-
ing: Dec. 16, 1821; m1. 325—377.

Baxter, H. F., On Organic Polarity: March 8, 1858;
xX. 248—260.

Bevan, B., Experiments on Percussion: Noy. 10, 1825;
Tr. 444.

Bonp, Prof., Statistical Report on Addenbrooke’s
Hospital for 1836: March 13, 1837; vi. 361
—3i77.
—— The same for 1837: Apr. 30, 1888; v1. 565—575.
Booug, Prof. George, Of Propositions numerically defi-
nite: March 16, 1868; x1. 396—411.
Brewster, Dr., On the Brazilian Topaz; its colour,
structure, and optical properties: May 6, 1822;
1. 1—9.,

Bropig, P. B., On Land and Freshwater Shells, and
Bones of Animals found near Cambridge:
Apr. 30, 1838; vim. 138—140.

Cay ey, Prof., On the Theory of Determinants: Feb. 20,
1843; vu. 75—88.

—— On the Theory of Involution: Feb. 22, 1864; x1.
21—38.

—— On a case of the Involution of Cubic Curves:
Feb. 22, 1864; x1. 39—80.

—— On the classification of Cubic Curves: Apr. 18,
1864; x1. 81—128.

Cay ey, Prof., On Cubic Cones and Curves: Apr. 18,
1864; x1. 129—144.

— On certain Skew Surfaces, otherwise Scrolls:
Nov. 11, 1867; x1. 277—289.

—— On the Six Coordinates of a Line: Noy. 11, 1867;
XI. 290—323.

— On a certain Sextic Torse: Nov. 8, 1869; x1.
507—523.

— On the Centro-surface of an Ellipsoid: March 7,
1870; x11. 319—365.

— On Dr Wiener’s model of a cubic surface with
27 real lines; and on the construction of a
double-sixer: May 15, 1871; x1. 366—383.

—— On the geometrical Representation of Cauchy’s
theorems of Root-limitation: Feb. 16, 1874;
xu. 895—413.

Ceci, W., On Hydrogen Gas, as a moving power in
Machinery, &c.: Noy. 27, 1820; 1. 217—239,

— On an apparatus for grinding Mirrors and Object
Lenses: Dec. 11, 1822; 11. 85—99.

CHALLIS, Prof., On the extension of Bode’s Law to the
distance of Satellites from their Primaries:
Dec. 8, 1828; m1. 171—183.

— On the small Vibratory Motions of Elastic Fluids :
March 30, 1829 ; m1. 269—320.

—— On the general Equations for the Motion of Fluids,
&e., &e.: Feb. 22, 1830; 1. 383—416,

— Researches in the Theory of the Motion of Fluids:
March 3, 1834 ; v. 173—203.

— On the Decrement of Temperature depending
on the Height above the Earth’s surface :
Feb. 13, 1837; vi. 443—455. :

—— On the motion of a small Sphere, acted on by
Vibrations of an Elastic Medium: Apr. 26,
1841; vil. 333—353.

— On the Differential Equations applicable to the
Motion of Fluids: Apr. 11, 1842; vi. 371—396.

—— On a new Equation in Hydrodynamics : March 6,
1843 ; vu. 31—43.

—— On the Theory of Luminous Rays on the -hy-
pothesis of Undulations: May 11, 1846; vir.
363—370.

—— On the Theory of the Polarization of Light, on the
same hypothesis: May 25, 1846; vim. 371—
378.

— On the Transmission of Light, and on Double
Refraction; on the same hypothesis: May 17,
1847; vill. 524—532.

— On the Mathematical Theory of Luminous Vibra-
tions: March 6, 1848; vit. 584—594.

— On the Aurora Borealis of Noy. 17, 1848: Nov. 27,
1848; vu. 621—632.

— On the Determination of the Longitude of Cam-
bridge Observatory by Galvanic Signals: May
15, 1854; 1x. 487—514.

I. NAMES OF AUTHORS. lil

Curistin, S. H., On the Laws by which Masses of Iron
affect Magnetic Needles: May 15, 1820; 1.
147—173.

Crark, Prof. W., On a case of Human Monstrosity:
May 16, 1831; Iv. 219—255.

CuiaRKE, Prof. E. D., Inaugural address at the first
general meeting of the Society : Dec. 13, 1816;

I. (1—7).
— On the Purple Precipitate of Cassius: May 15,
1820; 1. 53—61.

—— On a deposit of Natron in the tower of a Church:
Noy. 27, 1820; 1 193—201.

— On the Crystallization of Water, &c.: March 5,
1821; 1. 209—215.

Cuirton, R. B., Note on Prof. De Morgan’s Memoir on
the history of Signs + and —; Feb. 18, 1865;
XI. 213—218.

Coppineaton, Rey. H., On the improvement of the
Microscope: March 22, 1830; 11. 421—428.

Cox, HomrersHam, On Impact on Elastic Beams: Dec.
10, 1849; 1x. 73—78.

— On the Deflection of Imperfectly Elastic Beams,
and on the Hyperbolic Law of Elasticity:
March 11, 1850; rx. pt. ii. 177—190.

Cumaine, Prof., On the connexion of Galvanism and
Magnetism: Apr. 2, 1821; 1. 269—279.

— On Magnetism as a Measure of Electricity: May
21, 1821; 1. 281—286.

— On a large Human Calculus in the Library of
Trinity College: Nov. 26, 1821; 1, 347—
349,

— On the development of Electro-Magnetism by
Heat: Apr. 28, 1823; m1. 47—76.

— See Alderson, J.

De Moreay, Prof., On.the general Equation of Curves
of the Second Degree: Noy. 15, 1830; Iv.
71—78.

—— On the General Equation of Surfaces of the

: Second Degree: Nov. 12, 1832; v. 77—94.

— On Discontinuous Constants, as applied to Infinite
Series: May 16, 1836; vr. 185—193.

— On a Question in the Theory of Probabilities:
Feb. 26, 1837; vi. 423—430.

—— On the Foundation of Algebra; Dec. 9, 1839;
vi. 173—187,

— Do. Do. Do. Nov. 29, 1841;
Vil. 287—300.

LO: Do. Do. Nov. 27, 1843;
vu. 139—142.

= Do. Do. On Triple Algebra : Oct. 28,

1844; vin. 241—254,

—— On Divergent Series, &., &c.: March 4, 1844;
vill. 182—203.

— On the Structure of the Syllogism, and its ap-
plication, &.; Noy. 9, 1846; vit. 379—408,

Dr Morean, Prof., On Integrating Partial Differential
Equations: June 5, 1848; vin. 606—613.

—— On the Symbols of Logic, the Theory of the Syllo-
gism, &c., &c.: Feb. 25, 1850; 1x. 79—127.

—— On some points of the Integral Calculus: Feb. 24,
1851; 1x. pt. ii, 107—138. -

— On some points in the Theory of Differential
Equations; March 27, 1854; 1x. 515—554.

—— On the singular points of Curves, and on Newton’s
method of Co-ordinated Exponents: May 21,
1855; 1x. 608—627.

— On the Solution of a Differential Equation: Apr. 28,
1856; x. 21—26.

—— On the Beats of Imperfect Consonances: Nov. 9,
1857; x. 129—145.

— On the Syllogism, No. mz, and on Logie in
general: Feb. 8, 1858; x. 173—230.

—— On the Syllogism, No. tv., and on the Logie of
Relations ; Apr. 23, 1860; x. 331—358*.

—— On the Proof of the existence of a Root in every
Algebraic Equation: Dec. 7, 1857; x. 261—270.

—— On the General Principles of which the Composi-
tion of Forces is a Consequence: March 14,
1859; x. 290—304.

— On the Theory of Errors of Observation : Noy. 11,
1861; x. 409—427.

— On the Syllogism, No. v., and on some parts of
the Onymatic System: May 4, 1863; x. 428—
487.

—— On Infinity : and on the sign of Equality : May 16,
1864; x1. 145—189.

—— Ona Theorem relating to Neutral Series : May 16,
1864; x1. 190—202.

— On the Early History of the Signs+and-—: Noy.
28, 1864; x1. 203—218.

— On the Root of any Function: and on Neutral
Series, No. 1: May 7, 1866; xr. 239—266.

— Note on the same: Oct. 26, 1868; x1. 447—460.

Denison, E. B., On Clock Escapements : Noy. 27, 1848;
VIII. 633—638.

— On Turret-clock Remontoirs: Feb. 26, 1849; vut.
639—641,

—— On some recent Improvements in Clock-Escape-
ments: Feb. 7, 1853; 1x. 417—430.

Donaxpson, Dr. J. W., On the Structure of the Athe-
nian Trireme: Noy. 6, 1856; x. 84—93.

— On the Statue of Solon mentioned by Aischines
and Demosthenes: Feb. 22, 1858; x. 231—239.

— On Plato’s Cosmical System: as exhibited in The
Republic: Book x.; Feb. 28, 1859; x. 305—316.

— On the Origin and Use of the word ArGuMENT:
Noy. 28, 1859; x. 317—326.

EarnsHaw, Rev. 8., On Fluid Motion, as expressed by
the Equation of Continuity: March 21, 1836:
VI. 203—233.

73—2

iv INDEX TO TRANSACTIONS I—XII.

EarssHaw, Rey. S., On the Diffraction of an Object-
Glass with a triangular Aperture: Dec. 12,
1836; vi. 431—442. See Airy.

—— On the Nature of the Molecular Forces of Lumi-
niferous Ether: March 18, 1839; vit. 97—112.

—— On the Values of the Sine and Cosine of an Infinite
Angle: Dee. 9, 1844; v1. 255—268.

—— On two great Solitary Waves of the First Order:
Dee. 8, 1845; vii. 326—341.

Exus, R. L., On the Foundation of the Theory of
Probabilities: Feb. 14, 1842; vir. 1—6.

— On the method of Least Squares; March 4, 1844;
vill. 204—219, ;

—— Remarks on the Theory of Matter: May 22, 1848 ;
vim. 600—605.

—— On the Fundamental Principle of the Theory of
Probabilities : Nov. 18, 1854; 1x. 605—607.

Farisu, Prof., On Isometrical Perspective: Feb. 21,
March 6, 1820; 1. 1—19.

FENNELL, C. A. M., On the First Ages of a Written
Greek Literature: Nov. 23, 1868; x1. 461—
480.

FisHER, Rev. OsmonD, On the Purbeck Strata of Dorset-
shire: Noy. 13, 1854; 1x. 555—581.

—— On the elevation of Mountains by lateral pres-
sure, &c., &c.: Apr. 27, 1868; x1. 489—506.

— On the Inequalities of the Earth’s Surface viewed
in connection with the secular cooling: Dec. 1,
1873; xu. 414—433.

—— On the same, as produced by lateral pressure, on
the hypothesis of a liquid substratum: Feb. 22,
1875; xm. 434—454.

GLaIsHER, J. W. L., Tables of the first 250 Bernouilli’s
Numbers, and of their logarithms: May 29,
1871; x1. 384—387.

—— Supplement to the same memoir: March 11, 1872;
x11. 388—391.

Goprray, Hue, On a Chart and Diagram for facilitat-
ing Great Circle Sailing: May 10, 1858; x.
271—282.

Goong, Henry, On a peculiar Defect of Vision: Noy. 9,
1846; May 17, 1847; vi. 493—496.
Goopwty, Rev. H., On the Connexion between Me-
chanics and Geometry: Feb. 10, 1845; vim.

269—277.

— On the Pure Science of Magnitude and Direction:
May 12, 1845; vim. 278—286.

— On the Geometrical Representation of the Roots of
Algebraic Equations: Apr. 27, 1846; vit. 342
— 360.

Green, GeorGe, On the Laws of Equilibrium of Fluids
analogous to the Electric Fluid: Nov. 12, 1832;
vy. 1—63.

—— On the Exterior and Interior Attractions of Ellip-
soids, &e., &c.: May 6, 1833; v. 395—429.

Green, GrorGE, On the Reflexion and Refraction of
Sound: Dee. 11, 18387; v1. 403—413.

— On the Motion of Waves in a variable Canal of
small depth and width; May 15, 1837; vr.
457—462.

— Note on the motion of Waves in Canals: Feb. 18,
1839; vil. 87—95.

—— Memoir on the Laws of Reflection and Refraction
at the common Surface of two non-crystallized
Media: Dec. 11, 1837; vir. 1—24.

—— Supplement to the memoir: May 6, 1839; vil.
113—120.

— On the Propagation of Light in Crystallized
Media: May 20, 1839; vir. 121—140.
Gregory, Dr. OLmrHus, On some Experiments to
determine the Velocity of Sound: Dee. 8, 1823;

i. 119—137.

Hattstone, Rey. J., On an extraordinary depression
of the Barometer in Dec. 1821, &c., &c.:
Feb. 25, 1822; 1. 453—458.

Havitand, Dr., On the solution of the Stomach by
Gastric Juices; Dec. 11, 1820; 1. 287—290.

Haywarp, R. B., On a direct method of estimating
Velocities, &c., &c., with reference to Axes
moveable in Space: Feb. 25, 1856; x. 1—20.

Henstow, Prof., On the Geology of Anglesea: Noy. 26,
1821; 1. 359—452.

— Ona hybrid Digitalis: Nov. 14, 1831; Iv. 257—
278.

—— On the Monstrosity of the Common Mignionette :
May 21, 1832; v. 95—100.

Herscuet, Sir J. F. W., Deviation in Crystals from
Newton’s scale of Tints, with Polarized Light :
May 1, 1820; 1. 21—41.

—— Planes of Polarization, as affected by Plates of
Rock Crystal: Apr. 17, 1820; 1, 48—52.

— Functional Equations, Reduction of, to Equations
of Finite Differences: March 6, 1820; 1. 77—87.

— Apophyllite, On the Refraction of coloured Rays
in: May 7, 1821; 1. 241—247.

— Ona Machine for resolving by Inspection Trans-
cendental Equations: May 7, 1832; rv. 425—
440,

Hiern, W. P., A Monograph of Ebenacex: March 11,
1872; xu. 27—300.

Horpitcu, Rev. H., On Rolling Curves: Dec. 10, 1838;
vil. 61—86.

— On Small Finite Oscillations: May 15, 1843; vir.
89—104.

Horxiys, W., On Aerial Vibrations in Cylindrical Tubes:
May 20, 1833; v. 231—270.

—— Researches in Physical Geology: May 4, 1835;

vi. 1—84.
—— On the Motion of Glaciers: May 1, 1843; vu.
50—74.

I. NAMES OF AUTHORS. Vv

Hopkins, W., On the Motion of Glaciers (second me-
moir): Dec. 11, 1843; vim. 159—169.

— On the Transport of Erratic Blocks: Apr. 29,
1844; vi. 220—240. ~

—— On the Internal Pressure of Rock Masses, &c., &e. :
May 3, 1847 ; vii. 456—470.

— On the External Temperature of the Earth, and
the other Planets of the Solar System: May 21,
1855; 1x. 628—672.

Humpury, Prof. G. M., On the growth of the Jaws:
Noy. 9, 1863; x1. 1—5.

Jarrett, T., On Algebraic Notation: Nov. 12, 1827;
Ii. 65—103.

Jess, Prof. R. C., On the place of Music in Education
as conceived by Aristotle in his “ Politics :”
May 17, 1875; x11. 523—530.

Jenyns, Rey. L., On the Ornithology of Cambridge-
shire: Nov. 28, 1825; 11. 287—324.

— On Pennant’s Natterjack; with a list of the Rep-
tiles of Cambridgeshire: Feb. 22, 1830; U1.
373—381.

— Monograph on the British species of Cyclas and
Pisidium : Noy. 28, 1831; Iv. 289—312.

KELLAND, Prof., On the Dispersion of Light, on the
theory of Finite Intervals: Feb. 22, 1836; v1.
153—184.

— On the Motion of a System of Particles, with
reference to Sound and Heat: May 16, 1836;
VI. 235—288.

— On the transmission of Light in Crystallized
Media: Feb. 13, 1837; vi. 323—352.

— Supplement to the same: May 1, 1837; v1. 353—
360.

— On Molecular Equilibrium: March 26, 1838; vu.
25—59.

— On the Quantity of Light intercepted by a grating
placed before a Lens; and on the effect of the
Interference: March 30, 1840; vir. 153—171.

Kemp, Guorce, On the Nature of the Biliary Secretion :
March 6, 1843; vir. 4449.

Kivg, J., A new demonstration of the Parallelogram of
Forces: Apr. 14, 1823; m1. 45—46.

Lex, Prof., On the Astronomical Tables of Mohammed
Abibeker Al Farsi: the mss. of which are in
the Public Library: Nov. 13, 1820; 1. 249—
265.

LesLt®, Prof., On the Sounds excited in Hydrogen Gas:
Apr. 2, 1821; 1. 267—268.

Lows, R. T., Primitiz Faune et Flore Madere et
Portus Sancti: Nov. 15, 1830; rv. 1—70.

— Piscium Maderensium Species (see M. Young):
Nov. 10, 1834; vr. 195—201.

—— Novitie Flore Maderensis: or Gleanings from
Madeiran Botany: May 28, 1838; vi. 523—
551,

Lupsock, Sir J., On the Calculation of Annuities, and
on some points in the Theory of Chances:
May 26, 1828; m1. 141—154.

— Comparison of various Tables of Annuities:
March 30, 1829; mr. 321—341.
Lunn, F., Phosphate of Copper from the Rhine, Analy-
sis of: March 5, 1821; 1. 203—207.
MANDELL, W., On the improved methods of proctring
Potassium: Noy. 26, 1821; 1. 343—345.

MAaxweELt, Prof. J. Ciherk, On the Transformation of
Surfaces by Bending: March 13, 1854; 1x.
445—470.

— On Faraday’s lines of Force: Dec. 10, 1855,
Feb. 11, 1856; x. 27—83.

— On Boltzmann’s Theorem on the average distri-
bution of energy ina system of material points :
May 6, 1878; x11. 547—570.

Mitter, Prof. On the Crystals of Boracic Acid:
Noy. 30, 1829; 111. 365—367.

— On Crystals found in Slags: March 22, 1830; 11.
417—420.

—— On the position of the Axes of Optical Elasticity
in certain Crystals: Dec. 8, 1834; v. 431—
438. March 21, 1836; vit. 209—215.

— On spurious Rainbows: March 22, 1841; vu.
277—286.

Moors, A. A., On a difficulty in Analysis noticed by
Sir Wm. Hamilton: May 1, 1837; v1. 317—322.

Morton, Pierce, On the Focus of a Conic Section:
March 2, 1829; 1m. 185—190.

Mosetey, Rey. H., On the Equilibrium of the Arch:
Dee. 9, 1833; v. 293—313.
— On the Theory of the Equilibrium of Bodies in
“Contact: May 15, 1837; v1. 463—491.
Munro, Rev. H. A. J., On a Metrical Latin Inscrip-
tion at Cirta in Algeria: Feb. 13, 1860; x.
374—408.

Murry, R., On the General Properties of Definite
Integrals: May 24, 1830; mm. 429—443,

— On the Resolution of Algebraical Equations:
March 7, 1831; tv. 125—153.

— On the inverse method of Definite Integrals, with
Physical Applications: March 5, 1832; iv.

353—408.

—— Second memoir on the same: Noy. 11, 1833; v.
113—148.

— Third memoir on the same: March 2, 1835; v.
315—393.

— On Elimination between an Indefinite number of
unknown Quantities : Noy. 26, 1832; v. 65—75.

—— On the resolution of Equations in Finite Dif-
ferences: Noy. 15, 1835; vr. 91—106.

O’Brien, Rey. M., On the Propagation of Luminous
Waves in the Interior of Transparent Bodies :
Apr. 25, 1842; vi. 397—437.

vl INDEX TO TRANSACTIONS I—XII.

O’Barey, Rev. M., On the Reflection and Refraction of
Light at the surface of an Uncrystallized Body :
Nov. 28, 1842; vir. 7—26.

— On the possibility of accounting for the Absorption
of Light, &c., &c.: Feb. 14, 1843; vir. 27—30.

—— Ona New Notation to be used in Geometry, &c.,
&e.: Nov. 23, 1846; vit. 415—428.

— On a System of Symbolical Geometry and Me-
chanics: March 15, 1847; vu. 497—507.

— On the Equation for the Vibratory Motion of an
Elastic Medium: March 15, 1847; vir. 508—
523.

Oxes, J., On the remains of a Fossil Beaver found in
Cambridgeshire: March 6, 1820; 1. 175—177.

— Ona dilatation of the Ureters, &c.: Nov. 12, 1821;
I. 8351—358.

Owen, RicHarD, F.R.S., Description of an extinct
Lacertian Reptile: Apr. 11, 1842; vir. 355—
369.

Pacet, Prof. G. E., On some remarkable Abnormities
in the Voluntary Muscles: March 8, 1858; x.
240—247.

Patey, F. A., On Homeric Tumuli: March 12, 1866;
XI. 267—276.

— On the Comparatively Late Date, and Composite
Character of our Iliad and Odyssey: Nov. 26,
1866; x1. 360—386.

Puear, J. B., On the Geology of some parts of Suffolk,
particularly of the Valley of the Gipping:
Feb. 27, 1854; rx. 431444.

Pierson, R., On the Theory of the Long Inequality of
Uranus and Neptune: 1852; 1x. Appendix,
pp. lxvii.

Porter, R., Mathematical considerations on the Prob-
lem of the Rainbow: Dec. 14, 1835; v1. 141—
152.

—— On a new correction in the Construction of the
Double Achromatic Object-Glass: Apr. 30,
1838; VI. 553—564.

—— On the Heights of two Aurore Boreales, &c.,
&e.: Dec. 8, 1845; vir. 320—325.

Power, Rey. J., On the principle of Virtual Velocities:
March 21, 1825; 1. 273—276.

—— On the Theory of Residuo-Capillary Attraction,

&c., &c. : March 17, 1834; v. 205—229.

On a Railway Accident; and on a Principle of
Motion involved in precautions against Col-
lisions: May 29, 1841; vu. 301—317.

—— On the Truth of a Theorem in Hydrodynamics:

May 9, 1842; vit. 455—464.

Rieavp, Prof., On the relative Quantities of Land and
Water on the Globe: Feb. 13, 1837; v1. 289
—300.

Rours, J. H., On the Oscillation of a Suspension Chain :
Dec. 8, 1851; 1x. 379—398.

Rours, J. H., On the Motion of Beams, and thin Elastic
Rods: Apr. 23, 1860; x. 359—373.

—— On the Strains to which Ordnance are subject,
and on the Vibrations of Solid Bodies: Apr. 18,
1864; x1. 324—359.

Rotman, R. W., On Variations of Magnetic Intensity,
as computed and observed: Noy. 10, 1825;
ir. 445.

— Onan Ancient Observation of a Winter Solstice:
Noy. 30, 1829; m1. 361—363.

—— An account of Observations of Halley’s Comet:
Dec. 11, 1837; v1. 493—506.

Sauter, J. W., On Crotalocrinus rugosus, Miller: a
Crinoid in the Woodwardian Museum: Feb. 8,
1869; xt. 481—484.

— Diagram of the relations of the Univalve to the
Bivalve: and of this to the Brachiopod:
Feb. 8, 1869; x1. 485—488.,

Srepewick, Prof., On the Primitive Ridge of Devonshire
and Cornwall: March 20, 1820; 1. 89—146.

— On the Physical Structure of the Lizard district
in Cornwall: Apr. 2, May 7, 1821; 1. 291—330.

— On some Trap Dykes in Yorkshire and Durham:
May 20, 1822; 1. 21—44.

— On the Association of Trap Rocks with Mountain
Limestone in Tees-Dale: May 12, 1823;
March 1, 15, 1824; 1. 139—195.

— Note on a memoir by Dr Brodie on Land and
Freshwater Shells, &c.: Apr. 30, 1838; vimI.
139—140.

SairH, ARcHIBALD, On the Equation to Fresnel’s Wave-
Surface: May 18, 1835; v1. 85—89.

Spimspury, F. G., On the Magnetism evolved by a single
Galvanic combination, &c., &c. : Noy. 25, 1822;
1. 77—83.

SrepHens, J. F., Description of Chiasognathus Grantii,
a Lucanideous Insect: May 16, 1831; 1v. 209
—216. s

Sroxes, Prof. G. G., On the Steady Motion of Incom-
pressible Fluids: Apr. 25, 1842; vir. 489—453.

— Memoir on some cases of Fluid Motion: May 29,
1843; vu. 105—137.

—— Supplement to this memoir: Noy. 3, 1846; vit.
409—414,

— On the Internal Friction of Fluids in Motion:
and the Equilibrium and Motion of Elastic
Solids: Apr. 14, 1845; vim. 287—319.

— On the Theory of Oscillatory Waves: March 1,
1847; vu. 441—455.

— On the Critical Value of the Sums of Periodic
Series: Dec. 6, 1847; vu. 583—583.

—— On the central Spot of Newton’s Rings: Dee. 11,
1848; vii. 642—658.

—— On the Variation of Gravity at the Earth’s Sur-
face: Apr. 23, 1849; vi1l. 672—695.

I. NAMES OF AUTHORS.

Stokes, Prof. G. G., On an Equation relating to the
breaking of Railway Bridges: May 21, 1849;
vill. 707—735.

—— On the Dynamical Theory of Diffraction: Noy. 26,
1849; 1x. 1—62,

— On the numerical Calculation of a class of Definite
Integrals and Infinite Series: March 11, 1850;
Ix. 166—187.

—— On the effect of the Internal Friction of Fluids
on the motion of Pendulums: Dec. 9, 1850;
Ix. pt. il. 8—106.

— On the Colours of Thick Plates: May 19, 1851;
Ix. pt. il, 147—176.

— On the Composition and Resolution of Streams of
Polarized Light from different sources : Feb. 16,
March 15, 1852; 1x. 399—416.

— On the Discontinuity of Arbitrary Constants in
Divergent Developments: May 11, 1857; x.
105—128.

—— Supplement to the same memoir: May 25, 1868;
XI. 412425.

THompson, Prof. W. H., On the genuineness of the
Sophista of Plato, and on some of its philoso-
phic bearings: Nov. 23, 1857; x. 146—165.

TopHuntTER, I., On the Method of Least Squares:
May 29, 1865; x1. 219—238.

— On the Arc of the Meridian measured in Lapland:
May 1, 1871; x1. 1—26.

— On the equation determining the form of the
strata in Legendre’s and Laplace’s Theory of
the Figure of the Earth: Oct. 16, 1871; xm.
301—318.

TozER, J., Mathematical Investigation of the effect of
Machinery on the Wealth of a Community:
May 14, 1838; vi. 507—522.

— On the effect of the Non-residence of Landlords,
on the same: March 16, 1840; vir. 189—
196.

—— J., On the Force of Testimony in Legal Evidence :
Noy. 27, 1848; vu. 143—158.

Wattace, Witt1AM, Geometrical Theorems and For-
mule, as applicable to Geodesy: Nov. 30, 1835;
vi. 107—140.

Wareurton, H., On the Partition of Numbers; Com-
binations and Permutations: March 1, 1847;
vill. 471—492.

— On self-repeating series: May 15, 1854; 1x. 471
—486.

Warren, Rey. J. W., Exercises in Curvilinear and
Normal Co-ordinates: May 22, 1876; May 7,
1877; XI. 455—522; 531—545.

Wenpewoop, H., On the Knowledge of Body and Space:
March 11, 1850; rx. 157—165.

WueweE tt, Dr, On the Apsides of Orbits of great
excentricity : Apr. 17, 1820; 1. 179—191.

vil

WuewE.t, Dr., On the double Crystals of Fluor Spar:
Nov. 26, 1821; 1. 331—342,

— On the Rotatory motion of Bodies: May 6, 1822;
I. 11—20.

— On the Angle made by two Planes, or two straight
lines, referred to three oblique Co-ordinates :
Noy. 24, 1823; m. 197—202.

— Note on Mr Crcrt’s memoir on Grinding Mirrors,
&c.: Dec. 11, 1822: 11. 100—103.

— On Crystalline Combinations: Noy. 13, 1826; 11.
391—425.

—— On a Notation to designate the Planes of Crys-
tals: Feb. 11, 1826; 1. 427—439.

— A Mathematical Exposition of some doctrines of
Political Economy: March 2, 14, 1829; 111.
191—230.

—— Second memoir on the same subject: 1x. Apr. 15,
1850; 123—149.

— Third memoir on the same subject : Nov, 11, 1850;
Ix. pt. il, 1—7.

— Ditto, Ditto, as applied to Ricardo’s Political
Economy: Apr. 18, May 2, 1831; Iv. 155—
198.

— On the Nature of the Truth of the Laws of
Motion: Feb. 17, 1834; v. 149—172.

— On the results of Observations with a new
Anemometer: May 1, 1837; v1. 8301—315.

—— Demonstration that all Matter is heavy: Feb. 22,
1841; vit. 197—207.

— Discussion whether Cause or Effect are simul-
taneous or successive: March 14, 1842; vil.
319—331.

— On the Fundamental Antithesis of Philosophy :
Feb. 5, 1844; vi. 170—181.

— Second memoir on the same subject: Nov. 13,
1848; vit. 614—620.

— On the Intrinsic Equation to a Curye, &e., &e. :
Feb. 12, 1849; vit. 659—671.

— Second memoir on the same subject: Apr. 15,
1850; 1x. 150—156.

— On Hegel’s Criticism of Newton’s Principia:
May 21, 1849; vu. 696—7U6.

— On Aristotle’s account of Induction: Feb. 11,
1850; Ix. 683—72.

— On the Transformation of Hypotheses in the
History of Science: May 19, 1851; 1x. pt. ii.

139—146.

— On Plato’s Survey of the Sciences: Apr. 28, 1855 ;
Ix. 582—589.

—— On his notion of Dialectic: May 7, 1855; Ix.
590—597.

—— On the Intellectual Powers, according to Plato:
Nov. 12, 1855; 1x. 598—604.

—— Of the Platonic Theory of Ideas: Noy. 10, 1856;
x. 94—104.

vii

Wits, Prof, On the pressure produced by a stream
of Air on a flat plate, &c. &e.: Apr. 21, 1828;
li. 129—140.

—— On Vowel Sounds; and on Reed Organ-Pipes:
Noy. 24, 1828; March 16, 1829; 111. 231—268.

— On the Mechanism of the Larynx: May 18, 1829;
Iv. 323—352.

INDEX TO TRANSACTIONS I—XIL

Youna, J. R., On the Principle of Continuity, in refer-
ence to Analysis: Dec. 7, 1846; viir. 429—
440,

Youne, M., Piscium Maderensium Species, Iconibus
illustrate: Nov. 10, 1834; vr. 195.

II, INDEX OF SUBJECTS.

Aberration in the Eye-pieces of Telescopes : May 14, 21,
1827; 1. 1—58.

Achromatic Eye-pieces, and Achromatism: May 17,
1824; 1. 227—252 ; m1. 59—63.

—— Object-Glass, New Correction for: Apr. 30, 1838 ;
VI. 553—564.

Addenbrooke’s Hospital, Report on, for 1836: March 13,

1837 ; VI. 361—377.

Ditto, Ditto,

1838; vi. 565—575.

Al Farsi, Astronomical Tables of: Noy. 13, 1820; 1

for 1837: Apr. 30,

249—265.

Algebra, Foundations of, Part I.: Dec. 9, 1839; vi.
173—187.

— Ditto, Ditto, II.: Nov. 29, 1841; vu.
287—300.

— Ditto, Ditto, III.: Nov. 27, 1843; vit.
139—142.

— Ditto, Ditto, IV.: Oct 28, 1844; vit.
241— 254.

Algebraic Equations, Geometrical representation of their
Roots: Apr. 27, 1846; vim. 342—360,

Notation: Noy. 12, 1827; m1. 65—104.

Algebraical Equations, Resolution of: March 7, 1831;
Iv. 125—153.

Analysis, on a Difficulty in, noticed by Sir W. Hamil-
ton: May 1, 1837; vi 317—322.

Analyzer, on a new: March 5, 1832; Iv. 313—322.

Anemometer, Observations with a new: May 1, 1837;
vi. 301—315.

Angle, Memoir on, as referred to three oblique Co-
ordinates: Nov. 24, 1823; 11. 197—202.

Anglesea, Geological description of: Nov. 26, 1821;
I. 359—452.

Animals, Occurrence of Extinct, near Cambridge:
Apr. 30, 1838; vir. 138—140.

Anhuities, Calculation of, with Tables: May 26, 1828;
mi. 141—154.

—— Comparison of various Tables of : March 30, 1829;
m1. 321—341.

Antithesis, Fundamental, of Philosophy: Feb. 5, 1844;
vill. 170—181,

Antithesis, Second memoir: Nov. 18, 1848; vir. 614
—620.

Apophyllite, Refraction in Rays from: May 7, 1821;
I. 241—247,

Apsides of Orbits of Great Excentricity ; April 17, 1820;
I. 179—191.

Arbitrary Constants, Discontinuity of, &c.: May 11,
1857; x. 105—128.

—— Ditto, Ditto, Supplement to this memoir: May
25, 1868; x1. 412425.

Arch, Equilibrium of the: Dec. 9, 1833; v. 293—313.

Argument, Use and meaning of the word: Noy. 28,
1859; x. 317—326.

Aristotle’s account of Induction: Feb. 11, 1850; rx.

63—72.

Atmospheric temperature, Decrement of : Feb. 13, 1837 ;
vi. 443—-455,

Attraction, Residuo-Capillary: March 7, 1834; v. 205
—229.

Attractions of Ellipsoids, Determination of: May 6,
1833; v. 395—429.

Aurora Borealis, Height of : Dec. 8, 1845; vir. 320—
325.

— of Nov. 17, 1848: Nov. 27, 1848; vir. 621—632,

Axes, Moveable, On Velocities, &c., &c., relative to:
Feb. 25, 1856; x. 1—20.

Barometer, Extraordinary Depression of: Feb. 25, 1822 ;
I. 453—458.

Beams and Elastic Rods, Theory of: Apr. 23, 1860;
X. 859—373.

Beayer, Fossil Remains of: March 6, 1820; 1.175—177.

Bernouilli’s Numbers, Tables of, &c., &.: May 29,
1871; x11. 384—391.

Biliary Secretion, Nature of: March 6, 1843; vim, 44
—49.

Bivalve, see Univalve.

Bode’s Law, Extension of, to Satellites: Dec. 8, 1828;
im. 171—183.

Boltzmann, see Material points,

Boracic Acid, on the Crystals of; Nov. 30, 1829; m1.
565—367.

Il. INDEX OF SUBJECTS. 1x

Brachiopod, see Univalve.
Brazilian Topaz, Colour, Structure and Optical Pro-
perties of: May 6, 1822; m1. 1—9.

Calculus of Functions, Notation employed in: May 1,
1820; 1. 63—76.

—— Human, specimen of : Nov. 26, 1821; 1. 347—349.

Cambridgeshire, On the Ornithology of ; Noy. 28, 1825;
I. 287—324.

—— List of Reptiles found in: Feb. 22, 1830; 1m.
373—381.

Cassius, Constituents of Purple Precipitate of : May 15,
1820; 1. 53—61.

Cause and Effect, simultaneous or successive : March 14,
1842; vu. 319—331.

Caustic, Intensity of Light near : May 2, 1836, March 26,

1838; vi. 379—402.

Ditto, Supplement to this memoir: May 8,

1848; vit. 595—599.

Chances, Some points in the Theory of: May 26, 1828;
i. 141—154,

Chiasognathus Grantii, Description of ; May 16, 1831;
Iv. 209—217.

Clock Escapements: Noy. 27, 1848; vit. 633—638.

—— Improvements in: Feb. 7, 1853; 1x. 417—430.

—— Turret Remontoirs: Feb. 26, 1849; vin. 6839—641.

Colours of Thick Plates: May 19, 1851; rx. [147—

176.]

Combinations and Permutations: March 1, 1847; vit.
471—492.

Comet, Observations of Halley’s: Dec. 11, 1837; v1.
493—506.

Composition of Forces, General Principles of : March 14,
1859; x. 290—304.

Conic Section, Focus of : March 2, 1829; 11. 185—190.

Consonances, Beats of Imperfect: Nov, 9, 1857; x.
129—145.

Continuity, Principle of, with reference to Analysis :
Dec. 7, 1846; vitr. 429—440,

- Co-ordinates, Six of a Line: Nov. 11, 1867; x1. 290—
323.

— Curvilinear and Normal; May 22, 1876; x11. 455

—522.

Ditto,

—d45,

Copper, Analysis of Phosphate of; March 5, 1821; 1.
203—207.

Cornwall, Fossil Shells: Feb. 16, 1838; v1. 415—422.

Lizard District of; Apr. 2, May 7, 1821; 1. 291—

330.

Primitive Ridge of : March 20, 1820; 1. 89—146.

Crotalocrinus rugosus: Feb. 8, 1869; x1. 481—484.

Crystalline Combinations, on their Classification, Nov.
13, 1826; 1. 891—425,

Crystallization of Water: March 5, 1821; 1. 209—215.

Woe, SOUL

Ditto, May 7, 1877; x11. 531

Crystallized Media, Propagation of Light in: May 20,
1839; vii. 121—140.
Ditto, Transmission of Light in: Feb. 13,
1837 ; VI. 323—352.
Ditto, Supplement: May 1, 1837; vi. 353
—360.
Crystals, Axes of Optical Elasticity in certain: Dec. 8,
1834; v. 431—438.
Position of Axes of Optical Elasticity in; March 21,
1836; vu. 209—215.
— Planes of, on a Notation to designate: Feb. 11,
1826; 1. 427—439.
as affecting Planes of Polarization: Apr. 17, 1820;
I. 43—52.
found in Slags: May 22, 1830; 11. 417—420.
— Variation in Tints developed by: May 1, 1820;
I. 21—41.
Cubic Cones and Curves: Apr. 18, 1864; x1. 129—144.
Cubic Curves, Involution of: Feb. 22, 1864; x1. 39
—80.
— Classification of: Apr. 18, 1864; x1. 81—128.
— Surface, with 27 lines, by Dr Wiener: May 15,
1871; x11. 366—383.
Curve, Intrinsic Equation to: Feb. 12, 1849; vit. 659
—671.
—— Second memoir: Apr. 15, 1850; rx. 150—156.
— Rolling: Dec. 10, 1838; vm. 61—86.
— of the Second Degree, General Equation to:
Noy. 15, 1830; rv. 71—78.
—— Singular points of: May 21, 1855; 1x. 608—627.
Cyclas and Pisidium, on the British Species of: Nov. 28,
1831; Iv. 289—312.
Cylindrical Tubes, Aerial Vibrations in: May 20, 1833;
v. 231—270.

Decrement of Atmospheric Temperature : Feb. 13, 1837 ;
vi. 443—455.

Definite Integrals, Inverse method of, with applications :
March 5, 1832; Iv. 353—408.

— Ditto, Ditto, Ditto, Noy. 11,
1833; v. 118—148.
— Ditto, Ditto, Ditto, March 2,

1835; v. 315—393.

— Numerical calculation of: March 11, 1850; Ix.

166—187.

Properties of: May 24, 1830; 111. 429—443.

Determinants, Theory of: Feb. 20, 1843; vi. 75—88.

Devonshire, Primitive Ridge of: Mar. 20, 1820; I.
89—146.

Differential Equations, Theory of: March 27, 1854; 1x.
515—554.

— Supplement to this paper: Apr. 28, 1856; x,
21—26.

Diffraction, Dynamical Theory of; Noy, 26, 1849; rx,

1—62.
74

x INDEX TO TRANSACTIONS I—XII.

Diffraction, Newton’s Experiments on: May 7, 1833; v.
101—111.

— of an Object-glass with Circular aperture: Nov. 24,

1834; v. 283—291.

Ditto, Ditto,

1836; vi. 431—442.

Digitalis, Hybrid: Noy. 14, 1831; Iv. 257—278.

Discontinuous Constants, Use of: May 16, 1836; vI.
185—193.

Dispersion of Light, Hypothesis for: Feb. 22, 1836;
VI. 158—184.

Divergent Developments, see Arbitrary Constants.

Series: March 4, 1844; vu. 182—203.

Double-Sixer, Construction of: May 15, 1871; XII.
366—383.

Durham, see Yorkshire.

Triangular Ditto: Dec. 12,

Earth and Planets, External Temperature of: March 21,
1855; Ix. 628—672.
Earth, Theory of the Figure of; Oct. 16, 1871; x11. 301

—318.

— Inequalities in the Surface of: Dec. 1, 1873;
xi. 414—433.

Ebenacex, Monograph of: March 11, 1872; x11. 27—
300.

Elastic Beams, Deflection of, &c.: March 11, 1850; Ix.
[177—190.]

—— Impact of: Dec. 10, 1849; 1x. 73—78.

—— Fluids, Vibratory Motions of: March 30, 1829;
In. 269—320.

—— Medium, Effect of Vibrations on a Sphere;
April 26, 1841; vu. 333—353.

—— Medium, Vibratory Motion of: March 15, 1847;
vu. 508—523.

—— Rods, see Beams.

—— Solids, Motion of: Apr. 14, 1845; vim. 287—
319.

Electric Fluid, Equilibrium of Fluids analogous to:
Nov. 12, 1832; v. 1—63.

Electricity, Origin of: Dec. 7, 1863; x1. 6—20.

Electro-Magnetism, Development of by Heat: Apr. 28,
1823; 1. 47—76.

Elevation of Mountains by Lateral Pressure: Apr. 27,
1868; x1. 489—506.

Elimination between Unknown Quantities: Nov. 26
1832; v. 65—75.

Ellipsoid, Centro-Surface of : March 7, 1870; x11. 319
—365.

Ellipsoids, Exterior and Interior attractions of: May 6,
1833; v. 395—429.

Endosmose and Exosmose, Explanation of: March 17,
1834; v. 205—229.

Equality, Sign of: May 16, 1864; xr. 145—189,

Equation, Algebraic, Proof of a root in every: Dee. 7,
1857; x. 261—270,

,

Equation, Algebraic, Another proof: Dec. 6, 1858; x.

283—289.

Ditto, Supplement to this memoir: Dee.

12,1859; x. 327—3830.

—— relating to the breaking of bridges: May 21, 1849;
vit. 707—735.

—— to a Curve, The Intrinsic: Feb. 12, 1849; vit.

659—671.

Ditto, Second memoir: 1x. 150—160.

— General, to Surfaces of the second degree : Noy. 12,
1832; v. 77—94.

—— Integration of Partial differential: June 5, 1848;
vir. 606—613.

—— Machine for resolving by Inspection: May 7, 1832 ;
Iv. 425—440.

Equilibrium of the Arch: Dee. 9, 1833; v. 293—313.

— of Bodies in Contact: May 15, 1837; vr. 463—491.

Molecular: March 26, 1838; vir. 25—59.

Erratic Blocks, Transport of: Apr. 29, 1844; VIII.
220—240.

Errors of Observation, Theory of: Nov. 11, 1861; x.
409—427.

Escapements, Theory of: Noy. 26, 1826; m1. 105—128.

Exponents, Newton’s method of: May 21, 1855; rx.
608—627.

Extinct Lacertian Reptile, Traces of: April 11, 1842;
VII. 355—369.

Eye, Change in the State of: May 25, 1846; vim. 361
— 362.

—— Defect in, and how cured: Feb. 21, 1825; 11.
267—271.

— Further observations on: Feb. 12, 1872; xm.
392, 3.

Fauna and Flora of Madeira: Nov. 15, 1830; rv. 1—70.

Figure assumed by a Fluid Homogeneous Mass:
March 15, 1824; 1. 2083—216.

Finite Differences, Resolution of Equations in: Nov. 15,
1835; v1. 91—106.

Flora of Madeira, Notes and Gleanings: May 28, 1838 ;
VI. 523—551.

Fluid Motion, on: March 21, 1836; vr. 203—233.

— Ditto, On some cases of: May 29, 1843; vm.
105—137.

— Ditto, Supplement to this memoir: Nov. 3,
1846; vit. 409—414.

Fluids, Equilibrium of Certain: Noy. 12, 1832; v.
1—63.

— General Equations of the motion of, &e., &c.:
Feb. 22, 1830; mr. 383—416.

— Motion of Incompressible: April 25, 1842; v1.
439—453.

— Theory of the motion of: v. 173—203.

—— in motion, Internal Friction of: Apr. 14, 1845;
vill. 287—319.

II. INDEX OF SUBJECTS.

Fluids, Effect on Pendulums of Internal Friction of:
Dee. 9, 1850; 1x. [8—106.]

Fluor Spar, Double Crystals of: Nov. 26, 1821; 1. 331
—342.

Focus of a Conic Section: March 2, 1829; mr. 185—190.

Force, Faraday’s Lines of: Dec. 10, 1855, Feb. 11, 1856;
x. 27—83.

Forces, Principles of the Composition of: March 14,
1859; x. 290—304.

Fossil remains of Beaver, found in Cambridgeshire :
March 6, 1820; 1. 175—177.

— Shells, New Genus of: Feb. 26, 1838; v1. 415—

422.
Fresnel’s Wave Surface, Equation to: May 18, 1835;
VI. 85—89.

Functional Equations, Reduction of, to Equations of
Finite Differences: March 6, 1820; 1. 77—87.

Galvanism, as connected with Magnetism: Apr. 2, 1821;
I. 269—279,

Gas, Hydrogen, as a moving Power in Machinery:
Noy. 27, 1820; 1. 217—239.

—— on Sounds excited in: Apr. 2, 1821; 1. 267, 268.

Gastric Fluids, Solvent effect of, on the Stomach after
Death: Dec. 11, 1820; 1. 287—290.

Geodesy, Geometrical Formule applicable to: Noy. 30,
1835; vi. 10O7—140.

Geology, Researches in Physical: May 4, 1835; v1.
1—84.

Geometry and Mechanics, Symbolical: March 15, 1847;
vill. 497—507.

—— Substitution of, for the doctrine of Proportions:
Dec. 7, 1857; x. 166—172.

Gipping, Geology of the Valley of: Feb. 27, 1854; 1x.
431—444.

Glaciers, Motion of: May 1, 1843; vr, 50—74.

Ditto, Dec. 11, 1843; vir. 159—169.

Globe, Relative Quantities of Land and Water on:

r Feb, 13, 1887; vi. 289—300.

Going-Fusee, New Construction of: March 2, 1840;
vir. 217—225.

Gravity, Variation of, at the Earth’s Surface: Apr. 23,

1849; vim. 672—695.

Great Circle Sailing: May 10, 1858; x, 271—282.

Greek Literature, First Ages of written : Nov. 23, 1868;
x1, 461—480.

Halley’s Comet, Observations of: Dec. 11, 1837; Yt.
A93—506.

Heat, see Motion of Particles.

Hegel's criticism of Newton: May 21, 1849; vimi. 696
—706.

Homeric Tumuli: March 12, 1866; x1. 267—276,

Human Monstrosity, Case of, with Commentary:
May 16, 1831; Iv. 219—255.

xl

Hybrid Digitalis, Examination of: Novy. 14, 1831; rv.
257— 278.

Hydrodynamical Theorem, Investigation of: May 9,
1842; vu. 455—464.

Hydrodynamics, New Fundamental Equation
March 6, 1843; vit. 31—43.

Hyperbolic Law of Elasticity: March 11, 1850; 1x.

in:

[177—190.]
Hypotheses, Transformation of: May 19, 1851; 1x.
[139—146.]

Ideas, Platonic Theory of: Noy. 10, 1856; x. 94—104.
Iliad and Odyssey, Late date, &., &¢., of: Nov. 26,
1866; x1. 360—386.
Induction, Aristotle’s account of: Feb. 11, 1850; 1x.
63—72.
Infinite Angle, Sine and Cosine of: Dec. 9, 1844; vuiL.
255—268.
Series, Use of Discontinuous Constants in, &c.,
&c.: May 16, 1836; v1. 185—193.
—— Ditto, General Term for a new Class of: May 3,
1824; m1. 217—225.
Infinity, On: May 16, 1864; x1. 145—189.
Inscription, Metrical Latin, from Algeria: Feb. 13,
1860; x. 374—408.
Integral Calculus, On some points of: Feb. 24, 1851;

1x. [107—138].
Integrals, General Properties of Definite: May 24, 1830;
Im. 429—443,

— Inverse method of, with Applications: March 5,
1832; Iv. 353—408.

Internal friction of fluids; Apr. 14, 1845; vim. 287—
319.

Inyolution, Theory of: Feb. 22,1864; xr. 21—38.

Involution of Cubic Curves: Feb. 22, 1864; x1. 39—80.

Jaws, Growth of: Nov. 9, 1863; x1. 1—5.

Knowledge of Body and Space: March 11, 1850; rx.
157—165.

Laminated Pressure of Rock Masses: May 3, 1847;
vill. 456—470,

Land, see Globe.

Laplace, on his Theory of the Attraction of Spheroids;
May 8, 1826; 1. 379—390.

Lapland, Are of the Meridian measured in: May 1,
1871; x11. 1—26.

Larynx, On the Mechanism of: May 18, 1829; ty.

323— 352.

Latitude of Cambridge Observatory: Apr. 14, 1834;
v. 271—281.

Least Squares, Method of: March 4, 1844; vir. 204—
219.

— Ditto, Ditto: May 29, 1865; x1. 219—238,

74—2

xil

Light, Nature of, from the Double Refraction of
Quartz: Feb. 21, 1831; rv. 79—128.
—— Nature of, from the Double Refraction of Quartz:
Apr. 18, 1831; Iv. 199—208.
— on the Dispersion of: Feb. 22, 1836; v1. 153—184.
—— Transmission of, in certain Media: Feb. 13, 1837;
VI. 323—352.
Ditto,
353—360.
—— Intensity of, near a Caustic: May 2, 1836,
March 26, 1838; vi. 379—402.
Ditto, Supplement to this memoir: May 8,
1848; vir. 595—599.
— Reflection and Refraction of, &c.: Dec. 11, 1837;
vir. 1—24,
Ditto, | Supplement to this memoir: May 6,
1839; vu. 1183—120.
—— Propagation of, in Crystallized Media: May 20,
1839; vir. 121—140.
—— Quantity of, &c., absorbed by a Grating placed
before a Lens: March 30, 1840; vi. 153—171.
—— Reflection and Refraction of: Nov. 28, 1842;
vil. 7—26.
—— Absorption of, &c.: Feb. 14, 1843; vi. 27—30.
—— Transmission through Transparent media: May
17, 1847; vir. 524—532.
—— Polarized: Dec. 8, 1851; 1x. 379—398.
Lines of Force, Faraday’s: Dec. 10, 1855, Feb. 11, 1856;
x. 27—83.
Liquid Substratum of the Earth, Theory of: Feb. 22,
1875; x11. 434—454.
Lizard district of Cornwall, Physical Structure of:
Apr. 2, May 7, 1821; 1. 291—330.
Logic, in general: Feb. 8, 1858; x. 173—230.
of Relations: Apr. 23, 1860; x. 331—358.
Symbols of, &., &c.: Feb. 25, 1850; 1x. 79—127.
Longitude of Cambridge Observatory: Noy. 24, 1828;

Supplement: May 1, 1837; v1.

mm. 155—170.

— Ditto, Ditto, May 15, 1854;
Ix. 487—514.

Luminiferous Ether, Constitution of: March 18, 1839;
vi. 97—112.

Luminous Rays, Theory of: March 11, 1846; vim.
363—378.

—— Vibrations, Theory of: March 6, 1848; vim. 584
—594.

—— Waves, Propagation of: April 25, 1842; vi. 397
—437.

Machine for resolving Equations: May 7, 1832; tv.
425—440,

Machinery, Influence of, on the Wealth of a Com-
munity: May 14, 1838; v1. 507—522.
Madeira, Fauna and Flora of: Nov. 15, 1830; rv. 1—70.

Fishes of: Noy. 10, 1834; v1, 195—201.

INDEX TO TRANSACTIONS I—XII.

Madeira, Flora of, Notes and Gleanings: May 28, 1838 ;

vi. 523—551.
Magnetic Intensity, observed Variations of: 1825; 11.
445.

—— Needles, as affected by Masses of Iron: May 15,
1820; 1. 147—173.

Magnetism, Connection of, with Galvanism: Apr. 2,
1821; 1. 269—279.

— as a Measure of Electricity: May 21, 1821; 1.
281—286.

—— evolved by a single Galvanic Combination, Ex-
tract from Memoir on: Noy. 25, 1822; 11.
77—83.

Magnitude and Direction, Pure Science of: May 12,
1845; vill. 278—286.

Material points, Energy in a system of : May 6, 1878;
xu. 547—570.

Mathematical Reasoning, Influence of Signs on: Dee. 16,
1821; 11. 325—377.

Matter, Demonstration that it is heavy: Feb. 22, 1841 ;
vit. 197—207.

—— Remarks on the Theory of: May 22, 1848; vi.
600—605.

Mechanics and Geometry, Connection between: Feb. 10,
1845; vin. 269—277.

Microscope, Improvement of: March 22, 1880; m1.
421—428,

Mirrors and Object-Lenses, Apparatus for Grinding :
Dec. 11, 1822; 11. 85—103.

—— Use of Silvered Glass for: Noy. 25, 1822; 1.
105—118.

Molecular Equilibrium: March 26, 1838; vi. 25—59.

Monstrosity, Human, Case of: May 16, 1831; Iv. 219
— 255.

—— of the Common Mignionette: May 21, 1832; v.
95—100, _

Motion of Fluids, on the: Noy. 24,1828; 11. 3883—416.

Ditto, Ditto, March 3, 1834; v. 173—208.

—— of Fluids, Differential Equations to: April 11,
1842; vil. 371—396.

—— Incompressible: April 25, 1842; vir, 439—453.

—— Truth of the Laws of: Feb. 17, 1834; v. 149—
172.

—— of Particles, as affecting Sound and Heat: May 16,
1836; vi. 285—288.

—— of Waves in a small Canal: May 15, 1837; v1.
457—462.

—— of Waves in Canals: Feb. 18, 1839; vu. 87—95.

Motive Power, Hydrogen Gas as a: Nov. 27, 1820;
I. 217—239.

Mountains, Elevation of, by Lateral Pressure; Apr. 27,
1868; x1. 489—506.

— Second memoir: Feb. 22, 1875; x11. 484—454.

Music in Education, place of, according to Aristotle :
May 17, 1875; xu. 523—5380.

II. INDEX OF SUBJECTS.

Natron, remarkable deposit of: Nov. 27, 1820; 1. 193
—201.
Natterjack, Habits and Character of: Feb. 22, 1830;
I. 373—381.
Neptune, see Uranus.
Neutral Series, Theory relating to: May 16, 1864; x1.
190— 202.
— Ditto, Note on this paper: Oct. 26, 1868; x1.
447—460.
Newton's method of Co-ordinated Exponents: May 21,
1855; 1x. 608—627.
—— Experiments on Diffraction: May 7, 1833; v.
101—111.
—— Principia, Criticism of: May 21, 1349; vi. 696
—706.
— Rings, Remarkable change in: Nov. 14, 1831;
Iv. 279—288.
On some Phenomena of: March 19, 1832;
Iv. 409—424.
———— Central spot of: Dec. 11, 1848; vir1. 642
—658.
See Hegel.
Non-Residence of Landlords, Influence of: March 16,
1840; vir. 189—196.
Notation employed in the Calculus of Functions: May 1,
1820; 1. 68—76.
—— Algebraic: Novy. 12, 1827; 11. 65—103.
to designate the Planes of Crystals: Feb. 11,
1826; 11. 427—439.
—— a New, in Geometry, &., &c.: Nov. 23, 1846;
vill. 415—428.
Numbers, Partition of: March 1, 1847; vit. 471—492.

Object-Glass with circular aperture, Diffraction in:
Noy. 24, 1834; v. 263—291.

— with triangular aperture, Diffraction in: Dec. 12,
18386; vi. 431—442.

— Achromatic, Correction for: Apr. 30, 1838; v1.
553—564.

Observatory at Cambridge, Longitude of: Nov. 24,
1828; mm. 155—170.

— Ditto, Latitude of: Apr. 14, 1834;
v. 271—281.
—_ Ditto, Longitude of, by Galvanic

Signals: May 15, 1854; 1x. 487—514.

Odyssey, see Iliad.

Onymatic System, on various points of: May 4, 1863;
xX. 428—487.

Optical Elasticity, Axes of in certain Crystals: Dec. 8,
1834; v. 431—438.

—— Ditto, Ditto, (second memoir): March 21,
1836; vu. 209—215.

Orbits of great Excentricity, Position of their Apsides:
Apr. 17, 1820; 1. 179—191.

Ordnance, Strains upon: Apr. 18, 1864; x1. 324—359.

xill

Ornithology of Cambridgeshire: Noy. 28, 1825; 1.

287—324.

Oscillations, on small Finite: May 15, 1843; vim. 89
—104,

—— of a suspension Chain: Dec. 8, 1851; 1x. 379—
398.

Oscillatory Waves, Theory of: March 1, 1847; vu.
441—455,

Parallelogram of Forces: New Demonstration of: Apr.
14, 1823; 1. 45—46.

Partial differential Equations, Method of integrating :
June 5, 1848; virr. 606—613.
Pendulum, Correction of, by a Ball suspended by a
wire: Noy. 16, 1829; 111. 355—360.
Pendulums, Disturbances of: Nov. 26, 1826; tr. 105—
128,

—— Effect of Internal Friction on: Dec. 9, 1850; 1x.
[8—106.]

Percussion, Experiments on: 1825; 1. 444.

Periodic Series, Critical Values of: Dec. 6, 1847; vi.

5383—583.

Perpetual Motion, How possible: Dec. 14, 1829; m1.
369—372.

Perspective, Isometrical: Feb. 21, Mar, 6, 1820; 1.
1—19.

Philosophy, Fundamental Antithesis of: Feb. 5, 1844;
vin. 170—181.

Second memoir: Noy, 13, 1848; vi. 614—620.

Phosphate of Copper from the Rhine: March 5, 1821;
I. 203—207.

Physical Geology, Researches in: May 4, 1835; v1.
1— 84.

Piscidium, see Cyclas.

Piscium Maderensium Species, &c., &c.: Noy. 10, 1834;
vi. 195—201.

Planets, see Earth.

Plato’s Survey of the Sciences: Apr. 23, 1855; Ix.
582—589.

— Notion of Dialectic: May 7, 1855; 1x. 590—597.

—— Ditto, of the Intellectual Powers: Novy. 12, 1855;
Ix. 598—604.

—— Genuineness of the Sophista of; Noy. 23, 1857;

x. 146—165.

Cosmical system: Feb. 28, 1859; x. 305—316.

Platonic Theory of Ideas: Novy. 10, 1856; x. 94—104.

Plumbago, on the artificial formation of: Feb. 21, 1825 ;
11. 441 —443.

Polarity, Organic: March 8, 1858; x. 248—260.

Polarization, Use of a new Analyzer in: March 5, 1832;
Iv. 318—322.

Polarized Light, Certain effects in Crystals exposed to:

May 1, 1820; 1. 21—41.
Ditto, as affected by Rotation: Apr. 17,
1820; 1. 43—52.

x1V

Polarized Light, Composition and Resolution of:
Feb. 16, March 15, 1852; rx. 899—416.

Political Economy, Mathematical discussion of: March

2, 14, 1829; m1. 191—230.

Ditto, as expounded by Ricardo. First

memoir: Apr. 18, May 2, 1831; Iv. 1550—

198.

Ditto, Mathematical Theory of, Second

memoir: Apr. 15, 1850; rx. 128—149.

Ditto, Ditto, Third memoir: Nov. 11,

1850; rx. [1—7.]

Potassium, Apparatus for procuring: Nov. 26, 1821;
I. 343—345.

Pressure on a flat Plate opposed to a Stream of Air:
Apr. 21, 1828; m1. 129—140.

Primitive Ridge of Devonshire and Cornwall : March 20,
1820; 1. 89—146.

Probabilities, Question in the Theory of : Feb. 26, 1837 ;
vi. 423—430.

—— Foundation of Ditto: Feb. 14, 1842; vir. 1—6.

—— Fundamental principle of the Theory of: Nov. 13,
1854; rx. 605—607.

Proportions, see Geometry.

Propositions numerically definite: March 16, 1868;
x1. 396—411.

Purbeck Strata of Dorsetshire: Nov. 18, 1854; 1x.
555—581.

Quartz, Nature of the Light produced by: Feb. 21,
1831; Iv. 79—123.
Ditto, Ditto,
1831; Iv. 199—208,

— Ditto, Apr. 18,

Railway Accidents, Causes of Fatal, &c.: Nov. 29, 1841;
vu. 301—317.

Railway Bridges, Equation relating to their breaking:
May 21, 1849; vim. 707—735,

Rainbow, Problem of, Mathematically considered:
Dec. 14, 1835; vr. 141—152.

Rainbows, Spurious: March 22, 1841; vu. 277—286.

Reed Organ Pipes: Nov. 24, 1828, March 16, 1829;
In. 231—262.

Reflection and Refraction of Light: Dec. 11, 1837;
vit. 1—24.

—— Supplement to this memoir: May 6, 1839;

vit. 118—140.

Ditto, Ditto, &c.:

vu. 7—26.

Refraction, Theory of Double: May 17, 1847; vu.
524—532,

Ricardo, see Political Economy.

Rock Masses, Internal pressure of: May 3, 1847; virt.
456—470.

Rocks, Weathering of: March 2, 1868; x1. 387—395.

Root of any Function: May 7, 1866; xi 239—266,

Noy. 28, 1842;

INDEX OF TRANSACTIONS I—XII.

Root-limitation, Cauchy’s Theorems of: Feb. 16, 1874;
xu. 895—414.
Rotatory Motion of Bodies: May 6, 1822; 11. 11—20.

Secular Cooling of the Earth: Dec. 1, 1873; x11. 414
—433.

Series, on Divergent: March 4, 1844; vim. 182—203.

— Critical Value of Periodic: Dec. 6, 1847; vin.
5383—583.

— Numerical calculation of Infinite: March 11, 1850;
1x. 166—187.

— Self-repeating: May 15, 1854; rx. 471—486.

— Theorem on Neutral: May 16, 1864; x1. 190—202.

—— Ditto, Part II.: May 7, 1866; x1. 239—266.

Note on Ditto: Oct. 26, 1868; x1. 447—460.

Sextic Torse, On a certain: Nov. 8, 1869; x1. 507—

523.
Shells, Occurrence of, in Gravel: Apr. 30, 1838; vii.
138—140.

Signs, Influence of, in Mathematical Reasoning : Dee. 16,
1821; 11. 325—377.

—— + and -, Early History of: Nov. 28, 1864; x1. -

203—212.

Note on this Memoir: Feb. 13, 1865 ; x1. 213—218.

Skew Surfaces, or Scrolls: Noy. 11, 1867; x1. 277—289_

Slags, Crystals found in: March 22, 1830; m1. 417—420,

Solid Bodies, Vibrations of: Apr. 18, 1864; x1. 324—359.

Solitary Wayes, Mathematical Theory of: Dec. 8, 1845;
Vill. 326—341.

Solon, Statue of: Feb. 22, 1858; x. 231—239,

Sound, Experiments on the Velocity of: Dec. 8, 1823;
mu, 119—137.

— see Motion of Particles,

— Reflection and Refraction of: Dec. 11, 1837;

vi. 403—413.

Spar-Fluor, Double Crystals of: Nov. 26, 1821; 1.
331— 342.

Spermaceti Whale, account of: May 16, 1825; 1.
253—266,

Sphere, Motions of, acted on by Vibrations of an Elastic
Medium: April 26, 1841; vir. 333—353.

Spherical Aberration in Eye-pieces of Telescopes: May
14, 21, 1827; ur. 1—63.

Spheroids differing little froma Sphere, on Laplace’s
Theory of; May 8, 1826; 11, 379—390.

Squares, Method of Least: March 4, 1844; vr. 204

—219,

Ditto, Ditto : May 29, 1865; x1, 219—238.

Suffolk, see Gipping.

Surfaces of the second degree, Equation to: Nov. 12,
1832; v. 77—94.

— Transformation of, by Bending: March 13, 1854;
Ix. 445—470,

Suspension Chain, Oscillations of: Dec. 8, 1851; 1x.
379—398,

II. INDEX OF SUBJECTS.

Switzerland, Tertiary Formations of: May 20, 1839;
vil. 141—152.

Syllogism, Theory of the structure of: Noy. 9, 1846;
Vit. 879—408.,

— Ditto, Ditto, Pt. Il.: Feb. 25, 1850;
Ix. 79—127.

— Ditto, Ditto, Pt, III.: Feb, 8, 1858;
x. 173—230.

— Ditto, Ditto, Pt. IV.: Apr. 23, 1860;
x. 331— 358.

— Ditto, Ditto, Pt. V.: May 4, 1863;
x. 428—487.

Symbolical Geometry and Mechanics: March 15, 1847;
vir. 497—507.

Tertiary Formations of Switzerland: May 20, 1839;

vit. 141—152.

Testimony, Measure of Force of: Nov. 27, 1848; vu.
143—158.

Theory of Probabilities, Question in: Feb. 26, 1887;
vi. 423—430.

Topaz, see Brazilian Topaz.

Transcendental Equations, Machine for resolving: May7,
1832; Iv. 425—440.

Trap Dykes in Yorkshire and Durham: May 20, 1822;
1. 21—44.

— Rocks, as associated with Mountain Limestone:
May 12, 1823: March 1, 15, 1824; 11. 139—
195.

Trinomial, Resolution of a certain: Nov. 9, 1868; XI.
426—443.

— Note on this memoir: Noy. 23, 1868; x1. 444,
445.

Trireme, Structure of the Athenian: Noy. 6, 1856;
x. 84—93.

Tumuli, Homeric: March 12, 1866; x1. 267—276.

Undulations, Theory of, applied to Luminous Waves:
May 25, 1846; vir. 371—378,

Uniyalve, Relations of, to the Bivalve, and to the
Brachiopod: Feb. 8, 1869; xr. 485—488.

Uranus and Neptune, Long Inequality of: 1852; rx.
Appendix.

Ureters, Dilatation of: Noy. 12, 1821; 1. 351—358.

XV

Velocities, &c., referred to Moveable Axes: Feb. 25,

1856 ; x. 1—20.

Velocity of Sound, Experiments on: Dec. 8, 1823; 1.
119—137.

Vibrations in Cylindrical Tubes: May 20, 1833; v.
231—270.

—— Theory of Luminous: March 6, 1848; vir. 584—
594.

—— of Solid Bodies: Apr. 18, 1864; x1. 324—359.

Vibratory Motion of Elastic Medium: March 15, 1847;
vil. 508—523.

Virtual Velocities, Demonstration of their principle:
March 21, 1825; 11. 273—276.

Vision, Peculiar defect in: Nov. 9, 1846, May 17, 1847;
vill. 493—496,

Voluntary Muscles, Abnormities in: March 8, 1858; x.
240—247.

Vowel Sounds, On the: Nov. 24, 1828, March 16, 1829:
11. 231—268.

Water, Crystallization of: March 5, 1821; 1. 209—215.

—— see Globe.

Wave Surface, Equation to Fresnel’s: May 18, 1835;
VI. 85—89.

Waves, Motion of, in a small Variable Canal: May 15,
1837; v1. 457—462.

—— in Canals, Motion of: Feb. 18, 1839; vir. 87—95.

— Theory of the two great Solitary: Dec. 8, 1845;
Vill. 326—341.

Wealth of a Community, Influence of Machinery on:

May 14, 1888; vr. 507—522.
Ditto, Effect of Non-Residence of
Landlords on: March 16, 1840; vir. 189—196.
Weathering of Rocks: March 2, 1868; x1. 387—395.
Wheels, On the Forms of the Teeth of: May 2, 1825;
11. 277—286,

Wiener, see Cubic Surface.

Winter Solstice, Ancient Observation of: Nov. 30, 1829;
II. 8361—363.

Written Greek Literature, First Age of: Noy. 23, 1868;
x1. 461—480.

Yorkshire and Durham, Trap Dykes in: May 20, 1822;
Ir. 21—44,

CAMBRIDGE ; PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS.

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