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TRANSACTIONS 



OF THE 



CAMBKIDGE 



PHILOSOPHICAL SOCIETY. 



VOLUME XIII. 




CAMBRIDGE : 

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

AND SOLD BY 

DEIGHTON, BELL AND CO. AND MACMILLAN AND BOWES, CAMBRIDGE; 

G. BELL AND SONS, LONDON. 

M.DCCCLXXXIII. 



Cambrifigf : 

PRINTED BY C. J. CLAY, JI.A. & SONS, 
AT THE UNIVERSITY PRESS. 



Q 

CI? 



^/ 



INDEX TO VOL. XIII. 



Catlet, Prof. A. 

= 20, 1—4 
Catlet, Prof. A. 

the Polyhedral 
Bibliography 

§§ 1-v. 

§§ 8-15. 

§§ 16-20. 

§§ 21-45. 

§§ 46-62. 
§§ 63—68. 

Part II. 
§§ 69—80. 
§§ 81—84. 
§§ 85—86. 

§§ 87—93. 

§§ 94-95. 

§§ 96—103. 
§§ 104—117. 

§§ 118—127. 
§§ 128—134. 



Table of A"'0"-MI('/h) up to m = n 

On the Schwarzian Derivative, and 
Functions, 5 — 68 

6 ; Part I. 8—35. Part II. 35—68 
The Derivative (s, x}, 8 — 9 
The Quadric Function of three or 

more Inverts, 10 — 12 
The functions P, Q, R, 12—13 
The PqR-TAhXa, and Annex, 14—15 
The Difl'erential Equations [x, z) and 

[s, .r}, 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 FormulEe, 39 — 40 
Investigation of the forms /5 and 

hb, 41-42 
Invariantive property of the Stereo- 
graphic Projection, 42 — 44 
Groups of homogi'aphio transfomia- 

tions, 44 — 45 
The Regular Polydedra, 45 — 50 
The groui^s of homographic trans- 
formations, resumed, 51 — 61 
The system of 15 circles, 61 — 65 
The Regular Polyhedra as soUd 
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 e.\pression for Ratios of Lines and Points, 
95 

FormuL-e in Coordinates, 97 

Relations between the Sides and Angles of a Tri- 
angle, 99 

The different kinds of Uniform Space, 101 

Imaginary Geometry of Tliree Dimensions, 104 

Spherical Geometry of Three Dimensions, 110 

Ordinary Geometry of Tliree 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. 



GuMSHta, J. \V. L. Tables of the Exponential Func- 
tion, i43— 272 

Tables of <*, e-', logoff' and logijO"^' 

Table I. From j. = 0-001 to .r = 0100 at intervals 

of 0-001, 254—255 
Table II. From .i=0-01 to .i-=200 at intervals 

of 001, 256—259 
Table III. From .r=01 to .r= lO-Q at intervals of 

01, 260—261 
Table IV. From .(•=1 to .r = 500 at intervals of 

luiity, 262—271 
Comparison with Schulze's table, no erroi-s, 272 

„ „ Vega's table, errors, 272 

Other existing Tables, Schulze, 243 
„ „ „ Vega, Koliler, Gudermann, 

244 
Grassniann's Aiisdehnungslehre, applications of (Cox), 
69, llo 



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

Part I. From ./■ = to .i=15-349, at intervals of 

001 to twelve decimal places, 151 — 227 
Part II. From .i= 15-350 to .;•= 17-298 at inter- 
vals of -002, and from ,r=17-300 to .i- = 27-635 
at intervals of -005 to fourteen decimal places, 
228—241 

Polyhedral Fimctions, the (Cayley), 35 

Quaternions (Cox), 69 

Schwarzian Derivative (Cayley), 5 



Table of A"'0"-^^('«) (Cayley), 1 
Tables of e^, c", log,oe"^ and log,„e-"^ (Glaisher), 243 
Table of e " (Newman), 145 

Hill, M. J. il. On Functions of more than two Tesseral Harmonics, functions analogous to (Hill), 
variables analogous to Tesseral Harmonics, 273 — 299 273 



CAMBBIDOE : PRINTED BT C, J. CLAY, M.A, AMD SONS, AT THE CNIVER8ITT PEES8. 



\ 



'i 



■J> 



CONTENTS. 



PAGE 

I. Tabl^ of A'"0"-^n (m) up to m = n = 20. % Peof. Cayley 1 

II. On the Schwarzian Derivative and the Polyhedral Functions. By Peop. Cayley 5 

III. On the Application of Quaternions and Grassmann's Ausdehnunyslehre to different kinds 

of Uniform Space. By Homersham Cox, B.A., Fellow of Trinity College 69 

IV. Table of the Descending Exponential Function to Twelve or Fourteen Places of Decimals. 

By F. W. Newman, Emeritus Professor of University College, London 145 

v. Tables of the Exponential Function. By J. W. L. Glaishee, M.A., F.R.S., Fellow of 

Trinity College, Cambridge 243 

VI. Un Functions of more than two variables analogous to Tesseral Ha/rmonics. By M. J. M. 

Hill, M.A 273 






ADVEETISEMENT. 



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



The Society takes this opportunity of expressing its grateful 
acknowledgments to the Syndics of the University Press for their 
liberality in taking upon themselves the expense of printing this 
Volume of the Transactions. 



I. Tabic of A^O^-^n^Hj) up to m = 7i=20. By A. Cayley, Sadlerian Pro- 
fessor of Pure Mathematics. 



[Read October 27, 1879.] 



The differences of the powers of zero, A^O", present themselves in the Calcuhis of Finite 
Differences, and especially in the applications of Herschel's theorem, /(e') =/(l-l- A) e'", 
for the expansion of the function of an exponential. A small Table up to A^O'" is 
given in Herschel's Examples (Camb. 1820), and is reproduced in the treatise on Finite 
Differences (1843) in the Encyclopwdia Metropolitana. But, as is known, the successive 
differences AO", A'O", A'O",... are divisible by 1, 1.2, 1.2.3,... and generally A^O" is 
divisible by 1.2.3...m, = n(»i) ; these quotients are much smaller numbers, and it is there- 
fore desirable to tabulate them rather than the undivided differences A^O" : ii is more- 
over easier to calculate them. A Table of the quotients A"'0"^n(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'- n(n)), and without 
using for their determination the convenient formula C,.'*' = nC„* -F (7..,' given by Bjorling 
in a paper Crelle, t. xxvili. (1844), p. 284. The formula in question, say 

= m . „ ,- - + 



U(m) n(m) n(w-l)' 

is given in the second edition (by Moulton) of Boole's Calculus of Finite Difference.^, 
(London, 1872), p. 28, under the form 

A'"0" = m (A'^-'O--' + A^O"-'). 

It occurred to me that it would be desirable to extend the table of the quotients 
A"'0'' ■- n (wi), up to vi = 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 column is A 
B W + A 

C SC + B 

D iD + C 

E oE + D 

+ E- 
Vol. XIII. Part I. 1 



2 Pkof. CAYLEY, table of A"'0' h- II («t) Ur to m = n = 20. 

ami then we obtjviu a good verification by taking the sum of the terms in the new 
cohimn, ami comparing it with the value as calculated from the formula, 

Sum = '2A + SB + 4C' + oD + QE : 

observe that in the two calculations wo 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"'-- n(m). 



< 






























1 


0> 

1 


OS 

1 


0» 

1 


0* 

1 


0« 

1 


00 

1 


0' 


08 


0» 


QIO 


0" 


0- 


013 


0" 


1 


1 


1 


1 


1 


1 


1 


1 


2 




1 


3 


7 


15 


31 


63 


127 


255 


511 


1023 


2 047 


4 095 


8 191 


3 






1 


6 


25 


90 


301 


966 


3 025 


9 330 


28 501 


86 526 


261 625 


788 970 


■i 








1 


10 


65 


350 


1 701 


7 770 


34105 


145 750 


611 501 


2 532 5.30 


10 391 745 


5 










1 


15 


140 


1050 


6 951 


42 525 


246 730 


1 379 400 


7 508 501 


40 075 035 


6 












1 


21 


266 


2 646 


22 827 


179 487 


1 323 652 


9 321312 


63 436 373 


1 














1 


28 


462 


5 880 


63 987 


627 396 


5 715 424 


49 329 280 


8 
















1 


36 


750 


11880 


159 027 


1 899 612 


20 912 320 


9 


















1 


45 


1 155 


22 275 


359 502 


5 135 130 


10 




















1 


55 


1705 


39 325 


752 752 


11 






















1 


66 


2 431 


66 066 


12 
























1 


78 


3 367 


13 


























1 


91 


14 




























1 


15 






























16 






























17 






























18 






























19 






























20 































<1 


013 


016 


01- 


018 


019 


020 


1 


1 


1 


1 


1 


1 


1 


1 


2 


16 383 


32 767 


65 535 


131071 


262 143 


524 287 


2 


3 


2 375 101 


7 141 086 


21 457 825 


64 439 010 


193 448 101 


580 606 446 


3 


4 


42 355 950 


171798 901 


694 337 290 


2 798 806 985 


11259 666 950 


45 232 115 901 


4 


5 


210 766 920 


1 090 190 550 


5 652 751 651 


28 958 095 545 


147 589 284 710 


749 206 090 500 


5 


6 


420 693 273 


2 734 926 558 


17 505 749 898 


110 687 251039 


093 081601 779 


4 306 078 895 384 


6 


7 


408 741 333 


3 281 882 604 


25 708 104 780 


197 462 483 400 


1 492 924 634 839 


11 143 554 045 652 


7 


8 


216 627 840 


2 141 764 0.53 


20 415 995 028 


189 036 065 010 


1 709 751 003 480 


15 170 932 662 679 


8 


9 


67 128 490 


820 784 250 


9 528 822 303 


106 175 395 755 


1 144 614 626 805 


12 011 282 644 725 


9 


10 


12 662 6.50 


193 754 990 


2 758 334 1.50 


37 112 163 803 


477 297 033 785 


5 917.584 964 655 


10 


11 


1479 478 


28 936 908 


512 0601)78 


8.391004 908 


129 413 217 791 


1 900 842 429 486 


11 


12 


106 470 


2 757 118 


62 022 324 


1 256 328 866 


23 466 951 300 


411 (116 6.33 391 


12 


13 


4 550 


165 620 


4 910 178 


125 8.54 63H 


2 892 439 160 


61068 660 380 


13 


14 


105 


6 020 


249 900 


8 408 778 


243 577 530 


6 302 524 580 


14 


15 


1 


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 


1 18 








1 


171 


15 675 


18 


; 19 










1 


190 


19 


20 












1 


20 



Prof. CAYLEY, TABLE OF A^O" - n (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) etc. 




1 

3 

6 

10 


1 

7 
25 


1 
15 


1 










15 


65 


90 


31 


1 








21 


140 


350 


301 


63 


1 






28 


266 


1050 


1701 


966 


127 






36 


462 


2 646 


6 951 


7 770 


3 025 






45 


750 


5 880 


22 827 


42 525 


34 105 






55 


1 155 


11880 


63 987 


179 487 


246 730 






66 


1705 


22 275 


159 027 


627 396 


1 323 652 






78 


2 431 


39 325 


359 502 


1899 612 


5 715 424 






91 


3 367 


66 066 


752 752 


5 135 130 


20 912 320 






105 


4 550 


106 470 


1 479 478 


12 662 650 


67 128 490 






120 


6 020 


165 620 


2 757 118 


28 936 908 


193 754 990 






136 


7 820 


249 900 


4 910 178 


62 022 324 


512 060 978 






153 


9 996 


367 200 


>< 408 778 


125 854 638 


1 256 328 866 






171 


12 597 


527 136 


13 916 778 


243 577 530 


2 892 439 160 






190 


15 675 


741 285 


22 350 954 


452 329 200 


6 302 524 580 





20 



19 



18 



17 



16 



15 



14 



13 etc. 



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 followins 



Table, No. 2 (explanation infra). 



< 
















-g 
















a 


1 


2 


3 


4 


5 


6 


7 





1 


1 


1 


1 


1 


1 


1 


1 




2 


6 


14 


30 


62 


126 


2 




1 


12 


61 


240 


841 


2 772 


3 






10 


124 


890 


5 060 


25 410 


4 






3 


131 


1830 


16 990 


127 953 


5 








70 


2 226 


35 216 


401 436 


6 








15 


1600 


47 062 


836 976 


7 










630 


40 796 


1 196 532 


8 










105 


21225 


1 182 195 


9 












10 930 


795 718 


10 












945 


349 020 


11 














90 090 


12 














10 395 



We have by means of this Table, the general expressions of A'^C, A'~'0', 
to A'~°0^ viz. the formulae are 



A'-\r 



1—2 



up 



4 Prof. CAYLEY, TABLE OF A'O' - n (m) UP TO m = n=^ 20. 

S'O' - U(r) =1, 
A-O' - II (,• - 1) = 1 + 2 ('■ - -)'+ 1 ('■ ~ -)' , 

Sec, &c. 

where tlio uuraorical coefficients are the numbers in the successive columns of the table ; 

and where for shortness I 1 is written to denote the binomial coefficient — p,,^7— . For 

instance, r= 10, we have 

AV- n (8) = 1 + G . 7 + 12 . 21 + 10 . 35 + 3 . 35, = 750, 

agreeing witii 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, 1.3.5.7.9, and 1.3.5.7.9.11. This 
is itself a good verification : I further verified the last column by calculating from it 
the value of A"0"^ 11(14),= G 302 524 580 as above. The Table shows that we have 
A'~"0'' -^ n (r — m) given as an algebraical rational and integral function of r, of the degree 
2m. But the terms from the top of a column, A0'' = 1, A'O'-r- 1.2 = 2'"' - 1, &c., are not 
algebraical functions of i: 

22 October, 1870. 



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



[Read March 8, 1880.] 



The quotient s of any two solutions of a linear partial differential equation of the 
second order, -y j + J^ ;/ + 52/ = 0> is determined by a differential equation of the third 
order 

,dp 




}'*^i-^). 



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



f^'^J'=V-t(?)'. 



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

Writing in general (a, b, c .'.JX, Y, Zf to denote a quadric function 

(a, b, c, i(a-b-c), i(-a + b-c), H-a-b + c)$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 x, the resulting differential equation 
of the third order is of the form 



., ^} = (a, b, c.-.)(^. ^, ^J; 



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. Nom, 
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, 18G9 and 1873; the 



6 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

latter of these (relating to the algehraic integration of the differential equation for the 
hyiiergeometric series) is the fundamental Memoir upon the subject, but the theory is in 
son\e 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 Eeilie 1+y -a;+.... Crelle, t. xv. (1836), pji. 39—83 

and 127—172. 
Schwarz, Ueber cinige Abbildungsaufgaben. Crelle-Borcliardt, t. lxx. (18G9), pp. 105—120. 
Ueber diejenigen Falle in welchen die (?aMssische hypergeometrische Eeihe eine alge- 

braische Function ilires vierten Elenientes dai-stellt. Do. t. lxxv. (1873), pp. 292—335. 
Cayley. Notes on Polyhedra. Quart. ]\[ath. Jour. t. vii. (1866), pp. 304—316. 

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

Fuchs Ueber diejenigen Differentialgleichungen zweiter Ordnung welche algebraische Integralen 

besitzen, imd eiue Anwendung der luvariantentheorie. Crelle-Borchardt, t. 81 (1875), pp. 

97—142. 
Klein, Ueber binare Formen mit lincarcu Transformationen iu sich selbst. Math. Ann. t. ix. 

(1875), pp. 183—209. 
Brioschi, Extrait d'une lettre k M. Klein. Math. Ann. t. xi. (1877), pp. 111—114. 
Klein, Ueber lineare Differential-Gleichungen. Math. Ann. t. xi. (1877), pp. 115 — 118. 
Brioschi, La theorie des formes dans I'Lntegration des equations diifereutielles hiieaires du second 

ordre. Math. Ann. t. xi. (1877), pp. 401—411. 
Gordan, Ueber endliche Gruppen linearer Transformationen einur Veranderlichen. Math. Ann. 

t. XII. (1877), pp. 23—46. 

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

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

Weitere Uutersuchungen iiber das Icosaeder. Math. Ann. t. xii. (1877), ])p. 503 — 560. 

Cayley, On the Con-espondence of Homographies and Rotations. Math. Ann. t. xv. (1879), 
j.p. 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 iu i3articular 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 

{s,^} = (a, b,c.-.)(^^--^, ^4^, ^J. 

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, x = rational function of s ; others of the 



AND THE POLYHEDRAL FUNCTIONS. 7 

form, rational function of a; = 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 s 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 cc and s respectively, viz. the equation 



and we have the theorem that the solution of this equation depends upon the deter- 
mination of P, Q, R rational and integral functions of z (containing each of them multiple 
factors) which are such that P+ Q + E=0: (using accents to denote differentiation in regard 
to 0, this implies P' -I- Q' + -R' = 0, and consequently QR' - Q'R =RP' - R'P = PQ' - P'Q): 
and are further such that the equal functions QR' — Q'R, RP' — R'P, 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 x, z is 
then expressed iu the symmetrical firm f{x — «) : g{x -h) : h{x — c)=P : Q : R. 

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 

[s, x} - (a, b, c .-.) ( , — --, , 

'■ ' ^ ' \x — a x — b X — c 

although the other form in [x, 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. 

B. Quadric functions of any three or more inverts ^ . 

X — L 

C. Rational and integral functions P, Q, R having a sum = 0, and whicli are 
such that QR'-Q'R, =RP'-R'P, =PQ'-P'Q, 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 x. 

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. 



Prof. CAYLEY, ON THE SCIIWARZIAN DERIVATIVE 



PART I. 
The Derivative [s, x], Article Nos. 1 to 7. 
d fy ds\ ^T.__ ,_ „, dp 



' ^f^'4=.^^»l)'*^^°5^'^5=^-^^'- 



2. The derivative [s, x} 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 8, and the accents 
as before for differentiations in regard to x, we have 

s' — S's^, 
whence, differentiating the logarithms, 



and again differentiating 


s s, 


S" 


'r-(S)''^' 


5, \sj 


+ s 


But -i(-] = S" 


[-*©'] 


-s 


and consequently 







S" 



fS"\ 



S"\' 



-n -KI-) 



s ^\s J 



^'3 _ . h'2 



+ 



S'" 






{^.^] = (^£f{^,s] + {s,x], 



that is 

the required formula. 

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

^dX' 



''^'=(£)^t'' -^i-t^>^'i)- 



8. If in this formula we write S for s, and substitute the resulting value of [S, x\ 
in the former formula, we have 

which is the formula for the cliaiigc of both variables, and it in fact includes the 
other two : viz. writing X = x, or S =s, and observing that {s, s] = [x, x] = 0, we have 
the other two formulae. 

4. By putting in the first formula X = s, we obtain 
a formula for the iuterchan"e of the variables. 



AND THE POLYHEDKAL FUNCTIONS. 9 

5. Writing S = — -—7, and using for a moment the accents to denote differentiation 
cs "T~ a 

in regard to s, we have 

ad — he S' — 2c 



S' = - 



{cs + df S' cs+d' 

*UV {cs + df 
Consequently {S, s] = (whence also {s, S}=0). 

Hence in the first formula {iS, a;} = {s, x], that is 



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

CUB + B 

6. Again if X = j; , then from the second formula 

7X + 



that is 






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

as + h ax + I3\ _ (yx + 8)* . . 

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

7. Let s be a given function of x, the equation {S, x\ — \s, x\ is a differential equa- 
tion of the third order in S, and by what precedes, its general integral is S=- — -,. 

S" s" 2cs' 

The direct process is as follows: we have a first integral -f77= — ,; a second 

^ '^ S s cs + d 

integral log /S'= logs'- 2 log (cs + cZ) + Const, that is S' = -, — 7— 7^2; and thence a final in- 

j^ 

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

CS T~ tt 



Vol. XIII. Paet I. 



10 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 



The Qtiadnc Function of three or more Inverts. Ai-t. Nos. 8 to 15. 
S. We consider a quadric function of any number of inverts - — - , — -^,... all of 

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

a 



the coefficients of the linear terms is = 0. We have therefore square terms „ , 

{X a.) 

product terms '■ ^, and linear terms , where the sum of the coefficients A 

^ x — a.x — p x- a 

is = 0. Any product term 3 is expressible in the form of a difference ^ _ o _ 

iJC ^~ fit * iC ~" p * r^ *^ ^ 

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

o — p X — p 

thus expressed the sum of the coefficients of the linear terms is still = 0. The function 

is thus always in expressible in the form 

a b A 5 

where the sura A + B + ... is = : this may be called the reduced form. 

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

form: this will be the case if a = (that is if the oridnal form contains no term in -;) , 

V (x-ayj 

and if also ^ = 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 {x — af{x — ^)^ (x — yY... 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 x may vanish, and the numerator will then be of a lower degree. 

But this numerator will for any non-essential invert contain the factor (.r — yf, 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. 



o 



ABC 

11. It is to be remarked that the linear terms H p;-) ..., where 

X — a x — p x — y 

A + B + C + ...=0, can be (and that in a variety of ways) expressed as a sum of dif- 
ferences r,, that is as a sum of product-terms -^. Hence the quadric 

X— a. x- p "^ x — a.x — p ^ 



AND THE POLYHEDRAL FUNCTIONS. 11 

fuuction can be (and that in a variety of ways) expressed as a homogeneous function 
Ta, ... 0^— —,—_-„, ...j ; we must have in the form all the essential inverts, and we 

need have these only. Supposing that this is so, and that tlie number of the essential 
inverts is =n, then the number of constants is =|m(«+1), whereas the number of con- 
stants in the reduced form is only =2n — l: 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 



x — ^.x — y x — y.x — a x — a.x—j3' 
In particular if the number of essential inverts is =3, then the quadric function is of 

the form (a, b, c, f, g, hu -, -j=, j , which contains one superfluous constant, 

and equivalent functions differ only by a multijjle of 

ig-7 I Y-a I °-/3 
X — 13 .x — y x — y.x — a x — a.x — (3' 

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." 

The conditions in order that the function (a, b, c, f, g, h () -, -, , ) may 

V Ax — ax — p x — y) ■' 

be curtate are easily found to be 

a+b+c + 2f+2g + 2h = 0, 

a (a + h + g) + /3 (h + b + f ) + 7 (g + f + c) = ; 

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

^a,b,c, J(a-b-c), J(-a + b-c), i(-a-b + c)|^, ~^, ~^ 

which, as already mentioned, we denote by 

1 1 1 



a, b, c .-. 



x — a' X — ^ x—y. 



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

only the three essential inverts , ^ , , is always expressible in the foregoing 

form 



a, b, c.-.() , 7s, ). 

AX —a X — p x — y/ 



2—2 




12 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

13. It may be remarked that tbe function (a, b, c .•.$A' Y, Zf is a function of 
the diflerences of the variables A', Y, Z\ and similarly in the case of four variables a 
function (a, b, c, d, f, g, h, 1, m, uJA' 1', Z, Tl')' for which a + li + g + 1, h+b + f+m, 
g+f+c + n, l + 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 vai-iablos any inverts whatever, a diaphoric function 
is always curtate. 



14. The function 



a 



, f a b c 1' 

( x — a X — p x—'y ) 



where the coeflScients a, b, c... satisfy the relation a + b + c + ... = — 2, is diaphoric, and 

therefore curtate. In fact forming the sum, coeff. -. r. + i coeff. ^ + . • ■ , this is 

— a— ^a'— ^ ab — |ac..., = — ^a(2 +a + b + c. ..) which is =0; and similarly the other 
conditions are satisfied. 

1.5. The fiinction 

/ i_ Y a a, h b, c c, 

a, b, c .-.0 + — '- + ..., a+ — -o-+--> + —+ ■■ 

\ A« — a x — a, ic — /3 X— p^ x—y x - y^ 

regarded as a function of the inverts , , . . . .... where 

a; — a x — a^ x — 13 

a + o, + ... = 6 + &, + ... = c + Cj + ..., = i suppose, 
is diaphoric, and therefore curtate. In fact the condition in regard to is 

a(a*+aa, + aa, + ...) + |(- a +b- c) (a6 + aZ), + ...) +i (- a -b + c) (ac + ac, + ...) = ; 

that is 

aA;{a + i(-a + b-c)+|(-a-b+c)} =0, 

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



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

16. We consider P, Q, R, rational and integral fractions of z, such that P + Q + i? = : 
hence, using the accent to denote differentiation in regard to z, we have also P' + Q' + K' = ; 
aad therefore QR' - Q'R= RF - RP= PQ - FQ, = % 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, eftecting upon a solution P, Q, R any linear substi- 
tution (aa 4- j8) -H (7^ + S), and omitting the common denominator, we have a solution; but 
this is regarded as identical with the original solution. The three functions, if not ori'nn- 
ally of the same order, can thus be made to be of the same order; or by taking account 
of the root s = oo , 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 mio-ht 
therefore be struck out. 

18. We may therefore write 

P = FU{z-lf, Q = (?n(3-m)', R=Hn{z-n)', 

where (z — If is taken to denote the distinct simple or multiple factors of P, and the 
like as regards Q and R; the factors z — l, z — m, z — n are thus all of them diiferent. 
And we have S = Sj9, ='^q, =S?'. 

19. It is at once seen that © is of the degree 28 — 2, and moreover that it con- 
tains the factors \l{z-l)^'^, 11(3 — w)'"', n(2-«)'~'; hence it contains the factor 

Il{z-l)^'{z-my-'{z-n)'-\ 

Suppose the number of distinct indices p is = o-,, that of distinct indices q is <t , and 
that of distinct indices r is = 0-3 ; then the degree of the factor is = 3S — cr^ — o- — o- ; 
and if this be = 2S — 2, then can have no other variable factor : viz. if the numbers 
o-j, a^, o-j of the distinct indices ^j, q, r respectively are such that o-^ -f cr^ + cr, = S -|- 2 (a 
relation which is henceforth taken to be satisfied), then we have 

e = A'n (^ - l)"-' iz - m)"-' {z - 7ir\ 

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 = Q here implies that two functions, say P, Q, 
are of the same degree, and the third function R of an inferior degree ; but, this 

being so, we have only to regard R as containing the factor f 1 J of the degree t proper 

for raising its degree up to that of P or Q. 

20. Solutions are given in the following P^i?-Table : in which, where required, 

the proper factor (1 j 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. 



/ 



14 



Prof. CAYLEY, ON THE SCHWARIAN DERIVATITE 







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2 « 

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a O) 

53 



7 



16 Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 

The Differential Equations [x, z] and {s, x]. Art. Nos. 21 to 45. 

21. In reference to what follows, it is convcuicnt to put P = XP^, P' = A',P(,, whore 
i'„ is written for 11 (j-?)""'. the G.C.M. of P and P'; and X is consequently =F into the 
product Il(z-l) of the several factors taken each with the index unity; and so for Q 
and R: viz. we write 

P, Q,R=XP,, YQ,,ZB„ 

p;q;r = x,p,,y,q,,z,b,. 

and the foregoing value of then is 

0=A'P„QA- 

We come now to the investigation of the leading theorem. Take a, b, c arbitrary, 
^ ,^_ /, _ ^ _ c_ c-a, a-h; P, Q, B functions of z as above ; and write 

f{x-a) : g(x-b) : h{x-c) = P: Q : R, 

equations consistent with each other, and which determine a; as a rational function of z. 
Usinw 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 

(..,.}+^'^(a,b, c .•.][^, -i^, -1-) . 

22. Calculation of the first term [x, z}. 

We have x=a, function [<^£+^)'^\yji+^)' ^^'^ thence {x,z]=\j^, zi, = [^, z] for 
a moment; then 

.,_/PY_^-^ _ P„<?A _P„(3„ 
^ ~\Rj ~ R^ ' zm: • z'^r; 

Substituting the values 

P=X\{z-ir\ q,= Xi{z-mr\ P„ = n(2-nr', Z^U{z-n), 



we have 
and thence 



or .sav 



f z — l z—m z — n 

{.,z}=\-t-p-^.-%j3^.+%r-^^ 

' ' ' [ i^-i) (^ - "0 (^ ~ ''') ) 

^ [ z—l z — m z — n) 

_ J '-^:^ + ^'"^ + ^^ + 2^^ ...- '^ - '-^^^ ...Y, 

'^ \ z- 1 z — h z — m z — 7/1, z — n z -n^ ) ' 



AND THE POLYHEDRAL FUNCTIONS. 17 

where it is to be observed that 

S{p-l) + S(q-l)--Z{r+l), =S-<r,+ S-<r,-(8 + cr,), =B-a,-a,-<r„ =-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 d priori, and it will be presently verified. 

23. Calculation of the second term 

„/ , Y 1 
a; a, b, c . . 



(a; — a' x — b' X — c) 
We have 

fix- a), g{x-b), h{x-c) = nP, QQ, HR, 

where fl is a determinate function of z. Hence 

x x X P' ft' q ft' E ft' 



x-a' X-h' X-C P ' ft' (3 ' ft' iJ ' ft' 

and then substituting these values, by reason that the function is diaphoric, the terms in 
•^ disappear, and we have 

x" fa, b, c .-.y ^ 






ix — a' x — b' x — cj 

(F q P!\ 

= I a,, u, u . . 1 

which is 

=fa,b,c.-.](<-i^,, s-^, s^y. 

V X z — l z—m z—nl 

We have 1p = 1.q = Xr, = B : and hence by what precedes, this function considered as a 
function of the inverts -,, &c., is diaphoric, and therefore curtate. 

24. We have therefore 



{.;,.}+ ^'^ (a, b,c.-.^^, -1-^,-1^) = 
tl - 2-1 ^^ z-m " z-n) 



'.-I 

+ fa,b,c.-.^2-i^,S-^,S^Y, 
V K z — m z-m, z-nj 



where the whole function on the right hand is curtate. 

Vol. XIII. Paet I. 



IS Prof. CAYLEr, ON THE SCHWARZIAN DERIVATIVE 

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

a A 

,+ ... + + ... 



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

^{z-if {z-if^ (z-lf+■■■ 



iz-lf'^^ (z-l.f 

r fi 
same set; and so in other like cases. 



viz. z — I here denotes any particular factor, and z — l^ represents any other factor of the 



26. The whole coefficient of -. ^„ is 

{z-lf 

- (i' - 1) - Hi' - 1)' + H^\ = HI -/) + a/ ; 

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

27. The whole term in ■, comes from 

Z — 6 

z — I \ z — l^ z—m z — nj 

+ -^, f2aS' -^,+ (- a - b + c) S — ^ + (- a + b - c) 2 -^V 
z—l\ z — I ^ 'z — m ' z-nj 

viz. each term such as ^ j is to be replaced by -, — r ( -, j ) , giving rise to 

the term -, — r -. ^ or contributing the term , — =- to the coefficient of ; . The whole 

I — l^z — l ° '■ — h •2~' 

coefficient thus is 

28. Suppose first that z — l is a multiple factor of P, viz. a factor with an index j) 

O' R' 
greater than 1 : then for z = 1 we have Q + R = 0, Q' + B' = 0, and thence fj = -p > that 

is S } = 2 i . We have therefore 



AND THE POLYHEDRAL FUNCTIONS. 19 

'^ \ l — m I — mj 

and moreover in the top line the terms S i— ^ — and — S j destroy each other. The whole 

coefficient of , when {z — I) is a multiple factor of P, thus is 

a form which is now symmetrical in recjard to the inverts ^ and ^ . 

•^ ° l — m I — 11 

29. The value just obtained is 

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

(l-^ + 2ap)jS'^4±i^--Siii+-^-^g-^-S ^^^7^""^^ Uo, 
\ r f ]^ i_i^ I- m l — n j 

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

30. We have as before S -r-^ — = S -; , or writing each of these quantities = ^, 

l — m I — VI 

the equation to be verified is 

We have • ^=,-^ + 2'^. =§. 

that. S'A = g.__^Jfor. = i, 

The first derived function of the numerator is X,' {z — I) + X^— pX', which for 
z = l is X^— pX', which is = ; and for the denominator it is X' (s - Q + X, which is 
also = ; passing to the second derived functions we find 

^, p, _ 2X;-p X" X.'-jpX' 
^ z-l, 2Z' ' ~ X' 

3—2 



20 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

X' 1 1 
From the equation ^=7-= i + 2' , 

2l Z — I Z — I 

we find in like manner, 

and we thence obtain {z being always = I) 

^ z-l, X" 
so that the equation to be verified becomes 

X^ '^ ' '^ l — m ^ I — n 

31. But from the equation 0, =PQ'-P'Q, = KP,Q^E„ we find X7,-X^T=KJR„ 
and then, dififerentiating, Xl\' + X'Y^ — X^'Y — X^Y' = EE^': writing in these equations 
J = ?, thev become — X Y = KB 

X'y,-x:y-xj' = kb,. 

and dividing the second by the first 

Y Q' 

or recollecting that X, = pX', and -y ~7) ' ^^ ^^^^ 

X' p{r, YJ^Q' 

that is ^'=p(s!— i-s,-^)+s^. 

A -^ V ' — « l — vil I- m 

= (p + 1) <& -;j2 j^-P^j^, 
l — m'L — n 

the required relation. 

32. The result is that z — I being a multiple factor of P the coefficient of the term 
1 



IS 



z-l 

= n _ ^.2 J. 9or>«^ i-s" ^?~ ^ _i_ •s? ^.ZlJ^ _ -s" ^ 



= (l-/.2a,»){2Mzi+X^_^^-r^4,]^ 



= 2(ai>) 






33. In the case where z — I is a simple factor of P we have p = l, and the 
coefficient is 

= 2a2'^^^+(-a-b+c)2^-X_^ + (-a + b-c)2^, 

= a(2S'-iV-2r2__2.-^)-(b-c)(s.-^-2^). 
\ t — (, l — m l — nl '\ l—.m l—nj 



AND THE POLYHEDRAL FUNCTIONS. 21 

34. Of course the formulai for the coefficients of j-„ and o-ive at once bv 

a mere change of letters those for the coefficients of ^ , , and 



{z-mf z-m' ""^ [z-nf z-n' 
and the function in question, 



[x, z]+x'U, b, c .-1^-, ~, -^y, 



is now obtained in the required form, 



Cap) , _M_ . (c*-) , ^ , B ^ C 



(z-lf" (z-mf" {z-tif'" z-l'" z-m'" z-n'" 

where (ap) denotes | (1 -p") +ap'', and the like for (bj) and (cr) ; and where z-l 
being a multiple factor of P, the coefficient A contains the factor (a^)); 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 — l being a simple factor of P, the expression contains 

T-j , or the invert is essential : and similarly z — m being a simple factor uf Q, 

or z — n a. simple factor of R, the inverts and are essential. But for z—l 

2— 711 Z —11 

a multiple factor of P, the coefficient (ajj) of the term n- — ^ may vanish, viz. this 

[z — I) 

will be the case if a=j(l ^j ; and when this is so the coefficient A of the cor- 
responding term , also vanishes ; that is , is a non-essential invert. And similarly 

Z ^ C Z ~~ V 

1 1 

for any multiple factor z — m of Q or z — n of R, the invert or may be non- 

z — m z—n ■' 

essential. 

3G. If P, Q, R contain each of them only multiple factors of the same index, 
say of the indices jj, q, r for the three functions respectively, viz. if the functions are 
F (U {z - 1})", (?(n (s -m))', // (11 (s -«))', the result contains only the six terms written 

down: and then if a, b, c are =^(l 2). 1(1 i) • ^(1 — :«) respectively the result 

is = : viz. we then have 

{x,.)+.-(a, b, c.-.][^, -1^, -L-)=0, 

or we in fact have for tlie values in question of (a, b, c) a soluticm 

f{x-a) : g{x-h) : h{x-c) = P : Q : R 
of this differential equation of the third order. 



22 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

S7. The reasoning applies directly to lines 2, 3, 4, 5 of the PQE-Table : and 
with a slight variation to line 1; viz. here the factors of E{=-l + z") arc all simple 
lactors, but in virtue of c = 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 PQE-TMe 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. 

Thus line 3 shows that the function x determined by 
f(x-a):g{x-h):h{x-c)={z'+2j^3z' + ir:-Uj^{z'-zf:-(z*-2j^Sz'^ + ir 

satisfies 

, , „/4 3 4 Y 1 1 1 V-n 

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 - — — , - — r- , - — - , 

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 



(a. K, c. .•.][^-4^^, ^^, ^j 



where a , b,, c, are the values of the three which do not vanish in the series of expres- 
sions (a/j), (b}), (cr). 

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

f{x-a) : gix-h) : h(x-c) = P: Q : R, 
and giving 

{X. z]+x^[^, b, c .•.][^4^. ^^, ^J =(a., b., c. ,1^^. -1^. -L.). 

39. For instance, from line IX we have 

f{x-a) : r,{x-b : h (x- c) = {z-4>y : -(s-l)(3+8)» : 27^^(1-^), 

the values of (a, b, c) are - , -, -„.; and since P, Q, R 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 

a3; bl, b2 ; cl, c2, 

which are = ; h . ; ^ , ?^ respectively. 

Hence writing a, , b, , c, = - , ^ , -ztt: the corresponding inverts are =- , , - , 

and the result is 

/4 3 12 Y 1 1 1 \' /3 12 21 X 1 1 IV 

\x, z\ + x' 



9' S' 25 " Xx-a' x-b' x-cJ \8' 25' 50 ' " Xz - 1 ' z-x ' z}' 

40. It is hardly necessary to remark that an expression 

1 1 1 



in fact denotes 



I,, b^, c, ■■y^^_^^> ^_j^> ^_^ 

a, t), -a,-b, + c, 

{z-a,r^ {z-h,f^ {z-a^{z-\) 



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

transformation upon the z make them to be , j- , with the (a, b, c) 

^ z — a^ z — b^ z — c^ 

arbitrary; or we can give to the a^, 5,, 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) 

11 1 



z ' z — x' z — 1 
to effect it. 



but the numerical work would be troublesome, and it is not worth while 



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

f{x-a) : g{x-b) : h{x-c) = P : Q : R 
of the equation 

{,r, .] + x^ (a, b, c .•.][^ , -i^ ,_!__) = (a, b, c. :\^^ , ^^ , ^^ , 

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 {x, z]. 

42. Recuniug to the results from the Arabic lines of the PQE-Tahlc, but for 
convenience writing s instead of s, we have 

f[x-a) : g{x-b) : h{x-c) = P : Q : R 

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

, , fdx\'f , Y 1 1 1 



dsj \' ' -'Xx-a' x-b' x-c. 



24 Prof. CAYLEY, ON THE SCUWAEZIAN DERIVATIVE 

But we have 



{^•^'=-(rf^) {^'^J' 



and the foregoing is therefore a solution of 



a differential equation of the third order; this is the Differential Equation {s, x}. 

43. From the Roman lines if we assume 

f{x-a) : g{x-h) : h{x-c) = ^ : © : H 

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

a. 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 
<if them. (See as to this the foot-note referring to the Annex to the PQi2-Table.) The 
last equation then becomes 

or since [x, ^] + {-,-) {s, 0(;] = {s, z], this is 

{.,.] = (a., b„c, .-.^^4^^, ^.' ^c)' 

the corresponding relation between s, z being of course obtained by the elimination of a: 
from the two sets of equations 

f{x-a) : g(x-l) : h{x-c) = F : Q : R, and/(x-a) : g{x-b) :h{x-c) = '^ : © : H ; 

viz. the required relation is 

P : Q : R = l& : © : m 

(where P, Q, R are functions of s ; "^ : d^ : 31 functions of z ; and in virtue of 

P + Q+R=0, ^H-(a-l-lX = 

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

^ : © : U = P: Q : ill 

as a solution of 



{s, 0.} =(a., b., c. .•.][^^, ^_, ^J. 



AND THE POLYHEDRAL FUNCTIONS. 



25 



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


Fxmctions of s. 






1 


fix -a) : ffi-v-b) : k(.v-c) 


= P : Q : R 


(1) 


Polygon 


I 


») 


= 


» 


(2) 


Double Pyramid 


II 


» 


= 




(3) 


Tetrahedron 


III 


4x : -{x + iy- : (x-iy 


— 




(3) 


>j 


IV 


f{x-a) : c,{x-b) : h{x-c) 


^ 




(4) 


Cube and Octahedron 


V 


{x-\f : -{x + \f: Ax 


= 




(4) 


» 


VI 


f{x-a) : g{x~h) : h{x~c) 


^ 




(5) 


Dodecahedron and Icosahedron 


VII 


4x -.-{x + iy- : (x-lf 


= 




(5) 


» 


VIII 


{x-lf -.-{x+iy : Ax 


= 




(5) 


55 


IX 


P -.Q : R (IX) 


= 




(5) 


)> 


X 


(X) 


=:= 




(5) 


n 


XI 


(XI) 


= 




(5) 


» 


XII 


(XII) 


= 




(5) 


»j 


XIII 


(XIII) 


= 




(5) 


» 


XIV 


(XIV) 


= 




(5) 


» 


XV 


(XV) 


— 




(5) 


)> 



The values of the P, Q, R as functions of x, or of s, are taken out of the PQH- 
Table : only in the lines III, V, VII, VIII, where P, Q, R are given as 

=4^, -{z+ir, {z-\r, 

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 

{s, x] = (a,, b,, c, .-. 



.x — a^' x — b^' x — c^ 

as a particular case of the equation {s, x] = Rational function of x, = R (x) suppose, then 
we have in 1, I, II, IV, VI solutions of the form a; = Rational function of s. 

Vol. XIII. Part I. 4 



26 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

Consider in general a solution of this form, x = F{s) a rational function of s : s is 
then an irrational function of x, and if s,, s., are any two of its values {s,, a:} = E (x), 

{«,, x} = Rix); that is [s,, a') = {s,, w], and therefore (ante, No. 7) «» = — '-7-7- And then 

x=^F(s) = f(— — J, =F(s,): viz. i^ (s) is a rational function of s transformable into 
^ - \c«, + dJ 

itself l)v the transformation s into ^ : and it is moreover clear that between any two 

cs + a 

roots s whatever of the equation x^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 F{s) is a rational function of s transformable into itself by the 

several homographic transformations of a group of such transformations: viz. taking x to 

be a rational function of x, it is onli/ in the case x = F{s}, a function of the form in 

question, that {s, x] can be equal to a rational function of x. 

47. We may in any equation between x and s consider these as imaginary variables 
p + qi and u + vi respectively; considering then {p, q) and (u, v) as rectangular co- 
ordinates of points in different planes, we have a first plane the locus of the points x, 
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 x 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 + hi) dx (a and b functions of x 
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 X, 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 dx and d^x be consecutive elements of the path of 
the point x, and ds, d^s the corresponding consecutive elements of the path of the point 
s, then the ratio of the lengths of the elements dx, d^x 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 x is 
a curved line without abrupt change of direction, then at the corresponding point the 
path of s is a curved Une without abrupt change of direction. In what precedes we 
have the relation at ordinary points, but there may be critical corresponding points (x, s), 
the relation at a critical point between the corresponding elements dx, ds being of the 
form ds = (a -I- bi) {dx)'', (\ a positive integer or fraction) : here the angle between two 
elements ds is = X times tliat between the two elements dx ; 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 tt) then the angle between the two 
consecutive elements ds is = Xtt : viz. there is in the path of the point s an abrupt change 
of direction. 



AND THE POLYHEDRAL FUNCTIONS. 27 

48. I consider the foregoing equation {s, x} = R {x), where R {x) is a rational func- 
tion, and is now taken to be a real function of a; : we may assume s' = p'd'e' , where 
the accents denote differentiation in regard to x, and where p', 9, and therefore also 9', 
are real functions of x. We have 

s" p" 6" .., 
-r = ^, + ^, + id', 
spa 

and thence 

and thence 

[s, x] = [p, x] + [9, x] + 1&--P^,- i P^ . 

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

{p,x] + {9,x] + ie'^-^ = R{x); O^iPj. 

The last equation gives p"9' = 0, that is 0' = 0, which gives s' = 0, and may be disregarded ; 
or else p" = 0, therefore p', a real constant, = 7 suppose, and {p, x] —0: hence for the solu- 
tion of the equation {s, x] = R (x), we have s' = yO'e'^, 6 a real quantity determined by 
[9, x} + .^^'^ = R{x): and then integrating the equation for s' we have s=a + fii + ye^^, a, /S, 7 
real constants. 

49. The conclusion is that if {s, x] = B (x), a real function of x, and if x be real, that 

is if the point x move along a right line (say the x-line) then s = (x + ^i + ye^ {9, and 

the constants a, /3, 7, being real), that is the point s moves in a circle, coordinates of 
the centre a, /3, and radius =7. 



c 

-+- 



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

4—2 



•28 Prof. CAYLEY, ON THE SCHWAEZIAN DERIYATIYE 

since {s, x]=R (.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- = a, b, c 
shall correspond the values s = A, B, C respectively. 

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

51. If however the points a, b, c are critical points, such that the element ds at the 
coiTesponding points A, B, C are equal to multiples of (dx)'', (rf^)", (dx)" respectively, then 
to the flat angles tt at a, b, c correspond in the path of s the angles Xtt, fj-tr, vtt at the 
points A, B, C respectively : and (assuming that a, b, c are the only critical points on 
the X-line) the path of s is made up of the three circular arcs CA, AB, BC meeting 
at angles Xt, fi-rr, vtt respectively. The arcs are completely determined by these conditions; 
for supposing the arc BG to make with the chord BC, at the points B and C, the angles 

f, f, and similarly the arcs CA and AB to make with the con-esponding chords the angles 

g, g and h, h, then the conditions give Xir, fiTr, vtt = i A+g + h, iB + h+f, iG+f + g, 
where the angles referred to are those of the rectilinear triangle ABC: we have thus 
the values of/, g, h; and the arc BC is the arc on the chord BC meeting it at angles//: 
and the like as regards the arcs CA and AB respectively. 

.52. The foregoing equation 

{s,x]=(a,h,c.:l^, ^^, ^J, 

where a, b, c have the values ^ (1 — X^, 4 (1 — fj^^), ^ (1 — f^) {\ IJ-, v beiug real and positive), 
has a; = a, b, c for critical points of the kind in question : in fact, writing x — a= h, the 
equation is of the form 

[s, h] = ~^~i — -' -I- ^^+ ff, + aji + ■•• 

which is satisfied by 

d . ds \ +\ , , , , , „ 

Th^'^dh—ir^^^+^^^'+^'J'^- 

and we thence obtain an integral of the form 

s = kh'''{l + lcji + k.ji-+ ...), =kcf> for shortness. 

This is a particular integral, but we have from it the general integral 

_ a -I- ffkcf) 
y + Sk<f> ' 

' Since there is no critical point on the x-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 > 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 /i = 0, then ^ = SA, and we find 



Vh^^' ~(^+»(^+JJ' -^+°4^^+- 



viz. reducing -7 to its principal term h , and then writing ds, dx for s — A, and h(=x — a) 

resjjectively, we have ds = IC {dx) , or x = a is a critical point with the exponent X ; and 
similarly x=b and x=c are critical points with the exponents /j, and f respectively. 

53. Hence in the equation 

s, «] = ( a, b, c ' " 



X — b' X — c. 

as the point x, passing successively through a, b, c describes the a;-line, the point s passing 
successively through A, B, G describes the sides AB, BC, CA of the curvilinear triangle 
ABC. To points x indefinitely near the a;-line correspond points s indefinitely near the 
boundary AB, BC, CA of the triangle, viz. to points x 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 X on the same upper side of the a;-line, corresj)ond 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 
X on the upper side of the a;-line correspond points s, one of them within each of the 
triangles ABC, &c., and to a point x on the lower side of the a;-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 «-line correspond the two sets of triangles ABC, &c., 
and ABC, &c., respectively. 

55. In order that the relation s and x 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 Xtt, fji-rr, vir 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. 323, obtained the series .of values I to XV of X, /i, v, giving for 
a, b, c, =^(1 — A,^), ^(1— /li"), ^[\—v'), the series of values mentioned in the Annex of 
the PQiJ-Table : and thus showed a j^riori that the equation 

{s, x] = (a, b, c .-.f 



.X— a' x — b' X - 

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



30 Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 

50. As au instance take the double pyramid form : tlic integral equation is 
/(.r-a) : g{x-b) : h{x-c) = 4:s" : -{s"-!)' : {s'+lf. 



or sav 



(c-a)(x-b ) ^ _ is^-iy 
{a-b)(x-c) (s" + l)' 



or 



is"-!) 



if for greater simplicity vre assume a, b, c = 1, 0, oo , this is a; = , or say 



1 — / 

— Is' — 1) = Jx (s" + 1), that is, s" = 7= , a solution of the differential equation 

1 t'Jx 



In particular if n = 3, we have x = 



s'+l 



or s' = 7^ an integral of 

l±Jx 



IS, ^i-\^8' 9' 9 --j^^' a,_i' a;-ooJ ■ 



57. We have here the spherical surface divided by the equator and three meridians 
into twelve triangles, each with the angles ^tt, ^tt, ^tt: 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 G triangles. The figure of the 
jr-plane is by the a;-line divided into two half-planes, one shaded, the other unshaded ; 
and we bave on the line the point c at x , a at the origin, and b at the distance 
unity. 



AND THE POLYHEDRAL FUNCTIONS. "• 31 

58. Take x real, then if x 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 x is 
positive and greater than 1, i is real and negative, and we have the infinite half-lines 
at the inclinations 60°, 180°, 300°. If x is real and negative, then s^ is of the form 

, — r-., =cos^4-z sin ^; whence s is of the same form, or the locus of the point s is a 
1. -T in 

1- /c 

circle radius unity. Writing i = ^, and supposing that the point x moves along the 

1 -f- V^ 

«-line from h through a to c at — x , and then from c at + co to 6, the point s describes 

the sides BA, A C, CB of the shaded triangle marked K. 

59. Suppose that the point x is at k, in the shaded half-plane at an indefi- 
nitely small distance from a; say we have x = — 2ic^i, (k small), then taking for Jx 

1—K (1 — t) 
the value «(l-i) we have s'=r3: — ) i _ ■( > =l-2«:(l-z) nearly, and hence a value of 

s is = 1 — |«-t- f /ci, 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 a;-plane corresponding 
to this shaded triangle: to the same value x = — 2k^{ 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 

{.,^} = (a, b, c.-.([^, ^, -l-J. 

that is ^-(b-c)(c-a)(a-b) f a ^ b _c N 

X — a.x — b.x — c \b — c.x — a c — a.x — b a — b.x — cj 

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, /3, 7, 0^, 6^, 6^ such that a, b, c are = a + ^i + ye''', a + ^i + ye^i\ 
a + /3i -^ 76*'' (this is in fact finding a and the coordinates of the centre, and 7 the radius 
of the circle through the three points a, b, c) : we then have x = (x + ^t + <ye^, a varia- 
ble parameter, the equation which expresses that the point x is situate on the circle in 
question. 

We have a; - a = 7 (e«'' - e'*«0, = 76* '«+«»' {eit^-W^-e-^t*-*"''}; the second factor is 
i?,m\{d — 6^, = iP suppose, or the equation is x— a = iPy . e-'^^^^'-^ say 

x-a = iPy. expi i (£1 -f e,). 

Similarly x-b= iQy expi ^{6 + 6 J, and x — c = iPy . expi ^(d + 6^ ; where P, Q, R denote 
sin \{d — 6J, sin ^{0 - 0^), sin ^ (0 - 6^ respectively : in like manner b — c, c — a, a - b, 
= iFy expi i(0, + 61,), .iGy expi i{0, + 0,), iRy expi i{0,+ 0,}, where F, G, H denote 
sin i (0, - 0^), sin \ {0., - 0,), sin \ (0, - 0J respectively. 



32 Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 

We have 

b — c . c — a . a— b — FGH ■ , in , a , a na\ 

with the like values for r and i ■ Hence the right-hand side of the 

c—a.x—b a—b.x—c 

equation is 



= fqr[i^^QG^W'"'^^^''^^^- 



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

To) 

or substituting for x its value = a+ ^i + 76*', this becomes 

{s,x] = -ke-"-^^({s, 6]~h), 
7 

that is = - 4 ([s, ^1 - i) expi (- 2^). 

Assume s= L + Mi + Ne^^ L, M, and N constants ; then using the accent to denote 
diflferentiation in regard to 6, we find without difficulty [s, 6\ = {©, 6] + 1 0'", and the 
value of [s, x] becomes 

= -i({e, e\ + \&'-ij expi (-2^). 

Hence, substituting the values of the two sides of the equation, the imaginary factor 
expi (— ■ 19) divides out, and the equation becomes 

FGlll & b c \ 



{e,^} + i0'=-i = -^^^^+^ + ^j, 



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



Connection vnth the differential equation for the hypergeometric series. Art. Nos. G3 to C8. 

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



AND THE POLYHEDRAL FUNCTIONS. 33 

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

d^v ■n.dv ^ 

and assume 

y = wv. 

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 x. 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 maimer. 

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 tliat the two equations are 
consistent, then there is nothing to determine the value of z, which may be regarded 
a& an arbitrary function of a;; y and v are then fiinctions of cr, 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 x. In particular if P and 
Q denote the same functions of z that p and q are of x ; and if we assume z = x, 
P, Q will become =p, q respectively : the given equation in v will be 

d'v dv 

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

d^i/ dii 

dx) 
viz. writing A^ = j9^+ 2 -^ — 4^, then by what precedes, the equation for w will be 

[w, x\=- \X. 

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^ = wv^, and therefore — = — ; calling each of these equal quantities s we have s denot 

^i '■I 
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'' + 2-j —4:q, and similarly 

dP 
Z=P^+2^-4:Q, we have 
dz 



and since in general 

Vol. XIIT. Part I. 



{s,x}=-hX, \s,z]=-\Z, 



34 Prop. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

we obtain 

as the required equation for the determination of r as a function of x. The process does 
not give the vdue of w, but this can be found without difiSculty, viz. 

frdi-fpdx (Je; 

10'= Ce -r~. 

ax 

If s, X are regarded each of them as a function of the new independent variable 
Q, then the equation is 

6G. Jacobi's differential equation of the third order for the transformed modulus X, 
Fund. Nova, p. 78 is 

3 (i'V' - \^h"^ - 2k'\' (//V" - XT') + k'X" K^^y Z^-" - (J^x)' ^'i " ^• 

where the accents denote differentiations in regard to an independent variable 9 : viz. 
dividing by 2k'\'^ this becomes 

which is thus a particular case of Rummer's equation, k, X con-esponding to x, z 
respectively, and the values of X, Z being 



^-(S)' ^-(^ 



G7. In the case of the hypergeometric series, the two differential equations of the 
second order are 

(Ty ^ 7 - (g + /3 + 1) a ; dy a^ij ^ ^ 
da? x.\—x dx X . 1 —X ' 

^ y-{a+l3'+l)s dv a/Sv _^ 
dz' z .1 — 3 dz 2 .1 — z 

Hence 7('^ + (1 -^)) - (^ + ^ + 1) ^^ 7^ 7 -«-/3-l -«/? ^ 

X.l—X X 1 — X ^ X.\ —X 

and hence 

.^.c,dp_,„_'/-2-^ , (7-a-/3-l)' + 2(7-a-/3-l) 4a/3 + 27 (7-a- ^- 1) 
^'^"dx ^~ x" + (1 - xy ^ x.l-x 

viz. writing 

X='=(l-7y^ a=.i(l-X'), 

M^ = (a-/9r, b = i(l-A 

i'' = (7-a-/3A c = i,{l-v"-). 



AND THE POLYHEDRAL FUNCTIONS. 35 

and putting in the formula x—1, =-{l—x), wc have 

^yP^ da; V~ x' ^{x-lf^ x.x-1 

a c — a+b— c 

^x''^{x-iy'^ x.x-1 ' 

= (a, b, c .'.O-, , ^) , 

\ AX a; — 00 X — 1/ 

with a Uke formula for i(p=' + 2^- 4q). We then have 

y — wv, 

vr = Cx-y (1 - x)v-<'-3-i 2v' (1 _ 2;)-v+a +3'+i '^ , 
and the differential equation of the third order for the determination of z is 

{., =rj + (a, b, c, .-.g, ^, _^J(|y_(a, b, c .".g , ^, ^) =0, 

where a,, b,, c^ are the same functions of a', /3', 7' which a, b, c are of a, /3, 7. 
This is in effect Kummer's equation for the transformation of the hypergeometric series. 

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

[s, «} = ( a, b, c 



,X X— QC X- 



PAET II. THE POLYHEDRAL FUNCTIONS. 

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

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

jl. Polygon or 

12. 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. 

5—2 



36 Prof. 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 x and y 
be drawn in the plane of the equator in longitudes 0" and 90" respectively, and the 
axis of s upwards througli the North Pole : the position of a point on the sphere is 
determined by means of its N. P. D. and longitude /: moreover we take X, Y, Z 
for the coordinates of the point on the surface, and x, y for those of its projection ; 

and we then have 

A", Y, Z=s\n dcosf, sin sin/, cos^; 

Y Y 

a;= j-^=tau^^cos/, y = ^-^ = tau ^ ^ sm/, 

and conversely, 

X, Y, Z=2x, -ly, \-x'-y\ ^{\+x'-vf). 

We represent the point (A', Y, Z) on the spherical surface by means of the magni- 
tude a; + ty (= tau^6(cos/+isin/)), or say by the linear factor, s—{x+iy): and similarly 
any system of points on the surface by means of the system of magnitudes x + iy, or 
say by the function IT [s — {x + iy)], 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 x, y as the coordinates of 
the new projection of the point, then x +{y' is a homographic function 

a{x + iy) + h ■~{c{x + iy) + d\ 

of x+iy: 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 mav be proper to remark that the figures in question are spherical figures 
havinc' summits which are points on the spherical surface, edges (or sides) which are 
arcs 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 x + iy are 

27r . . 27r (n - I) 27r , . . (n - 1) 27r 

I, cos \-isin — , ... cos ^^ \-i sm . 

n n n n 

The corresponding function of .9 is s" — I. 

The values of x->riy for the mid-points of the sides are 

•n- . . TT Stt . . Stt (1n-\)v . . (2n-l)7r 

cos - -)- 1 .sm - , cos + I sm - , . . . cos + 1 sm , 

n n 11 n n n 

and the corresponding function is s"+1. 



AND THE POLYHEDEAL FUNCTIONS. 



37 



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



as the function corresponding to the double pyramid. 

72. (3) 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 13.5", 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^ — 2i ^J^s^ + 1. 

In fact the equation s' — 2x^35^+ 1 = gives s'' = i(V3 + 2), and hence the values of 
s are the four values of x +iy shown in the annexed table for the values of X, Y, Z, 
and x+ iy for the summits of the tetrahedron, 

long. X Y Z 



45° 
135° 

225° - 
315° + 



V3 V3 



+ 



~V'3 

+ 



+ 



x + iy 



1+i 


V3-1 

-1+; 


V3 + 1 

-1-; 


V3-1 

i + ; 



V3+1" 



Corresponding to the centres of the faces, or summits of the opposite tetrahedron 
we have the function s* + 2i t^/S s' + 1. 

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



viz. the points in question are the North Pole s = 0, the South Pole s = oo , and the four 
points s= + l, s=±i 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, oo , +1, ±i, and the function is s(l J (s*- 1). 



The function for the centres of the faces, or summits of the cube is s''+14s^ + l. 

The function for the mid-points of the sides of the octahedron or of the cube is 

s"- 33s' -33s* -1-1. 

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 Prof. CATLET, OX THE SCHWAEZIAN DERIVATIVE 

The function for the summits of the Icosahedron is sfl j (4'"+ ll/ — 1). 

The function for the centres of the faces of the Icosahedron, or summits of the 
Dodecahedron is s* - 22Sa" + id-is" + 228/ - 1. 

The function for the mid-points of the sides of the Icosahedron or the Dodecahedron is 

5^- 5225" + lOOOos" + Os" - 10005s"' + o22s' + I. 
I give for the present these results without demonstration. 

75. Writing - for s so as to obtain homogeneous functions {*'^.v, y)" — it will be 

recollected that the x, y of these functions have nothing to do with the x, y of the 
foregoing values x-\-iy — the forms which have thus presented themselves may be denoted 
as follows : 

(3) /3 = (1, -2iV3, \1x\ yr, 

/i3 = (l, +2iV3, 1$^', 2/T. 
t3 = xy{x*-y*), 

(4) fi = ary(_x*-y% 

hi = (l, 14, lj,x\yr. 

<4 = (I, -33, -33, l$a;^ y*)', 

(5) fo = xy{l, 11, -IJA rff, 

ho = {l, -228, +494, +228, -l$x^, ^)\ 
^5=(l, -522, 10005, 0, -10005, 522, l\x\ y'Y, 

where observe that /4 is the same function as i3. In each set of functions /, /(, t, 
we have h and t covariants of/, viz. disregarding numerical factors, 

h is the Hessian, or derivative (/, /), and t is the derivative (/, /;). 

76. Since /4 is the same function as t3, we have of course /4, hi and ti them- 
selves covariants of /3 : but it is convenient to separate the two systems. 

77. It is to be observed that /3 is a quartic function having its quadrinvariant 
(7')=0; but independently of this, that is qua quartic function, it has only the covariants 
hS and <3 the (Hessian and the cubicovariant respectively), viz. every other covariant is 
a rational and integral function of f'3, h3 and t3. In particular hi and ti are rational 
and integral functions of /3, h3 and <3; but inasmuch as /3 and hS are not covariants 
■>f fi this is not a property of /i4 and ti considered as covariants of /4, and the rela- 
tion in question need not be attended to. 

78. It has just been stated that /3 qui quartic function has (in the sense explained) 
only the covariants hS and tS : fi qui special sextic function and fo qah sjpecial dode- 
cadic function have the like property, viz. fi has only the covariants hi and ti ; fb 
only the covariants ho and to. Hence /3, fi, fo are "Prime-forms" in the sense defined 



AND THE POLYHEDRAL FUNCTIONS. 3.9 

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 J7, V are functions {*~^x, y)" of the same order n, 
then using the accent to denote differentiation in regard to w, UV'—U'V and (U, V) 

differ only by a numerical factor : and further that writing as before s = - , 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 (fT, V) differing by a numerical 
factor only. We have in the PQR-tahle, lines 3, 4, 5, P, Q, R equal to given numerical 
multiples of M, P, /", the indices a, /3, <y being siich as to make these to be functions 
of the same degree: hence neglecting numerical multipliers FQ' — P'Q is equal to a function 
{¥, P), which is =M-Hy-^Ji, t): and the theorem that PQ'-P'Q, =QR'-Q'B, =RP'-R'P 
contains only factors of P, Q, R is in fact the theorem that {h, t), {h, /), and {t, f) 
axe each of them equal to a term or product of /, h, t: which is a result included in 
the theorem that / has only the covariants h and t. And by this last theorem we 
know already how from R, assumed to be known, we can derive P and Q: viz. J? is a 
power of/; and we thence have /» = (/,/) and t = (k,f), equations giving the functions 
h and t, upon which P and Q depend. 

Covariantive Formula}. Art. Nos. 81 to 84. 

81. The various covariantive formulse will be given with their proper numerical 
coefficients. 

Tetrahedron function. /, h, t stand for the before-mentioned values, /3, hS, tS 
(P, Q,R = K -m^s.f, -f% 

For /3. (a, b, c, d, e) = 1, 0, ^ , 0, 1. 



I (/> fT = - 96i V3 . h, I {h, hj- = dGi V3 ./, 1 {t, ty- = - '25fh. 

V3 



(/, /0= S2i^S.t, (/,/y=576/=0, (/, /0*=1152/=1152.-|*, 



{h,t)= 4./^ 
fh={i, 14, l$^^ 2/V(=/t). 



40 Prof. CAYLEY, ON THE SCHWAEZIA.N DERIVATIVE 

It is convenient to remark that f, /', A' being of the same order we have 

that is t\ 3 . 3/*A' (/, k) +f\ 3 . 2 Irt {/,, t) + h\ 2 . 3 f/' {t, f) = 0, 

an equation, which substituting for (/, A), (/(, t), [t, f) their vahies, reduces itself to the 
before-mentioned relation A'— /'— 12i v'3<* = 0; and we have thus a verification of the 
values of (/, h), {h, t) and {t, f). The like remark applies to the other two cases, which 
follow. 

82. Hexahedron function. /, A, t stand for the before-mentioned values /i, A4, i4 
(P. Q, R = h\ -e, -108/^). 

For /4. («, b, c, d. e. / ^r) = (0, i 0, 0, 0, -i 0). 

^{f,fr = -2oh, K/./r = 0, I (/,/)' = (720)'. I, 

(/ t) = -12h\ ^{t, ty =2\2\lV.fVi, 

(A, t)=-172Sf\ 
h'-f -108/^ = 0. 

83. Dodecahedron function. /, A, t stand for the before-mentioned values fo, ho, t5 
{P. Q, ie = A», -f, -1728/^). 

For/5, (a, 6, c, d, e, f, g, A, z, j, /.■, ?, ;h) = (0, Jj- 0, 0, 0, 0, J,, 0, 0, 0, 0, -^V, 0). 

H//y = o, H/./r =H024)M720r.iA/* 

i (//)'"= 0, !(//)" = i (924)' (720r . M *. 

\ (A, A)" = 173280/', 
\[t, if =9082800/ 'A, 





h(f,ff = - 


-121A, 
0, 




{/.h) = - 


-20<, 




if.t)=- 


- 30A', 




(A, <) = - 


- 86400/^ 




A»-<'-: 


1728/' = 0. 


84. We have 


t^ix'" 


Write 






then 







t = {x"'+y"')(l, 522, -10006, -522, l$x^ f)*. 

^={x' + f).il. 2, 6, -2, lj,x,yy, 

t = ^(l, -10, 455r,/). 

o **• ? (a;'4-^')(l, 2, 6, -2, l$a;, vV 

Or puttmg p = 4>, = ^^ - / X ' ^ ' 

^ V/ y:ry(a;'»-Hla;y-y'') 

that is, ^^p-Jf, then ^j» _ i o/ + 45^} = -^^ . (Klein.) 

* The numerical coefficients -/, and |5 are Klein's B factor X) the form is 
and ii : the latter of them the ordinary quadrinvariant of a A (f /)5 = 1 (921)- (720)-. - " \./ 

dodecadic function; the former is an invariant linear as _v u • v " i ^ -nr i. i 

which 13 linear as regards X. We have also 

regards the coefficients of/, and existing only for the special 

form/ in question: viz. writing for a moment * (/./)'- = 4 (924)= (720)*. |J X» : 

say A = \\\', B = -Jt\; or 8iB- = A. Of coiirse in the 
■' \ * T^ "• » *y /• jagg of a general dodecadic function /, we have (/, /)», an 

then (/, /)• contains the factor X', and (/ containing the irreducible covariant, not breaking up into factors. 



A.ND THE POLYHEDRAL FUNCTIONS. ^^ 

Investigation of the forms fo and ho. Art. Nos. 85 and 86. 

/5 — 1 

85. Writing for shortness * i = tau a = "^^ — , and g = cos 36° + i sin 36", then the values 

of x + iy corresponding to the summits of the Icosahedron are 

0, 
k, kg\ kg\ kf, kf, 

k-'g, k-y, k-y, k-y, h-y, 

oo; 

and the function fb is thus 

= s{\-^){i-k'){s-k-% 

where the product of the Last two factors is s'" + (i"° - i^) s^ - 1 . We have 

k-' = ^ (80 V-5 + 176), = h (5 ^Jb + 11), 
k" = Jj (80 VS - 176), = i (5 V5 - 11), 

and consequently k"^ — U=\\; or the function is s f 1 — — j (5'° + lOs^— 1). 

86. Similarly wTriting for shortness * Z = tan ^7, 1! — tan iy', where 

„ 5 + 2 V5 .J 10-2v/5 iM r cos 7 3 + Vo . 

cos 7 = — , . , sm 7 = zrz ; and tnereiore -. — = — ;; — 

'15 'Id sin7 4 

, , 5 — 2 v/5 .,,10 + 2^/5 cos 7' 3 - V5 

cos 7 = — — , sm''7 = 7^ ; -^ — ; = — 7 — ; 

' 1.3 lo sin 7 4 

and g = cos 36" + i sin 36° as before, then the values of a; + iy for the summits of the 
dodecahedron are 

lO' ¥' W' ¥. ¥. 

I'g. ly, ly, l'g\ lg\ 

i'-\ I'-y. ry. ry ry, ' ■ 

i-\ ry, ry i-y ry 

and the function ho is therefore 

= s'o + s' {r - n + 1 . s'° + s= (l" - I-') - 1. 

,,- , 7-^ 75 (1 + cos 7)'' — (1 — cos 7/ 2cos7,, , ^,-. J , 4 , 

We have Z ' - Z" = ^ ^^V — = ■ 5 (o + 10 cos'7 + cos*7) 

sm 7 sin 7 ^ 

^2^os7 384 + 64V5^128 c_os7 ^^ =114 + 50 V5. 

sin''7 4o 4o sin°7 

(viz. this last identity depends on f| (3 +\/5) (6 + ^5) = (114 + 50 V^) sin'7, that is 

160 (3 + V5) (6 + V5) = (114 + 50 -Jb) (120 - 40 ^b), 

or 2 (3 + V5) (6 + Vo) = (57 + 25 ^Jb) (3 - V5), 

or finally (7 + 3 V5) (6 + ^Jb) = 57 + 25 V5, which is right). 

* a is the a, 7 is the 7, and 7' the a-^ of the Table, No. 99. 

Vol. XIII. Part I. 6 



42 Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 

Similarly r' - T = ll-t - 50 ^5, 

and observing that the sum and product of ll-i + oOVo, 114 — 50 V'^J are =228 and 496 
respectively, the required function of s is 

(«'» - 1 )' - 228 (s" - s') + 49Gs"<', 
= s"" - 228s'° + 494s"' + 228/ + 1, 
which is the required value of ho. 

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 6 and azimuth /, and 




next in regard to B by its distance 6' and azimuth /', the azimuths from the gi-eat 
circle ABx which joins the two points A and B, then we have 

tani5(cos/+r sin/), and tan ^5' (cos/' +i sin/'), 

homographic functions one of the other: calling them s, s, and putting the distance 
AB=c, the relation between them in fact is 

, s — tan Ic 
1 -I- 5 tan ^c ' 

or, what is the same thing, tan {c (1 + ss) = s — s' ; 

or observing that ss' = tan J^ tan i^ (cos (/+/')+ i sin (/+/')], we have the two equa- 
tions 

tan ic {H- tan ^0 tan | ff cos (/+/)) = tan ^dcosf- tan \6' cos/', 

tan I c { tan ^ ^ tan 1 6' sin (/+/')} = tan l ^ sin/- tan ^ 9' sin/'. 



AND THE POLYHEDEAL FUNCTIONS. 43 

88. If we denote the angles of the spherical triangle hj C, A, B and the opposite 
sides by c (as before), a, h, then 6, 6' = b, a; f, f = A, tt—B, -whence 

s, s' = tan ^ b (cos A +i sin ^4), — tan ha (cos B — i sin B) : 
or we have between the sides a, b, c and angles A, B oi a, spherical triangle the 

relations 

tan Jc (1 — tan ^a tan |6 cos {A — B)\ = tan hb cos A + tan |a cos B, 

tan^c{ - tan Jatan-J Jsin (J. — £)} = tan it sin J. — tan |asin5 ; 

equations which may be verified by means of the ordinary formulte of Spherical 
Trigonometry. 

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

Taking (A', Y, Z), (A,, F,, Z,) for the coordinates, referred to two different sets of 
axes, of a point on the spherical surface : also x, y, x^, y^ for the coordinates of the 
corresponding stereographic projections, we have 

(A„ F„ ^J = (« , B , 7 )(A; Y,Z), 

a , P , 7 
X -.Y : Z : l = 2a; : 2y : X-a? -f : \-\-x' + y\ 

A, : r. : Z^ : 1 = 2r, : 2y^ : l-x^-y;: l+x^' + y^, 
and thence 

iTj : 2/, : 1 = 2aa; + 2/3j/ +7 (1 — a;" - y") 

: 2a'a; + 2^'y + 7' {\-x^-f) 

: \+x^ + 7f+ '2.'x"x + 2/3"y + 7" (1 - x" -/). 

90. Introducing i^, z for homogeneity, or writing - , =- and ^ , =-^ in place of x, y 
and a-j, y,, respectively, we have 

a-j = ^'xx + 2^y + 7 {z- - x" - ?/'), =(-7, -7, 7,/3,a, 0$a;, ?/, ^j", 

2/.= 22'a; +2,S>+7' («'-a:^-/). =(-7. -7. 7',^','^',0$ „ )^ 

s^=2^ + .r^ + /+2-/'.r+2y3"2/ + 7"(3^-*-'-r). = (1 -7", 1 - 7". 1 + 7", /3", «", 0$ „ f, 
and thence without difficulty 

z, = -. ^-^, {(1 + 7") ^ + (a" + «^") (•« - '■.!/)/ {(1 + 7") -' + ( a" - '■^") (-f + »^)}, 
i + 7 

^. + '>. = ^-^2 {(1 + 7") ^ + (a" + «vS-') (a; - iy) ]{{\- 7") ^ + (- a" + i-^") (x + iy)], 
7 + 7 

^, - iUr = ^-^ !(1 - 7") ^ - (a" + in (^ - »»} {(1 + 7") ^ + ( a" - i^") (■« + '»}, 
7-7 

viz. the form is z^ : x^ + iy^ : x^ — iy^ = MN : NL : LM (L, M, iV linear functions of 

z, X + iy, X — iy) : showing that the relation between two stereographic projections of the 

same sj)herical 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 Mobius. 

6—2 



a , P . 7 



44 Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 

91. The actual values are 

:r, + r>, ^ l+7" (I-7" )^-(a"-t-;5 - ')(a; + ij^) 

-, 7 + 7'» ' (1 + 7") - + (a" - i/3") (a; + iy) ' 

x,-iy, ^ 1+7 " {\-^")z-(o."+iff-){x-iy) 
z^ 7 - 7'* ' (1 + 7") ^ + (a" + */S") (•« - »» ' 

viz. attending ouly to the former of these, we have — — a horaographic fimction of 

, which is the before-mentioned theorem. 

92. Supposing that the transformation from {X, Y, Z) to (A",, 1',, Z^ is made by 
a rotation the coordinates of which are \, fi, v (that is, if f, g, h are the inclinations 
of the resultant axis to the axes of x, y, z respectively, and Q the angle of rotation, 
jrutting X, /i., r = tan i cosy, tan i^ cos ^r, tan i(^ cos A), then the coefficients of trans- 
formatiou are 

(2, ^, 7 ) = ( l+V-/i'-i'', 2(V + r) , 2(\i/-/i) )-(l + \^ + /i'+0, 
l{iLX-v) , l-X'-\-fi.''-v\ 2{,j.v + X) I 
2(i'\+/x) , 2{fiv-X) , l-\--,i'+v"-\ 

and substituting these values, the formulse become after an easy reduction 

g, + t'y, _ — (v + i) (x + iy) + (X + ifx) z 
2, (\ — ifi) (x + iy) + {v — i)z ' 

g.-^'y, ^ -{v-i){x- iy) + (X - ifj.) z 
s, (\ + t^) (a; — i?/) + (1/ + i) 2 ' 

attending to the former of these, and writing for greater simplicity -* =-i , 

z, 2 

= s,, s respectively, we have 

_ - (v + r) a + (X + 1» 
*'~ (X-i/x)s + {i^-i) ' 
or writing this 

then ^ : .B : C7 ; D = — v — i : X + z/t : X — I'/u : v — i. 

93. I call to mind that the condition in order that the hoinograpliic transformation 
», = {As + /J) -r (C's + D) may be periodic of the order n is 

(A + D)' - 4 (4i) - BCT) cos' ?^— = 0, 

in particular n = 2, it is A+D=0: n = 3, it is .4^ + ^D + Z)'+ 5C= : n = i, it is 
^4' + z>« + 2BC = : and n = 5, it is {A + Z?/ - ^ (3 ± V5) (.4Z) - BC) = 0. 

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

94. The formula' just obtained servo to connect the theory of the rotations of a 
polyhedron with that of the hrimographir transformations s into As + li ^ {Cs + D) : and. 



AND THE rOLYHEDRAL 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 CO homographic transformations s into As + B ^ (Cs + B). The 
group of GO 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 ahcd, the group of 24 as that of all the substitu- 
tions upon the four letters, and the group of CO as that of the positive substitutions 
upon five letters abcde. 

95. I call to mind that a group of functional symbols 1, a, /3,... can always be 
expressed in the equivalent form 1, ^a^"', ^/3^"V-- where ^ is any functional symbol what- 
ever: clearly, a, /3,... being homographic transformations, then, ^ being any homographic 
transformation whatever, the new symbols ^a^"', ^^~\.. 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 2 
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 l", Pole at a point ©; 2°, 
Pole at a point A ; 3°, Pole at a point B. 

The regular Poli/hedra. Art. Nos. 96 to 103. 

90. We require a theory of the regular Polyhedra considered as systems of ptoiuts 
on a sphere. I refer to my two papers (186G) 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 0, and 60 points $. 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 <1> 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 <t> 
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 intei'sects ^40 at right angles. 

97. The points <I> are comparatively unimportant, and it is proper in the first in- 
stance to attend only to the 12 points A, the 20 points B, and the 30 points 0: these 
form 6 pairs of opposite points A, 10 pairs of opposite points B, and 15 pairs of opposite 
points 0. Considering the diameters through each pair of opposite points 0, 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 



Prof. CAYLEY. ON THE SCHWARZIAN DERIVATIVE 



coui-se the other two axes: it contains also two axes each through a pair of opposite points 
^4. and two axes each through a pair of opposite points B. If instead of the plane we 
consider its intersection with the spliere, we have thus on the sphere 15 circles each 
containinff 4 points 0, 4 points ^4 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 .4, B, being =36°, GO" and 90° respectively. The whole number 15.14, = 210 
of the intei-sections 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 o times, and the 30 points 
each counting once; 210= 120 + (10 + 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 
bv a side at the centre is =72°, and the angle between two adjacent sides is =120°. 




We write Ae = <x, .60 = ^, AB = y, BJi^ = x, iB^B^B = 0, 05, = ^, ^BB^B = <}>. 

From the triangle A&B, the angles of wliich are 36°, 90°, 60° and the opposite sides 
13, y, a, -we find the values of a, /3, 7, and these are such that 0:4-/3 + 7=1. 

From the triangle B,BB, where the sides B.B, BB,, and the included angle arc 2/3, 
2/8, 120°, we have the opposite side x, and the other two angles each = 6. 

From the triangle B^BS, where the sides BJi, B@, and the included angle are 2^, 
/8, 120°, we find the opposite side ff, the angle BB^, — j>, and the angle BfiB, = 45". 

Hence each of the angles Bfi>B, BfiB, being =45°, the angle BJdB is =90°: in this 
triangle the hypothenuse B.^B^ is =x, and each of the other two sides is =cf: whence 
we have cosjc=cos'5f as is in fact the case, and moreover the values give a; +2// = 180°. 
Also each of the other angles is found to be = 00° ; that is we have ^ B^Bd^ = 60°, 
or the whole angle at B^ being = 120°, the sum of the remaining angles B^B^B^ and 
BBfi is = 60' : that is + <f> = 60°. 



AND THE POLYHEDRAL FUNCTIONS. 



47 



From the triangle ©yS, 0' wliere the two sides and the inckxded angle are ^,0,120", 
we find ©0' = 36". 

And from the triangle @B^®", where the two sides and the included angle are </, f/ 
and (120<'-2(^=)26l, we find @©"=G0». 

99. We thus arrive at the foUowins Table : 



Ae 
Be 

AB 

[BB) 

{Be) 

BBB 

BeB 



ee 



a 


31" 43' 





20" 55' 


y 


37° 22' 


X 


70» 32' 


9 


54 44 


e 


37» 46' 


«^ 


22 14 


2a 


63 26 


2/3 


41 50 


27 


74 44 


a-^ 






18" 




36" 



/5-V5 

V ~To~ 

^/5-l 
2^3 



-2^5 



/l0-2 

-V 15 
2^/2 
3 

J2 

,v/3 

V3_ 

2V2 

x/3 (v/5 - 1) 

4V2 

J_ 

x/5 

2 

3 

2(x/5 + l) 
3^/5 

/o - 2 v/5 
V 15 



^/5-l 
4 



^/ 



5-^5 



-2^5 



Where as above 

a+/3+7 = 90', 
x + lg = 180", 
^l + c/) =G0. 

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 of 
opposite points on the equator tlie 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. 8 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 E of the centre 
left-handedly) : and in the three figures the positions are as follows. 



43 



Prop. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 



101. Ficr. 1. Pole at 0. 




Longitudes. 



2x1 


a= 310 43' 


QO, 


ISO" 


2A 


90" -a= 58 17 


90, 


270 


4A 


90 


(0, 


180)^=0 = 31" 43' 


2A 


90» + a = 121 43 


90, 


270 


2A 


180»-a=148 17 


0, 


180 


2B 


/3= 20" 65' 


90", 


2700 


4B 


5-= 54 44 


45, 


135, 225", 3159 


2B 


90»-/3= 69 5 


0, 


180 


iB 


90 


(90, 


270)±/3=20''55' 


2B 


90'' + ^=110 55 


0, 


180 


4B 


180-5r=125 16 


45, 


135, 225, 315 


2B 


ISO" -/8= 159 5 


90, 


270 


le 


0» 


— 




46 


36 


(90°, 


270»)±a=31»43' 


49 


60 


(0, 


180)=fei3 = 20 55 


46 


72 


(90, 


270)±a = 31 43 


46 


90 


0, 


90, 180", 270" 


46 


108 


(90, 


270)±a=31 43 


46 


120 


(0, 


180)±/3=20 55 


46 


144 


(90, 


270)±a=31 43 


16 


180 


— 





AND THE POLYHEDRAL FUNCTIONS. 
102. Fig. 2, Pole at A. 



4S 




N.P.D.'s 



Longitudes. 



A 





- 


5A 


20 = 63° 2G' 


00 720 1440 2160 288" 


54 


180' -2a = 116 34 


30 108 180 252 324 


A 


180 


- 


bB 


y= 37 22 


36 108 180 252 324 


5B 


900-0 + 3= 79 12 


36 108 180 252 324 


55 


90 +a-3=100 48 


72 144 210 288 


5B 


ISO -y = 142 38 


72 144 216 288 


5e 


a= 31 43 


72 144 216 288 


5e 


900-0= 58 17 


36 108 180 252 324 


lOe 


90 


(36 108 180 252 324) ± IS" 


5e 


90 +a=121 43 


72 144 21G 288 


59 


180 -a=144 17 


36 108 180 252 324 



Vol. XIII. Part I. 



50 Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 

103. Fig. 3, Pole at B. 







N.P.D.'s 



Longitudes. 



3.4 


y= 3-«22' 


30n50»270» 


3^ 


9O»-a + 0= 79 12 


90 210 330 


34 


90 +a-^=100 48 


30 150 270 


ZA 


180 -y = 142 38 


90 210 330 


B 





- 


ZB 


2^= 41 50 


90 210 330 


m 


x= 70 32 


(30 150 270)±a = 37»46' 


6B 


180"- a;=109 28 


(90 210 330)±a = 37 46 


3B 


180 -2/3=138 10 


30 150 270 


B 


180 


~ 


3e 


^= 20 55 


90 210 330 


6e 


<7= 54 44 


(90 210 330)±^ = 220 14' 


3e 


90" -0= 69 5 


30 150 270 


66 


90 


GO 120 180»240»300<> 


3e 


90 +^=110 55 


90 210 330 


66 


180 -g=no 16 


(30 150 270) ±(^ = 220 14' 


36 


180 -j3=159 5 


30 150 270 



AND THE POLYHEDRAL FUNCTIONS. 



51 



Tlie groups of homographic transformations, resimied. Art. Nos. 104 to 117. 

104. The axes of rotation for the dodecahedron and the icosahedrou are 15 axes 
each through a pair of opposite points 0, 6 axes each through a pair of opposite points A, 
and 10 axes each through a pair of opposite points B; or say 15 0-axes, 10 .B-axes and 
6 J.-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 
0-axis 1 position, in respect of each J.-axis 4 positions, and in respect of each JS-axis 2 posi- 
tions; in all, including the original position, 1 + 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 

X, /A, i- = tan i ^ cos/, tan|^cos<7, tan ^^ cos A, 

and thence the values of 

A, B, C, D = — v — i, X+i/j,, \—ifi, v — i, 

viz.: in the case of a 0-axis ^ is = 180", (so that here tan^^=oo, or the values of 
A, B, C, D are = — p, X+t'f^, X — i/x, v, that is — cos A, cosy -fi cos ^, cosf—icosg, cosh); 
in the case of a 5-axis the values are S^ = 120°, 240", and therefore tan^^= + \/3; and in 
the case of an A-a.xis, they are ^=72", 144", 216", 288", and therefore 



tan h^=± 



7l0-i-2V5 V10-2V5 



V5-1 



V5-I-1 



105. The 0-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 abode, but No. 37 
of the group having been taken for this substitution abode, 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 0. 





{A.C 


+ B) H- 


{Cx 


+ D) 


1 


1 








1 


2 
3 

4 
5 
6 


-1 


2 
2 




1 
1 

-2 + ^5 + i{ 1-V5) 
-3 + ^5 + i{-l+^5) 




1 
-1 

-3-Fv'5 + i(-l + v'o) 
- 3 + ^5 + 1 ( 1-V5) 


1 



_ 2 
-2 



1 

ah.cd 
ac.bd 
ad. be 
bc.de 
ae.bc 



7—2 



52 



Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 



*9 
4 


2 


3-v'5 + t(-l + v'5) 


3-V5 + t( l-v/5) 


-2 




ad.ce 


8 


2 


3-V5 + t( l-x/5) 


3-V5 + i-(-l+^/5) 


-2 




ad.be 


9 


2 


-l-V5 + i'( 1-V5) 


-1-V5 + i(-1 + n/5) 


-2 




ae.cd 


10 


2 


-l-v/5 + t'(-l+V5) 


-l-V5 + i( l-v/5) 


-2 




ab.de 


11 


2 


l+x/5 + t(-l + ^/5) 


l + V5 + i( 1-V5) 


-2 




be.cd 


12 


2 


l + N'5 + t( 1-V5) 


l+V5 + i(-l+v/5) 


-2 




ab.ce 


13 


2 


-l-V5 + i(-3-V5) 


-1-^5 + 1 ( 3+V5) 


-2 




ac.be 


14 


2 


-l-V5 + i( 3 + ^'5) 


- 1-^5 + 1 (-3- V5) 


-2 




bd.cc 


15 


2 


l+N'5 + i( 3 + V5) 


l+V5 + j(-3-V5) 


-2 




ae.bd 


16 


2 


l+x/o + t(-3-v'5) 


l+V5+t( 3+V5) 


_ 2 




ac.de 


17 


_ i 




1 


1 




abc 


18 


- 1 




1 


i 




act 


19 




-I 


1 


i 




ode 


20 


-t 


-t 


1 


-1 




acd 


21 






1 


-1 




adb 


22 






1 


-i 




abd 


23 


-1 


-i 


1 


-i 




bed 


24 




-t 


1 






bdc 


25 


-l-V5 + !( 3 + v/5) 


2 


-2 


-l-V5 + i{ 


-3-v/5) 


ace 


26 


1+n/5 + i( 3 + V5) 


2 


-2 


H-V5 + i( 


-3-v/5) 


ace 


27 


l+s/5 + i(-3-V5) 


- 2 


-2 


H-v/5 + i( 


3 + v/5j 


bed 


28 


-l-V5 + t(-3-V5) 


2 


-2 


-l-V5 + j( 


3 + V5) 


bde 


29 


-3 + s/5 + i( 1-^5) 


2 


2 


3-v/5 + i( 


1-V5) 


bee 


30 


-3 + V5 + i(-H-V5) 


2 


2 


3-v/5 + ?( 


-l+x/5) 


bee 


31 


3-V5 + t(-l+N/5) 


2 


2 


-3 + v/5 + i( 


-1+V5) 


aed 


32 


3 - V5 + 1 ( 1 - V5) 


2 


2 


-3 + V5 + !:( 


1-V5) 


ade 


33 


2 


-l-v'5 + i(-l+V5) 


l+v/S + iC-l+VS) 


2 




cde 


34 


2 


l+s'5 + {( 1-V5) 


-l-sjb + i{ l-s/5) 


2 




ced 


35 


2 


-l-s/5 + i( 1-V5) 


l+v/5 + 2( 1-V5) 


2 




aeb 


36 


2 


H-x/5 + i(-l+>'5) 


-1-^/5 + 1 (-1+^5) 


2 




abe 


37 


-l-^'o + i{-3-V5) 


2 


2 


1+^5+1 ( 


-3-v/5) 


abede 


38 


-l-^5 + i( 1-V5) 


2 


2 


l+x/5 + i( 


1-^5) 


acebd 


39 


-l-V5 + t(-l+v'o) 


2 


2 


l+N/5 + t( 


-1+V5) 


adbec 


40 


-l-s'5 + i( 3 + ^5) 


2 


2 


l+v/5 + i( 


3 + V5) 


aedcb 


41 


l+V5+t( 3+V5) 


2 


2 


-l-v/y + t( 


3 + V5) 


adeeb 


42 


l+^'5^-^(-l+^/5) 


2 


2 


-l-N'5 + i( 


-1+V5) 


aebde 


43 


H-N/5+i( l-Vo) 


2 


2 


-l-v'.'' + '( 


1-n/5) 


aedbc 


44 


l+v/5 + t(-3-^/5) 


2 


2 


-l-v'5 + J 


-3-V5) 


abecd 


45 


-l-V5 + i(-l+x'5) 


2 


-2 


-1-V5 + J 


l-v/5) 


acbed 


46 


-3 + s/5+i(-l+V5) 


2 


-2 


-s+va+t 


1-n/5) 


abdce 


47 


3-v/5 + i(-I+V5) 


2 


-2 


3-^5 + !' 


l-v/5) 


aecdb 


48 


l+V5 + i(-l+V5) 


2 


_ 2 


1+V5 + 1 


. 1-V5) 


adebc 


49 


l+v'5 + i( I-^^5) 


2 


-2 


1 + v/5 + / 


'-l+v/5) 


acebd 


50 


3-j5 + il 1-V5) 


2 


-2 


3-v/5 + !' 


(-l+v/5) 


acdeb 


51 


-3 + V5 + i( l-^/5) 


2 


-2 


-3+V5 + i 


.-1+V5) 


abede 


52 


-l-V5 + i( 1-V5) 


2 


-2 


-l-V5 + i 


(-l+v/5) 


adbee 


53 


2 


-3 + V5 + i(-l+V5) 


3-v/5 + t(-l + v'5) 


2 




acbdc 


54 


2 


-l-V5 + i( 3 + V5) 


l+x'5 + j( 3 + ^5) 


2 




abced 


55 


2 


l+V5 + t(-3-V5) 


-l-V5 + t(-3-V5) 


2 




adecb 


56 


2 


3-V5 + t( l-^/5) 


-3 + V5 + !'( 1-V5) 


2 




aedbe 


57 


2 


-3 + V5 + ;( 1-^/5) 


3-v/5 + i-( 1-V5) 


2 




abdec 


58 


2 


-l-V5 + i(-3-v'5) 


H-V5 + t(-3-V5) 


2 




adebe 


59 


2 


l+^5 + i{ 3 + V5) 


-l-v'5 + t( 3 + V5) 


2 




aebcd 


60 


2 


3-s'5 + t(-H-V5) 


-3 + V5 + t(-l+V5) 


2 




acedb 



AND THE POLYHEDEAL FUNCTIONS. 



53 



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

Groups of 12 and 24. Pole at 0. 





{Ax 


+ B) - 


{C.V 


+ D) 




1 


1 








1 




2 


-1 








1 




3 














4 
5 







-1 







— i 








1 




6 


-1 








i 




7 


1 


-I 






i 




8 


— i 


-I 






-1 




9 


I 








-1 




10 


1 








-i 




11 


-1 


-I 






-i 




12 


i 


-i 




1 




13 


i 








1 




14 


— i 








1 




15 





i 


1 







16 
17 







-1 







1 


-1 




1 




18 


— i 


-1 




i 




19 


i 






i 




20 


1 






-1 




21 


-1 


-1 




-1 




22 


i 


-1 




~i 




23 


— i 






— i 




24 


-1 






1 





1 

ab.cd 
ac.bd 
ad. be 

abc 
ach 
ade 
acd 
adb 
abd 
bed 
bde 

adbc 
acbd 
cd 
ab 

acdb 

bd 

abed 

be 

abdc 

ac 

adcb 

ad 



108. The group of 60 was obtained in the ^-form by Gordan, in his paper. The 
passage from the ©-form to the J. -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 ^-form : we may write 

X,Y,Z^ bX, - a^, : F, : aX, + hZ, where a, b = /y/^-=/^ , \/^^ ' 

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, : F, : - a A', + cZ^^L :M: N ; or taking these to be A', : I", : Z^ = L^: M^ : N^ , 
we may write L^, M^, X, = bZ-f aX, M, -aL + bA'': these values are such that L^-\- M^+N'' 
= U + AP + N^, and hence \, fx,, v and X, , fx^, v^ being the rotations, we may write 



54 Prof. CAYLEY, ON THE SCH^yARZIAN DERIVATIVE 

i. J/, T = Sf\, V' ^" ; i,, J/., iV^=&\, V,.^",; "^^'licrc ^ has the same value ia each set 
of equations. From the equations 

^ : .B : C : D = — v — i:\ + ifi:\— ii^.v — i, 

we have 

i?+C:i?_(7:Z)-.'l:Z> + ^ = X:?>:// :-« 

and simiharly 

i?,+ (7,: 7?.- C,: X>.- J,: A + ^^ = A-- ^K- N,--%^. 

Hence we may write 

5. + C,= h{B+C) + ^{D-A), 

B,-C^= B-C, 
D^-A^ = -ti{B+C)-\-h{D-A), 
n^ + A^= D + A. 
Or say, 

^,= a{B+C)-h{D-A) + {D + A), 

i?,= b(i?+C') + a(Z)-^) + (5-(7), 
(?,= h{B+C) + a{D-A)-iB- C), 
D^ = -a{B + C)+h{D-A) + {D + A), 

which are the values for a transformation (J.j, 5,, C,, D,) in the ^-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 e an imaginary fifth root of unity : taking 
€ = cos 72" + i sin 72", we have 



J 3 Vo + 1 , . /o - Vo 

= ,6= —+t^—^; 



•where the upper signs belong to e, e' and the lower to e^ e^ It may be remarked that 

1 _ / o + V5 1 _ /S^yS b _ V5 + 1 a _ Vo- 1 
a~V 2 ' b~V 2 ' a~"2"' b 2 ' 

For instance, we have in the ©-group {A, B, C, Z>) = (- 1, 0, 0, 1); ab.ccl: and thence in 

the il-group A^, 5,, C\, i), = (- 2b, 2a, 2a, 2b); ab.cd: or say this is f - 1, ^, ^, 1 j , 

= (-1, e+e*. £+e*, 1); which is in the Table given as (-e', e" + e', e' + e', e') ; ab.cd. 

By effecting the passage to the -4-group in this manner we of course obtain the 
proper substitution corresponding to each transformation : but I found it easier starting 
from two transfonnations and the corresponding substitutions, to obtain thence by successive 
compositions the entire group. 



AND THE POLYHEDEAL FUNCTIONS. 



55 



110. 



Homograph ic 
G No. (As 



Transformations. 

+ B) 



The group of 60. 



Pole at A. 



1 

2 




1 
4 


1 






1 





-1 


1 





3 




13 





-£* 


1 





4 




9 





-63 


1 





5 




10 





-6- 


1 





6 




14 





- e 


1 





7 




6 


e + (- 


6* 


1 


-(e + 63) 


8 




5 


e + fS 


1 


** 


-(6 + 63) 


9 




16 


C + f3 


e 


63 


-(6 + 63) 


10 




3 


e + e3 


e- 


f2 


-(6+63) 


11 




15 


e + f3 


f3 


e 


-(6 + 63) 


12 




12 


-1 


6 + e3 


6^ + 6* 




13 




11 


-€ 


.3 + 1 


62 + 6* 




14 




7 


-.2 


1+^2 


6- + 6* 


j2 


15 




2 


-£' 


(2 + c* 


62 + 6* 


j3 


16 
17 




8 


-fl 


f^ + € 


6^ + 6* 


^4 




21 


£3+1 


€ 


1 


-(6 + 63) 


18 




35 


fS+l 


€= 


€* 


-(6 + 63) 


19 




30 


e3 + l 


63 


6^ 


-(6 + 63) 


20 




34 


c3 + l 


6* 


€2 


-(6 + 63) 


21 




19 


e^ + l 


1 


C 


-(6 + 63) 


22 




33 


6 + (* 


6^ 


1 


-(6 + 63) 


23 




20 


e + e4 


63 


e* 


-(6+63) 


24 




22 


€ + ,* 


6* 


63 


-(6 + 63) 


25 




36 


£ + 6* 


1 


6^ 


-(6 + 63) 


26 




29 


f + e^ 


6 


€ 


-(6 + 63) 


27 




31 


- f 


6^ + 6' 


62 + 6« 


1 


28 




17 


-c^ 


6< + 6 


62 + 6* 


6 


29 




27 


-€' 


€ + 6= 


62 + 6* 


6- 


30 




25 


-f* 


63+1 


6= + 6* 


j3 


31 




23 


-1 


1 + 6^ 


62+6* 


j4 


32 




24 


-6« 


l+e2 


62 + 6* 




33 




32 


-1 


6^ + 6' 


6^ + 6^ 




34 




18 


— e 


6^ + 6 


62 + 6* 


,:i 


35 




28 


-62 


6 + 63 


62 + 6* 


j3 


36 
37 




26 


-£' 


63+1 


62+6* 


j4 




44 


e 










38 




43 


e2 










39 




42 


.3 










40 




41 


€' 










41 




38 


e^ + e* 


1 


1 


-(6 + 63) 


42 




46 


e^ + e* 




t* 


-(6 + 63) 


43 




58 


f2 + f4 


j2 


6» 


-(6 + 63) 


44 




55 


e^ + f* 


j3 


^2 


-(6 + 63) 


45 




50 


c2 + €« 


j4 


e 


-(6 + 63) 


46 




51 


1+6^ 


j3 


1 


-(6 + 63) 


47 




39 


l+e^ 


j4 


6* 


-(6 + 63) 


48 




47 


l+f2 




63 


-(6 + 63) 


49 




59 


l+(- 




62 


-(6 + 63) 


50 




54 


l+f2 


j2 


6 


-(6 + 63) 



abecd 
aedbc 
achde 
adceb 
acebd 
ahdce 
adcbe 
adecb 
acdeb 
abedc 
adbee 
aecdb 
aebcd 
abced 



56 



Pkof. cayley, on the schwarzian derivative 



51 




56 


-.» 


52 




49 


-6' 


53 




37 


-<« 


54 




45 


-1 


55 




57 


-t 


56 




48 


-^3 


57 




60 


-(* 


58 




53 


-I 


59 




52 


- e 


60 




40 


— f '" 



«'+l 

e + €' 

(3 + 1 

1 + .2 

€= + (* 



c- + c' 
t» + e« 

«2 + €' 
€2 + fl 
."- + e* 

c- + f* 

€2 + . 4 

f= + €* 
f= + (* 
(= + €< 



acdhe 
aecbd 
abode 
acbed 
ahdee 
adehc 
acedb 
aebdc 
adbce 
aedcb 



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

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



{As 



+ B) 



■i-{Cs 



+ D) 



1 




1 








1 


1 


2 







-1 


1 





ad . be 


3 




€ + ^ 


£« 


€= 


-(. + .3) 


ac . bd 


4 




-^ 


^+€* 


£= + €« 


e3 


ab . cd 


5 




- (- 


f + f« 


£2 + €* 


€ 


abc 


6 




- f 


f + e* 


e^ + fl 


^^ 


acb 


7 




€ + €* 


«3 


€* 


-(. + .3) 


acd 


8 




€=+1 


1 


f 


-(e + e3) 


ode 


9 




e + e« 


fi 


(3 


-(^ + ^^) 


abd 


10 




f3 + l 


e 


1 


-(. + .3) 


adb 


11 




-1 


l + e2 


£=+£< 


6* 


bed 


12 




-f' 


l+£2 


€2 + f« 


1 


bdc 


13 




1 


1+26^ 


l + 2f 


-1 


ab 


14 




-€= + €' 


l + e + 3f* 


-l-3f- * 


,2_,3 


cd 


15 




e^-e* 


3 + e + c5 


- 1 - 3f - f3 


-f3+f* 


ac 


16 




-l+€= 


-l-c2 + 2e4 


l+c--2e3 


1-^2 


bd 


17 




2 + c' + 2f« 


-2-2€2-€3 


2f + €3 + 2c4 


2e + 2f- + «3 


ad 


18 




2 + 2e- + e3 


2 + c3 + 2f< 


-2e-2E2_t3 


2e + c3 + 2€* 


be 


19 




-2 + € + e3 


-. + c3 


-c + e' 


e + f3-2€* 


abed 


20 




1 


-1 


1 


1 


abdc 


21 




1 


1 


-1 


1 


acdb 


22 




l + f + Sf* 


€=-(3 


,2_,3 


l+3f+€« 


acbd 


23 




l+2e* 


-1 


-1 


-l-2€ 


adbe 


24 




3 + ( + (^ 


-6= + t* 


-€^ + €* 


l+3f + 63 


add) 



As an example of the calculation we have {A, B, C, D) = (0, i, — 1, 0) ; ab. Hence 
J„ £., C., A = (a(^' - 1). b (t- 1) +i+ 1, b (i -l)-(» + l), -a (/+!)), = (l,^, ^ , -l)- 

The second and third coefiBcients are — i a/ — ^ — , — ^' * V — s^^ > which in 

virtue of the values of e and e' are = 1 + 2e* and 1 + 2e respectively : or the result is as 
above (1,1 + 2e\ 1 + 2e, - 1). 



AND THE POLYHEDRAL FUNCTIONS. 57 

112. In like manner for the passage from the ©-form to the 5-form, if X, Y, Z 
be the coordinates of a point on the spherical surface in regard to the 0-axes, X^, Y„, Z,^ 
those of the same point in regard to the i?-axes, we may write 

X -.Y-.Z^X,: bi; + ^Z, : - ar; + b^, where a, b = ^^ , ^^ . 

Hence A': Y : Z=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 
Z, :bF, :a^, :-ai; + bZ^ = i:il/: JV, or calling these X^:Y^:Z^ = L^:M^:N^, we have 
L„,M^,N^ = L:hM-a.N -.SiM+hN: these values are such that L.^ + M^ + N^'=U+ ]\P+N\ 
or X, fi, V, \, fi.,, I'j being the rotations, we have L, M, iV = '&\, '^yu, ^v; L.,, M„, iV^ = ^\, ^fji.^^v^, 
where ^ has the same value in the two sets of equations. We have thus 

B + C : B-G : D -A : D + A = L : 2M : N : -i"^, 

B^ + a^: 5,- C; : D^-A.^ : D^+A.=L^ : 231, : N^: - 1% 
and hence 

B, + C\= B + G, 

B.^~G,^ h{B-G)~M{D-A), 

D„_- A, = -s.i{B -G) +h{D - A), 

D„ + A.^= D + A; 

A,= B.i{B-G)-\> {D-A) + {D + A\ 
£,= h{B-G)-Ai{D-A) + {B + G), 
C; = - b (5 - C) + ai {D-A) + {B + G), 
D„ = -m{B-C)+ h{D-A)Jr{D + A). 

113. As an example of the transformation take 

{A, B, C, D) = (2, - 3 + Vo + i (1 - Vo). - 3 + Vo + 1 (- 1 + V5), - 2) [he . rfe] : 



and thence 



then B-G, B+G, D - A, D + A = i{l- ^/r)), -3+^5, -2, 0; and thence 

"^^=2^3^ *^~^^^^ + 273^ 2 + 2V5), 

B, = 2 ^3 (- 40 + 273 (''■ (1 + V5) + (- 3 + V5)^ 

^^=2V3< *'■) +2^(2^(l-Vo) + (-3+V5) 

^■^=2-73^-'^ + '^') + 273(-2--^-')' 
viz. multiplying by 2^/3, these are 

8, ^•(-6 + 2V•5) + 2V3(-3 + ^/5), i (6 - 2^5) + 273(-3 + v/5), -8, 
that is, 8, (-6 + 2V5)(« + V3), (- G + 2 V5)(- i + ^3), -8, 

or since 2 + ^3 = — 2i"(y and — 2 + ^3 = 2iV, dividing by 4 these are 

2, { (3 - V5) w, i (- 3 + V-^) w', - 2, 
as in the table. 

Vol. XIII. Paet I. 



58 



Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 



114. Homographic Transformations. The group of 60. Pole at B. 

(o = h{-l+i V3). 



(As 



+ B) 



MCs 



+ I>) 



1 

2 




1 








1 


1 





1 


1 





ac. bcl 


3 







o> 


1 





ae.hd 


4 







«.3 


1 





hd.ce 


5 




2 


t{ 3-V5) 


i( -3 + V5) 


-2 


ab .ed 


6 




2 


i(-3-V5) 


i( 3 + Vo) 


-2 


ad. be 


7 




2 


t ( 3 - V5) o> 


z( -3 + ^/5)0.2 


-2 


beds 


8 




2 


I ( - 3 - V5) B 


i{ 3 + v'.')) 0)2 


-2 


be. cd 


9 




2 


t( 3-V5)<a^ 


i( -3 + V5)<o 


-2 


ad. be 


10 




2 


t(-3-V5)<i.2 


i( 3 + ^'.5)q, 


-2 


ab . de 


11 




2 


(-v/3-iV5)<» 


(-V3 + u/5)o.2 


-2 


ab . ce 


12 




2 


-V3-tV5 


-s/3 + j\/5 


-2 


ac. be 


13 




2 


(-V3-iV5)<»* 


( - v/3 + 1\/5) (» 


-2 


ae.bc 


14 




2 


V3 - 1\'5 


x/3 + i\'5 


-2 


ac . de 


15 




2 


(V3-!;\/5)a) 


(^'3 + 1\/5) o>2 


-2 


ad.ce 


16 




2 


(V3-2\/o)u.2 


(s/3 + t\/o)(i) 


_2 


ae.cd 


n 




a> 








1 


ace 


18 




0.2 








1 


aec 


19 




V3-i\/5 


2 


-2 


^/3 + j-V5 


bed 


20 




-^/3-tV5 


2 


-2 


-x/3 + i\/5 


hde 


21 




-V3-tV5 


20.2 


-2a. 


-^3 + iiJo 


bdc 


22 




V3-tV5 


2a.2 


-2a. 


V3 + 1 sjb 


bed 


23 




-V3-iV5 


2« 


-2a>2 


-V3 + ?V5 


aid 


24 




v/3-iV5 


2(0 


-2a>2 


^'3-i\/5 


adh 


25 




2«2 


-v/3-i\'5 


-v'3 + i\/5 


-2a) 


abc 


26 




2<a 


-V3-i\/5 


-V3 + t\/5 


-2o.2 


acb 


27 




2«2 


-v/3-2\/5 


(-V3 + 2v'5)a.2 


-2 


abe 


28 




2 


-^/3-j\'5 


( - V3 + 1\/5) 0)2 


-2o>2 


aeb 


29 




2a 


V3-iV5 


v/3 + iV5 


-2o.2 


acd 


30 




2«* 


«/3-iV5 


^/3 + tV5 


-2o> 


adc 


31 




2o.2 


V3-iV5 


( v'3 + !\'5)o.2 


-2 


ade 


32 




2 


V3-iV5 


( ^/3 + 1^5) 0.2 


-2o.2 


aed 


33 




2 


-s'3-t\/5 


(-V3 + i\/5)o> 


-2o, 


bee 


34 




2a> 


-V3-2\'5 


(-V3 + !.\/5)o) 


-2 


bee 


35 




2a> 


v/3 - i ^/5 


( x/3 + 1\/5) 0. 


-2 


cde 


36 
37 




2 


x/'3-i\/5 


( V3 + j\/5)o) 


-2o. 


eed 


2 


i( 3-Vo)'B^ 


f(-3 + V5) 


-2o.2 


adeeb 


38 




-V3-»Vo 


+ 2<a2 


-2 


(-V3+jV5)o.2 


acbde 


39 




^'3-i\'5 


2 


-2o) 


( V3 + i v'o) 0) 


aedbc 


40 




2 


t( 3-V5) 


t ( - 3 + s/5) 0) 


-2o, 


abecd 


41 




2 


»■( 3-Vo)<» 


i(-3 + V5) 


-2o. 


aedcb 


42 




-V3-tV5 


2o> 


-2 


(-x'3 + i\/5)o) 


adbec 


43 




v'3-j\'5 


2 


-2o)2 


( V3 + i.'V5)o)2 


acebd 


44 




2 


t( 3-V5) 


t(-3 + V5)o)2 


-2o>2 


abcde 


45 




2 


»■( 3-^5)0.2 


i(-3 + V5)»' 


-2o. 


xtdebc 


46 




V3-i\/5 


2a>2 


-2o.2 


( V3 + i\/5)tt> 


aecdh 


47 




-V3-iV5 


2a, 


-2o) 


(-V3 + i\/5)o.2 


abdce 


48 




2 


t ( 3 - V5) » 


t(-3 + V5)<» 


-2o.2 


ached 


49 




2 


t ( - 3 - VS) <» 


i( 3 + V5)o. 


-2o.2 


acdeb 



AND THE POLYHEDRAL FUNCTIONS. 



59 



50 




V3-i\/5 


2<a 


-2a 


(V3 + j\/5)a)2 


adbce 


51 




-V3-i\/5 


2a,2 


-2,^2 


{-\f3-i\fb)(o 


aechd 


52 




2 


i(-3-V5)<a2 


l( 3 + V5)a)2 


-2m 


abedc 


53 




2 


l(-3-V5)<- 


i( 3 + V5) 


-2(0 


aehcd 


54 




-v'3-!V5 


2a> 


-2 


(-VS+iV^)!* 


abdec 


55 




V3-2V5 


2 


-2co2 


( x/3 + tV5)o)2 


acedb 


56 




2 


t(-.3-V5) 


i( 3 + V5)<.>2 


-2u3 


adcbe 


57 




2 


i(-3-V5) 


I ( 3 + V5) 0) 


-2<a 


adecb 


58 




-V3-2V5 


2 


-2<o 


(-x/3 + iV5)<B 


aebdc 


59 




v'3-iV5 


2o)2 


-2 


( x/3 + iV5)«2 


acdbe 


60 




2 


2'(-3-V5)a)- 


2-( 3 + V5) 


-2a,2 





115. We hence derive 

nomographic Transformations. The groups of 12 and 24. Pole at B. 



(As 



+S) 



-i-iCs 



+ ^) 



1 




1 












1 




1 


2 




2 




i( 3-v/5) 




i( -3 + V5) 


-2 




ab.cd 


3 









1 




1 







ac. hd 


4 




2 




t (-3-^/5) 




t( 3 + V5) 


-2 




ad. be 


5 




2a)2 




-v/3-;V5 




-V3 + iV5 


-2o. 




aba 


6 




20) 




-s/S-is/5 




x/3 + iV5 


-2a)2 




acb 


7 




-V3-i\/5 




2<» 




-2o.- 


-V3 + tV5 




ahd 


8 




VS-iVa 




2<o 




-20.2 


llZ + Ob 




adb 


9 




2a> 




V'3-!'V5 




Vs+iVs 


-2o.2 




acd 


10 




2co2 




V3-iV5 




V3 + iV5 


-2o. 




adc 


11 




^/3-i\/5 




20.2 




-2o. 


V3 + iV5 




bed 


12 
13 




-V3-i\'5 




20.2 




-2o> 


-v'3 + iV5 




bdc 


2 




^3( l+x/5)+ (- 


-3-V5) 


v/3( 1+^/5) + 2 ( 3 + ^5) 


-2 




ab 


14 




2 




^/3(-l-V5)+ (- 


-3-V5) 


V3( -l-V5) + i( 3+V5) 


-2 




cd 


15 




V5 




— i 




I 


-^/5 




ac 


16 




1 




ijb 




-tV5 


-1 




hd 


17 




2 




^/3(-l+^/5) + ^( 


3-v/5) 


V3 (-1+^/5) + ? (-3 + ^/5) 


-2 




ad 


18 




2 




V3( 1-V5) + J( 


3-V5) 


v/3( l-V5) + i(-3 + V5) 


-2 




ho 


19 




1 




I 




i 


1 




abed 


20 




1 




— I 




— i 


1 




adch 


21 




V3( l-sf5) + i{ 


3 + V5) 


2 




-2 


v/3( l-v'5) + i(- 


-3+V5) 


abdc 


22 




V3( H-V5) + z(- 


-3+V5) 


2 




-2 


V3( l4-v'5) + i( 


3+^5) 


acbd 


23 




V3(-l + ^/5) + ^•( 


3-^/5) 


2 




-2 


V3(-l+V5) + i(- 


-3 + v/5) 


acdb 


24 




V3(-l-x/5) + t(- 


-3-V5) 


2 




-2 


V3(-l-v/5) + ^( 


3 + s/5) 


adbc 



IIG. I give also the group of 12, [ahce), slightly modifying the form: viz. I \vi-ite 
first V3 + iv''5 = 2\/2^, and therefore VS — «V5 = 2\/2 . t: then for x I write Xx, and divide 

the A and 2? by X : the A and B then contain - , and the C and D contain — , and 

assuming - = i, we have j = — i. For instance in the transformation corresponding to 

abc, the Ax + B and Cx + D, = ^ul'x — (V3 + «V5) and (— \/3 + 2 V5) a; - 2a) become fii'st 

8—2 



60 



Prof. CAYLEY, ON THE SCHWAEZIAN DERIVATIVE 



2(o'.r - 2 J-2k. and -2\f2yx-2o), and tlicu (omitting also the factor 2) w^i- - ^2 -' and 

— v'2 7 a- — o), viz. when = {, they arc co^x — i \/2 and x .i\J2 — w, that is the values of 
k A. 

A, B, C, D are ta', —in/2, i>J2, -to. The group is 

Group of 12. Pole at B. 



1 








1 


1 


<» 








1 


ace 


»« 








1 


aec 




-(•<oV2 


im v'2 


— 0)- 


ahc 




-t<aV2 


iW2 


— 0) 


acb 




-i-a>^2 


iV2 


— 0) 


abo 




-iV2 


10) V2 


— ft) 


aeb 




-tVv'2 


1^2 


-0)2 


bee 




-t\/2 


iaij2 


— 0) 


bee 




-i<i)x/2 


ia>- s'-2 


-1 


ab . ce 




-ta>\/2 


to) v/2 


-1 


ae. be 




-tV2 


iV2 


-1 


ac . be 



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

1 1 i(a; — 1) —i{x—\) {{x-\-l) —{{x+Vj x + i x — i —{x + -i) —{x — i) 

x,-,-x,-- • 



x' x+1 



a;+ 1 



x—l' x—\'x — i'x-^i' X- 2 ' x + i ' 



and if we proceed to form the product of the twelve factors s — x, s — , s + x, &c., we 



have fii-st the three products 



s'-x'.s'--^; s' + 

XT 

= s* + as^ + l; 
if for shortness, 

•,/3.r--(:.'+~) 



.S-- + IBs' + 1 ; s'+ys'+l; 



c' + Gx^ + l ^x'-Gx^+l 



{x'-iy • {cc'+lf ' 

The product of the three quartic functions is 

= (s* + If + (s* + If s^a + /3 + 7) + (*' + 1) ■^■' (/87 + 7^ + a/3) + / . a'^y ; 
and we have 



^+7 = 



32x' {x* + 1) 

{x*-ir ■ 



a + /3 + 7 



-(.r"-33/- 3 3s'' + l ) 
x'ix*-!)' 



^ -4(a;'-34a;*-l-l) ,„ , - 32a^ (*•' + Ij' „ , ^ ^ -36x'(x*-iy 



X'ix'-lf 



9^7 = 



4(x"-33^'-33s^4-l) 

x'{x'-\)' 



AND THE POLYHEDEAL FUNCTIONS. 61 

Hence the product is found to be 

= (s'' - 33s' - 33s' + l)-s' (s' - ly . .;: , f,t , 

^ ^ \ / a; (a; — 1) 

which is 

- 2^4 ^^^ ( ^" - '^ 3g" - 33s^ + 1 co'' - SSx' - mx' + 1 ) 

We thus verify that the twelve transformations x into x, into - &c., give each of them 

cc 

a transformation of the function 

a;"-33a;'-33a;^+l 

x^x'-iy 

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 — i : \ + {fi : \— ifi : v— i, 
and thence 

X:/x:i/:l= B + C : - i {B - C) : D - A :i{B + A), 

that is 

\, f^, v=-i{B+C), - (B-C), -i(B-A); -^{B + A). 

The equations of the axis thus are 

a; _ ly _ z 
'B+C~B'^~B^^' 

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

-{B + C)x + i{B-C)y+{A-B)z = 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 jjlanes, 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 Lx + My + Nz = 0, it is at once seen that the 
equation of the projecting cone (vertex at the South pole) is 

N{x' + 7;' + z"--l)-2{s+l) (Lx + My + Nz) = 0, 

and hence, writing z = 0, we find 

Nix' + f - 1) - 2 {Lx+My) = 

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. 



n = x' + y'- 


-I. 


z=0, (ab.cd) 


n = o, 


a; = 0, (((C . bd) 


X =0, 


y=0, {ad . he) 


y = o, 



62 Prof. CAYLEY, ON THE SCHWARZIAN DEEIVATIVE 

120. Taking the 0-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 



the equations are 



(3-V5).i-+( l-^o)y+2z=0, [ae.lc) fi - [( 3- ^5) «+ ( l-^5)i/] = 0. 

-l-V5)a' + (-l + V5)2/+23 = 0, (ab.ce) ^-[{-l - ^o).v + {- 1+^/5) y] = 0, 

(l + V'o)« + ( o + >J5)y + 22 = 0, (etc. be) n-[( l+>^5)x+{ 3 + V-5)2/] = 0; 

- 3 + Va) x + {—l+ v'5) 1/ + 2z = 0, (ad . be) and similarly for the other circles. 
(l + V'5)a;+( l-^5)y + 2z = 0, {ab.de) 

-1--v/5)x+(-3-a/5)?/ + 25=0, {ae.bd) 

- 3 + ^Jo) x + { 1 - Vo) y + 2z = Q, {ad . ce) 
{l + s/o)x+{-l + ^5)y + 22=0, {ae.cd) 

-l-^/o)x+{ 3 + V5)y + 2z = 0, {ac.de) 

{2-'^o)x+{-l + ^/5)y + 2z=0, {hc.de) 

- 1 - V5) « + ( 1 - \/5) y + 2z = 0, {he . cd) 
(1 + V5) X + (- 3 - V5) ?/ + 2^ = 0, {hd . 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 fl, =x' + y^ — l, =0 corresponding to the 
equator, and two of them are the right lines a; = and y=0. The equations of the 
remaining 12 circles may be written in the somewhat different form 

£l + {^o-l)[i/-h{s/o-l)x] = 0, 
fi-(V5-l)[3/-i(V5 + 3).r] = 0, 
n - (V5 + 3) [y + i (Vo - 1) x] = 0, 
fi-(V5-l)[y-i(V5-l)«;] = 0, 
n + (V5 - l)[y-i(V5 + 3) a;] = 0, 
n+(V5 + 3)[3^ + KV5-l)a]=0, 

n + (V5 - 1) 0/ + J (V-5 -!)«] = 0, 
n - (V5 -!)[!/+ h (V5 + 3) x] = 0, 
n-(V5 + 3)[y-i(V5-l)a;] = 0, 

n-{>J5-l)[y + h{^o-l)x] = 0, 
n + (V5 - 1 ) [y + KV5 + 3) a;] = 0, 
n + {>J5 + S)[y-l{>/5-l)x] = 0, 

and it hence appears that 4 and 4 circles have with fl = the common chords 
y + J(V5 — 1) a; = 0, ^— ^(\/5 — 1) a; = respectively: and that 2 and 2 circles have with 
n = the common chords y + i (V5 + 3) a; = 0, y — i ( V5 + 3) a; = respectively. 



AND THE POLYHEDRAL FUNCTIONS. 63 

122. The equations of the 12 circles are ia fact 

n±{^/o-l)y±h (Vo - 1) a] = 0, n±{^/o + s) [?/ + i ( Vo - 1) .'■] = o, 

n ± (V5-l)[(/±i(v'5 + 3)«] = 0: 

hence the radii are = \/5 — 1, 2 and >Jo + 1 respectively. 

The construction of the 12 circles is as follows: starting with a circle radius 1, 

Lay down the diameters y ± ^{\/o — 1) x = {A A in the figure), and through the 
extremities of each describe 2 pairs of circles with the radii i/o — l, V5 + 1 respectively. 

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

12.S. For the ^-form, the equations of the fifteen planes are at once found to be 

y =0, ad . he 

— X + (e + e") 3 = 0, ac.hd 
{e + e*) X + z = Q, ah. cd 

(e-e')x -i(e'' + e'')y =0, ache 

- (e- + e') X + i {^ -i)y+1{e + e') a = 0, ae . Ic 

- a; + i (e^ + e' - € - e") y + 2z = Q, ah . ce 



(e — e*) X — i (e + e*) y = 0, ab . de 

- (e + e') a: + i (e - e") y + 2 (e + e') s = 0, ae . hd 

+ (e^ + e' + 2) a; - i {e~ - e^) y + 22 = 0, ad . he 

(e — e^) X + i (e + e'^) y = 0, ae . cd 

-[e + e*)x - i (e - e') y + 2 (e + e') 3 = 0, ac.de 

(e- + i + 2)x +i (e' -e^)y+ 2z = 0, ad . ce 

{/-e^)x +i{e' + e')y =0, hd . ce 

-{e'+€')x -i{e'-e')y + 2{e+e')z = 0, hc.de 

-a;-i(e' + e*-e-e')y+ 2^=0, he . cd. 

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 fl = 0, are the diameters incliued to the axis of x 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 —x + {e + e*)z = 0, (€ + e*)x + z = 

respectively. The equations of the circles thus are (e + €*} Q. +2x = 0, fl — 2 (e + e*) x = 0, 

2 

or, recollecting that 2 (e + e^) = V5 — 1 and therefore j = \/o + 1, the equations are 

e + e 

x^ +y'' — {^/5 — 1) X — I -0, x' -^-y" + {\/5 + 1) X = 0; hence for the first circle the a^-co- 



64 pkof. cayley, on the schwarzian derivative 

oi\liuato of the centre is ^(\/5 — I) and radius is =^ »J{lO — 2 \/o) ; for the second circle 

the .r-coordinate of the centre is =^(\/5 + l), and radius = | v'(10 + 2\/5). We have thus 

the oonstniction of these two circles, and consequently the construction of all the 12 
circles. 

125. For the B-iorm after some easy reductions, and attending to the relation 
w — w'=i'\/3. the equations of the 15 planes become 



a: 




= 0, 


ac . hd 


(- 


■ 3 + V5) y + 


23 = 0, 


ah . cd 




(3 + V^)y + 


23 = 0, 


ad . be 


^/Sx + 


V5 2/ + 


23 = 0, 


ac . be 


-(l + V5)V3^+( 


3-V5)2/ + 


43 = 0, 


ab . ce 


(-l+Vo)V3a;4(- 


-3-V5)2/ + 


43 = 0, 


ae . be 


x + 


V3y 


= 0, 


ae . bd 


-'JSx + 


2/+(3+V5)3 = 0, 


ad . be 


V3a;- 


2/+(3- 


- a/5) 3 = 0, 


ab . de 


-^Sx + 


Vo y + 


23 = 0, 


ac. de 


(I - ^/o) V3a; + (- 


-3-V5)2/ + 


43=0, 


ad . ce 


(1 + V5) V3a' + 


(3-V5)2/ + 


43 = 0, 


ae . cd 


X — 


V3y 


= 0, 


bd . ce 


V3:c + 


t/+(3 + V5)3=0, 


be . de 


-^3^- 


2/ + (3- 


-V5)3 = 0, 


be . cd. 



12G. Of the 15 circles 3 are the lines x-y\/'i = 0, x = 0, a:+y^/S = 0, viz. these 
are lines at inclinations 30°, 90°, 150" to the axis of x. The equations of the remaining 
12 circles are 

ft + (3 - VS) y = 0, 

n- (.3 + V5)2/ = 0, 
(3 + v'5)ft-2(2/-a;V3) = 0, 
(3-v'5)n + 2(2/-a;V3)=0, 
(S + >Jo)n-2(y + x V3) = 0, 
(3-V5)n + 2(y + ,xV3) = 0, 

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



2n - ['- 1 + V5) \/3a; - (3 + V5) y] = 0, 

n-[ -^/3x+ >Joy] = 0, 

2n + [( l + V5)V3x-(3-V5)2/] = 0, 



2a + [(- I + V5) \/3x + (3 + V5) y] = 0, 

fl-[ v'3«+ V5y] = o, 

2n - [( I + V5) V3a; + (3 - V5) y] = 0. 



127. The first three of these have for common chords with 11 = 0, the diameters 
whose equations are 

(_ 1 + V5) VSx - (3 + V5) y = 0, - V3x + V5y = 0, (1 + \/5) V3*- - (3 - \/5) ,7=0: 



AND THE POLYHEDEAL FUNCTIONS. 



65 



viz. these equations are y = {—2-{- s/5)x^/S, y = -^ x, y = {2 ■{- ^5)xfJS. If, as in a fore- 

going table 6= 37° 46', sin = -— ^ , cos = -— — ; and therefore tan ^ = — ^ , then the inclinations of 

these diameters to the axis of x are respectively 60°—^, 6 and 120°—^, or say 30°— (0 — 30°), 
30° + (51 -30°) and 90" - (61 - 30°), where 61 - 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 ^-case in question is thus not so simple as in the ©- and -i4-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 

Rad. of circum. sphere, JB 

Rad. of inters, sphere, p 

Rad. of inscribed sph., r 

Rad. of circle circum. to 
face, R' 

Rad. of circle inscribed 

to face, / 
Incl. of adjacent faces 

Incl. of edge to adjacent 
face 



1 

'2V2 
1 



2V2V3 
1 

1 
* • 2 V3 
cos-i J = 70»28' 

cos-' -^ = 54" 46' 



«-W3 
1 

1 

90« 

90" 



1 



v/2^/3 
1 
^V3 
1 
*2V3 
008-1-^=109° 32' 

cos-i-4;i = 125<'44' 



V3(V5 + 1) 
4 
3 + ^/5 






'2 5 + 11 v'5 
40 
+ V5 



10 
2^5 



•J' 

. 1+V5 
' ~4~ 
. 3 + ^5 

1 

'V3 
1 
'2^3 



S + v'S 



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 . \ (\/5 + 1), 

altitude, k „ = s . ^V(5 + 2 \/5), 

segments of do. e „ =s .\'^{\Q — 2^Jb), 

f „ =s.W(10 + 2V5), 

where ^■ = e +/ = iJ' + ?-'. 

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 00, = p, =s.|(3 + V5), 
meeting at right angles in 0; then drawing the double ordinates BB, each = s, through ©^ 

Vol. XIII. Part I. 9 



66 



Prof. CAYLEY, ON THE SCHWARZIAN DERIVATIVE 



and ©3 respectively, and joiuiug their extremities with &„ aud ©^ : the sides @^B and ©^/? 
are then each = k, = s .^'^{5 + 2'J5); aud inserting upon them the points A, O from the 
tigure of the pentagon, we have several geometrical relations; viz. the line Aji. cuts the 




G- 



parallel sides B&^, B&^ at right angles, and when produced passes through the inter- 
section of jB0, and B&^ : we have OA, OB, 0& = r, E, p respectively : the four points <I> 
form a square, the side of which is g, =s A (\/o + 1). 



131. We find also 



, /2.5 + 11 Vo „ ,„ 



0. = ./-^8 



11 V5 



= r. V5, 



40 



MB 



/2 (5 + 2 



2V5) 



AND THE POLYHEDRAL FUNCTIONS. 



67 



It may be remarked that in the figure ^0^, B@^ are the projections of pentagonal 
faces, at right angles to the plane of the paper, having their centres at the points A, A, 
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 points 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- 
Bpective 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 . i (VS + 1) : 
the slope-breadths of the triangular faces are e, =s.^\/(10 — 2 Vo), and those of the 
quadrangular faces are /, =s. JV(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 = ^, other sides each =/; and the square of the altitude is thus =/^ — :J(/' 




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



68 



Prof. CAYLEY, ON THE SCIIWARZIAN DERIVATIVE, ic. 



133. For the icosahedron the scctioa through two opposite edges is a hexagon, as 
sliown in the figure : to construct it we take the four distances 00 each = p 
= s.J(l + V5) meeting at right angles; and then the distances A&.,. A&^ each =^s; 
and complete the hexagon. This gives the sides A@„ A@, each = s . i n/3, the altitude 

1 
of the triangular face, side = 5 ; and then taking 0,B one-third of this, = s ^-^, we 

have OB at right angles to ^0., and OA, OB, 00 = B, r, p respectively. 

-^1 (§>a 




I 
I / 



-.M 



-:4^: 



,'■-1/ 



®4 




Moreover, joining ^,0,, and OA^ we have these Unes cutting at right angles in a 
point M : we find 

^,03 = s.iV(5 + 2v/o), 
nffi. /5 + 2V5 



A 






+ \/5 
10 ' 

5+ \/5 _ J 



13i. It may be remarked that A^^, ^,0, are the projections of two pentagons in 
planes perpendicular to that of the paper, their centres being M, M : producing DM, 
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 .T + 5 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. 



tn 



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



[Read February 20, 18S2.] 

IXTROUUCTIOX. 

The object of this paper is, following Grassmaan, to establish a pure algebraical calculus, 
the laws of which will coincide with those of actual geometry. Ordinary algebra may 
be considered a calcukis 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 V — 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 waj's in which distance may be introduced, and consequently 
three different kinds of uniform geometry. These are, ordinary geometrj', 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 j^oints. 

Vol. XIII, Part II. 10 



70 Mk cox. on the application of quaternions and GRASSMANN'S 

The expressions 2.1, S.l, 4.1, &c., which have no distinct meaning except in combination 
with other symbols, will be called mnltiplcs of the point A. 

Every expression of the form /).l + (jIJ, where p and q are numbers, is defined to be 

some multiple of a point. 

s n 

pA +qB and vA+sB will be considered different points if - is not equal to '■ , hut if 

* = - then thev are multiples of the same point. 
r p 

Hence all the points included in the series pA + qB (neglecting tlie 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. 
P 

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 2}A+qB = rA+sB 

involves the equations p = r, q = s, for otherwise we should have B = ~ A, 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) + (vA +sB)={p + r)A + (q + 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 xA + ijB can be derived from any two points pA + qB, rA + sB on the 
same line by means of the equation 

xA +yB = x'( pA + qB) + i/'{rA + sB), 
which gives x = ^Ja;' 4- rt/', y — qx + 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 pA + qB. 

Then all the points included in tlie expression jjA + qB + i'C, leaving out of con- 
sideration the number of times each point is taken, form a doubly infinite number. They 
will be said to con.stitute a plane. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 71 

If X =lA + mB + nC, Y = I A + vi B + n C, 

be any point in the plane, tlien 

Z = 2)X +qY= {Ip + I'q) A + {mp + m'q) B + {up + n'q) 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 2:>lane. 

The condition for three points lying in a straight line may be written symmetrically 

\X+ fiY +vZ= 0, 

where \, n, v are any numbers. 

Suppose P=xA-\-yB-^ zC, then x, y, z or their ratios may be called the co-ordinates 
of the point P. If Q = x^A+ yfi + zfi, R = x^A + y^B + z^C be two other points; then 
applying the condition that the three points P, Q, R should lie on a straight line, and 
equating the coefficients of A, B, C to zero, we have 

\X + jJLX^ + I'X^ = 

Xz + /jL.Z^ + vz^ = 0. 
Therefore 

= 0. 

If X, y, z be considered variable this is the equation to a straight line and may be 
written in the form Ix + my + nz — where I, m, n are constant. 

Now suppose P = X,^ + a,B, Q = \A+ii^B, then the ratio ^ : '^- or '^ will be 

A., \ fi.-,X^ 

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*. 

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

P^'^pA + '^'aB, q = '^pA+'"'aB, 

and this gives for the new anharmonic ratio 

PM, . P^.=^A., as before. 

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-Euclidisclie Geo- als Strecken-Verhaltuisse definirt werden, da iliess die 
metrie," Mathematische AnnaUn, Vol. iv. p. 624, says Kenntniss eiuer Massbestimmung voraiissetzeu wiirde." 
" Die Doppel-Veihaltnisse diirfen dabei iiatuiiich niclit 

10— -2 



a^, 


y. 


z 


oo„ 


2/.. 


«1 


X,, 


y-v 


^2 



Mr fOX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 



Now let bo any new point, join OA, OP, OB, OQ, and let a new line cut these 
Hues iu -1', F, B', Q' ; put 

A' = pO + A, F = aO + B; 

then F (the intersection of OP and A'B') = X^A' + /j.,B' = {\p + fi^a) + P, 

Q = \A' + fi.Ji' = {Xj} + p..fj) + Q; 

therefore the anharnionic ratio of xV, F , B' , Q is -J-^- , the same as that of A, P, B, Q. 

This ratio ma}' 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. 

If F = .r^A + i/^B, Q = x„A + i/^B, R=x^A+y^B, S = x^A+y^B be four points on the 
line AB, and we wi^h to tind the anharmonic ratio of the points P, Q, B, S, we must 
solve the equations 

3-, = X,.i-, + fi.jr,, y, = \,y, + ix^y^, 

M, \ __ (a:,y, - x^y,) {x^i - x,y^ 
\ M2 {-^ai/s - ^3^/2) (•^43'i - -^■1^4) ' 



whence 



The quantities x^j^—x.2i^ which occur in this expression are invariants, for if 

P^xlA^ylB, (2 = ,r;.4+y;i?, 
where A! = p^ + qB, B' = rA +sB; 

then a-,y„ - xj/^ = ( jjs - qr) (jc^'y,^ - <;/,'). 

To find the anharmonic ratio of four straight lines l^x + m^y + ?!,z = 0, &c., inter- 
secting in a point, we may take the points n^B—m^C, n.^B—m^G where they cut the line 
BC. This gives for the ratio 

Wc shouW have also the expressions 

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

X ^ y ^ i_ 

X _ y _ z 



m^n^-m^n^ vj^-nj^ l^m^-l/n^' 



We may find now the anharmonic ratio of the four lines joining tlie point x, y, z 

to (x., y,, «,) (a:,, y,^, z^ {x^, y^, 2^ {x^, y,, zj ; ?,, n\, n^, /,, m,, n, are now the minors of 

X, y, z ' 

^v 2/1. «. 
x^, 2/j. z^ 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 



73 



Therefore 



and the required ratio is 



3:, y, z 

^2> 2/2. ^2 



X, 



«. y, z 


«, y, z 


«i. y.. «^i 


^3. ^3. ^3 


^2. 3/». ^2 


^4. y*' ^4 



«, 3^, 3 


«. y. 


«2. y^. ^2 


^4. y4. 


^3. y,. ^3 


^1. Vv 



If we put this equal to a constant h, the locus of the point x, y, 



becomes 



a;,, y,. ^1 



^3' y3' ^3 



= i 



«, y, « 


X, y, z 


a-2. y2. ^2 


^4. y4. « 


^3. y3- 23 


a^i. yi. 2 



^4. y4. ^ 

au 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 Jc, 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, &c. 

It may be shewn in the usual way (by taking a point X+\x, Y+\y, Z->rX3 on the 
line joining (A', T, Z) and {x, y, z)) that 

ax ay dz 

is the tangent at any point x, y, z of the curve F{x, y, z) =0. 

Similarly, as usual, are proved all descriptive or projective theorems concerning higher 
plane curves, such as Pllicker's equations, the Hessian, the nine points of inflexion of 
a cubic 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 = 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,, D^D^ intersect DC, CA, AB, OA, OB, OC in six points 

E,,^hB-cC, E^^ = cC-aA, E^^ = aA-hB, 

E,^ = + aA E,, = + hB E,^=0 + cC 

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



74 mk cox, on the application of quaternions and GRASSMANN'S 

They lie by threes on four new straight lines {E.,^E^^EJ, {E^^E^EJ, &c. The ratios 
E^BDfi, E^D^L\^1\, Sec. are harmonic. 

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

E,,, = 2aA + cC, E,,,= 2aA-l>B + cC, and L\^^ = '2aA +ShB + cC ; 

iJ£'„,£'„„£'„„ form an harmonic range. 

In this way an indetinite number of points can be found, including every point of 
the form xaA + i/bB +scC, where x, y, z are whole numbers, and an indefinite number 
of lines including every line the coefficients of whose equation are multiples of a, h, c. 

The quantities a*, y, s are called by Hamilton the anharmonic co-ordinates of a point. 
If P be the point xaA + i/hB + zcC, and Q^, Q^, Q, be the points where AP, EC; BP, CA; 
CP, AB intersect; then 

- is the anharmonic ratio of the points BB,CQ,, 

y 

- CB,AQ„ 

z 

I ^D,BQ,. 

X 

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 xA + yB + zC + tvD. 

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^,...A^^ unconnected. Every 
point x^A^->r x„A^...-^xJi^ will belong to a space of (?i — 1) dimensions, which may be 
called an n pofiit space. (See H. D'Ovidio, Mathematische Annalen, Vol. xi.) 

A point can be determined by the ratios of its n co-ordinates, that is by (w— 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 
\a necessary to fix the straight line. Therefore a straight line requires 2 (w — 1) - 2 = 2 (« — 2) 
quantities to fix it. 

A plane can be determined by tlirce points, but in determining each of these the 
two quantities necessary to fix its position in its plane will have been determined in 
cxces-s of what is needed; therefore a plane requires 3 (n — 1) -3. 2 = 3 (n — 3) quantities 
to fix it. 

In general an r point space requires r{n-\) — r{r—\)—r(n — r) quantities to 
determine it 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 75 

Or we may begin with the (« - 1) point space which is represented by the general 
equation 

This is determined by (n-l) quantities the ratios of a,, a^, ... a„. An (« - 2) point space 
is determined by two equations, 

b^x^ + h^x.,+...+b„x\ = 0, 

but as a-, may be eliminated from the first equation and x^ from the second, these may 
be put ill the form 

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 — r) point space. 

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

If, however, p + q — 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 — pi), 

the q point space in the p+q — r is q{p+ q — r — q), 

and p + q — r point space in the n 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 - li) + q{n - q) - p {p + q - r - pi) - q{p + q - r - q) - {p + q- r) {n + r - p - q) 

= r {n + r—p —q). 

The n points A^, A^...A^ will constitute what may be called an w-hedrou. 

It will contain n points, and n, («— 1) point spaces; 

n{n—\).. , n(n — \) , 

^ — - Imes, and ^ ^ , (re - 2) pomt spaces ; 

n(»-l)(ft-2) , n(«-l)(re-2) , „. ■ ^ 

^r^ planes, and — ^ ^^ , [n - 3) pomt spaces. 



7t; Mu I'OX, ON THE APrLICATION OF QUATERNIONS AND GRASSMANN'S 

It wo take another point = (7,^1, + «./i.,+ ... +«,/!„ we shall be ahle to construct a 
geometrical net. 

The intersections of OJ, and {A.J^...A„), &c. give n points B^=a^A^+ ... + a,A„. 
B^B„ intersects A^A.^ iu a^A^ — a.^A„= C\^, 
and OA^ intersects B^B,...B,^ in B.^ + B^+ ... B„ = {71 - 2) + a^A^ = C „. 

We then get - — , — new points which lie on straight lines in threes. 

In this way every point of the form a-,ajJ, + a-^a,^^ + ... + .r_0„^„ can be constructed, 
where x^a-^...x^ are whole numbers. 

The equation to any ('( — ]) space will be 

F{x^x,...a-J = 0, 

and the equation to the tangent space derived from the (n-1) points near ' j?,^'^ . . . .r„ 
will be 



I 



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 2)A + gB will be a 
mi.xture of the two, and rA + sB will be the same or a different mixture according as 



s 



S . Q 

- is equal or different from - , since the quantity is not considered. — j)A + B would in 

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 
])A+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 .il, Ji denote two conies in the same plane, and therefore intersecting 
iu four (ordinary) points. ])A + qB may represent any conic passing through those four 
l)oints, and all these conies are equivalent to what has been called a straight line. 

All the conies in a plane form a manifold of 5 dimensions, and if .?,.<! ,?, denote 

six of them, any other may be represented by a-^s, +a-./.^+ ... + .'r..S|j. 

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, p)-^ + 5-^ can be written p{A + B) +{q— p) B. The 
point p)A + qB then lies between the middle point of AB and B. 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 G = pA-^qB. AC is therefore less, equal or greater than CB, according 

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

1 p 

then, if we put —pA + qB=C, we shall have B = ~ G +*- A, so that B lies between A 

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

We make these assumptions, 1st, that if C lie between A and B, 
distance ^(7 + distance C5 = distance AB. 

2nd. If A, B, G be single points (not multiples of point) and A', B', C other 
single points, also A+B = \C, A' + B' = \C', then the distance between A and B is equal 
to the distance between A' and B'. 

The distance AB is thus some function of X, or we may say inversely X is some 
function of the (unknown) distance. Put AC= CB = x, and \ = (j}{x). Take a point A^ 
between C and A and a point A^ in CA produced, and let A^A= AA., = i/. 

Similarly take points B^ , B^ so that B^B = BB.^ = y ; then 

A^G=CB^ = x-y, 
A^G=GB, = x + y. 

Therefore A, + B^ = <f>{x-y) C, ^^ + B,= <j>{x + y) G, 

A, + B, + A, + B, = {4>{x-y) + ^x + y)]G 

But A^ + A^^<l>{jj)A, B^ + B^=<j}{y)B, d + B = <j>{x)C; 

.■.A, + A^ + B,+B,= cf>{x)cj>(y)C. 

Hence (}>{x + y) + (f>{x-y) = <f){x)4>{y). 

This is Poisson's functional equation. 
Vol. XIII. Part II. 11 



78 Mr cox, on the APPLICATION OF QUATERNIONS AND GRASSMANN'S 
Its solutious are 

(f) (.(■) = 2 sinh T , 

<f>{x) = 2, 

OS 

</)(.c) = 2sin^, 

where k is some constant. 

We may next determine in an equation ^)P + qQ = rR the connection between the 
([uantities p', q, r and the distances PR, RQ, PQ. 

Take a point iZ, between Q ^'^^ P> such that PR^ = RQ; also points iS and T in 
QP and PQ produced, such that SP = PP., P,Q = Qr. 



Ill II i 

S P E^ R Q T 

Also suppose PR = m9, RQ = nO, where m and ti are whole numbers. Divide 8P, 

PR^, R^R, RQ, QT into equal parts d. Consider for the moment the first case only, and 

6 . 6 . 

take each point of division sinh^_ times, excepting R^, which must be taken 2 sinh ^_ times. 

a 

The points of division from S to P., each taken sinh-^ times, are equivalent to a point 
at P taken sinh ^ f 1+ 2 cosh t + 2 cosh -r- + . . . + 2 cosh ^j = sinh ^^^ — ^ times. 

a 

The points of division from P, to T, taken sinh -^j times, are equivalent to a point 
at Q taken smh ^ ( 1 + 2 cosh -r + . . . + 2 cosh -77)-= ^mli — s — A; *^'^^^- 

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

pomt at R taken smh ^^. ( 1 + 2 cosh t + ... + 2 cosh — ^p ^j = smh ^ ^ times. 

We have then the equation 

smh. — ^ — 7. • -^ + ^^°^ ^o — T.-Q- ^^^^ oT. ■ ^^i + ^'"^ ~^ — 2is ' 

Let PP = a. RQ = I3, PQ = y, « + /3 = 7. 

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

8 OL y 

sinh J . P + sinh 7. ^ = sinh j R. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 79 

Or, if pP + qQ = rR, 

P '/ >■ 

sinh 7; sinh ^ sinh y 

iC Ic fC 

These equations may be replaced by 

q'' = ])' + q" + 2pq cosh j , 

p sinh -f =q sinh j . 

Ih rC 

In the other two cases the equations are respectively 

r' = p + q, 
2)7. = # j 
and . }■" = ;/ + g' + 2pq cos | , 

. a . /3 

J) sin ,- = q sm - . 

We may also determine the connection between the distances of points and the 
quantities p, g, r by a method almost identical with that used by Klein in his article 
" Ueber die sogenannte nicht Euclidische Geometrie." 

SujDpose, as before, A, B, and G are unit points, and A + B = XC; also suppose that 
by a transformation A and C become altered into G and B respectively, and any point 
P^=-x^A + yfi into P.^ = x^C + yji. 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 C or G and B. 

We have P^ = xjC+y^B = xJCl + y^{\C - A) = x^A + ?/,C, 

where a;, = - y„, 

y, = a'o + ^2/o- 

Repeating the transformation we get a point P,^ = x.^A + yJ3, 
where ^2 = ~yi. 

2/2 = *. + ^y, ; 

so that a-j, — \x^ + «„ = 0, 

y. - ^y, + 2/0 = 0- 

In this way we get a point P„ = x„A + y^fJ with the equations 

oc„ - \a^„_, + x,,^, = 0, 

2/,. - ^Z/„-i + 2/n-= = 0. 

11—2 



80 Mr cox, on the APPLICATION OF QUATERNIONS AND GRASSMANN'S 

If J, - be the roots of the equation z''—\2+l = 0, 

we shall have x„ = Az' + Bz-", 

ami y„=-^„,, = - (J.'"- + &--). 

Now if we put p {.r,A + rjfi) + q {x^A + y,fl) = r {x^A + y^C), 

p q r 



then 



or 



P _ g _ ^ 



^r^n+l ~ '^i.^'r+l '^O^r+1 ~ ^r^l ^O^.i+l *l^n 



so that ^^ — ^ = — - — 

We will suppose as before the distance between P^ and P. to be 9, that between 
P^ and P„ to be /3, and that between P^, P„ to be 7. 

Let the distance between ^C be 6, then a = r6, ^=(n — r)6, y = n6. 

There are three cases, according as the roots of 2^ — \s + 1 = are real, equal or 
imaginary, that is, according as X > = < 2. 

6 * -* 

In the first we may put X = 2 cosh -r , s = p'' , s~^ = p * where k is some constant ; then 

- -- a 

z' — z"^ = p'' — p *■ = 2 sinh -^^ . 

sinh -7 sinh ^ sinh j 
In the third we may put \ = 2 cos ^, z= p*', z'^ = p"*', 

r«i rfli 

z^ — z^ — p^ — p ^' = 2l sin y ; 

th t P _ 7 _ r 

. I3 . % .7* 
SID y sm J sin ^ 

Lastly, when z = z'^=\, 

z'-z-^ = {z- z-') (r'-' + 3-= + . . . + 2--'^ = r (3- - s"') 

a 



in the limit = - (z — s"') ; 
7 

and therefore £ = 2 = - . 

P a 7 



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 x^A + 7,(7 = ^ {x^A + %C), 

or zx, + 2/0 = 0, 

so that z' — \z+\=Q, 

the same equation for determining z as before. 

In the first case then there are two real points that remain unaltered by a translation. 

-- t 

These are J.-p * C and C — p~''A. 

If R be the latter point then // is in AC produced, and its position is given by 

sinh CH -» 





sinh AH 




H 


» 


en 

P* - 


en 

■p'" 


9 

'/I 


= 


CH-AB 

p * ; 


pAU _ 


p-AII- P 



or 

but this is the case when CH and AH are infinite, so that ZT is a point at infinity- 
This clearly ought to be the case, since the distance of all points at a finite distance 
is diminished. 



/ _29 _e n 

The number of times H is taken is A/l+p *— 2/} * cosh r = 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, q, r in the equation pP + <iQ — 1'R ''^taA 
the distances PR, RQ is then determined. Whether these relations be considered to 

give the distances in terms - , - , or, on the other hand, to £five - , - in terms of the 

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 ;jP, qQ represent portions of different fluids 
that mix without condensation and p, q be the volumes of the fluids, then r = p + q, 
the second of the three possible laws of combination. We can define then the distance 

PR to be —2 — multipHed by some constant, and the distance RQ to be — , — multi- 
p-\-q ^ ^ P + 'l 

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 



82 Mr cox. ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

takiiio three fluids auJ 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, acconliug 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, q, r would 
naturally be taken to mean the intensities of the ditferent colours. But the idea of 
distance cannot always be introduced. For we found that r, p, q are not independent 
but are connected by the relation 

r = f + q^ + ^Cpq, 

where C is some quantity independent of j) and q, 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" = <j> {p, q) where (f)(p,q) is a homogeneous function of degree 
II is J) and q, and involves besides only the distance between P and Q. 

Put F = x^A+ y^B, Q = x^A + y.JS, 

and let ■<^{x,y) be a function corresponding to (^ for A and B; 

then '>/r(a:..v,) = I, -v/r (a;^2/„) = 1 ; 

and if rR = pP +qQ = {px^ + qx.^ A + {py^ + qy„) B, 

then r" = -v/r (px^ + qx^, py^ + qy^ 

The ciu titles x,-^ ■\-y,~- must be independent of .r,?/,, x„v. except in so far as 
' an ^dx^ dy^ '• 

they involve one single quantity, the distance. This can only be the case if n is 2, for 

otherwise we should get a new relation between a-,y,, xjj^. 

Therefore »•" = // + <i + ^cpq. 

If the relation between r, p, q is not of this form, no meaning can be attached to 
the distance. The theorems then that have been proved before, about intersection of lines, 
anharmonics, ratios, conies, &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 = \A->ru..B, Q = \A + /J;B then ^^^ was defined to be the anhannonic ratios 
of APBQ. With the first form of measurement of distance this becomes 

. . AP . , QB 

smh — snih -y- 

sinh —r- sinh -^— 
k k 

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 t = 1. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 83 

The expression for any jDoint was P = ccA +yB + zC. 

AP intersects BC in a point D which is a multiple of yB + zC, 
therefore y sinh BD = z sinh BG\ 

PB intersects CA in E where 2 sinh C^ = a;sinh^^, 

CP intersects AB in F where xs^vth AF = y smh FB. 
•r ,. ,1 1 . sinh BD . sinh CE . sinh AF 

It follows that ■ x ,,n ■ ■ c . ■ 1 no = 1. 

sinh l)G . sinh EA . smh FB 

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 oi A + B + C. 

A straight line whose equation is la; + my + iiz = will cut BC in a point D fir 

, . , sinh BD sinh DC 

wluch mil = — nz, or = . 

Ill — n 

sinh CE sinh EA 



Similarly BC cuts CA in E where 

AB in F where 



n — I ' 

sinh AF sinh FB 



I —m 



sinh BD . sinh CE. smh A F 
iheretore ^.^^^ ^^, ^.^^^ ^^ ^.^^^ ^.^ - 1, 

and this is the necessary and sufficient condition that D, E, 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 ,, 

P' = aP+^. 

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 JlR COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

Now square the equatiou jiP-t qQ = rli. 

Wo have ;r {olP+ I3) + q\aQ + ^) + pq{P. Q + Q . P) = r' (aR + /3), 

or pq{P. Q + (2.P) = {r'-p'- q')^ + a{{-2p-jy) P+{rq-q')Q}. 

Now if the iiuiltiplicatiou be uniform tlie relation between P . Q, Q . P, P and Q 
nuist be independent of the quantities p and q. But this can only be tlie case if a = 0. 

Supposing then 6 the distance PQ we shall have 

and P. (2 + Q.P = 2^cosh6', 

or P.Q+Q.P = 2I3, 

or P.Q+Q.P=2^cose, 

according as the first, second, or third law of addition liolds. 

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^=(^ = ^, P .Q + Q.P= 2^ cosh are 
sufficient to ensure that iJ^ = /3 where R is any otlier point, but it has to be proved 
that they are sufficient to ensure that P" . Q' + Q' . P' = 2^ cosh <f), where P', Q' are two 
other points, the distance between which is <f). Let PP' = w, PQ=y, then <f> = y — (r. 

We are supposing, of course, that the first law of addition holds. 

„, p, _ sinh(^ — a;)P +sinh a; ^ ^, _ sinh (^ — y) P + sinh 2/Q 

men i- - ^^^ , (^ -. ^^^g- , 

and P' Q' = gj^jp^ (sinh {0 - x) sinh {6 -2/)P^ + sinh (6 - x) sinh ;/ . PQ 

+ sinh X sinh (6 — y) QP + sinh x sinh y Q'] ; 

hence P' Q' + Q'P' = ■ , a {sinh {6 — x) sinh {d- y) + sinh (6 - x) sinh y cosh 6 



^0 
" sinlV^ 6 '^"'"^^ ^^ ~ "^^ 5*"'"'* (^ - 2/) + sinh 2/ cosh 6"} 



+ sinh a; sinh {6 — y) cosh 6 + sinh a; sinli y] 

f sinh 2/ cosh 6] 

+ sinh {.s sinh y + sinh (0 — y) cosh ^}] 



2/3 
= ■ , ^ [sinh (0 — x) cosh ?/ + siidi x cosh (^ — ^)} 

= 2/3 (cosh a; cosh y — sinh x sinh 3/) = 2/3 cosh <^. 
The laws P'= f/ = B, PQ+ QP=20 cosh ^, 

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. 

1st. We can put /3 = 0, P''=Q'=0, PQ = -QP. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 85 

This applies to all three addition laws. With the first we shall have 

P' . Q' = ~^^\rh [sinh {d — x) sinh y — sinh x sinh [6 -y)] P . Q 

P-Q , ■ . 

= . , ^ (sinh y cosh x — sinh x cosh y). 



Therefore 



P. Q' P. Q 



sinh (f) sinh 6 ' 

or, for any two points on the same straight line the product P .Q 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 = 9 x some constant peculiar to the line. 

In the third P .Q = sin6 x some constant peculiar to tlie line. 

The second and third cases are what Grassmann calls the outer (aussere) multiplication 
of points and strokes (strecken). He has not considered the first case. 

2nd. We can put P'=Q-=.l, PQ = QP = cosh d; 

in the 2nd case P' = Q'=l, PQ = QP = 1; 

„ 3rd case P' = Q' = 1, PQ=QP = cos 6. 

This is in the third case what Grassmann called the inner (innere) multiplication of 
strokes. He dismisses the inner multiplication of points as useless. 

3id. We can combine these two forms of multiplication and obtain the following 
general form, 

P^ = Q'' = /3, PQ = ;8 cosh 0+7 sinh 61, QP= /3 cosh0 -y sinh 9. 

where /? and 7 are constant for the special line under consideratinn. 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 — 1st, the associative 
principle ; 2nd, that /f is a mere number. 

Then multiplying PQ and QP we have 

/3'=/3'cosh'6'-7sinh=6l; 

therefore /3^ = 7^. 

We will introduce a quantity i peculiar to the line under consideration such that 
t"=l. Then we may put 7 = /St, and therefore 

PQ = I3 (cosh e ± L sinh 0), QP=/3 (cosh + t sinh 0). 
Dividing the second equation by /3 or P^ we have 

QP"' = cosh + I sinh 0. 

Vol. XIII. Part II. 12 



86 Mk 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 6 + i sinh 0, 
and therefore PQ = ^^ (cosh - i sinh 0), QP = ^' (cosh + sinh ^). 

In the second case P'=Q'=^, PQ = /3 + 7^, Q7' = /3 - 7^ ; 

.-. 13' = I3°- - y'd\ 
Hence 7' = (), and we may put y=±i3i with the condition t" = 0. 
With the same convention as before, 

. PQ = /3'{l-i0), QP = l3'(l + c0), QP-'=l+i0. 
In the third case P^= Q- = ^, PQ = ^ + y sin 0, QP= ^cos0-y sin 0. 

^ = 1^ eos' - y'' sin' 0, 

We may put 7 = + /3t where 1" = — !, and with the same convention 

PQ = ^ (cose -t,sin0), QP= ^{cos0 + ism0}, and QP"' = cos ^ + t sm 6*. 

Comparing the three cases we have, 

in the first QP'' = cosh ^ + t sinh ^, t''=I, 

second QP-'=l + i.0, t' = 0, 

third QP-' = cos d + i,Hm0, i' = -l. 

In all three cases QP"' = e'* (i* = 1, 0,-1 respectively). 

This result might have been arrived at directly by assuming — 1st, that QP~' is the 
operation of transferring P to Q 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{0)^ EQ-'=f{<f>), 

where is the distance between P and Q, j> between Q and R. 

Then f{0)f{cl,) = EQ-\QP-'=PP-'=f{0 + ct>). 

This functional equation gives f{0) = e"*, where v is some constant connected with the 
special line. 

We need not in this method assume the addition formulae, 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=Pe"''*, Q = Re"', if = distance Pii, e"" + e'"' = some number. 

Since this is true for all values of it follows that i/^ 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 

It follows that either v^ is positive = ^ where h is a number ; 

v' is null = 0, 
J/" is negative = — 71. where k is a number. 

In all three cases we may put v = y, (i"=l, or —1). 
Then, in the first case, 

P + Q = {pk + p" *) i? = 2 cosh I . B. 

More generally, if pP+qQ = rE, and a be as before the distance from P to B, (3 the 
distance from E to Q, 7 the distance from P to Q, 

la. i|3 

fp '' + qp'' = r. 

n R 

This gives the equations p cosh t + <? cosh 1; = ^, 

J) sinh -, = q sinh -j . 
Squaring and adding these we have, since 

cosh Y cosh Y — sinh y sinh ~r = cosh -, , 

k K K K K 

r" = p' + q" + 2pq cosh -r , 

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-PI 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 6, and that between Q and E to be (/>, and put 

Q-P =/{&), E-Q = f(4>), 
then m+f(i>)=f{0 + 4>). 

Therefore f{d) = v6, where v is some constant depending on the particular straight line. 
Hence we have with the notation used a little while before, if pP+qQ = rE, 
p{E-va) + q(E + v^) = r ; 
.: p + q = r, 

1)2 = q^. 

12—2 



88 Mu COX, ON THE APPLICATION OF QUATERNIONS AND GRASSM ANN'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 
liivision ; 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^=^' = /8 = — 1, 
we have PQ = -cos d + i, sin 0. 

This is the Quaternion multiplication. It can be, as Grassmann has shewn in the 
Mathematische Annalen (Die Ort der Quaternionen in der Ausdehnunglelire), derived from 
the general form PQ = ^ oosd + j sin 6, by introducing the associative principle. 

The corresponding forms for the other two systems will be 

PQ =- cosh + I sinh 0, 
and PQ = -1 + i0. 

It remains only to prove the distributive principle. That is, to prove if pP+ qP= rR 
and be any point on the same line, 

pO.P + qO.Q = rO.R. 

Wo will take the first system and the most general form of multiplication ; that is, if 
distance OR = a, PR = 0, RQ = 4,, PQ = x, 

then we will put . P = /3 cosh (cr — ^) + 7 sinh (cr — 0), and so on. 

It has to be shewn then that 

p cosh {a- — 0) + q cosh (a- + <f>) = r cosh a, 
p sinh {(T— 0) + q sinh (a + (f>) = r sinh a. 

But these equations follow at once from the equations 

p cosh + q cosh (f> = 7; 
and p sinh = q sinh <f>. 



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 xA +yB + z 0, we shall have a number of 
different quantities i connected with the difftrent lines that can be drawn in the plane. 

Now all the lines {I, m, n) that can be drawn through a point x, y, z satisfy the equation 

Ix + my + «3 = 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 + t. sinh 0, we will put SPQ = - cosh, d, and call it the scalar 
part of PQ, and FPQ={ sinh ^, 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 t,, i^ be the quantities corresponding to the lines joining 

xA+yB + zC to x^A+y^B + zJJ and x^A+y.Ji + z^G respectively, 
then t, = some multiple of V [xA +yB +zG) (a;, J. + yji + z^ C) 

= some multiple of [yz^ - zy^ VBC + {zx^ - xz^) {VGA + {xy^-ys^) VAB 
= some multiple of IJBC + mJGA+nJAB, 
if l^m^n^ be the coefficients of the equation to the line. 

Similarly i^= s.omQ mMlii-^lQ oi l^VBG + m^VGA + nJ''AB. 

And if tj be another line passing through the point (we may say line instead of 
(quantity connected with line) l^m^n^ coefficients of its equation, 

^3 = some mult, of J^VBG+m^VCA +7i^VAB. 

Now (I^m^n^), [l^m^n^], (l^m^v,) are connected by the equation 



h' ^2' "2 



= 0, 



and therefore we may put I^=\l^ + fil.,, m^ = \m^ + fim^, n^ = Xn^ + fj.11,^. It follows that we 
may put rt3=pti + qi^, so that all the lines 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 0{x,y,z) to be within the triangle ABC, so that x, y, z are all 
positive. Then the equation Ix + my + ?!3 = requires that one of the quantities I, m, n 
should be negative if I, 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 = Q, x = Q; 7JZ + Ix = 0, y = ; 
Ix + ny = 0, z = 0. That is, in points nB — mC, IG — nA, mA — lB. 

If ?, be negative, the second and third points must lie between G and A and A and B 
respectively. So that every line drawn out from the point in either direction cuts one 
of the sides of the triangle. 

If then we start with the line OA, and draw lines from to successive points of 
AB, then to successive points of BG, then to successive points of CA ; the line OA will 
have returned to its old position. It follows that the equations connecting jy, q, r with the 
angles must contain only periodic functions of the angles, and therefore the third kind of 



90 Mu COX, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

addition laws must hold. We have then, if a be the angle between t, and t^, /8 between 
«, and tj, and 7 between t, and l^, while i'i.^ = pi^ + qi^, 

r,' = ^j' + 5' + 2pq cos 7, 

p sin a = sin yS. 

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 mifltiplication. Then, 
if p, (7 be two lines making an angle 0, 

pa = cr°(cos ^ — / sin 6), 

Avhere 7 is a quantity whose square is — 1. 

Suppose to be the point where the lines meet ; then we may identify the quantity 
/ with the point 0, and write pa = cr" (cos 9—0 sin 6). 

In the first kind of geometry a'=\, 

therefore pa = cos ^ — sin 6. 

Therefore Spa = cos 6, Vpa = — sin 6, TVpa = sin 6. 

In Grassmann's notation p\a = cos 6, pa = sin 6. 

In the second kind a" = 0, 

therefore pa = 0, Spa = 0, Vpa = 0. 

In the third kind o-'=-l, 

therefore pa = — cos ^ + sin 6, 

Spa = - cos 0, Vpa = sin 6, TVpa = sin 6. 

In Grassmann's notation pa = cos 6, pa = s'm 6. 

The inner multiphcation of Grassmann corresponds (neglecting sign) to the scalar 
multiplication of quaternions; and the outer multiplication to the vector multiplication of 
<luaternions. 

If p and a be hnes at right angles, we have in the first case 

per — — 0, ap = 0. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 91 

Therefore = - , or may be interpreted as the operation of turning a line through 

r 

a right angle in the positive direction. Multiplying by cr and by p we have also the 
equations 

Op = a; pO = — (T, 

Oa- = — p, crO = p. 

If A be another point on the line o- and p be the distance OA, then 

—^ = coshp + o-sinhj:), 

or A =0 cosh 2^ + 9 sinh p. 

Let 5 be a third point at a distance s from 0, then we have 

B — cosh s + p sinh s. 

This gives 

^ sinh s — B sinh p = sinh (s — 2') = sin 8, if B = s —p, 

an equation equivalent to those already found, and also 

A cosh s — B cosh p = — p sin S. 

The first equation assigns a real point for XA — p.B for all values of ^ from to 

e~'. The point gradually moves from A to infinity along the line BA produced. When 

- is greater than e the first equation ceases to give a value for XA — fj,B, 
\ 

sinhn e" — e^ .s , , ... 

smce . ■ ^ = < e when p and s are positive. 

sinh s e — e 

But the second equation gives a value for XA—fiB, since 

coshp _e'' + e~'' -5 
cosh s e' + e~' 

As then - increases from e"* to — r— k , XA — u,B is a line perpendicular to AB, and 
\ • cosh b 

this line moves from infinity in BA produced up to A. When the line passes A, 

p becomes negative, but the sign of ^ is unchanged. When - = 1, then 

A. X 

cosh H (^ — -B) = - p sin S, or A — B=-2p sinh ^ , 

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 ^ becomes greater than 1, we can consider the expression 

X 

IX.B - XA and the ratio - , and everything will occur as before, but in inverse order. 



92 Ml! COX. ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

The line will be at an infinite distance in AB produced when - = e~* or ^ = e'. 
Tiic expression will then represent a point which will move up to B and reach it when 
- = or ^ = x . 

fj, A. 

Therefore the expression XA — /j,B, as - varies from to x , 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 25roduced to B. 

Since siuh" jj + sinh's — 2 sinh p sinh s cosh B = sinh° B, 

2 cosh^ cosh s — cosh'p — cosh" s = sinh' S, 



the magnitude of \A — fiB when it represents a point is J\^ + A'" ~ 2X/i cosh B, and when 
it represents a straight line JiX/j, cosh S — X," — /i", so that it can never represent a real point 
and straight line at the same time. 

Returning to the equation 

A = cosh p + p sinh p, 

aud multiplying by a we have, if we call p' the perpendicular to OA at A, 

p' = p cosh p + sinh p 
— (cosh p + cr sinh j)) p, 



so that — = cosh p + a sinh p, as ought to be the case, since p'p"' represents the operation 



P 
of transfen-ing a line along a in the positive direction. 

Also Sp'p = Spp = cosh ]}, 

Vp'p = - Vpp' = a sinh p. 

Now if the lines p, p' were to intersect at an angle 6 we should have, by what has 
gone before, Spp = cos 6, but it is impossible that cos 6 = cosh p, since cos 6 is always less 
than 1 and cosh^ always greater than 1. It follows that tiuo lines at rigid angles to 
the same line can never intersect. 

If the equation A= cosh^ + p sinhp be multiplied by p, we find 

SAp = SpA = sinh p, 
VAp = - VpA = a- cosh /). 

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 (7 at a distance ^ along p, then 

C = (cosh 4> + p sinh (/>) 
= cosh (p — a sinh cf). 

If js' be the distance of G from p, then, since C is on the negative side of p, 

— sinh 2) = SCp' = S{0 cosh (f) — a sinh <^) (0 sinh^J + p sinh^)) 
= — cosh (f> sinh p. 

Therefore sinh^ 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 drai^Ti at right angles to the same line diverge from one another. 

In the third kind of geometry, if p and a- be at right angles, we shall have equations 

per = 0, crp = — 0, 

Op = cr, pO = — (7, 

crO = p, Oa = — p ; 

and also, if the letters have the same meaning as before, 

A sin s — .B sin ^j = sin B, 
A cos s — B cosp = — p sin S. 

But if (7 be a point at a distance ^ from in the positive direction 

^ sin (s — ^ tt) — £ sin (/J — ^ tt) = sin B. 

It follows that 0' = p. 

Now (7 may be any line perpendicular to p, therefore all these lines pass through a 
point 0' at a distance ^tt from p. 

This may be confirmed by the other equations analogous to those obtained above. 
Thus sin y = cos <^ sin ^j shews that p' is less than p and the perpendiculars converge, and 

also that p' = when 4' — ~a 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 6 their angle of intersection, then 

Spp' = cosp and Spp' = cos 0; 
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. 

Vol. XIII. Part II. 13 



A 



= 1 + 


crp 


A = 


--0 + 


pp. 


B- 


-A = 


-pS, 


B- 


-A = 


-pB. 



04 Mk cox, on the application OF QUATERNIONS AND GRASSMANN'S 

It" at points distant i ir, pcrpendiculai-s be raised to the line joiuiug tbem, these per- 
pendiculars will intersect at right angles, and we shall have a triangle all of whose sides 
iuid angles are equal to i tt. 

If i, j, k be the points or sides of this triangle they may be identified (except in 
sign) Anth 0, p, a, and it follows that 

jk = i, kj=^-i, ki=j, ik = -j, ij = k, ji=-k. 
These are the symbols of Quaternions. 

In the intermediate kind of geometry 0, p, a may be taken to satisfy the follow- 
ing laws 

po- = 0, o-p = 0, Op = a, pO = — a, 0<r=—p, aO = p. 

The equation 
gives 

We have also 
but if p be the perpendicular at A 

Therefore all lines drawn at right angles to the same line are to be considered 
identical. 

K be a point distant from 0, and on the line drawn through perpendicular 
to 0-, and D he a. point at the same distance on the line through A, then 

C=0-<t4>, 

D = A-(T<j}. 

Therefore D-G = A-0 = pp. 

But if p be the imaginary quantity perpendicular to the line CD, and p be the 
length CD, then D- G^p'p. 

Therefore p'p = 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 xA+yO represents x + y times a point P such that xAP = yPO, 
A — will represent a small point at an infinite distance. But D— C= A — 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 
0, A, C expressed literally in terms of 0, p, a. But by definition any point in the 
plane can be expressed lineally in terms of three points 0, A, C. Therefore any point 
in the plane can be expressed Kneally in terms of 0, p, <r. 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 p cos a + o- sin a, 

and P0~^ = cosh a + (p cos a + <t sin a) sinh a. 

Therefore P—0 cosh a — cr sinh a cos a + p sinh a sin a. 

If then xp + ya + zO be identified with any multiple rP of a point P we must 
have x = 7' sinh a sin a, >/ = — '>' sinh a cos a, z = r cosh a. 

These equations give 

r^ = z'' — x'' — v^ tan a = — , tanh a = . 

y z 

It is always possible then to find values of r, a, a in terms of z, x, y \i ^ — x- — y^ 
be positive. 

Next take a line drawn through P perpendicular to OP and let \ be the cor- 
responding imaginary quantity. The imaginary quantity for a line at perpendicular 

to OP is <j cos a — p sin a. 

Therefore \ = {cosh a+{p cos a -|- cr sin a) sinh a\ {a cos a — p sin a) 

= — sinh a + p cosh a cos a — p cosh a sin a. 

If then xp + ya + zO be identified with rX, 

x=-—r cosh a sin a, y = t cosh a cos a, z = — r sinh a. 

So that r^ —x^ + if — z^, tan a = — , tanh a = ■ 



It is always possible to find values of r, a, a in terms of x, y, z,\i 'J? -\-if — :^ be positive. 

It foUows then xp ■\-ya-\- zO is a point on a line according as s° > < a;'' + 2/^ 

If xp + ya + zO be a point then ^u + xp + ya + zO = iu + rP may be identified with 
some multiple of the ratio of two lines meeting at the point P. Let 6 be the angle 
at which they meet, s the multiple of their ratio. Then s (cos 6 + P &in6) = iu + rP, 

s COS 6 = tu, s sin ^ = r, s^ = w'^ + r^, tan 6= - . 
' w 

Substituting the value of r" 

s- = iif + z^ — x^ — y'. 

13—2 



96 Mr cox, on the APPLICATION OF QUATERNIONS AND GRASSMANN'S 

If xp + ya +zO be a line theu we may identify w + xp + i/a + zO 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 + xp + ya +zO = w + r\ = s (cosh + X sinh 0). 

Therefore w = s cosh 0, r = s sinh 0, 

T 

tanh 0= - , s'=iv^ — r-=w^ + z' — x'' — w'. 

The condition that this substitution may be possible 

is w^ > r' or tu' + 2^ — x' — y" > 0. 

In the second case we must put 

'w + rX = s (sinh ^ + \ cosh 0) 

tanh = - , s' = r" — w' = x^ + y'' — w^- s". 

This will be possible if r' > w^ or x^ ■{■ y'^ — w^ — z' > 0. 
Thus a meaning can always be found for ^u + xp -{■ ya + zO. 

If we form the product of two such expressions 

(tv + xp + ya + zO) (w' + x'p + y'a- + z'O) 
= low' + XX + yy — zz' 
+ (wa;' + w'x + yz' — y'z) p 
+ {'wy'+ w'y + zx — z'x) a- 
+ {wz + w'z + yx — y'x) 0, 

then since the product of the magnitudes of two ratios is the magnitude of their product, 
(w'' + ^ - af - y') {10" + z" - x" - y") 

= (vjw' + xx' + yy' — zz')" + (wz' + w'z + yx' — y'xf 
— {wx + w'x + yz — y'zY — (wy + w'y + zx' — z'xY, 
a formula analogous to Euler's for the product of two sums of four squares. 

Since the multiplication of the quantities 0, p, a- 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. 

Id the third kind of geometry every point can be expressed in terms of the three 
points i, j, k. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 97 

If we put rF=on+yj + zk and call a, j3, y the distances of P from the points 
i, j, k, then multiplying by i, and taking scalars 

r cos a = — rSPi = x, 

r*cos/3= y, 

r cos 7 = z. 

Also squaring the equation rP = ad + yj + zlc, 

r' = a? + y'' + z\ 
and therefore cos"" a + cos^ /3 + cos° 7 = 1- 

If w + xi + yj + zk be the ratio of two points at a distance d, we may put 
w+oci + yj + zk = 'w + rP = s (cos + Psm6), 
where P is the pole of the line on which the points are. 

Then s' = w' + r' = w^ +x' + f + z", 

T 

tan 6 = — . 

IV 

We arrive at Euler's formula in the way shewn above, as is pointed out in Hamilton's 
Elements of Qtmternions. 



Formulae in Coordinates. 

If OX, OF 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 = sinh OP cos POX, y = sinh OP sin POX, z = cosh OP. 
These quantities are not independent, but are connected by the equation 

If p, a be the imaginary quantities corresponding to the lines OX, OY, then, by the 
preceding section, 

P = zO ~ xa- -\r yp. 

Now if PL, PM be the perjiendiculars from P on OX, OY, 

sinh PL = SPp = y, sinh PM = - SPa = x. 

Since sinh PL = 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 

I = sinh OQ, m = cosh OQ cos QOX, n = cosh OQ sin QOX. 



98 Mk cox, on the application of quaternions and GRASSMANN'S 
They are connected by the relation 

If \ be the imaginary quantity corresponding to the line 

\z= — nO + la— yp. 

Supposing PR to be the perpendicular from P on the line \, 

sinh PE — — SPX = Iw-^ my — ns. 

If P be on the line, PB = and therefore its equation is 

Iv + my — nz = 0. 

The distance between two points P = zO — xa—yp, P" = z'0 — xa + y'p is given by 

cosh 9 = — SPP' = zz — XX —yy. 

The angle or shortest distance between two lines is given by 

cos Q — SXX' = ir + mm' — nn, 
or cosh 6 = 11' + 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 <^, cosh ^) (— sinh <p, o cosh <^) be the points, so that they are at equal dis- 
tances <^ on either side of 0, {x, y, z) the points whose locus is required ; then, if 
(?, m, n), (V, m, n) be the lines 

I m n 



y cosh ^ z sinh ^ — x cosh ^ y sinh <^ ' 

Zf t r 

m —n 



y cosh <ft — z sinh <^ — x cosh <^ y sinh ^ ' 

and, since W + mm! — nn = 0, 

if cosh' <f> + (x cosh <}> — z sinh (p) (x cosh (f) + z sinh </>) + y sinh'^ </> = 0, 
a^ cosh'^ + y^ cosh2<^ — z^ sinh'c/) = 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 + xi + yj, 
where, if PL be the perpendicular on OX, then OL = x, PL = y. 

It follows that L-0 = ad, P-M = yj. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 99 

P —0 

If r be the length of OP then represents a unit line perpendicular to PO; 

so that if OP makes an angle 6 with OX, then 

P-0 n ■ ■ n 

= I COS c^ + 7 sin ^. 

r •' 

Therefore x = r cos 6, y^i" sin 6, 

and hence x'+ tf = 1^, 

relations connecting the angles and sides of the right-angled triangle POL. 

If 6' be the angle OPL, then in the same way 

X = r sin 0', y = f cos 9', 

so that cos d = sin 6', sin 6 = cos ff. 

The angles 6, & 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 {x — x'Y + {y — 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, /3, 7 the corresponding angles ; 

p, cr, T the imaginary quantities belonging to the opposite sides ; 

a, h, c the length of these sides. 

Then C5"' = cosh a+ p sinh a, BC'^ = cosh a — p sinh a, 

AC'' = cosh b + a- sinh b, CA'^ = cosh b — a sinh b, 

BA'^ = cosh c -h T sinh c, AB'^ = cosh c — r sinh c. 

Therefore cosh a — p sinh a = (cosh c+t sinh c) (cosh b + a sinh b). 

Take the scalar parts : then, since Srcr = cos (tt — a) = — cos a, 
cosh a = cosh b cosh c — sinh b sinh c cos a. 

Take the vector parts : 

— p sinh a = T sinh c cosh b + a sinh b cosh c + A sinh b sinh c sin a, 
since Vto- = A sin (tt — a) = A sin a. 



100 Shi cox, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

Multiply by A and take scalars, 

SAp . sinh a = sinh b sinh c sin a. 

Now, if jD be the perpendicular from A on BC, SAp = sinh p. 

Also, SAp sinh a = SAVBC = ^^£0, since S{ASBC) = 

= 5f/)^ sinh a = /SF^C . ^ = SBCA = SB VGA 
= SBa = ,S(7t. 

Hence the quantity' SABC is unaltered by the interchange in cyclic order of A, B, C, 
and we have the equations 

sinh b sinh c sin a = sinh c sinh « sin j8 = sinh a sinh b sin y 

= sinh a sinh p = sinh b sinh 5' = sinh c sinh r 

= ^sinh^ b sinh* c — (cosh b cosh c — cosh a)* 

= ^1 + 2 cosh a cosh 6 cosh c — cosh" a — cosh" b — cosh''' c, 
by the preceding equation. 

We have thus thi-ee independent equations between a, b, c, a, /3, 7, and can determine 
any three in terms of the others. The equation giving a in terms of /3, ya can however 
be obtained directly in the same form as that giving a in terms of b, c, a. 

For To-'^ = — cos a + ^ sin a, ctt"' = — cos a — ^ sin a, 

pr"' = — cos (3 + B sin /3, rp'^ = — cos /S— Bsin /S, 

ap~^ = — cos y + C sin 7, pcr'^ = — cos y—C sin 7. 

Therefore — cos a — .4 sin a = (— cos 7 + C sin 7) (— cos /3 + B sin /3). 

Taking scalars cos a = — cos /S cos 7 + sin /3 sin 7 cosh a. 

From this last equation it is seen that 

cos (tt — a) < cos (j3 + 7), 
TT - a > /3 + 7, 
7r>a + /3+7; 
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 b, 
cos a = sin /3 cosh a, 

cota= . - cosh a = sinh 6 cosh a. 
sm a 

Now let B move off to infinity while A is unaltered, then AB will be said to be 
parallel to CB, and cot a = sinh b. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. lUl 

This gives the angle the line drawn from any point parallel to a given line maki-s 
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 jjurely 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 
— j1 as distinct or identical points. 

In the first sub-case since if the equation ?a; -h mt/ -f- ks = is satisfied by (a;, y, z). 

Vol XIII. Part II. 14' 



10-2 Mr cox, on THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

it is also satisfied by (— o', — y, — ~) \ all the Hues drawn through a given point pass 
also through another point which may be Ciilled the opposite point. The distance from 
r to j is iTT, from j to —i Jtt, from —i to —j Itt, from —j to i Jtt; so that the whole 
length of a line returning into itself is lir. Every line, for instance that joining j, k, 
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 
{Grelk, Vol. 83, for the year 1877, p. 293) ; Killing {Grelle, Vol. 8G, for the year 1879, 
p. 72) ; and Frankland " On the simplest continuous Manifoldness of two dimensions " in 
Xatufe, 1878. 

The distance from i to j is lir, and from j to —i is ivr, and therefore as —i coincides 
with i; the length of all straight lines is tt. As in the sphere, all the points distant 
Att from i 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 _;' and walk along the 
line jk till he returned to j, he would then be in the position P'j, 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 (cosh a, sinh a), or (cos a, sin a) respectively. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 103 

Or else reintroducing the constant k we have only to put in the equations 

.a . h . c 
sin , sm J sin ,- 

sin a sin/rf sin 7' 

k = x after having multiplied by k to obtain the corresponding plane equations. Similarly 
those of imaginary trigonometry can be obtained by making k 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 linear manifolds have been considered, but a non-linear manifold such as 
that of the points p^A+pqB + q^C will obviously lie in a linear manifold xA+yB + zC 
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 jjA -i- qB and a plane of all the points ^^J. + 5'B+ r(7. 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 obser\"ation 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 three lines at right angles, and let i, j, k be the correspond- 
ing imaginary quantities. Let also / denote the operation of turning the line j round 
till it coincides with k, and J, K the operations of turning k round to i and i to j. 
If we consider the points where i, j, k cut the sphere, then these operations will move 
the points along three arcs, distances equal to right angles. I, J, K then must be 
identical in their properties with the quantities before found for the spherical calculus. 

Therefore JK = I, KI=J, 1J=K, r = -l, J'=-l, K'=-l. 

But since / turns _;' to k and therefore k to — j, 

Ij = k, Ik = -j, 
and therefore since f = k'' = l, kj = I, jk = — I, jl = — k, kl—j. 

Similarly Jk = i, kJ= — i, Ji = — k, iJ = k, ik = J, ki=—J, 

Ki=j, iK=—j, Kj = — i, jK=i, ji=K, ij = — K. 

The equations may be written thus 

I^JK^kj, i=jK=Jk, 

J = KI = ik, j = hi = Ki, 

K= IJ =ji, k = iJ = //, 

/•= J'' = A'^=-1, ^»=/ = ^'' = l; 

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 hj I and assuming the associative principle 

Ii = kK==Kk. 
Therefore the quantities li, JJ, Kk, il, jj, kK must be all equal. 

The equation I = —jk shews that il= — ijk. 

Put ijk = v, il=—v, then i° = 1, P = — l, 

vl = i, vi = — I, D° = — 1 , 

and since li = — v, Iv = i, iv = — I. 

The quantity v is therefore commutative with each of the six quantities /, J, K, i, j, k. 

A unit line through the origin will be li + mj + nk if {I, m, n) be its direction- 
cosines. A rotation through a right angle about this line will be ll+mJ +nK. The 
product (li + mj + nk) {11+ mJ+ nK) is equal to v as it ought to be. 

If r times the line It + mj + nk be transferred through a distance 6 along a line at 
right angles whose direction-cosines are (I', m, n) we shall obtain the line 

r {cosh 6 + (I'i + mj -\- nk) sinh 6} {li + mj + nk) 

= rl cosh 6i + rm cosh 6j + rn cosh 6k 

+ r {mn — m'n) sinh 61 -^-r {71T — n'l) sinh 6J+r {hi — I'm) sinh 6K. 

If we write this Xi + Yj +Zk + LI+3fJ+ NK, 

then XL + YM+ ZN = 0, 

and X'+Y'+Z^-L^- ]\P - N' = r' a positive quantity, 

since {mn' - mnf + {nl - nlf + {Im - I'mf = {P + m' -1- 71') {P + ni'' + n") - {11+ mm' + nn) = 1. 

If these two conditions hold Xi+ Yj + Zk+ LI+ 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, 

r = JX'+Y-' + Z^-U-iP-N\ tanh 6 = ^^L+^p^^,, 

X ZM-YN 



^{X' +Y-' + Zy " ~ •^{X^+ Y" + Z') ^{L' + HP + N') • 

When r becomes and 6 infinite while X, Y, Z, L, M, N still keep finite values, 

X'+Y' + Z'-U-M'-N'=Q, 

and Xi+ Yj + Zk + LI + MJ+ NK represents a null translation along a line at an infinite 
distance. 

If r times the rotation lI+mJ+nK be transferred through a distance 6 along the 
line whose direction-cosines are {V, m', n) where W + mm' + nn = 0, we shall obtain the 
rotation 

r (cosh 6 + {I'i + mj + nk) sinh 6} {11+ mJ+ nK] 

= rl cosh 61 + rm cosh 6 J + rn cosh 6K 

+ r {m'n - mn') sinh di + r {n'l - nl') sinh 6j + r {I'm — I'm) sinh 6k. 



106 Mb cox, on THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

We may put this equal to 

- Li - iMj - Nh + XI + YJ + ZK, 

ami if this be multiplied by v, it gives the corresponding line as it ought to do. It 

follows that, if 

XL+YM+ZN=Q, 

then Xi + Yj + 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 

+ Xi + fij + vk + pi + a J + tK. 

This may be written (p + X V^)/ + (a + /li V- f) J+ (t + j/ V- 1) A' 
for the quantity v used above has all the properties of the algebraic V - 1 , and may be 
identified with it. 

The two conditions p" + o-' + t" — X' — /i'' — j/" = I 

and pX + fT/i + TV = Q 

are equivalent to (p + X V- 1)° + (o- + /a V- 1)' + (t + 1- V- 1)^ = 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— I. Hence a force F along a 
line and couple G about that line may be identified with a bivector of magnitude 
G + Fj—1. Now any expression 

Xi+ Yj + Zk + LI+MJ+XK= {L + X'J^I+iM+Y'/^^)J+(N + Z^/^^l)K 
may be equated to 

(G + J^ V^) {(p + X V^) 7 + (o- + /t V^l) J + (t + 1/ V^) A'j. 

For we have only to solve the equations 

z + xv^ = (G+i?'\/-i)(p+xv-i), 

N+Z-J^l = (G + F^/^ir + v^-l), 

and this is analytically the same problem as in ordinary space finding the length and 
direction-cosines of a line whose rectangular co-ordinates arc given. Squaring and adding 
the three equations, we have 

{G + F'J^f = (L + X -j:^)' + {M + Y ^/ ^y + {N + Z\/^i)' ; 

and this determines F, G in a, single way if we agree tliat the sign of F shall always 
be positive. Moreover Xi+ Yj + Zk + LI + 3IJ+ 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, A', Y, Z may be called the components 
of the resultant force at the point 0, and L, M, N the components of the resultant 
couple. Then it is found from the above equations, that 

X' + Y' + Z'=F' ip' + a'- + t') + G' (X' + fL'+ v'}. 

Now the method by which the expression for a line was obtained shews that, if 6 be 
the distance of line {pcTT, \fiv) from the origin, 



cosh' e = p^ + a-+ T-, sinh' 61 = \' + ^- + v\ 

Therefore X' + Y' + Z' =: F' cosh" 9+0' sinh'' 0. 

This shews, first, that X'+Y'+Z' is always greater than F', 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 9, 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 is 

r + iP + N- = F' sinh" 9 + G' cosh" 9, 

and the same theorems are true with respect to it. 

Let a, /3 be two unit lines and let 9 be their shortest distance, <f> the angle the 
planes passing through either and that shortest distance make with one another. Then 
the operation of transferring a to /3 is equivalent to turning a through an angle <}) about 
the shortest distance, and moving it through a distance 9 along the shortest distance. 

If S be a unit line along the shortest distance, D unit rotation about the shortest 
distance, then we may put 

/3a"' = (cosh 9 + B sinh 9) (cos ^ + Dsm <j)) 

= (cos 9 J - 1 + Z> sin ^ J^) (cos <f> + D sin (p) 

= cos {<}} + 9j^^) + D sin (<^ + 9j^). 
This is a biversor with angle ^ + 9 J — I. 
If we identify 

F + QI + RJ+ SK +p J^ + qi + rj + sfc 

= {P+pJ^^) + {Q+qJ^I+{R+rJ^)J+{S+sJ^)K. 
with n times the ratio of two lines, we must have 

ncQa{(l)-9^P^) = P + Ps,^l, 
and n'sin''(<f>-9j^) = (Q+qJZ:if+{R + rj::^y + (S + sJ^y. 



108 Mk cox, ox the application of quaternions and GRASSMANN'S 

Therefore n* = (P +pj^^y +{Q + q V^)' +{R + r J^f +{8 + 8 J-iy, 
giving n' =P'+Q' + E'+ S'- / - <f - v- - s\ 

Pp+Qq+J^r+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 coui)le ; that is, some multiple of the ratio of 
a screw to a couple. 

If /S, o be two rotations, we still have 

/32-' = cos ((f>+0 J-1) + D sin (^ + ^ J^) ; 
and, as with real quaternions, 

Soi/3 = -cos(<f, + 6j'^). 

But if (p, a, T, X, fi, v) [p, ff, T , X', fjf, v) be the co-ordinates of the lines a, /3 

S2^ = -{p + \ J^) ip + X' 7^1) -{<T + ^ J^) (<r' + yx' 7^ -{r + v 7^) (r + v J^). 

Equating the real and imaginary parts in the two expressions 82^, 
cosh cos <p = pp' + aa' + TT — XX' — /x/x' — vv, 
sinh 6 sin<f> = p\' + cr/u,' + tu' + p'X + cr'/n + t'v. 

More generally, if the system of forces (A", Y, Z, L, M, N) reduce to a force F and 
a couple G, while {X', Y', Z , L', M', N') reduce to a force F and a couple 0', and <^, 6 
have the same meaning as before, and are referred to the central axes 

(G + Fj^){G' + F' J^ cos (<l> + ej^) 

= {L+Xj^) {L' + X' J^l) + (3I+YJ'^)(3r+ Y'J^) + (N+Zj^){N' + Z'J^i). 

Therefore (FF' - G G') cosh ^ cos </> - (FG' + FG) sinh 6 sin ^ 

= A' A" + YY' + ZZ' - LU - MM' - NN', 

{FG' + FG) cosh 61 cos ^ + {FF' - G G') sinh 6 sin <^ 

= XL' + 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 
coefiScient of two screws. 

If be the origin and I, m, n the direction-cosines of the line OP, 6 the length of 
OP, then li + my + nk is the imaginary quantity corresponding to the line OP ; we have 

PO'^ = cosh 6 + {li + mj + nJc) sinh d 
= w+xi + yj + zk 
we may say, where w^— x'— y" — z' = 1, 

{w, X, y, z) may be taken for homogeneous co-ordinates of the point P. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 100 

Also, OP"' = cosh - (li + mj + 7ik) sinli 9 

= w — xi — yj — zk. 

If {w, X , y, z) be the co-ordinates of any other point Q, 

QO'^ = w' + x'i + y'j + z'h. 

Therefore QP"' = {w + x'i + y'j + z'h) {w — xi — yj - zk) 

= WW — xx' — yy' — zz' 
+ {yox' — w'x)i + (wy — w'y)j + {wz' — w'z) k 
— {yz' — y'z) I — {zx — zx) J — (xy' — x'y) K. 

But if {p, cr, T, \, fi, v) be the co-ordinates of the line PQ, yjr its length, 
QP'' = cosh i/r -I- {pi + aj +Tk — \I — XJ— vK) sinh a/t. 

Therefore cosh y^ = ww — xx' — yy' — zz, 

_ wx' -t- w'x _yz — y'z 

^ sinh ifr ' sinh -^^ ' 

sinh" •«/r = (wx' — w'x)'' + {wy — w'yf + {wz' — lo'z)'' 

- in^' - y'zf - i^-e' - ^'^y - Q^y - x'yf- 

We may use these formulae to find the equation to the cylindroid. 

Suppose (A', 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, 0, L, 0, 0) will represent 

any multiple of a screw about the axis of x, but the ratio -„ must be taken constant, and 
we will put it equal to p„ in accordance with Prof. Ball's notation. Also, -t> = /'^- 

If F, G, be the force and couple of the resultant screw, {p, a, r, X, /x, v) its co-ordinates, 

if -h 7 7 :^i = ( G -f p y^i) (a + /. y :^1), 

o=(G-i-pv-i)(T+i'y^). 

The last equation shews that t = 0, v = 0, or the axis meets the axis of Z at right 
angles. 

Multiplying the first by p — \ ^— 1 , 

(i-KZ7=T)(p-\7-:T) = {G^F^^\) {p' + \'). 



Therefore 

Similarly 



G _ X\ + Lp _ X -I- Pap 
F~ Xp -LX~ p-jJa\' 



G _ Yji+Ma^ _ fi+Pf^a- 
F ~ Ya- Mp. ~ a-p^p. ' 

Vol. XIII. Part II. 15 



110 Mk cox, on the application of quaternions and GRASSMANN'S 

Now, if w, X, y, z, be any point on the axis of tlie screw, lu', o, o, z the point 
where it meets the axis, 

p = — w'x, o" = — w'y, X = yz, /^ = — xz'. 

But T = lo: — w'z = 0, so that —; = -,, and as only the ratios of p, <t, X, fi are required, 

M) z 

we may ^^Tite p = wx, a = xuy, \ = —y:, p, = xz. 

— yz+ p„wx _ xz+ p^wy 
wx + j)oyz wy — p^z ' 

That is 

(P<i - Pp) (">" -2')xy = {l+ p^Pp) (a;' + y") wz. 

This is the equation to the cylindroid and it is a surface of the fourtli degree as 
has already been shewn by Lindemann. 



Spherical Geometry of Three Dimensions. 

In this geometry we shall have three imaginary quantities, i, j, k represent transla- 
tions along lines, such that f =f = k" = - 1, and three quantities /, J, K representing 
rotations about the lines, such that 

r- = r^K' = -l, JK^I, KI=J, IJ=K. 

Also i = Jk = — kJ, j= Ki = — iK, k = Ij = —jL 

"\Ve write the equations thus 

i = Jk =jK, I = JK =jk, 

j = A'i = kl, J = KI = ki, 

k=Ij=iJ, K=IJ = ij. 

It follows that li = Jj = Kk = il =jJ = kK= — a say, 

and w'=li.il=l. 

Hence coI = i, o)J=J, a)K = k. 

The general vector expression is 

(X +Leo)I+{ Y+ 2Ico)J+ (X + K(o) 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. 

Eesults exactly similar to those of the last section can bo obtained by using this 
imaginary quantity a, which is commutative with /, J,.K in the place of J — I. It 
must be noticed that a rotation about a line is always equivalent to a translation along 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. Ill 

another real line which may be called the conjugate line. In fact, the rotation / moves 
all points in the plane j, k along circles with for centre. One of these circles, namely, 

that of radius ^ , is 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 

(l'« -Pn) ('"' + -') «3/ = (1 -PaPfi) {x^+y^) wz. 

Ordinary Geometry of Three Dimensions. 

In this case i? =f = P = 0, P = J' = IP = - 1. 

We may take a quantity to such that w'^ = 0, and put 

ail = i, Q)J=j, a>K = 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 {I, m, n) will be expressed by lI+inJ+nK. If 
this be transferred to a point x, y, z along a perpendicular line we have the rotation 

{1 +xcoI+y(oJ+za)K) {lI+i7iJ+nK) 

= 11+ mJ+ nK + w [{yn — zvi) I + [zl — xn) J+ {xl — yni) K]. 

In general the quantity 

{X + Zo)) /+ ( F+ Ma,) J+ {Z+ Nco) K 

will represent a rotation F and a translation G, or we may say a rotation of magnitude 
F+Gw. 

Putting p = l, <T = m, T = n, \ = yn — zm, fi = zl — xn, v=xl—ym where {I, m, n) are 
the direction-cosines of the rotations and x, y, z a, point on its line of application, we 
may call p + Xco, a + fiw, t + vu> the direction-cosines of the line considered in position 
as well as magnitude. 

Since the two equations 

p' + ct' -t- t' = I, pX + (7/i + TZ' = 

are equivalent to the equation 

[p + Xoof 4- (t7 -f- /io))' -h (t + vaif = 1. 

To find then the magnitudes of F and G and the lines about which they act, we 
have to solve the equations 

X + Lo)={F+Gw){p + \u,), 
Y+ M(o= {F+ Gw) (a-+fi(o), 
Z +Xw = {F+ Geo) (t + vw). 

15—2 



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+Lat, &c., for 
the co-ordinates of the extremity of the line, p + Xw, &c., for its direction-cosines, F+Go> 
for its length, where in the first kind of geometry w'' = — 1, in the second to* = 0, and 
in the third a>'= 1. 

In spherical geometry if {p, a, t, X, /x, i^) are the co-ordinates of a unit force along 
a line, (\, fi, v, p, cr, 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' + x^ + y'^ + z^= 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 

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 i, j, k. 

They could be considered as the commutative product of two systems 

1. i j, a, 

1, ijk; 
which may be written more shortly 

1, (', j, K, 

1, V, 
where 1, i, j, K form an imaginary plane system and v' — — !. 



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 I or instead by /i = ijl. 

ft" = — 1 and /x is commtitative with i, j and therefore with k but not with p. In fact 

lj,v = ijUjk = lijijk = — lk = kl = — v/m. 

Hence v, ft, vfi form a quaternion system. 

The system of four dimensions is therefore the commutative product of the two systems 

1. i, j, k, 

1, V, fl, Vfl, 

the one an imaginary system of two dimensions, and the other a spherical system of two 
dimensions. 

The system of five dimensions is obtained by multiplying this by a new unit m 
or else by &> = ijklm, 

la' — ijklniijklvi = ijklmmijkl = ijklijkl = — ijkllijk = — ijkijk = 1 , 

w is commutative with each of the units i, j, k, I and therefore with their products. 

Hence the system of five dimensions is the commutative product of 

1, i, j, k, 

1, V, fl, Vfl, 

1, w. 

The system of six dimensions is obtained by multiplying the system of five by a 
new unit n or else by tt = ijkln. 

TT^ = 1 and TT is commutative with each of the units formed out of i, j, k, I but not 
with o). In fact 

WIT — ijklmijkln = mijklijkln = mn — — nni = — ttw, 

so that (wtt)^ = — WTTTTW = — I. 

Hence w, tt, wtt form an imaginary plane system, and the system of six dimensions 
is the commutative product of the three systems 

1, i J, k, 

1, V, fi, Vfl, 
1, W, IT, WTT, 

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 

v = ijklmno, 
v = yklmnoijklmno = jklmno-i^jklmno = jklmnoklmno = — klmnojklnmo = — 1, 
and v' is commutative with all the former units i, j, k, I, 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 

1, i, i, A-, 

1, V, fl, Vfi, 
1, W, TT, WIT, 
1, V. 

If p be another unit aud fi' = ijklmiip, the system of eight dimensions will be the 
product of 1, i, j, k; 

1, V, fJ., Vfl, 
1, W. TT, WTT, 

1, v\ fi, v'n', 

that is of two imaginary systems and two quaternion systems. 

We arrive then at the following results : 

The system of ^vi 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 im dimensions by 
the system 1, w, where «d^=1. • 

The system of 4»i + 2 dimensions is the product of m spherical systems and m + 1 
imaginary systems. 

The system of ^in + 3 dimensions is the product of the system of 4m + 2 dimensions 
by the system 1, v where v' = — \. 

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 in spherical and m imaginary systems. 

The system of 4?n + 1 dimensions is the product of the system of hn dimensions by 
], V where v^= — I. 

The system of 4?n + 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 ^m + 2 dimensions 
by 1, to where oj' = 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 oif 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 + qSy = A^=B'^ for all values of p and q. 

Hence (p" + f/ - 1)^' +pq (A . B -{■ B.A) = 0. 

This can only be the case if A'^O, 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 = 2coshd.A\ 

A.B + B.A=2e\ 

A.B + B.A = 2cose.A\ 

and these can only be collected in one by putting 

A.B + B.A^O, 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 

^= = ^5^=^ = 0, BC = -CB, CA = -AC, AB==-BA. 

Assuming the associative principle to determine the products of three factors, we have 
ABC = - ACS = - BAG = BCA = - CBA = CAB, 
or the product is the same when the cyclical order is unchanged. 

If P = xA+yB + zC, Q = x'A+y'B + z'C, R = x"A+y"B + z"C 

be any three other points in the plane of ^, B, G, then, by the distributive law of multiplication, 

P^ = Q' = R-=Q, QR = - RQ, BP = - PR, PQ=- QP, 
and PQR = - QPR= ...= x, y, z 

x, y, z ABC. 

«", y", z" 

If P lie on the line QB, then P = \Q + /xR, 
and therefore PQR = XQ'R - h^R'Q = 0. 

* Die Ausdehnungslehre, Ed. of 1862, pp. 6 — 30. "Die verschiedenen Arten des Produktbildung." 



116 Mr cox, on THE APPLICATION OF QIJATEENIONS AND GRASSMANN'S 



Hence the equatioa to the line QR is 



37, 2/, z 

ff tr f 

00 ,y , z 



= 0. 



The product of any two points may be called a Hue, and every line in the plane may 
be expressed as the sum of three lines, BC, CA, AB. 



For 



QR = O/'r" - y'z') BC + {z'x" - z"x') CA + {x'y" - y'x") AB. 



Conversely, every expression of the form IBC+mCA + nAB may be considered a line, 
for we may write j-{lB-mA){lC—7iA). 

Bv multipl}'ing two new points on the same line we only obtain a different multiple 

of the line, since 

{\Q + fiR) (K'Q + fi'R) = (x^' - \» 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 = xA + 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 



X, y, z, w 

^1. Vv ^1. W'l 

^i. y^' ^2. w's 

^3. ^3. ^3. ^3 



= 0. 



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 

lBCD + mCDA + 7iDA B + rABC. 

Conversely, every expression of this form is the product of three points; for 

IBCD + mCDA = CD {IB + mA), nDAB = j DA {IB + vxA), 

since A^ = 0, IBCD + mCDA + nDAB = jD {,iA - IG) {IB + mA), 

rABC = ^ ^ ijiA - IC) [IB+mB). 



Hence IBCD + mCDA + nDAB + rABC= j,{lD-rA)inA-lC)(lB + mA). 

It follows that the sum of any number of planes in a four point space is itself 
a plane. 



AUSDEHNUNGSLEHEE 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 + YA C + ZAD +LCD + MDB + NBC, 

where X, Y, Z, L, M, N are numbers. But such an expression need not represent a Hue; 
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'' = to be 

LX + MY + XZ =: 0. 

In general, let e^e^e^e^ ... e^ be n unconnected points, then the expression for an 
(n — l)-point space is 

p,e,e,...e„ + p,e,e^...e,^ + p,e^e^e,...e^+...; 

and, conversely, such an expression will always be the product of ()i — 1) points. For 

P,e,--- e„ +P,e,e,... e„ = {p,e^+p,e^) e,e,... e„, 

Pz^i^e, ••• ^n = jj (d\^i + PA) e^«, ■■■ e„, 

PA-- e„ +P2e,e, ... e„ +p,e,e,e^ ••• ^^ = « 'd\e, + 2^ej ip,e, +p,e,) e, ... e„, 

P2 

and so term after term may be joined together. 

The general expression of the form 

a„^e,e, ... e, + a^...r^^e,e,... e,^, + ... 

1 /J- 1- 1.- T •. X n{n—l)...(n — r+l) 
involves (disregardmg multiphcity) -^-^ - 1 constants, and a r-pomt space in 

an n-point space requires r (n — r) constants to determine it. Therefore 

w(w-l)...(w-r+l ) 

_^-_- r{n-r)-l 

conditions are necessary that the two may coincide. For instance, in a 5-point space, if 

F = a^„e^e, + a„^e^e^ + ... 

5 4 
is equivalent to a single line, we must have -^—2x3-1=3 conditions satisfied. In 

fact, putting i^ = we obtain the five equations, 

«46«l2 + «41«2G+«42a51 = 0, 
«5,«.23 + a5=«3i + "53«n = 0> 
«.2«34 + «13«42 + «»,4«23 = ^' 

of which it is easily seen only three are independent. 

Vol. XIII. Part II. IG . 



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^ + l\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 I be any line and X a number the equation 

{F-\lf = 0, or F'-\Fl = Q, since Z==0, 
always gives a value of \ for which F —\l is a straight line. 

If fim be this straight line (where yu. is a number) then 

F = \l + fim. 

Thus wc can in general take any straight line and find another corresponding to it, 
such that F wiU be the sum of the two. The exception is when Fl = 0, for then the 
equation gives no finite value of X. All the lines satisfying the equation Fl = form 
what is called by Pllicker 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 i^ = and F be consequently a straight line, the equation Fl = 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 F into two lines so that F = \l' + ^111, the equation Fl will be satisfied 
if I'l = 0, 7rt7 = ; and, conversely, if Fl=0 and I'l = 0, then also ml = 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 I = xy where x, y are two points, then 
keeping x fixed the equation Fxy = is satisfied by all the points y which lie on the 
plane Fx. 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{\x^ + \^c^ = \Fx^ + \Fx^, 

and \x^ + \^x^ is any point on the line joining x,, x^, while \Fa\ + \Fx^ is a plane pass- 
ing through the planes Fx^, Fx^. 

If F^, F^, F^, F^, F^, Fg be any six given sums of lines not connected by any 
linear relation, then the quantities e„e,, e^e^ ... may be expressed in terms of F^, F^, &c. ; 
and therefore any system of line F may be expressed in the form 

F = XJ'\ + X./, + \F, + \F, + \F, + \F,. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 119 

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 
(« — l)-point space. The most general relation of the first degree between the coefficients 
of F may be expressed by the equation F'F = where F" is another line-system. 

If F=Xe,e^+...+Le„_e^+..., F' ^ X'e,e^ + ...-^^ L'e^e.^+ ..., 

then FF' = F'F = XL + YM + ZN' + LX + MY' + NZ', 

since e„<^,e.^e^ = e^e^e.e^, e^e^e^e^ = 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 F^, F^ ... F^, so that each is 
reciprocal to the rest. The lines of a complex Fl = belong to the screws (we may call 
the quantities F screws for shortness) reciprocal to F. 

The lines common to two complexes FJ = 0, FJ = are said to form a congruence. 
If F^, F„ are real, the congruence will always contain real lines. For, taking any real line 
I not belonging to it, F^ may be put equal to Xl + /j,l' and F^ to Xl + fjll", so that a line 
intersecting the line I, I', I" 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^...F^ be co-reciprocal, three of the quantities 
F^F.^ ...F^ must be positive and three negative. For the lines of the congruence FJ,=:i\ 
F„l=0 can be expressed in the form 

\K+\F, + \F, + \F^, 

since they are a particular case of screws conjugate to two given screws. 

Squaring this last expression, and remembering that F^F^ = 0, &c., 

\^F' + \;F; + XJ'F' + \'f; = 0. 

33 44' 35' 6b 

But this last equation can only be satisfied by real values of X3, &c. if one at least of 
the quantities F^^, F^, &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 J— I. 

IG— 2 



120 Mr cox, on the application of quaternions and GRASSMANN'S 

The lines belonging to two complexes F^l = 0, F^l=0 belong also to any complex of 
the system (X,F, + \F,) Z = 0. 

Now \F^ + \F,^ is a straight line when 

{\F^ + KF^y = 0, or \'F; + 2WF^F.^ + \;F^' = 0. 

This equation gives in general two real or imaginary values of -, and thus in general 

a congruence consists of all the lines intersecting two given lines. 

Tlie lines belonging to three complexes FJ = 0, FJ=0, FJ = 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 \F^ + X^F^ + X^F^ be a straight line 

\:F: + Vi^/ + KF,' + 2\\F^F, + 2\\F,F, + 2\\F,F, = ; 
and this equation leaves one of the ratios -- , --^ arbitrary. 

All these lines intersect the lines contained in the expression \^F^ + \Fl + X^F^. 

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 X are the intersection of the planes F^x = 0, F^x = 0, F^x = 0, and these must 
intersect in a straight line if x lie on the surface. 

There are two lines common to four complexes FJ = 0, FJ, = 0, F^l = Q, FJ=^0. 
They are in fact the two lines included in the expression \F^ + X^Fi^, where F^, F^ are 
reciprocal screws to F^FJF^^. 

The ratio ^-» is obtained from the equation X^F^ + 2\^^F^F^ + \^F^ =Q. In particular 

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, FJ=0, FJ — O 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 1= X^F^ + X^F^ gives only one line, that is to say, if 
the equation X'F* + 2X^X^F^F^ + X^^F^^ = has equal roots, we must have 

Vi?;' + X,F,F, = 0, X,F^F, + X,r = 0, 

or ^^^^^ = 0, FJ,= 0. Hence I can be derived from the screws F^F^F^F^, and we may put 

Substituting in the equations 

F^l = 0, FJi = 0, F,l = 0, FJ = 

we have X^F,' + X,F,F, + X,F,F^+\F,F, = 0, 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 121 
with three others. Hence 



P2 rrrr p p pp 

FF F'' FF FF 

•* 1 2' 2 ' ■* 2 3' 2 4 

FF FF F^ FF 

1 3' 2 3' a > -^3 4 

FF FF FF F^ 

^ 1^ 4' ^2 4' -^ 3^ 4' -^ i 



= 



is the condition that four complexes should have only one line in common. In particular 
the condition that the four lines l^, l„, l^, I^, should only be intersected by one line, reduces to 

M Jkh + -JiT, M ± M JU, = 0- 

If the five complexes FJ, = 0, FJ,=^0, F,l = 0, FJ, = 0, FJ, = have a line in common, 
then since I is reciprocal to itself, it must belong to the screws derived from the five 
F F F F F 

-^ 1> -'21 -'3' -^ 4' -^ 5' 

Hence \F^ + \F^ + \F^ + \F^ + \F^ = I, 

and since FJ = \F: + \F,F„_ + \ F, F^ + \F^F^ + \F^F^, = 0, 

with four others. Eliminating X^X^W^- 

F^ FF FF FF FF 
FF F^ FF FF FF 

■^ I-' 2' -^ 2 ' -^ 2-' 3' -^ 2-* 4> -^ 2-' 6 

F F FF F^ F F F F = 0. 

■' l-* 3' -* 2-' 3' -^ 3 > •* 3^ 4' -^ S-* 5 

FF FF FF F^ FF 

-f l-t 4> ■* 2-* 4' ■* S-* 4' -^ 4 ' -^ 4^ 5 

FF FF FF FF F^ I 

Lastly the condition that there should be a linear relation among the screws 
F^FJF^F^F^F^ is found by putting 

X,F, + \F^ + X^F^ + \F^ + \F^ + \F^ = 0, 

and multiplying by F^, &c. to be 

F'' FF FF FF FF FF 
FF F^ FF FF FF FF 

1 2' -^ 2 ' -^ 2'' 3' •^ 2 4' 2 6' -^ 2 6 

FF FF F' FF FF FF 

I-' 3> ■'2 3' -^ 3 ' ■^3 4' ■* 3-' 5> ■* 3'' 6 

FF FF FF F"" FF FF 

^ I-' 4' -^ 2-' 4> 3 4' 4 > -^ 4 5' -^ 4-' a 

FF FF FF FF F' FF 

■* l-* 5> ■' 2-' 6' -^ 3-^ S' -^ 4^ 6' ■' 5 ' -^ 5^ 6 

FF FF FF FF FF F'' 

■^ I-' 6' ■'■ Hr e> ■* 3 6> ■* 4-' 6' -^ 6 a' -^ 6 I 

If F', F^F^, &c. be interpreted in ordinary geometry these are identical with the 
equations Prof. Ball has given in his Theory of Screivs, in generalisation of those of 
Profs. Cayley and Sylvester*. 



= 0. 



* Sturm "Sulle Forze in Equilibrie" in the Annali di Matematica. 



1-- mk cox, on the application of quaternions and GRASSMANN'S 



The Kegressive Multiplication. 

If there be only three units e^, e^, e^, or in other words if only points iu a plane 
lire being considered, we may put e^es^=^, since no higlier products can occur. Witli 
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 = x^e^ + xfi^-\-xj!^, and so if we put £,=6/,, E^ = e^e^, E^ = e^e,, 
any line can be expressed in the form 

X = X,E^ + Xj:, + X,E,. 

We may carry this reciprocity further, and introduce a multiplication of line exactly 
corresponding to the multiplication of points. 

We may put E,E,^e=-E,E,, E,E=e„ = -E^E^, E,E.=e, = - E.,E„ 

e; = o, ^/ = o, £/=o. 

It must be observed that the associative principle cannot hold when the two kinds 
of multiplication are combined. 

For CjCj . e^e., = c, and e^e^e„ = 0, since e^' = 0. 

The product of two lines X= X^E^ + X.,E.^ + X,E^, Y= 1\E^+ Y.^E^_+ 1\E^ is defined 

by XT= {X\E^ + X\E^ + X,E,) {Y^E^+ Y,E„_+ Y^E,) 

= (xi; - a;i:) e,e, + (a;f. - x, rj e,e^ + {Xj.^ - xj\) e^e,, 

and this definition involves the distributive principle, 

or X{Y + Z)=XY + XZ. 

Now the equation —E^E^ = e, may be WTitten e^e„ . e^e^= e^ = {e^e„e^) e^, and we will 
shew that if x, y, z be any points 

xy . xz = {xyz) x, 

<ir 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 x = x^e, + x.^e^ + x^e„ y = y^e, + y/.^ + y^e, , z = z^e, + 2/2 + ^3^3 . 

yz = A;^, + X,E, + X^E^, zx = r.A', + Y.,E.^ + Y,E,, xy = Z,E, + ZJ!:„_ + ZJi,, 

then A',, }',, &c. are the minors of 

^2. y-i' ^2 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 123 

and zx . a:ii = {Y,Z, - \\Z^) e^ + {l\Z^ - l^,^^) e, + {\\Z^ - Y,Z^) e. 



^1. 2/,. 


1 


^2. y-^' 


"2 


^3. ^3- 


^3 



= («y~") 



a;. 



Just as the mviltiplication of not more than three points is associative, so the 
multiplication of not more than three lines is associative. 

For yz {zx . xy) = xyz . yzx = {xyzf, 

since xyz is a number 

{yz . zx) xy = {xyz . z)xy = {xyz)\ 

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 x be the 
point whose locus is required, a, b, c the fixed points, A, B the fixed straight lines, 
then xa is the line joining x and a, and therefore one of the sides of the triangle, 
xaA is the point where xa intersects A and one of the angles, xaAb another side, 
xaAhB the next angle, xaAbBc the third side, and as this must pass through x, 

xaAhBcx = 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 Xx + fxy be a point on 
the line joining x, y and also a jjoint on the locus 

(Xa; + fxy) aAbBc (Xcc + /x?/) = 0, 

or \^ [xaAbBcx] + \/j, [xaAhBcy + yaAbBcx} + ^[yaAbBcy] = 0. 

The coefficients of X^ \fi, fj,' arc numbers, and this equation gives two values of - 

X 

determining the points of intersection. It is clear that in general the number of times x 

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, 

xaAbBcy + yaAbBcx = 0, 

will represent the tangent to the curve in x, and from the symmetry of tliis equation 
in x 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 Crellc'e 
given in the Ausdehmingslehre of 1862, p. 195, or in that of Journal. 



124 Mr cox, ON THE APPLICATION OF QUATEPvNIONS AND GRASSMANN'S 

The condition that the opposite sides of the hexagon formed by a; a, b, c, d, e 
should intersect in points lying on a straight line is 

{xa . cd) {ab . dc) (be . ex) = 0, 

and the equation shews that the locus of a- 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 a and e since or = 0, e" = 0, and using tlie sign 
= to mean, is congruent to, is a multiple of 

(ba . cd) {ab . de) = ab, (be . eh) = b, 

and ah . b= 0, 

ca . cd= c, be . ec = c, 

c {ab . de) c= 0, 

da . cd = d, {ah . de) {be . ed) = ed, 

d . e(Z = 0. 

Therefore it passes through b, 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, 6, c be the fixed points. A, B, C the fixed straight lines, its locus is 

(xa . A) {xh . B) {xc . 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 BCbB = BC, BCcG = BG, and lastly through the points be . A, ca . B, ab . c. 

For if a; be any point on the line be, xb = be, xe = be, 

{xb . B) {xe . C) = {be . B) {be . C) = be, 

and xa . A . bc = 0. 

xa, A, he 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, G 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 degree can be found by mere counting. The degree is 
equal to the number of times the variable point is introduced in the construction. 



AUSDEHNUNGSLEHEE TO DIFFERENT KINDS OF UNIFORM SPACE. 125 

If a, \a + fih, b, X'a + /j,'b are four points on a straight line their anharmonic ratio 

was defined to be r — . Similarly the anharmonic ratio of four lines A, XA + jj-B, 
A. ft 

B, yJA + ij! B is defined to be ^ — , . 

\ /i 

If be any other point the lines joining it to a, Xa + /xb, b, Xa + fib are 
oa, \oa + iJLob, ob, X'oa + fi'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 = Xa + ixb, d=X'a + fib, then the anharmonic ratio of the points a, c, b, d 

is -5^ . — , where it must be remembered that -p , — , are numbers, and that ac, db are 
CO ad CO 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, bd in f, and ad, be in g, then since 

{a + c){b + d)=-{a + by = 0, 
ab + cd = — (ad + be) = some multiple of line eg since it passes through e and g; 
again (a+ d) (6 + c) = 0, 

ab — cd=— {ac + db) = some multiple of ef. 

Hence since the Unes 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, G, D will be 

AB DC 
BC • AD' 

but if these be the lines joining x to four points a, b, c, d 

AB = xa . xb = (xab) x, BC~ (xb . xc) = (xbc) x, 

, ^, ,. , xab xdc 

and the ratio becomes ,— . — i. 

xbc xad 

If this be constant 

(xab) (xdc) = k (xic) (xac), 

a curve of the second degree passing through a, b, e, 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. 

It must be noticed that though the product ab corresponds to Va^ in Quaternions, 
the product abc coiTesponds to Sa^y. 

Vol. XIII. Part II. 17 



126 Mr cox, ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

The theorem corresponding to ab . ac={ahc)a is 

F . Va^Vay = ~aSa0y, 

but in Quaternions a, y9, 7 could, if taken to be points, only be points on a sphere. 
In Grassniann's system if a, b, c were points on an ordinary plane, abc is double the 
area of the triangle they enclose. The Quaternion expression is VjSj + Vy2+ 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 conies, or the cones 
joining them to the centre of the sphere, and thus indirectly to plane conies. If p, a, ^, 
y, B, €, be vectors SVVpaVyB, VValSVBe, VV^yVep = 

is the equation the expression of that theorem leads to, and it is identical with 

(oca . 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 ?i-point space. 
So long as the number of points multiplied together is not greater than n and so long 
as the points are comprised in a space of lower dimensions, the laws of the progressive 
multiplication wnll 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 ee....e be n points then since no product can have a higher term than 
e,ej...e„ we may put e^e^...e„ = l. 

If follows that the product of any other n points unconnected by a linear relation 
is some number dififerent from 0. 

If E, F, G be different products formed from e^, e.,,...e^, such that EFG contains 
all the points e^e^... e„ without repetition and is therefore + 1, we will say that 

EF . EG = (EFG) E. 
For example e,e„ . e.e, ... e„ = (e.e^ ••• O ^1 = ^i' 

This assumption, with the distributive principle, will form the definition of the 
regi'essive multiplication. 

♦We will now prove that if E' , F', G' be products of any points such that E'F'G' 
is a number E'F' . E'G' = (E'F'G') E\ 

For this purpose let E = efi^...e,y F=e^^^...e,, G = e.„...e,., and change successively 
the points e,, e, ...e„ into a,, a.^...a„. 

• This proof ia given in nearly the same form in Auidehnungslehre of 18G2, p. 68. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 127 

Let a^ = x^e^ + x.^e.,+ ... x^fi^^ then 

E' = a,e„ ... e^=x^E + x^^^e,._,^e.^... e, + a",^„e^^„ . . . e, + . . . + a-„e„e, ... e^, 

since e.,e.^ ... e,, = 0, e^e^ ...e^ = 0, &c. 

E'F= x^EF + x^^^e^^^e^ . . . e^ + . . . + x^^e„e.^ . . . g^, 
E'G-= x^EG + av^/,^,c, . . . e, e,.,, . . . e„ + . . . + a;//, . . . e^e^^, . . . e„ . 

Now EF . EG = (J?i^G) E 

EF . e^^.e, . . . ee^^, . . . e„ = e^e.^ . . . e/,^, . . . e, . e,^,e, . . . e,e^^. . . . e„ 

Similarly EF . e^e, . . . e/^^, • • • ^„ = (^i^j • • • ej e/^ ...e^. 

Also e^^;e, . . . e^ . iS'G = e,_^^e^ ...e^ . e^e^ 

= - (e«««2 ••• e/A^. ••• O 6,^162 ■•• e. 

and e.^iC, •••e^ • «r+A ••• ^r«,+i ••• ^„ = 0> since one of the points e is wanting. 

Hence collecting all the terms 

E'F . E'G = X, ie,e, . . . ej {x^E, + x,^^e,^^e, . . . e, + . . . + x„e„e, . . . e,} = {E'FG) E' , 
since a;,eje2 . . • e„ = (x^gj + a-,/„ + . . .) a„ . . . e„ = a^e^ ■■■ ^„ = E'FG. 

Likewise a point in F for instance e,.^j may be changed into a^^.,. 

If a,„ = .r/. + .r/, + . . . a-,^/^^^ + . . . 

then EF' = e,e, . . . e, (.r^e, + . • . a;,« e^.^, ■ • • xej e,„ . . . e, 

= a;,^.ii'f +^,^^e,e, ... e^e^+,e,« .•■ e.+ ••■ +a'„eA .•• e/„e,^, ... e,, 

but e,e^ . . . e,e^^, e,.^^ . . . e, . e^e, . . . e/,^, . . . e„ = 0, since e,^, is wanting. 

Therefore EF'. EG = x^^^EF. EG = a;,^, (EFG) E = (^i'"G) £" ; 

and the same reasoning will ajiply to a point in G. 

Thus point by point all the points in E, F, G may be changed, and we shall have 
generally E'F' .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 ^,=e./3...e„, E^ = -e^e^... e^, E, = e,e^e^...e^. £■„ = ± e,e., . . . e„., . 
where the signs of E^, E^... are taken so that 

e,E^ = l, eA = l. e^E^ = \,&c. 

17—2 



r2S Mr cox, on THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

Then E^E^ ==-e,e,... e„e^ . e^e, . . . e„e. = (e/^ . . . e„e,e,) e,e^ ...e„ 

= (e,e, ...eje,e,...e„ = e/,...e„, 
E^E^ . E^ = {e,e, . . . e,) e,e,e^ ...e„ = ±e,... e,^e, . e, . . . e/,e, = ± (e, . . . e„e,e,e,) e, . . . e„ 

and E„E, = - e,e.,e. ...e... e,e„e, . . . e. 



2 3 



= -{e^e^...e,,e,e,)e^e,...e^ 
= («i«/3«4 • • • «J e,e, ...«„ = e,e, . . . e„, 
^1 ■ ^2-2; = e„e/4 ■■■«„. e,e^ ...<>„ = + e, .. . e/./^ . e, . . . e„e, 
= ± (^-i • • • e,.e/3^.) e, - • . e„ = (e,e, . . . e„) e, . . . e„ 
= e,...e„ = E^E,.E,. 

Thus the multiplication of the quantities E^, E^, E^ is associative. Proceeding in the same 
way £,i7,....e,= e,„...e„. 

We can arrange the quantities e^e^... so that e^e.e, ... = 1, where e^e, are any two of 
them; then in the same way 

.f^,.^', = e, ...= product of all other units so arranged that e^e,.E^E,= l 

And for any three of the quantities E^E^E,... 

E^.E, E^= E^E,. E^ = i:)roduet of all the units not containing e^e,e„ so arranged that 

eAe^.E,E,E, = l. 

It follows that any product of three terms is associative, so long as the multipli- 
cations involved are either all progressive or all regi'essive. 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 r, s, t quantities E. 
None of the quantities E 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 A BC 
is n~r — s — t, and hence r-^s + t is less than n. Therefore there are less than n quan- 
tities E and their product is therefore associative. A product in general including both 
progressive and regressive multiplications will not be associative. 

Grassmann writes the quantity E in the form | e, and calls it the complement (Ergiin- 
zung) of e,. In general if A be any product of the units e,, i\..., \A is the product of 
the remainining units so arranged that A\ A= \. 

li B = \A, then AB = 1 and therefore BA = ±\, and \B=±A. 

BA can only be — 1 when B and A both contain an odd number of points, and 
therefore n is even. 



AUSDEHNUNGSLEHEE TO DIFFERENT KINDS OF UNIFORM SPACE. 129 

For example in a two-point space, if e^e^ = l, e/^ = — l, e,= \e, e, = - 1 e^. 
If A = e^e,^ . . . e^ then it has been shewn that \A=E^E.^...E,, let also 

then \A\B=E,E,... EE^, ...E=\ e,e„... e. = I AB. 



r r+l 



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 Avrite 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 j^lane 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 jjlanes 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, C is xABCx = 0. 

For xA is the jilane containing x and A, xAB the point where it meets B, so 
that the line joining x to xAB meets both A and B : and xABCx = expresses that 
xAB, C, X lie iu the same plane so that the line joining x to xAB meets C. Hence x 
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 xA . xB . xC=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 Ml! COX. OX THE APPLICATION OF QUATERNIONS AND GRASSM ANN'S 

but .vA . .iB . aC is lUways some nuiltiple of the point x = mx say, where m is a 
number of the second degree in .v. 

Therefore the equation xA .xB.xC = reduces to w = again an equation of the 
second degree in x. In fiict in every equation employed the left-hand side must, if x 
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 jwints that are altogether midtipled together and see that it is divisible by four. 
Thus in xABCx, x and x each give one point, A, B, C two points each ; so that the 
wliole 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 hne 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, L the straight line, its equation is 

(xaA) (xbB) L = 0. 
This surface, as is easily seen from the equation, passes through the points a, b, and the 
jwints LA, LB 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 L, 
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 joining the 
points of intersection will always meet a certain fixed line. 

The equation [xaA) {xbB) (xcG) {xdD) = 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 IXNEU MULTIPLICATION. 

If a = a,e, + w/.^ + %c^ + ... -1- a„e„, 

where a,aj...are numbers and e,e.^...e„ a system of points such that e,e, ...e„ = l, then 
I a is defined to be a, | e^+a, \ e.^ + a^ | e^+ ... +a^ \ e„. 
It is clear that | (6 + c) = ] 6 4- | c. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 131 
Now e, I e, = 1, ej e, = l, &c. 

f. \e, = e^.e^e^.. .6,^ = 0, e^ 1^3 = 0, ...; 
therefore a | a = a,^ + a/ + . . . + o(^^ 

It will be assumed that when a is a simple point and not a multiple of a point 
a I a = 1 just as e, | e, = I, &c. 

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^ — l, e^ \ e^ = is called by Grassmann 
a normal system. 

If f^, J\---J\ form another normal system, and if 

/2 = «2,ei + «22«2 +•••+«■. A. 
&c., &c. 

then since /■ I /, = 1 > ^n + "i^' + • • • + «„.' = 1. 

and since " /J /, = 0, a„a„, + a,,o(„., + . . . + a,„Q(„„ = 0, 

&c., &c. 
From these equations 



«2, «.2---«2, 



= ±1, 



and we will take it equal to + I. 

Then /,/.•••/.= 1- 

Now with reference to the system f\, f„, -../„, 

I J I J -iJ^ ' ' ' Jn 



But 



J tJ % ' " Jn 



a,, ...a,. 



e/3 ...€„ + ... 



= «U 1 «1 + «12 I ^2+ •••+«!. 1 «..' 

and this is tbe meaning of | /, referred to the system e^e.^ ... e„. Similarly the meanings 
of I j^ , &c. are identical. 

Now if x = xj^ + xj.-^ +...+ .tJ,^ , 

then I « = a;, I/. +a;J/,+ ... + .r„ |/„; 

and this is the same whether x be referred to the system e^e^ ... e„ or to f,/^..-/,,. 
Hence | x does not depend on any special set of points, but is the same for any 
normal system. 



13-2 Mr cox, on the application OF QUATERNIONS AND GRASSMANN'S 

As auy poiut can be expressed in terms of e, , e.^...€„, so any quantity of the 
j-th order can be expressed iu terms of the products e^e.^...e^, e^e^ ... e^^^r ... . If 
J,, ^4,... ^4^ be these products and ■ 

A = a^A^+ci„A.^ + ... + a„A„, , 
then I A is defined to be o, | A^+ a,, \ A,,+ ... + a^\ yl,„. 

Let \B= ^^\A^ + ^^\A,+ ...+/3„\A^. 

Then I A \ B = i2fi^-a,3,) \ A^ \ A^+ 

= (a.i3,-a,/S,) \A^A^ + 

= I AB, 

since it was shewn iu the hxst section that | A^ 1 -^2~ 1 ^K^^^ 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 I ^ I 5 I C &c. = | ABC, &c. 

and in particular \ AA ■■■ fr = \fi\ A ■■■ \fr' 

and therefore i/i/^.--/. is independent of the particular normal system to which it is 
referred since |/, , \ f^ ... \/^ are independent. Since auy quantity A of the ?-th order can 
be expressed in terms of the quantities /,/^ ■■•fr> fifs ■ ■ • /r+i > ^'^- i*- follows as before 
that I A or the complement of A is independent of the special normal system to which 
it is referred. 

Since " \ {B+ C) = \ B + \ C, 

A\{B+C} = A \B + A I C, 

and therefore the quantity A j 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 ?-th order formed from the original units 
e,e.. ... e„ then by the definition of \ A^, \ A„, &c. 

A, I ^1, = I, A^ \A^ = 1, &c., 
and if ^, = e,e, ... e,, A^ = e.^e, ... e,^^, 

so that I A.^=±e^e,^^...e„, 

then A/\A^=± efi.^ . . . e,e/,^, . . . e„ = 0, 

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 = a^A^+a,A^+... + a^A^. 
B=0,A, + 0^A, + ...+3^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 e.e, ... e^e^,^ ... e,, efi,...e, which are not of the same order, 
then e,e. ... e/,^, ...e^. | e,e, ... e, 

= ±e.«...e/,e,...e,.e,„...e^...e„ 
= ± (e,„ . . . efifi.^ . . . e./,^, . . . e„) e^^, ...e^ 

so that £• I F when i? contains F is equal to the product of the remaining limits so 
arranged always that 

F[E\F\=^E. 

Again, e.e, . . . e J e,e^ . . . e^ = e^e, . . . e,e,^, . . . e„ 

= product of units except e^^, ...e^ 

= (-ir"-'U..,...e,. 

That is to say if E be of higher order than F and 5, r be their orders 

F\E={-Vr-'^\{E\F). 

From the distributive principle it follows that tliis result must be true for any 
quantities ^, i^ of orders s, r. 

If both E and F contain quantities -which do not occur in the other, then E | F= 
for \F will contain quantities which occur in E since F does not contain all the quan- 
ties in E, hence in E\ F some points will occur twice over; also E \ F will not contain 
all the points, for since E does not contain all the points in F, there will be some 
points neither in E nor in | F. 

If, in the equation F{E\F\=E, for i^ we substitute any quantity B of the same 
order where £ = /3,F, + ^^F, + ... and i^„ F^ are only made up of points contained in E^ then 
since F^ [E \ F^\ = E, F, {E \ F^} = E, 

and it is easily seen that F^ {E \ F^\ = 0, F, {E \ F} = 0, 

^'"^ h^ve B{E\E] = {^A + ^.P'.+ ■■■} {/3,E I F+/3.^E | F^ + ...} 

= (/8.= + /3/ + ...)£=(5| B]E. 
Also with the notation before used for F and E, 
F\ E = efi^...e^e^^^...e^, 
E[F\E] = e^e^... e^...e, . e^e^... e^e,^^... e„, 
= e^e^...e^=F, 
if F be entirely contained in the points of E. But if not, and F=F' + F", 
then F" \ E=0, so that F \ E = F' \ E, 

and E{F\ E} = E[F' \ E}= F'. 

Vol. XIII. Part II. Ig 



l.U Mk eOX, ON THE APPLICATION OF QTTATERNIONS AND GRASSMANN'S 

Hence when E is a product of normal points and F any (luantity of lower degi'ee, 
E{F\ E] expresses the part of F which is contained in E. If E were some nmltiplc 
of a product instead of the product itself, we should have E \F\ E] = [E\ E] F'. 

For example, if .r = pr^ + qe^ + re^, where j), q, r are numbers, 
then X I c/3 = {qc, + 7^3) | e.f^ = qe,<\ + re/.„ 

and e/., [x \ e/j) = qe^-\-re^. 

The ([uantity F' may be called the projection of F on E. Also x \ e.f^ = xe^ passes 
through tlie point x and is normal to e.fi^, since .re, | e^e^ = a;e^e^= 0. It may therefore be 
called the perpendiculars from x on e.,e.^ ; or the inner product of a point into a line 
re})resents the perpendicular from the point on the line. Again e.,e^ \ x is the point 
where the complement of x or its polar intersects e.,e^. 

If a^(i„...o^ be any points and x a point belonging to the system 

fl,o,... (7„ I ,r = (fl, I x) {a.fl^... a J - («, | x) a^a^... a. + (a, | a-) {o,n„n^... a J - &c. 

For let e,e„...e,, be a normal system including a^a^..M^^ and therefore x. 
Let «, = o',A + =',/.+ ■•• «':/..• 

&C. &.C. 

a; = x,e,+ ;c/, ...+a-„e„; 
and let ^,,,.1,.. be the minors of i) = a,,, a,,^, ...a,. 



Then w.a, ... «„ = .4„ | e, + .4,.^ 1 f, + ... ^,„ | e„, 

«i I «; = «„*•. + «.=^''=+--- 
Thus the coefficient of x, | e, on the right-hand side is s„J„ + a„, ^4^, + ... = />, and of 
;/■, I r, Ls s,, J,^, + a„, J^,.^ + o,,A,„_ + ... = U. 

Hence the right-hand side become D[x^ \ e, + -'.2 I ''2+ ■■■]-' '^\"----'^n I ^'' 
.since "i^r-- '^f, = ^' ^""^ I •''' — ''^'1 i ^1 + •'"2 I ''a + ••• 

As particular cases ab \ c = {a \ c)h — {h \ c) a, 

ahc I (/ = (a I d)hc+ {h \ d) ca+ (c \ d) ah, 
abed I e = (a I e) bed— {b \ e) eda + (c | e) dab — {d \ e) abc*. 

From this first equation and two similar 

tc I a + cc I 6 + '(i I c = 0. 
Hence, as a \ hc= — \ be \ a], a \ he + h \ cu + c \ ah = 0. 

This equation expresses that the perpendicvdars from the angles of a triangle on the 
opposite sides meet in a point. 

• Amdelmungsh'hre, pp. 131, 134. 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 135 



Again, 
and since 



hcd I a — cda | b + dah | c — ahc | rf = 0, 

a I bed— I [bed \ a), 
a I hcd — b I cda + c I (Zrt6 — fZ I abc = 0. 



That is to say, the four perpendiculars from the angles of a tetraliedron in 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 s^'steiu, in a three jmint 
space, is a conic when the other two points lie on fixed straight lines. For if x be 
the variable point, A, B the fixed straight lines, then A \ x, B \ x, are the other two 
points and since these are normal 

{A\x)\[B\x)= 0, 

an equation of the second degree*. This is MacCullagh's theorem. 

If a, b, c... a', b' , c\... be two sets of points of equal number, 

then abc... \ a'b'c'...= a \ a, a\b', a | c'... j 

b I a, b \ b', Z> I c'... I 
c I a, c \ b', c \ c... ! 



For we may write 



a' = a/e, + aX + a,>„ + • • • + a^„'e,, 



Then 



abc... = 



a,, a„ ... a„ 



e,e,...e„, 



u'b'c = ! a 'a' ... a' 

I ' - " 



Hence 



abc... I u'b'c... = 




a,a,' + ttjO-,' + . . . , a,^,' + 3„^„' + .. 



In particular 
and therefore 



a I a', a \ b'... 
b \ a', b \ b'... 



ab 1 ab' = (a \ a) {b | b') - {a \ b') (b \ a'), 
ab \ cd + ac \ db + ad \ be = 0. 

* Tait's Quaierriions, p. 146. 



18—2 



io6 mk 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 6, and call the distance between 
(I and b. 

Since ab \ ab = {a \ a) {b \ b)- {a \ bf=l -cos-B = sm-d, 

we may put ab = sin Oe^e.,, where e^e.^ are twt) normal points on the same line. The dis- 
tance between two normal points is always - , since e, | e, = = cos 5 . 

TT 

Again if A, B be two intersecting lines of length -, we will put A \ B = cos4>, and 
call ^ the angle between A and B. 

TT 

If be the point of intersection of ^1, B; a, b, the points distant - from 0, so that oa = A, 

ob = B, then 

cos (j>=oa \ ob = {o\ 0) (a \ b) - (0 i a) (0 | i) = a | 6 ; 

and thus (j) is equal to the distance between a and b, or the angle between two lines is 
equal to tbe distance between the points where they cut the polar of their point of inter- 
section. 

Attain if a, b be the poles of A and B in a three point space 

A ] B = \ a , \ b=\ a.b =b \ a, 

or the ancle between two lines is the distance between their poles. Since 
AB\AB=={A \A)(B\B)-(A j 5) (S | ^1) = 1 -eos> = sin'^; 

it follows that AB = sin (f) . 0, where is their point of intersection. If a, b, c be points 
of a triangle whose sides are a, /S, 7 and angles 6, cji, X' then 

ab 1 ac = sin /3 sin 7 cos 6, 
l)ut ab ] ac = (b \ c) —{a\b) (a | c) = cos a — cos j3 cos 7, 

so that cos a = cos /8 cos 7 -f- sin /3 sin 7 cos 6*. 

Again ab.ac = sin /3 sin 7 sin 0a, 

so that abc = sin /3 sin 7 sin ^ = sin 7 sin a sin </> = sin a sin /3 sin ^, 

sin 6 sin cf) sin •)(, 
and hence ii^ = iliT^ = sTn 7 " 

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\ x, 

• Wlien d = ir, = ^ + 7, 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 genei'al 
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 ou 
two straight lines is constant, the straight lines being the polars of the former points, and 
then in the usual way it may he 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 

xa\xh = 0, or {x \x) {a\h) — (x\ a) {x\h) = 0, 

and is a particular case of the former curve. 

TT 

If «, h, be any two points distant — from c and d in a space containing more than 

three independent points, then 

{ia + /36) I (7c + M) = 0, 

and therefore every point in the line ah is distant ~ from every point in the line cd. 

Also ca\cd = {c\ c) {a\d) — {c, a) {c\d) = 0, 

or ca is at right angles to cd. 

Again, ca\cb == a\h = 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 jjlanes at right angles to the line of inter- 
section from any point in that line. 

If 6 be this angle, and (f> be equal to the distance cd, then 

abed I abed = 1, a b, 0, c ' = sin" 9 sin" (f), 
a\h, 1, 0, c 
0, c, 1, c\d 
0, c, c\d, 1 

so that in a four-point space, where abed is a number, 

abed = ± sin d sin <f). 

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 -^ , be of lengths yjr, X' then 

acbd = + sin yjr sin •)(, sin sin cj), 



138 Mr cox. ON THE APPLICATION OF QUATERNIONS AND GRASSMANN'S 

or the product of two Hues is equal to the product of the sines of their lengths multiplied 
bv the sines of their perpendicular distances. 



TT 

Making again ac, hd of lengths -^ , 



ac\hd = \ a \ h, 
I 0, c\d 



= cos 6 cos </), 



and thus the inner product of two lines of length ^ is the product of the cosines of their 

perpendicular distances. 

If L, L' be two lines the equation LL' = 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[L, whore a is a number called the pitch of F. 

For \F=\L + oiL. 

Hence F-a'^F = (V -a')L; 

and therefore (F-a\F)- = (), 

or F'{l + a')-22F\F=0. 

This f'ives a quadratic equation for determining a the roots of which are a, . The 

meanino- of this is that if L + a\L be one solution, then a'i + -'(a!i) is an identical 

o ' a ■ 

solution. 

Either L or \L may be called the axis of F. 

Suppose we have two screws e^e.^+ ae^e^, ^i^a + Z^Vi' where e/.e^e, form a normal 
system, so that e^e^=\e^e.^, e^e.^=\e^e^, and wish to find the cylindroid or the locus of 
the axis of a screw compounded of these. 

Put Z + 7 ; Z = \ ((?,e, + ae^e,) + /u. (e.e, + ^e,e,), 

then e^e.^L + '^e^e^\L = X'x, 

e,e.^ i L + 7e,e.^Z = \, 

since i I e.Cj = Cje.^ ' i and \e^e.^\L = \Le^e^ = Le^e^. 

Hence e,e/> - ae,e.^ ' Z = 7 {ae^e„L - e,e.^ i L) ; 

similarly ^1^3 ^ - /3e,g, 1 L = y^^e^e^L - e^e, I L). 

Eliminating 7, 



AUSDEHNITNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 13'J 

Now, since ie,e^ = 0, Le^e^ = 0, L meets the Hues e,e,, e.,e^, and we may put 
L = axe^ + Txe^ where x is any point on L, 

Le.,e^ = gives — crx\e^ -\- ra; j e, = 0, 

and the previous equation gives 

(t'x e^.x\e.^+a-x e.,.x \ e^ (a - /3) = (a/3 + I ) {(^ ej + {x | e^)'j o-t. 

And therefore 

(a -^){x, e„) {x : e,) [{x \ e,)= + [x \ e,)=) = (a/3 + \){x\ e,) (x ej [{x \ e^f + {x j e.f]. 

The product of two screws L + 'y L, J/ + S J/ is 

il/(l+78) + (7 + S)i|ilf. 

If J/ meets i 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 31 perpendicular to the line passing through that point. It will al.so meet at 
right angles the polar line, and it will meet two other lines of the cylindroid in addition. 

We may suppose a screw with pitch equal and opposite to one of these last two 
lines and witli a.\is M. Then M + B M is reciprocal to two screws of the cylindroid, and 
therefore to all. 

Hence, if L + y L be the last line it meets since LM = 0, 

(y + S)L M = and 7 = - 8, 

that is, a line perpendicular to a line of the cylindroid meets it again in two lines 
corresponding to screws of equal pitch*. 

dx 
If X be any point, .r + -, dt its consecutive position, 

then * 777 ^* ^^ ^ small portion of its path, 

and X -J, its velocity ; 



« / dx\ d'x . , , , 

T, *~r = «-n7 IS the acceleration. 
■ dt\ dtj dt 

If F be the force acting, >u the mass, 



d'x „ 



* 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 „ 

dt 



and this includes the six equations required to determine the motion. 
For a particle F = xl, since F must pass through x, 



d'x , 
mx -n = xl. 

de 



dx 
Multiply by ! x -^^ , then 



since 



and therefore 



d'x 

""-de 


dx , , 
^dt = ^'^' 


.d\T 

""^de 


dx 

dt''^ 


dx 

di ■"" 


d'x 

dt' 


dx 
~ dt 




x 


-,- = 0, and x' x = 1, 
dt 






dx i ^x J dx 
'^dt df=^-dt' 






, dx 

^-""'dt 


~ = Illdx. 









drx 

He' 



If I depends only on x, tliis is the equation of vis viva 
For a mass 



^--SS=^(^^^^^'+^^'^ +•••)• 



Suppose ?,', /,' the parts of Z,, /, wliich arise from the mutual actions of a-, and .t^. 
Then, if ax^x^ be this mutual action where a does not depend on the time, 

l^' = 0.x.,. I,,' = ax, 
l^ \ dx^ + ?.,' rfx^ = a (xj , dx^ + a\ \dx^) = 0, 
so that the mutual action disappears. 
Considering only the external forces, 

dx I dx 



iS 



m 



dt dt 



= Spdx, 



and this is the equation of vis viva, which is therefore tnie for any space allowing the 
measure of distance. 

The motion of a particle under the action of a central force is given by 

mx -ji = Pxa, if P be a number. 



AUSDEIINUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 141 

Hence max -j-^ = 0, 

dx 
max -rr = const. 
at 

If V be velocity, p perpendicular on tangent, 

viv siu p = h, a constant. 



Imaginary and Flat Geometry. 
Instead of putting a|6 = cos^, we may put a!6 = cos^, where k is any number; 

K 

and we shall have three distinct cases according as k is real, infinite or imaginary. 

If k be imaginary, we may take it equal to the algebraic J —I, and put 

a\b = cosh 6 ; 
then ab\ah = 1 — cosli^ 6 = — sinh^ $. 

Take a line E of length 7 on ab such that sinh 7=1; 
then ab = sinh B.E, and E\E == - I. 

For any two lines E^, E^ of length 7 we may put 

EJE^ = - cos (j), 

where cf) is the angle between them. Then 

E^E^ = sm <f>. a, if a is the point of intersection. 

For hnes ab, ac of any lengths 0,, 6^, 

ab . ac = sinh 6^ sinh 6.^ sin <j>a, 
and therefore abc = sinh 6^ sinh 6, sin <f). 

The points | a, | 6 are imaginary. 

In ordinary geometry, making k infinite, a\b = I, and therefore this quantity does not 
give a measure of distance. 

But if a = /36 + 7c, 

the equation a 1 a = /3a | 6 + 7a | c 

gives 1 = /3 + 7, 

and hence (a — b) = y(c — a). 

For points on the same line therefore a-b, c-a are always in a numerical ratio, 
and we may take a — h proportional to the distance between a and b. 

Vol. XIII. Part II. 19 



14-2 Mk cox, on the APPLICATION OF QUATERNIONS AND GRASSjMANN'S 

Since ab = a(b — a), ac = it (c — a), 

ah. uc are also proportional to these distances. 

The quantity b— a is cjxlled by Grassmann a stroke (streike), and tlie quantity ab a 
line (Linientheil). 

A stroke may also be considered a point at an infinite distance, for since jSb + 7c 
represents a point dividing Ic internally, so that the distance ab is to ac in the ratio 
7 to /3; c — b will represent outside be, whose distance from c is equal to its distance 
from b, 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 6, a^ \ b^ = cos 6, so 
that a, a, = 1, t, t, = 1- Therefore a,6j | 0,6, = sin- ^ ; and, if ajj^ be of any length, and 
U be the product of two unit strokes in the same plane at right angles, 

a^b^ = a/S sin 6 U, 

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 

abc = a (6 — a) (c — a) = b [b — a) (c — a) = c [h — 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 pai'allelepiped with the planes for opposite faces. 

If a,, b^, c, be three strokes forming the sides of a triangle so that c, = i, — «,, then 

t'i i c. = «, I «, + ^-i ' /'. - 2a, i i, ; 



AUSDEHNUNGSLEHRE TO DIFFERENT KINDS OF UNIFORM SPACE. 143 

or, if «, I, c be the sides in magnitude, a, /3 the angles, 

c^ = a" + h" - 2ab cos c. 

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 hnes as well as their direction*. 

* A great portion of the "Ausdehnungslehre" is also analogous to Hamilton's linear and vector functions, and 
devoted to the algebraical mnltiplication of different quan- caUed by Grassmann fractions, are considered, 
titles which has not been mentioned here; also expressions 



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- 
cyclic functions which become prominent in Elliptic Integrals. 

Mode of Fo^-mation. This must be explained, in order to give confidence in their 
use. First, a table was constructed of e""" (to 16 decimals) with x an integer, by a 
method which afforded a systematic verification of the results. Put generally 

^±1+1:2*1:2:3 + *^' 
in which we may make h = l, or h=l, or h = '01, or A ='001 or any convenient small 
fraction. At starting, take h = l. To find e"^' when we know e""^' and e'", calculate 
the separate terms of 

^ 1^ " 1 + 1 72 ~ rr273 +1.2.3.4" '^°-j ' 

each term being derived by a simple division. Add in separate sums the odd terms 
(= il/) and the even terms {=N). If the work is done correctly, it ought to yield 
M + N=e'^'^ 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 M and iV 
are attained accurately for those 16. Any small error in the two last decimals (for some 
must exist) will be less in M — N than in M+N. We may then trust the result 
e'^^=M—N'. Thus from e~' we calculate e"^ by making x = l, and the test of accuracy 
is 31 + N=e'' = l. For the next step we make x = 2, and seek for e'^ from e'^, and our 
new test is M+ N = e~\ This being fulfilled, we attain e'* = M — N; and so on con- 
tinually. 

After this first skeleton table was finished in 37 entries, from e"' to e"" (since e'" 
does not yield any digits to 16 decimals), I proceeded to make h = '\, so as to calculate 
e"', c"'*, e'^ ... up to e""', with the same method of verification. But I found it con- 
venient first to halve the intervals, by making h = \, then from e""" I deduced simul- 
taneous e~^~'- and e-^+i. For instance, having deduced e"^'^ and e'^'^ from e"', I proceeded 
to deduce e'^'" and e'" 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. 

Vol. XIII. Part III. 20 



146 Mu F. W. NEWMAN'S TABLE 

Since I worked with 18 decimals, in order to got 16 always accurate, tlie 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. 

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, 
ami X for sum of even rows. Then M+N ought to make e""'^ accurately, if we are to 
trust M-X=e-^\ 

The error in M+N is naturally greater than in M—N; hence M+N varying from 
f"*' (as given) by only 4 in the ISth place, we have just confidence that M — N gives 
e~*^ correct to the 16th place. 

•0907 179.5 3289 4124 98 = 6"") 
•1002 5884 3722 8037 29 = 6""' 

10 ) 907 179.5 3289 4125 00 

20 ) -90 7179 o328 9412 50 

30 ) 4 5358 9766 4470 62 

40) -1511 96.58 8815 68 

50 ) 37 7991 4720 39 

60 ) - 7559 8294 40 

70 ) 125 9971 57 

80 ) - 1 7999 59 

90 ) 224 99 

100) -2 49 



3^ trustworthy for 16 places. 



i/ = 911 7192 1173 3512 59 
JV= 90 8692 2549 4524 66 



il/+iV=^1002 5884 3722 8037 25=6"", which is sufficiently correct. 
M-N= 820 8499 8623 8987 93=6''', which was souglit. 

A third step was, to make A ='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. Bat this table remains in MS., being 
superseded by a fourth (also carried to 12 decimals), in which h, the increment of x, 
is only •OOl. 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 
furmula, 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 x = 43, after which I fell back on my 
second table, and worked from it with 14 decimals, checking myself from the tliird table 
after every ten entries, and by my second after every hundred. 

Mode of using the tables. When x does not exceed 15'349, and the decimal part 
contains only tenths, hundredths and thousandths, the value of e"' wiU bo 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 
a; = 15'350 to a; = 17298 the values of a; 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 e""'", we have e-'""''=lC62 1229; e"'''" = 1658 8020 (each to 14 decimals), 
sum = 3320 9249, half sum = 1660 4624. Therefore e'""', true to 12 decimals, is 166046 
(six zeros must be prefixed). Proof of this rule. Given consecutive entries A = e'"*", 
3 = 6'°'", to find the intermediate U^e'" which is not in (this part of) the table. Here 
h = -00l, A and B begin with 6 zeros, therefore each is less than 10~°; so is /i". Now 

A = e-'.e^=u(l+l + J^;), B^e-.e- = u{l-\ + ^;\, 

since Uh' is less than 10"*"', or does not affect the loth decimal ; therefore 

K^ + -B) = CT (1 +ih^), whence U=i{A + B)(l + ^ A")'' = i(A + B){1 -ih' + ih + &c.). 
But fi' being < 10"^ l(A+B).^h:' is less than ^10"". Thus, true to 12 decimals, U=l(A + B), 
as was asserted. To obtain accuracy to 14 decimals, we must take account of the small 
factor |/(^ For this, divide the half sum by 2.10* and add the quotient to the half sum. 

After a; = 17-298, x increases by -00.5 at each step, as 17-300, 17305, 17310, 17-315... 
and the new question rises how to find an intermediate x. 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 x the form —a±h; where e"" is in the table, and h = '00l. Or again 
the X may end in 2 or 3, else in 7 or 8, so as to have the form —a±2h. Thus 
either by assuming y = a; + -001, else by assuming u = x±'002, y ot u will be in the 
series of the table (ending either in or in 5), and e~" or e"" will be known to us by 
the table. Either then x = y + k, and e'' = e'^ {1 + h), else x = u + 2k, e~' = e""(l + 2A). Each 
will be true to 16 decimals. 

Hence the Rule. If a; is a unit greater than y, subtract from e'" its thousandth 
part, to find e'^; or add, if x 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 x is less than y, add the 
five hundredth part. 

Examples. Given in the table 6-" = 104 6740, 6"^°°' = 104 1519, where y = 18-2,75. It 
is required to fill up for the intervals between and 5. Divide by 1000, and we get 1046 
and 1041. Subtract the former from e"" and add the latter to e-^'^, then e"""*=104 5694; 
e-"-"»=l04 2560; else, divide by 500, which yields 2093 and 2083. Subtract the former 
from e"*, and add the latter to 6-^°^ then 6-''"' = 104 4647, e""'"=104 3602. In these 
four new results, only the last (14th) figure is in error. 

20—2 



148 



JIr F. W. NEWMAN'S TABLE 



Table of e ' to eighteen decimal places {sixteen exact). 



X 


e- 


X 


e'' 


X 


e-" 


•I 


9048 3741 80359 59545 


5-1 


60 9674 65655 15637 


lo-i 


4107 95552 25302 


•2 


8187 3075 30779 81848 


5-2 


55 1656 44207 60774 


IO'2 


3717 03186 84128 


■3 


7408 1822 06817 17871 


5-3 


49 9159 39069 10218 


IO-3 


3363 30951 85721 


■4 


6703 2004 60356 39307 


5-4 


45 1658 09426 12670 


io'4 


3043 24830 08403 


•5 


6065 3065 97126 33423 


5-5 


40 8677 143S4 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 


57 


33 4596 54574 71272 


107 


2254 49379 13206 


■s 


4493 2896 41172 21599 


5-8 


•30 275s 47453 75S13 


10-8 


2039 95034 1 1 166 


"9 


4065 6965 97405 99120 


5-9 


27 3944 48187 68370 


io'9 


1845 82339 95777 


10 


3678 7944 11714 42321 


6-0 


24 7875 21766 66358 


II'O 


1670 17007 90246 


i-i 3328 710S 36980 79553 


6-1 


22 4286 77194 85802 


ii'i 


1511 23238 19857 


I-2 3OII 9421 I9I22 02096 


6-2 


20 2943 06362 95735 


1 1 -2 


1367 41960 65685 


i'3 2725 3179 30340 12603 


6-3 


18 3630 47770 28910 


11-3 


1237 29242 61791 


14 


2465 9696 39416 06475 


6-4 


16 6155 72731 73937 


11-4 


1119 54848 42595 


1-5 


2231 3016 01484 29829 


6-S 


15 0343 91929 77572 


"•5 


1013 00935 98631 


1-6 


2018 9651 79946 55407 


6-6 


13 6036 80375 47893 


II-6 


916 60877 36245 


17 


1826 8352 40527 34648 


67 


12 3091 19026 73481 


117 


829 38191 60755 


1-8 


1652 9888 82215 . 86535 


6-8 


II 1377 '51478 44802 


II-8 


750 45579 15075 


19 


1495 6861 92226 35054 


6-9 


10 077S 54290 4S510 


ivg 


679 04048 07381 


2-0 


1353 352S 32366 12691 


7-0 


9 1188 1965s 54515 


12-0 


614 42123 53327 


2 'I 


1224 5642 82529 81909 


7'i 


8 2510 49232 65905 


I2-I 


555 95132 41665 


2-2 


1108 0315 83623 33881 


7-2 


7 4658 58083 76681 


12-2 


503 04556 07114 


2 '3 


1002 5S84 37228 03731 


7-3 


6 7553 87751 93846 


12-3 


455 17444 63084 


24 


907 1795 32894 12500 


7-4 


6 1125 27611 29574 


12-4 


411 85SS7 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 4061 1 


12-6 


337 20152 34153 


27 


672 0551 27397 49761 


77 


4 5282 71828 86790 


12-7 


305 11255 58050 


2-8 


608 1006 26252 17961 


7-8 


4 0973 49789 79781 


12-8 


276 07725 72053 


29 


550 2322 00564 07225 


7-9 


3 7074 35404 59080 


12-9 


249 80503 25884 


3-0 


497 8706 83678 63943 


8-0 


3 3546 26279 02501 


13-0 


226 03294 06997 


31 


450 4920 23935 57806 


81 


3 0353 91380 78857 


13-1 


204 52306 24491 


3 "2 


407 6220 39783 66216 


8-2 


2 7465 35699 7213s 


13-2 


185 06011 97553 


33 


368 8316 74012 40006 


8-3 


2 4851 68271 07947 


i3'3 


167 44932 09446 


3 '4 


333 7326 99603 26081 


8-4 


2 2486 73241 78S44 


134 


151 51441 12156 


3 5 


301 0738 34223 18502 


8-5 


2 0346 83690 10644 


13-5 


137 09590 S6393 


3-6 


273 2372 24472 92561 


86 


I 8410 57936 67577 


13-6 


124 04950 79965 


37 


247 2352 64703 39390 


87 


I 6658 58109 87632 


137 


112 24463 65241 


3-8 


223 7077 18561 65595 


88 


I 5073 30750 95474 


138 


loi 56314 71020 


1 3'9 


202 4191 14458 04390 


8-9 


I 3638 89264 82038 


139 


91 89813 57913 


4-0 


183 1563 88887 34179 


9-0 


I 2340 98040 86675 


14-0 


83 15287 I9II9 


4'i 


165 7267 54017 61246 


91 


I 1166 58084 90111 


I4"i 


75 23982 99227 


4-2 


149 9557 68204 77705 


9-2 


I 0103 94018 37091 


14-2 


68 07981 34408 


' 43 


135 6855 90122 00932 


9'3 


9142 42314 78171 


14-3 


61 60116 26191 


j 4"4 


122 7733 99030 68440 


9"4 


8272 40655 56631 


144 


55 73903 69323 


1 ' 

1 4-5 


III 0899 65382 42306 


9-5 


7485 18298 87702 


! 14-5 


50 43476 62588 


4-6 


100 5183 57446 33583 


9-6 


6772 87364 90855 


i4'6 


45 63526 36810 


47 


90 9527 71016 95819 


97 


6128 34950 53224 


147 


41 29249 41607 


4-8 


82 2974 70490 20030 


9-8 


5545 15994 32180 


14-8 


37 36299 38007 


1 4-9 


74 4658 30709 24342 


99 


5017 46820 56176 


i4'9 


33 80743 48400 


1 S-o 


67 3794 69990 85467 


lO'O 


4539 99297 62485 


15-0 


30 59023 20519 



OF THE DESCENDING EXPONENTIAL. 
Table of e'" to eighteen decimal places {sixteen exact). 



149 



X 


e-" 


X 


e^ 


X 


e-' 


X 


e-' 


X 


e-' 


iS'i 


27 67918 65864 


20-I 


18650 08918 


25-1 


125 66332 


30-1 


84671 


35-1 


571 


15-2 


25 04516 37241 


20-2 


16875 29854 


25-2 


113 70489 


30-2 


76612 


35-2 


517 


15-3 


22 66180 12790 


20-3 


15269 40156 


25 '3 


102 88446 


30-3 


69323 


35'3 


467 


15-4 


20 50524 57575 


20'4 


13816 32570 


25-4 


93 09369 


30'4 


62725 


35 '4 


423 


15-5 


18 5S39I 36271 


20-5 


12501 52863 


25-5 


84 23462 


30-5 


56757 


35-5 


383 


rS-6 


16 78827 53003 


20'6 


11311 85098 


25-6 


76 21864 


30-6 


51356 


35-6 


346 


157 


IS 19065 96759 


207 


10235 38612 


257 


68 96548 


307 


46469 


357 


313 


iS-8 


13 74507 72802 


20-8 


9261 36038 


25-8 


62 40260 


30-8 


42047 


35-8 


283 


15-9 


12 43706 02371 


20'9 


8380 02554 


25-9 


56 46419 


30-9 


38044 


35-9 


256 


160 


II 25351 74726 


2I-0 


7582 56070 


26-0 


51 09089 


31-0 


34424 


36-0 


232 


i6-i 


10 18260 36937 


2I-I 


6860 98471 


26-1 


46 22985 


3I-I 


31149 


36-1 


210 


i6-2 


9 21360 08336 


21-2 


6208 07569 


26-2 


41 82968 


31-2 


28184 


36-2 


190 


i6-3 


8 33681 07883 


21-3 


5617 29937 


26-3 


37 84905 


31-3 


25502 


36-3 


172 


i6-4 


7 54345 83479 


21-4 


5082 74242 


26-4 


34 27424 


31-4 


23075 


36-4 


156 


i6-S 


6 82560 33757 


21-5 


4599 05558 


26-5 


30 98820. 


3i'S 


20878 


36-5 


141 


i6-6 


6 17606 13351 


2 I '6 


4161 39757 


26-6 


28 03927 


31-6 


18S91 


36-6 


128 


167 


5 58833 13920 


217 


3765 38823 


267 


25 37102 


317 


17094 


367 


n6 


168 


5 05653 13478 


21-8 


3407 06418 


26-8 


22 95663 


31-8 


15466 


36-8 


105 


i6-9 


4 57533 87708 


21-9 


3082 83916 


26-9 


20 72200 


31-9 


13994 


36-9 


95 


17-0 


4 13993 77202 


22-0 


2789 46822 


27-0 


18 79528 


32-0 


12662 


37'o 


86 


17-1 


3 74597 05575 


22-1 


2524 01519 


27-1 


17 00667 


32-1 


11457 






17-2 


3 38949 43271 


22"2 


2283 82340 


27-2 


15 3S828 


32-2 


10367 






17-3 


3 06694 12954 


22-3 


2066 48887 


27'3 


13 92387 


32-3 


9381 






17-4 


2 77508 32429 


22-4 


1869 83647 


27-4 


12 59884 


32-4 


8487 






17-5 


2 51099 91571 


22-5 


1691 89802 


27-5 


II 39991 


32-5 


7680 






17-6 


2 27204 59942 


2 2*6 


1530 89264 


27-6 


10 31506 


32-6 


6949 






177 


2 05583 22310 


227 


1385 20895 


277 


9 33346 


327 


6288 






17-8 


I 86019 39278 


22-8 


1253 38887 


27-8 


8 44526 


32-8 


5689 






17-9 


I 68317 30706 


22'9 


1134 11313 


27-9 


7 64157 


32-9 


5149 






i8-o 


I 52299 79752 


23-0 


1026 18800 


28-0 


6 91435 


33-0 


4658 






i8-i 


I 37806 55555 


23-1 


928 53333 


28-1 


6 25638 


33"i 


4215 






l8-2 


I 24692 52791 


23-2 


840 171 7 1 


28-2 


5 66101 


33'2 


3812 






i8-3 


I 12826 46525 


23-3 


760 21882 


28-3 


5 12231 


33'3 


3450 






i8-4 


I 02089 60750 


23-4 


687 87436 


28'4 


4 63485 


33"4 


3122 






i8-5 


92374 49702 


23-5 


622 41450 


28-5 


4 19376 


33-5 


2825 






186 


83583 90136 


23-6 


563 18394 


28-6 


3 79466 


33-6 


2556 






187 


75629 84148 


237 


509 58993 


287 


3 43356 


337 


2313 






i8-8 


68432 71049 


23-8 


461 09586 


28-8 


3 10681 


33-8 


2093 






i8'9 


61920 47706 


239 


417 21690 


28-9 


2 81116 


33"9 


1894 






19-0 


56027 96459 


24-0 


377 51347 


29-0 


2 54364 


34-0 


1715 






19-1 


50696 19869 


24-1 


341 58S31 


29-1 


2 30158 


34-1 


1552 






19-2 


45871 81754 


24-2 


309 08189 


29-2 


2 08255 


34-2 


1404 






i9'3 


41506 53683 


24-3 


279 66885 


29-3 


I 88442 


34'3 


1270 






19-4 


37556 66761 


24 '4 


253 05484 


29-4 


I 70511 


34'4 


1150 






i9'5 


33982 67815 


24-5 


228 97350 


29-5 


I 54280 


34-5 


1040 






19-6 


30748 79877 


24'6 


207 18380 


29'6 


I 39598 


34-6 


941 






197 


27822 66367 


247 


187 46766 


297 


I 26313 


347 


852 






198 


25174 98715 


24-8 


169 62776 


29-8 


I 14293 


34-8 


771 






19-9 


22779 27037 


24-9 


153 48556 


29'9 


I 03418 


34-9 


698 






20 '0 


20611 53619 


25-0 


138 87944 


30-0 


93576 


35'o 


631 







150 Mil F. ^V. NEWMAN'S TABLE. 



TABLE OF e-'. 

Part I. From x = to a;=15'349 at intervals of 001 to twelve decimal places; 
pp. 151—227. 

Part II. From a;=15350 to a: =17-298 at intervals of -002 and from ar= 17-300 to 
X = 27-633 at intervals of -005, to fourteen decimal places ; pp. 228 — 241. 



[ooo— 199] ^fR F. W. NEWMAN'S TABLE OF THE DESCENDING EXPONENTIAL. 151 



X 

ooo 


a-x 


X 


g-X 


X 


g-X 


X 


g-X 


I -oooo 


0000 0000 


■050 


•9512 


2942 4501 


•IO(^ 


•9048 3741 8036 


■150 


•8607 


0797 6425 


"OOI 


•9990 


0049 9833 


•051 


02 


7S67 0533 


•lOI 


39 33°3 2886 


■151 


■8598 


4769 8659 


•002 


80 


0199 8667 


•052 


•9493 


2886 6843 


•102 


30 2955 1669 


•152 


89 


8828 0741 


■003 


70 


0449 SS°4 


•053 


83 


8001 2482 


•103 


21 2697 3481 


■153 


81 


2972 1811 


•004 


60 


0798 9344 


■054 


74 


3210 6502 


•104 


12 2529 7421 


■154 


72 


7202 lOII 


■005 


5° 


1247 9193 


■055 


64 8514 7953 


•i°S 


03 2452 2586 


■155 


64 


1517 7483 


■006 


40 


1796 4054 


•056 


55 


3913 589° 


•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 1 


■009 


10 


4037 8773 


■059 


27 


0676 9157 


•109 


67 3041 7498 


•159 


29 


9635 8969 


010 


00 


4983 3749 


■060 


17 


6453 3584 


•iio 


58 3413 5296 


■160 


21 


4378 8966 


"Oil 


•9890 


6027 8774 


■061 


08 


2323 9777 


■III 


49 3874 8928 


•161 


12 


9207 1107 


•012 


80 


7171 2861 


•062 


•9398 


8288 6792 


•112 


40 4425 7500 


•162 


04 


4120 4540 


•013 


70 


8413 5018 


•063 


89 


4347 369° 


•113 


31 5066 0115 


•163 


•8495 


9118 8414 


•CI4 


60 


9754 4261 


•064 


80 


0499 9531 


■114 


22 5795 5882 


•164 


87 


4202 1880 


•015 


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 


■017 


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 


•019 


I I 


7936 2244 


•069 


Z2, 


2668 0079 


•119 


78 0780 0763 


■169 


45 


0890 3386 


■020 


01 


9867 3307 


•070 


23 


9381 9906 


•120 


69 2043 6717 


•170 


36 


6481 6596 


•021 


•9792 


1896 4570 


•071 


14 


6i8g 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 


42 6366 2561 


■173 


II 


3761 4844 


■024 


62 


857° 9758 


•074 


86 


7169 3842 


•124 


33 7984 0883 


•174 


02 


9689 765S 


•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 


40 


3992 4446 


•129 


89 7396 5546 


•179 


61 


0589 9396 


•030 


04 


4553 3548 


•080 


31 


1634 6388 


•130 


80 9543 0921 


•180 


52 


7021 1411 


■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 


03 


5114 7221 


•133 


54 6509 2108 


■183 


27 


6815 5737 


•034 


65 


7150 4637 


•084 


•9194 


3125 6096 


■134 


45 9006 4603 


■184 


19 


3580 3827 


•03s 


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 


I-037 


36 7613 5349 


•087 


66 


77°9 5634 


•137 


19 7022 6131 


•187 


•8294 


4373 6385 


•038 


27 


1294 0891 


•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 


•090 


39 


3118 5272 


•140 


•8693 5823 5399 


■190 


69 


5913 3943 


•041 


•9598 


2912 9947 


•091 


30 


1 7 71 0900 


•141 


84 8931 1699 


•191 


61 


3258 8151 


■042 


88 


6978 0572 


■092 


21 


0514 9546 


•142 


76 2125 6487 


•192 


53 


0686 8491 


•043 


79 


1 139 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 


•°45 


59 


9748 1833 


•095 


•9°93 


7293 4469 


•145 


50 2229 3111 


•195 


28 


3465 8056 


■046 


50 


4196 2191 


•096 


84 6401 6069 


•146 


41 5770 3185 


•196 


20 


1223 4678 


•047 


40 


8739 759° 


•097 


75 


5600 6134 


•147 


32 9397 7417 


•197 


II 


9063 3312 


•048 


31 


3378 7077 


•098 


66 


4890 3754 


•148 


24 31 1 1 4942 


•198 


03 


6985 3137 


•049 


21 


8112 9699 


•099 


57 


4270 8024 


•149 


15 69 1 1 4899 


•199 


■8195 


4989 3332 



15: 



MR F. \y. NEWMAN'S TABI,E 



[•200— -399] 



X 


g-X 


X 


e-x 


X 


f,-X 


X 


e-x 


•200 


•81S7 3075 307S 


•250 


•7788 0078 3071 


•300 


•7408 1822 0682 


•350 


•7046 8808 9719 


•201 


79 1243 1553 


•251 


80 2237 1559 


•301 


00 7777 2747 


1 -351 


39 8375 3856 


•202 


70 9492 7942 


•252 


72 4473 8o6g 


•302 


■7393 3806 4890 


■352 


32 8012 1977 


•203 


62 7S24 1425 


•253 


64 6788 1823 


•303 


85 9909 6371 


"353 


25 7719 3378 


•204 


54 6237 1187 


•254 


56 9180 2046 


•304 


78 60S6 6451 


•354 


J8 7496 7356 


•20s 


46 4731 6411 


•25s 


49 1649 7961 


•305 


71 2337 4392 


■355 


" 7344 3209 


•206 


38 3307 6282 


•256 


41 4196 8792 


•306 


63 8661 9456 


•356 


04 7262 0236 


•207 


30 1964 9987 


•257 


33 6821 3765 


■307 


56 5060 0907 


•357 


•6997 7249 7735 


•208 


22 0703 6711 


•258 


25 9523 2106 


•308 


49 1531 8009 


•358 


90 7307 5007 


•209 


13 9523 5643 


■259 


18 2302 3043 


■309 


41 8077 0026 


■359 


83 7435 1352 


•210 


05 8424 5970 


•260 


10 5158 5803 


•310 


34 4695 6224 


•360 


76 7632 6071 


•211 


•S097 7406 68S1 


•261 


02 8091 9614 


•311 


27 13S7 5869 


•361 


69 7899 8467 


•212 


89 6469 7567 


•262 


•7695 1102 3707 


•312 


19 8152 8228 


•362 


62 8236 7842 


•213 


81 5613 7217 


•263 


87 4189 7310 


•313 


12 4991 2569 


•363 


55 8643 3499 


■214 


73 4838 5023 


•264 


79 7353 9656 


■314 


05 1902 8159 


•364 


48 9119 4743 


•215 


65 4144 0177 


•265 


72 0594 9975 


•315 


•7297 8887 4269 


•365 


41 9665 0878 


•216 


57 3530 1873 


•266 


64 3912 7501 


•316 


90 5945 0167 


•366 


35 0280 1209 


•217 


49 2996 9305 


•267 


56 7307 1465 


•317 


83 3075 5126 


•367 


28 0964 5044 


•218 


41 2544 1666 


•268 


49 0778 1103 


•318 


76 0278 8415 


•368 


21 1718 1688 


•219 


33 2171 8153 


•269 


41 4325 5648 


•319 


68 7554 93°6 


•369 


14 2541 0450 


•220 


25 1879 7962 


•270 


33 7949 4337 


•320 


61 4903 7074 


•37° 


07 3433 0637 


•221 


17 1668 0290 


•271 


26 1649 6405 


•321 


54 2325 0990 


■371 


00 4394 1558 


' '222 


09 1536 4334 


•272 


18 5426 1090 


•322 


46 9819 0330 


•372 


■6893 5424 2524 


•223 


01 14S4 9294 


•273 


10 9278 7629 


■323 


39 7385 4368 


•373 


86 6523 2843 


•224 


■7993 15 13 4369 


•274 


03 3207 5261 


•324 


32 5024 2380 


•374 


79 7691 1828 


•-'25 


85 1621 8759 


•275 


•7595 7212 3225 


•325 


25 2735 3642 


•375 


72 8927 8790 


i '226 


77 1810 1665 


•276 


88 1293 0761 


•326 


18 0518 7432 


•376 


66 0233 3041 


1-227 


69 2078 2290 


•277 


80 5449 71 1 1 


•327 


10 8374 3027 


•377 


59 1607 3895 


•228 


61 2425 9835 


•278 


72 9682 1514 


•328 


03 6301 9705 


•378 


52 3050 0665 


•229 


53 2853 3505 


•279 


65 3990 3215 


■329 


•7196 4301 6747 


■379 


45 4561 2667 


•230 


45 3360 2503 


•280 


57 8374 1456 


•330 


89 2373 3432 


•380 


38 6140 9212 


•231 


37 3946 6035 


•281 


5° 2833 5481 


■331 


82 0516 9040 


•381 


31 77S8 9620 


■232 


29 4612 3307 


•282 


42 7368 4534 


•332 


74 8732 2854 


•382 


24 9505 3205 


•233 


21 5357 3524 


•283 


35 1978 7S61 


•333 


67 7019 4156 


•383 


18 1289 9286 


•234 


13 6181 5895 


•284 


27 6664 4707 


■334 


60 5378 2227 


•384 


II 3142 7179 


■23s 


05 7084 9628 


•28s 


20 1425 4319 


•335 


53 3808 6352 


•385 


04 5063 6204 


•236 


•7897 8067 3932 


•286 


12 6261 5946 


•336 


46 2310 5816 


•386 


•6797 7052 5680 


237 


89 9128 8018 


•287 


05 1172 8837 


•337 


39 0883 9903 


•387 


90 9109 4926 


■238 


82 0269 1094 


•288 


•7497 6159 2239 


•338 


31 9528 7898 


•388 


84 1234 3263 


■239 


74 1488 2373 


•289 


90 1220 5402 


•339 


24 8244 9089 


•389 


77 3427 0013 


•240 


65 2786 1066 


•290 


82 6356 7578 


•340 


17 7032 2763 


•390 


70 5687 4498 


•241 


58 4162 6388 


•291 


75 1567 8018 


•341 


10 5890 8206 


•391 


63 8015 6039 


•242 


5° 5617 7551 


•292 


67 6853 5974 


•342 


03 4820 4709 


■392 


57 °4ii 3960 


■243 


42 7151 3771 


■293 


60 2214 0697 


•343 


•7096 3821 1560 


•393 


50 2874 7586 


•244 


34 8763 4262 


■294 


52 7649 1443 


•344 


89 2892 8049 


•394 


43 5405 6240 


•24s 


27 0453 8241 


•29s 


45 3158 7466 


•345 


82 2035 3468 


•395 


36 8003 9249 


•246 


19 2222 4925 


•296 


37 8742 8020 


•346 


75 1248 7107 


•396 


30 0669 5937 


•247 


n 4069 3530 


•297 


30 4401 2362 


•347 


68 0532 8258 1 


•397 


23 3402 5633 


•248 


03 5994 3277 


•298 


23 0133 9748 


•348 


60 9887 6215 


•398 


16 6202 7662 


■249 


■7795 7997 3384 


■299 


15 5940 9435 


•349 


53 9313 0270 ; 


•399 


09 9070 1353 



[■400— -599] 



OF THE DESCENDING EXPONENTIAL. 



153 



X 


Q-X 


X 


g-X 


X 


Q-X 


X 


g-X 


•400 


•6703 2004 


6036 


■45° 


■6376 


2815 1622 


•500 


•6065 3065 9713 


•55° 


•5769 4981 0380 


•401 


•6696 5006 


1038 


■451 


69 


9084 2178 


■5°i 


59 


2443 2218 


•551 


63 7314 8948 


•402 


8g 8074 5690 


■452 


63 


5416 9725 


■502 


53 


1881 0647 


•552 


57 9706 3890 


•403 


83 1209 


9323 


■453 


57 


1813 3626 


•503 


47 


1379 4395 


•553 


52 2155 4629 


•404 


76 4412 


1269 


•454 


50 


8273 3246 


•5°4 


41 


0938 2856 


•554 


46 4662 0589 


■405 


69 7681 


0858 


•455 


44 


4796 7948 


■505 


35 


0557 5427 


•555 


40 7226 1196 


■406 


63 1016 


7425 


•456 


38 


1383 7098 


•506 


29 


0237 1504 


•556 


34 9847 5876 


•407 


56 4419 


0301 


■457 


31 


8034 0063 


■507 


22 


9977 0483 


•557 


29 2526 4053 


•408 


49 7887 


8822 


•458 


25 


4747 6207 


•508 


16 


9777 1762 


•558 


23 5262 5157 


•409 


43 1423 


2322 


•459 


19 


1524 4899 


■509 


10 


9637 4739 


•559 


17 8055 8612 


•410 


36 5025 


0136 


•460 


12 


8364 55°7 


•510 


04 


9557 8812 


■560 


12 0906 3849 


•411 


29 8693 


1600 


•461 


06 


5267 7398 


•511 


•5998 


9538 3381 


•561 


06 3814 0294 


•412 


23 2427 


6052 


•462 


CO 


2233 9942 


•512 


92 


9578 7845 


■562 


00 6778 7378 


•413 


16 6228 


2827 


■463 


•6293 


9263 2508 


•513 


86 


9679 1605 


•563 


•5694 9800 4529 


•414 


10 0095 


1265 


■464 


87 6355 4467 1 


■514 


80 


9839 4062 


•564 


89 2879 1178 


•415 


03 4028 


0705 


•465 


81 


3510 5190 


■515 


75 


0059 4618 


•565 


83 6014 6757 


•416 


•6596 8027 


0484 


•466 


75 


0728 4048 


•516 


69 


0339 2674 


•566 


77 9207 0695 


■417 


90 2091 


9944 


•467 


68 


8009 0413 


■517 


63 


0678 7634 


•567 


72 2456 2426 


■418 


83 6222 


8425 


•468 


62 


5352 3658 


■518 


57 


1077 8900 


•568 


66 5762 1381 


•419 


77 0419 


5268 


■469 


56 


2758 3157 


•519 


51 


1536 5877 


■569 


60 9124 6994 


•420 


70 4681 


981S 


•470 


5° 


0226 8283 


•520 


45 


2054 7970 


•570 


55 2543 8699 


-421 


63 9010 


1409 


•471 


43 


7757 8412 


■521 


39 


2632 4583 


•571 


49 6019 5928 


•422 


57 3403 


9393 


•472 


37 


5351 2918 


•522 


ij 


3269 5123 


•572 


43 9551 8118 


•423 


50 7863 


3112 


■473 


31 


3007 II 78 


■523 


27 


3965 899s 


•573 


38 3140 4704 


■424 


44 2388 


1909 


■474 


25 


0725 2568 


•524 


21 


4721 5607 


•574 


32 6785 5121 


•425 


37 6978 


5130 


■475 


18 


8505 6465 


•525 


15 


5536 4366 


•575 


27 0486 8806 


•426 


31 1634 


2120 


•476 


12 


6348 2247 


•526 


09 


6410 4681 


•576 


21 4244 5196 


•427 


24 6355 


2228 


•477 


06 


4252 9293 


•527 


03 


7343 5960 


•577 


IS 8058 3728 


•428 


18 1141 


4798 


•478 


00 


2219 6982 


■528 


■5897 8335 7612 


•578 


10 1928 3841 


■429 


II 5992 


9181 


•479 


•6194 


0248 4693 


■529 


91 


9386 9048 


•579 


04 5854 4974 


■430 


05 0909 


4723 


•480 


87 8339 1806 


•530 


86 


0496 9678 


•580 


•5598 9836 6565 


■431 


•6498 5891 


0774 


■481 


81 


6491 7703 


■531 


80 


1665 8912 


■581 


93 3874 8054 


■432 


92 0937 


6685 


•482 


75 


4706 1764 


■532 


74 


2893 6164 


•582 


87 7968 8882 


■433 


85 6049 


1804 


•483 


69 


2982 3373 


•533 


68 


4180 0844 


•583 


82 2118 8490 


■434 


79 1225 


5485 


•484 


63 


1320 1912 


•534 


62 


5525 2366 


•584 


76 6324 6319 


•435 


72 6466 7078 


•48s 


56 


9719 6764 


•535 


56 6929 0144 


•585 


71 0586 1811 


■436 


66 1772 


5935 


•486 


50 


8180 7313 


•536 


5° 


8391 3591 


•586 


65 49°3 4410 


•437 


59 7143 


1410 


•487 


44 


6703 2944 


•537 


44 


9912 2122 


•587 


59 9276 3557 


■438 


S3 2578 


2857 


•488 


38 


5287 3042 


•538 


39 


1491 5152 


■588 


54 3704 8697 


•439 


46 8077 


9629 


•489 


32 


3932 6993 


•539 


3i 


3129 2097 


•589 


48 8188 9274 


•440 


40 3642 


1083 


•490 


26 


2639 4184 


■540 


27 


4825 2374 


•59° 


43 2728 4734 


•441 


zz 9270 


6573 


•491 


20 


1407 4001 


■541 


21 


6579 5399 


•591 


37 7323 4521 


•442 


27 4963 


5455 


•492 


14 


0236 5832 


•542 


15 


8392 0589 


•592 


32 1973 8081 


■443 


21 0720 


7087 


■493 


07 


9126 9065 


•543 


10 


0262 7363 


•593 


26 6679 4858 


■444 


14 6542 


0826 


■494 


or 


8078 3090 


•544 


04 


2191 5140 


•594 


21 1440 4306 


■445 


08 2427 


6032 


■495 


■6095 


7090 7296 


•545 


■5798 


4178 3339 


•595 


15 6256 5867 


■446 


oi 8377 


2061 


•496 


89 


6164 1073 


•546 


92 


6223 1380 


•596 


10 1127 8990 


•447 


■6395 4390 


8274 


■497 


83 


5298 3811 


•547 


86 


8325 8683 


•597 


04 6054 3125 


■448 


89 0468 


4032 


•498 


77 


4493 49°2 


•548 


81 


0486 4670 


•598 


•5499 i°3S 7721 


•449 


82 6609 8694 


•499 


71 


3749 3739 


•549 


75 


2704 8761 


•599 


93 6072 2226 



V.L. XIII. Part III. 



21 



1j4 



MR F. W. NEWMAN'S TABLE 



[•600—799] 



X 


g-X 


X 


Q-X 


X 


(,-X 


X 


Q-X 


•600 


•5488 1 1 63 6094 


•650 


■5220 4577 6761 


•700 


•4965 8530 3791 


•750 


•4723 6655 2741 


•601 


82 6309 8772 


•651 


15 2399 1920 


■701 


60 8896 6697 


•751 


18 9442 2293 


•602 


77 1510 9714 


■652 


10 0272 8603 


•702 


55 9312 5692 


■752 


14 2276 3739 


•603 


71 6766 8370 


•653 


04 8198 6289 


•703 


50 9778 0281 


■753 


09 5157 6608 


•604 


66 2077 4194 


•654 


•5199 6176 4457 


•704 


46 0292 9967 


■754 


04 80S6 0429 


•605 


60 7442 6639 


•65s 


94 4206 2587 


•705 


41 0S57 4256 


•75s 


00 I061 4730 


•606 


55 2S62 5159 


•656 


89 22S8 0159 


•706 


36 1471 2653 


■756 


•4695 4083 9043 


•607 


49 8336 9207 


•657 


84 0421 6654 


•707 


31 2134 4666 


•757 


90 7153 2896 


•608 


44 3865 8238 


•65 s 


78 8607 1553 


1 708 


26 2846 9800 


•758 


86 0269 5820 


•609 


38 9449 1709 


■659 


73 6844 4338 


•709 


21 3608 7562 


•759 


81 3432 7348 


•6io 


33 5086 9074 


•660 


68 5133 4492 


•710 


16 4419 7461 


•760 


76 6642 7010 


•611 


28 0778 9790 


■66r 


63 3474 1497 


■711 


II 5279 9004 


•761 


71 9899 4338 


•612 


22 6525 3314 


•662 


58 1866 4837 


•712 


06 6189 1699 


■762 


67 3202 8865 


•613 


17 2325 9103 


•663 


53 0310 3995 


•713 


01 7147 5057 


•763 


62 6553 0125 


•614 


11 8i8o 6615 


•664 


47 8805 8457 


■714 


•4896 8154 8586 


•764 


57 9949 7650 


•615 


06 40S9 5309 


•66s 


42 7352 7707 


•715 


91 9211 1796 


•76s 


Si 3393 0974 


•616 


01 0052 4644 


•666 


37 5951 1230 


•716 


87 0316 4199 


■766 


48 6882 9633 


•617 


■5395 6069 4080 


•667 


32 4600 8513 


•717 


82 1470 5305 


•767 


44 0419 3160 


•618 


90 2140 3076 


•668 


27 3301 9042 


•718 


77 2673 4626 


•768 


39 4002 1092 


•619 


84 8265 1094 


•669 


22 2054 2304 


•719 


72 3925 1673 


•769 


34 7631 2963 


•620 


79 4443 7595 


•670 


17 0857 77S7 


■720 


67 5225 5960 


•770 


30 1306 8311 


•621 


74 0676 2040 


•671 


II 9712 4978 


•721 


62 6574 6999 


•771 


25 502S 6672 


•622 


68 6962 3892 


•672 


06 8618 3367 


•722 


57 7972 4304 


•772 


20 8796-7584 


•623 


63 3302 2613 


•673 


01 7575 2442 


■723 


52 9418 7388 


•773 


16 2611 0583 


•624 


57 9695 7668 


•674 


•5096 6583 1692 


•724 


48 0913 5767 


■774 


II 6471 5208 


•625 


52 6142 8519 


•675 


91 5642 0608 


•725 


43 2456 S955 


•775 


07 0378 0999 


•626 


47 2643 4632 


•676 


86 4751 S681 


•726 


38 4048 646S 


•776 


02 4330 7493 


•627 


41 9197 5471 


•677 


81 3912 5401 


•727 


ZT, 568S 7821 


•777 


■4597 8329 4230 


•628 


36 5805 0503 


•678 


76 3124 0261 


•728 


28 7377 2531 


•778 


93 2374 0751 


•629 


31 2465 9193 


•679 


71 2386 2752 


•729 


. 23 9114 0115 


•779 


88 6464 6595 


•630 


25 9180 1007 


•680 


66 1699 2366 


•730 


19 0S99 0090 


•780 


84 0601 1305 


•631 


20 5947 5413 


•681 


61 1062 8598 


•731 


14 2732 1974 


•781 


79 4783 4420 


•632 


15 276S 1879 


•682 


56 0477 0940 


■732 


09 4613 5285 


•782 


74 9°ii 5483 


•633 


09 9641 9872 


•683 


50 9941 8S87 


•733 


04 6542 9543 


•783 


70 3285 4037 


•634 


04 6568 8862 


•684 


45 9457 1934 


•734 


■4799 8520 4267 


■784 


65 7604 9623 


•63s 


■5299 3548 8318 


•685 


40 9022 9575 


•735 


95 0545 8975 


•785 


61 1970 1785 


•636 


94 0581 7709 


•686 


35 8639 1307 


•736 


90 2619 3189 


•786 


56 6381 0067 


•637 


88 7667 6506 


•687 


30 8305 6625 


•737 


85 4740 6429 


•787 


52 0837 4013 


•638 


83 4806 4179 


•688 


25 8022 5026 


•738 


80 6909 8216 


•788 


47 5339 3168 


■639 


78 1998 020I 


•689 


20 7789 6007 


•739 


75 9126 8073 


•789 


42 9886 7075 


•640 


72 9242 4043 


■690 


15 7606 9066 


•740 


71 1391 5521 


•790 


38 4479 5282 


•641 


67 6539 5178 


•691 


10 7474 3701 


■741 


66 3704 0083 


•791 


3i 9117 7334 


•642 


62 3889 3077 


•692 


05 7391 94 I I 


•742 


61 6064 1282 


•792 


29 3801 2776 


•643 


57 1291 7216 


•693 


00 7359 5696 


■743 


56 8471 8642 


•793 


24 8530 1157 


•644 


51 8746 7068 


•694 


•4995 7377 2053 


•744 


52 0927 1686 


■794 


20 3304 2023 


•645 


46 6254 2107 


•695 


90 7444 7985 


•745 


47 3429 9940 


•795 


15 8123 4922 


•646" 


41 3814 1809 


•696 


85 7562 2991 


•746 


42 5980 2928 


•796 


II 2987 9403 


•647 


36 1426 5649 


■697 


80 7729 6573 


■747 


37 8578 0176 


•797 


06 7897 5013 


•648 


30 9091 3103 


•698 


75 7946 8232 


•748 


33 1223 1209 


■798 


02 2852 1302 


•649 


25 6808 3648 


•699 


70 8213 7471 


■749 


28 3915 5556 ' 


•799 


■4497 7851 7820 



[•8oo— '999] 



OF THE DESCENDING EXPONENTIAL. 



155 



X 


Q-X 


X 


e~x 


X 


g-a; 


X 


C-x 


■800 


•4493 2896 


4117 


•850 


■4274 1493 


1949 


•900 


•4065 6965 9740 


■950 


•3867 4102 3454 


•801 


88 7985 


9742 


•851 


69 8773 


0653 


•901 


61 


6329 3298 


■951 


63 5447 5737 


•802 


84 3120 


4248 


•852 


65 6095 


6345 


•902 


57 


5733 3018 


•952 


59 ('831 4374 


•803 


79 8299 


7184 


•853 


61 3460 


8598 


•9°3 


53 


5177 8496 


•953 


55 8253 8979 


•804 


75 3523 


8104 


•854 


57 0868 


6986 


•904 


49 


4662 9326 


•954 


51 9714 9167 


•805 


70 8792 


6559 


•855 


52 8319 


1082 


•905 


45 


4188 5103 


•955 


48 1214 4552 


•806 


66 4106 


2102 


•856 


48 5812 


0462 


•906 


41 


3754 5421 


■956 


44 2752 4749 


•807 


61 9464 


4286 


•857 


44 3347 


4700 


■907 


37 


3360 9877 


•957 


40 4328 9374 


•808 


57 4867 


2665 


•858 


40 0925 


3371 


•908 


33 


3007 8067 


•958 


36 5943 8043 


•809 


53 0314 


6792 


•859 


35 8545 


6052 


•909 


29 


2694 9587 


•959 


32 7597 0371 


•810 


48 5806 


6223 


•860 


31 6208 


2318 


•910 


25 


2422 4034 


•960 


28 9288 5975 


•811 


44 1343 


0512 


•86r 


27 3913 


1746 


•911 


21 


2190 1005 


■961 


25 1018 4472 


■812 


39 6923 


9214 


■862 


23 1660 


3914 


•912 


17 


1998 0098 


•962 


21 2786 5479 


•813 


35 2549 


1885 


■863 


18 9449 


8398 


"913 


13 


1846 0911 


■963 


17 4592 8613 


•814 


30 8218 


80S2 


•864 


14 7281 


4776 


•914 


09 


1734 3042 


•964 


13 6437 3494 


•815 


26 3932 


7361 


•86s 


10 515s 


2628 


■915 


05 


1662 6091 


•965 


09 8319 9739 


■816 


21 9690 


9280 


■866 


06 3071 


1530 


•916 


01 


1630 9656 


•966 


06 0240 6967 


•817 


17 5493 


3395 


•867 


02 1029 


1064 


•917 


•3997 


1639 3338 


•967 


02 2199 4798 


•818 


13 1339 


9266 


•868 


•4197 9029 


0808 


■918 


93 


1687 6736 


•968 


•3798 4196 2851 


•819 


08 7230 


6450 


•869 


93 7°7i 


0343 


•919 


89 


1775 9451 


■969 


94 6231 0746 


•820 


04 3165 


4506 


■870 


89 5154 


9248 


•920 


85 


1904 1084 


•970 


90 8303 8103 


•821 


■4399 9144 


2994 


■871 


85 3280 


7105 


•921 


81 


2072 1236 


•971 


87 0414 4543 


•822 


95 5167 


1473 


•872 


81 1448 


3494 


•922 


77 


2279 9509 


■972 


83 2562 9688 


•823 


91 1233 


9505 


•873 


76 9657 


7998 


•923 


73 


2527 5505 


•973 


79 4749 3158 


•824 


86 7344 


6648 


•874 


72 79°9 


0199 


•924 


69 


2814 8826 


•974 


75 6973 4575 


•825 


82 3499 


2465 


•875 


68 6201 


9679 


•925 


65 


3141 9°75 


•975 


71 9235 3563 


■826 


77 9697 6517 1 


■876 


64 4536 


6021 


•926 


61 


3508 5856 


•976 


68 1534 9743 


■827 


73 5939 


S366 


•877 


60 2912 


8808 


•927 


57 


3914 8771 


•977 


64 3872 2738 


•828 


69 2225 


7574 


•878 


56 I3.^o 
51 9-90 


7624 


•928 


53 


4360 7426 


•978 


60 6247 2172 


•829 


64 8555 


3705 


•879 


2054 


•929 


49 


4846 1425 


•979 


56 8659 7668 


•830 


60 4928 


6321 


•880 


47 8291 


1682 


•930 


45 


5371 0372 


•980 


53 1109 8851 


•831 


56 1345 


4987 


•881 


43 6833 6093 


•931 


41 


5935 3873 


■981 


49 3597 5345 


■832 


51 7805 


9266 


•882 


39 5417 


4872 


•932 


37 


6539 1533 


•982 


45 6122 6775 


■833 


47 4309 


8723 


•883 


35 4042 


7605 


•933 


33 


7182 2958 


•983 


41 8685 2767 


•834 


43 0857 


2924 


•884 


31 2709 


3878 


■934 


29 


7864 7756 


•984 


38 1285 2945 


■835 


38 7448 


1433 


•885 


27 1417 


3279 


•935 


25 


8586 5532 


•985 


34 3922 6936 


•836 


34 4082 


3816 


•886 


23 0166 


5394 


•936 


21 


9347 5894 


•986 


30 6597 4367 


•837 


30 °759 


9641 


■887 


18 8956 9811 


•937 


18 


0147 8449 


•987 


26 9309 4863 


•838 


25 7480 


8472 


•888 


14 7788 


6117 


•938 


14 


0987 2806 


•988 


23 2058 8053 


•839 


21 4244 


9879 


•889 


10 6661 


3901 


•939 


10 


1865 8573 


■989 


19 4845 3563 


•840 


17 1052 


3429 


■890 


06 5575 


2752 


■940 


06 


2783 5358 


•990 


15 7669 1022 


■841 


12 7902 


8689 


•891 


02 4530 


2259 


•941 


02 


3740 2772 


•991 


12 0530 0057 


•842 


08 4796 


5228 


•892 


•4098 3526 


2011 


•942 


•3898 4736 0423 


■992 


08 3428 0298 


•843 


04 1733 


2615 


•893 


94 2563 


1598 


■943 


94 


5770 7921 


•993 


04 6363 1373 


•844 


•4299 8713 


0419 


•894 


90 1641 


o6n 


•944 


90 


6844 4877 


•994 


00 9335 2912 


•845 


95 5735 


8211 


•895 


86 0759 


8640 


•945 


86 


7957 09°2 


■995 


•3697 2344 4544 


■846 


91 2801 


5560 


•896 


81 9919 


5277 


•946 


82 


9108 5606 


•996 


93 539° 5899 


•847 


86 9910 


2036 


•897 


77 9120 


0114 


•947 


79 


0298 8601 


■997 


89 8473 6609 


•848 


82 7061 


7212 


•898 


73 8361 


2741 


•948 


75 


1527 9499 


•998 


86 1593 6303 


•849 


78 4256 


0659 


•899 


69 7643 


2752 


•949 


71 


2795 7913 


•999 


82 4750 4613 



21—2 



156 



MR F. W. NEWMAN'S TABLE 



[I'ooo — I 199] 



X 


c-^ 


X 


(,-X 


1 
X 


g-.r 


X 


e-x 


i"ooo 


-367S 7944 1171 


1-050 


■3499 3774 91" 


i-ioo 


-3328 7108 3698 


1-150 


•3166 3676 9379 


I"OOI 


5 1174 5608 


1-051 


5 8798 6272 


i-ioi 


5 3837 8995 


1-151 


3 2029 0875 


I 002 


I 4441 7557 


1-052 


2 3857 3°22 


1 1-102 


2 0600 6830 


1-152 


0412 8691 


1-003 


-3667 7745 6651 


I '053 


•3488 8950 9010 


, I-I03 


-3318 7396 6870 


1-153 


-3156 8828 2512 


1-004 


4 1086 2522 


1-054 


5 4079 3887 


1-104 


5 4225 8785 


I-IS4 


3 7275 2021 


1-005 


4463 4804 


I "055 


1 9242 7306 


ii-i°5 


2 1088 2242 


1-155 


5753 6903 


1-006 


•3656 7877 3130 


1-056 


-3478 4440 8917 


i-io6 


-3308 7983 6910 


1-156 


•3147 4263 6842 


1-007 


3 13-7 7136 


I •05 7 


4 9673 8372 


1 i'io7 


5 4912 2458 


1-157 


4 2805 1525 


1-008 


-3649 4814 6454 


1-058 


I 4941 5324 


i-io8 


2 1873 8555 


1-158 


I 1378 0635 


1-009 


5 8338 0721 


1-059 


-3468 0243 9426 


1-109 


-3298 8868 4871 


1-159 


•3137 9982 3859 


i-oio 


2 1897 9571 


I -060 


4 5581 0330 


i-iio 


5 5896 1075 


1-160 


4 8618 0883 


I -oil 


•363S 5494 2640 


1061 


1 0952 7690 


iiii 


2 2956 6839 


1-161 


I 7285 1393 


I-OI2 


4 9126 9565 


1-062 


■3457 6359 1159 


1-112 


-3289 0050 1832 


1-162 


-3128 5983 5075 


I -013 


1 2795 9980 


1-063 


4 1800 0392 


1-113 


5 7176 5725 


1-163 


5 4713 1618 


1-014 


•3627 6501 3524 


1-064 


7275 5°43 


1-114 


2 4335 8191 


1-164 


2 3474 0708 


1-015 


4 0242 9832 


1-065 


-3447 2785 4767 


1-115 


-3279 1527 8900 


1-165 


-3119 2266 2033 


I-0I6 


4020 8543 


1-066 


3 8329 9219 


1-116 


5 8752 7524 


1-166 


6 1089 5280 


1-017 


-3616 7834 9295 


1-067 


3908 8054 


I-II7 


2 6010 3735 


1-167 


2 9944 0138 


IOI8 


3 1685 1724 


I -068 


■3436 9522 0928 


1-118 


•3269 3300 7207 


i-i68 


-3109 8829 6296 


I-0I9 


•3609 5571 5471 


1-069 


3 5169 7497 


1-119 


6 0623 7612 


1-169 


6 7746 3442 


1-020 


5 9494 0173 


1-070 


0851 7419 


1-120 


2 7979 4623 


1-170 


3 6694 1266 


I -02 1 


- 3452 5470 


1-071 


-3426 6568 0348 


I-I21 


-3259 5367 7914 


1-171 


5672 9456 


1-022 


■3598 7447 1002 


1-072 


3 2318 5944 


1-122 


6 2788 7158 


1-172 


-3097 4682 7703 


1-023 


5 1477 6408 


1-073 


•3419 8103 3862 


1-123 


3 0242 2031 


I-I73 


4 3723 5697 


1-024 


I 5544 1329 


1-074 


6 3922 3762 


I-I24 


-3249 7728 2206 


1-174 


1 2795 3129 


1-025 


■3587 9646 5406 


I -075 


•3412 9775 5301 


1-125 


6 5246 7358 


1-175 


-3088 1897 9688 


I 02 6 


4 3784 8279 


1-076 


-3409 5662 8137 


1-126 


3 2797 7163 


1-176 


5 1031 5066 


1-027 


7958 9590 


1-077 


6 1584 1931 


1-127 


0381 1296 


1-177 


2 0195 8955 


1-028 


■3577 2168 8980 


1-078 


2 7539 6340 


1-128 


-3236 7996 9433 


1-178 


•3078 9391 1046 


1-029 


3 6414 6093 


1-079 


-3399 3529 i°25 


1-129 


3 5645 1249 


1-179 


5 8617 1030 


1-030 


0696 0569 


i-o8o 


5 9552 5645 


1-130 


3325 6422 


1-180 


2 7873 8601 


1-031 


•3566 5013 2052 


1-081 


2 5609 9860 


1-131 


■3227 103S 4628 


i-i8i 


•3069 7161 3451 


1032 


2 9366 0186 


1-082 


-3389 1701 3332 


1-132 


3 87S3 5545 


1-182 


6 6479 5272 


I '033 


■3559 3754 4613 


1-083 


5 7826 5721 


1-133 


6560 8850 


1-183 


3 5828 3758 


I •034 


5 8178 4978 


1-084 


2 3985 6688 


I-I34 


-3217 4370 4220 


1-184 


5207 8602 


1035 


2 2638 0925 


1-085 


-3379 0178 5895 


1-135 


4 22X2 1334 


1-185 


-3057 4617 9499 


1-036 


■3548 7133 2098 


1-086 


5 6405 3004 


1-136 


I 0085 9870 


1-186 


4 4058 6141 


1037 


5 1663 8142 


1-087 


2 2665 7676 


1-137 


-3207 7991 9507 


1-187 


1 3529 8225 


1-038 


I 6229 8704 


1-088 


-3368 8959 9576 


1-138 


4 5929 9924 


1-188 


■3048 3°3i 5443 


1-039 


•3538 0831 3427 


1-089 


5 5287 8365 


I-I39 


I 3900 0801 


1-189 


5 2563 7492 


1-040 


4 5468 1959 1 


1-090 


2 1649 3707 


1-140 


-3198 1902 1816 


1-190 


2 2126 4067 


I -04 1 


I 0140 3945 1 


1-091 


-3358 8044 5266 


1-141 


4 9936 2650 


1-191 


-3039 1719 4863 


1042 


-3527 4847 9033 


1-092 


5 4473 2705 


1-142 


I 8002 2984 


1-192 


6 1342 9576 


1043 


3 9590 6870 


1-093 


2 0935 5688 


1-143 


•3188 6100 2498 


1-193 


3 0996 7902 


1-044 


4368 7102 


1-094 


-3348 7431 3881 


1-144 


5 4230 0873 


1-194 


0680 9539 


I -045 


-3516 9181 9378 


1-095 


5 3960 6949 


1-145 


2 2391 7790 


I-I95 


•3027 0395 4182 


1-046 


3 4030 3346 


1-096 


2 0523 4556 


1-146 


•3179 0585 2931 


1-196 


4 0140 1530 


I -047 


•3509 8913 8654 


1-097 


•3338 71 19 6368 


1-147 


5 8810 5978 


1-197 


9915 1278 


I 048 


6 3832 4952 


1-098 


5 3749 2052 


1-148 


2 7067 6613 


1-198 


-3017 9720 3126 


1-049 


2 8786 1887 


1099 


2 0412 1273 


1-149 


•3169 5356 4519 


1-199 


4 9555 6771 



•399] 



OF THE DESCENDING EXPONENTIAL. 



157 



X 


e-x 


X 


g-a; 


X 


C-x 


X 


e-x 


I'200 


•3011 9421 1912 


1-250 


-2865 0479 6860 


1-300 


■2725 3179 3034 


1-350 


-2592 4026 0646 


I'20I 


-3008 9316 8247 


1-251 


2 1843 5268 


1-301 


2 5939 7462 


I -35 1 


•2589 81 14 9962 


I'202 


5 9242 5475 


1-252 


■2859 3235 9894 


1-302 


-2719 8727 4149 


1-352 


7 2229 8260 


I-203 


2 9198 3296 


1-253 


6 4657 0453 


i-3°3 


7 1542 2823 


1-353 


4 6370 5279 


1-204 


•2999 9184 1409 


1-254 


3 6106 6658 


1-304 


4 4384 3212 


1-354 


2 0537 0763 


1-205 


6 9199 9513 


1-255 


7584 8224 


1-305 


I 7253 5046 


1-355 


•2579 4729 4452 


1-206 


3 9245 7310 


1-256 


-2847 9°9i 4867 


1-306 


•2709 0149 8052 


1-356 


6 8947 6088 


1-207 


9321 4499 


1-257 


5 0626 6300 


1-307 


6 3073 1959 


I -35 7 


4 3191 5414 


1-208 


-2987 9427 0781 


1-258 


2 2190 2239 


1-308 


3 6023 6498 


1-358 


1 7461 2171 


1-209 


4 9562 5858 


1-259 


-2839 3782 2401 


1-309 


9001 1396 


I-3S9 


•2569 1756 6104 


I-2I0 


I 9727 943° 


1-260 


6 5402 6500 


1-310 


•2698 2005 6385 


1-360 


6 6077 6953 


I-2II 


•2978 9923 1199 


1-261 


3 7051 4253 


1-311 


5 5037 1194 


1-361 


4 0424 4464 


1-212 


6 0148 0868 


1-262 


8728 5377 


1-312 


2 8095 5553 


1-362 


I 4796 8379 


I-2I3 


3 0402 8138 


1-263 


•2828 0433 9588 


1-313 


1180 9193 


1-363 


•2558 9194 8442 


I '2 14 


0687 2713 


1-264 


5 2167 6603 


1-314 


-2687 4293 1845 


1-364 


6 3618 4397 


1-215 


•2967 1001 4294 


1-265 


2 3929 6140 


1-315 


4 7432 3239 


1-365 


3 8067 5988 


I-216 


4 1345 2585 


1-266 


•2819 5719 7917 


1-316 


2 0598 3109 


1-366 


1 2542 2960 


1-217 


1 1718 7290 


1-267 


6 7538 1651 


1-317 


-2679 3791 "84 


1-367 


■2548 7042 5057 


I-218 


•2958 2121 8112 


1-268 


3 9384 7060 


1-318 


6 7010 7197 


1-368 


6 1568 2025 


I-219 


5 2554 4755 


1-269 


1 1259 3863 


1-319 


4 0257 0880 


1-369 


3 6119 3608 


I-220 


2 3016 6924 


1-270 


•2808 3162 1778 


1-320 


I 3530 1966 


1-370 


I 0695 9553 


I-22I 


-2949 3S°8 4323 


1-271 


5 5093 0525 


1-321 


-2668 6830 0187 


1-371 


-2538 5297 9604 


1-222 


6 4029 6657 


1-272 


2 7051 9823 


1-322 


6 0156 5277 


1-372 


5 9925 3509 


1-223 


3 4580 3631 


1-273 


-2799 9038 9392 


1-323 


3 3509 6968 


1-373 


3 4578 1013 


1-224 


5160 4951 


1-274 


7 1053 8951 


1-324 


6889 4994 


1-374 


9256 1862 


1-225 


-2937 5770 0323 


1-275 


4 3096 8221 


1-325 


-2658 0295 9089 


1-375 


-2528 3959 5805 


1-226 


4 6408 9453 


1-276 


1 5167 6922 


1-326 


5 3728 8987 


1-376 


5 8688 2587 


1-227 


1 7077 2047 


1-277 


-2788 7266 4774 


1-327 


2 7188 4422 


1-377 


3 3442 1955 


1-228 


-2928 7774 7811 


1-278 


5 9393 1500 


1-328 


0674 5130 


1-378 


8221 3659 


1-229 


5 8501 6453 


1-279 


3 1547 6819 


1-329 


■2647 4187 0844 


1-379 


-2518 3025 7444 


1-230 


2 9257 7681 


1-280 


3730 °453 


1-330 


4 7726 1300 


1-380 


5 7855 3060 


1-231 


0043 1201 


1-281 


•2777 594° 2125 


1-331 


2 1291 6233 


1-381 


3 2710 0254 


1-232 


•2917 0857 6721 


1-282 


4 8178 1557 


1-332 


-2639 4883 5379 


1-382 


7589 8776 


1-233 


4 1701 395° 


1-283 


2 0443 8470 


1-333 


6 8501 8474 


1-383 


-2508 2494 8373 


1-234 


I 2574 2596 


1-284 


-2769 2737 2587 


1-334 


4 2146 5255 


1-384 


5 7424 8795 


I-23S 


-2908 3476 2368 


1-285 


6 5058 3632 


1-335 


I 5817 5456 


1-385 


3 2379 9792 


1-236 


S 4407 2975 


1-286 


3 7407 1328 


1-336 


-2628 9514 8816 


1-386 


7360 1112 


1-237 


2 5367 4125 


1-287 


9783 5398 


1-337 


6 3238 5071 


1-387 


-2498 2365 2506 


1-238 


•2899 6356 5529 


1-288 


-2758 2187 5565 


1-338 


3 6988 3958 


1-388 


5 7395 3724 


1-239 


6 7374 6897 


1-289 


5 4619 1555 


1-339 


1 0764 5215 


1-389 


3 2450 4516 


1-240 


3 8421 7939 


1-290 


2 7078 3090 


1-340 


■2618 4566 8580 


1-390 


7530 4632 


1-241 


9497 8365 


1-291 


-2749 9564 9896 


1-341 


5 8395 3791 


I -39 1 


-2488 2635 3823 


1-242 


-2888 0602 7886 


1-292 


7 2079 1698 


1-342 


3 2250 0586 


1-392 


5 7765 1841 


1-243 


5 1736 6213 


1-293 


4 4620 8221 


1-343 


6130 8703 


1-393 


3 2919 8437 


1-244 


2 2899 3057 


1-294 


I 7189 9190 


1-344 


•2608 0037 7881 


1-394 


8099 3362 


I-24S 


■2879 4090 8131 


1-295 


•2738 9786 4331 


1-345 


5 3970 7860 


1-395 


•2478 3303 6368 


1-246 


6 53" 1145 


1-296 


6 2410 3370 


1-346 


2 7929 8379 


1-396 


5 8532 7207 


1-247 


3 6560 1813 


1-297 


3 5061 6033 


r-347 


1914 9176 


1-397 


3 3786 5631 


1-248 


7837 9846 


1-298 


7740 2047 


1-348 


-2597 5925 9993 


1-398 


9065 1393 


1-249 


•2867 9144 4957 


1-299 


•2728 0446 1138 


1-349 


4 9963 0570 


1-399 


•2468 4368 4246 



158 



MR F. W. NEWMAN'S TABLE 



[1-400— I '599] 



X 


^-.r 


X 


Q-X 


X 


C-x 


X 


(,-X 


I -400 


•2465 9696 3942 


1-450 


-2345 70218 S094 


1-500 


-2231 3016 0148 


1-550 


-2122 4797 3S27 


I "40 1 


3 5049 0^35 


1-451 


3 3583 5052 


1-501 


-2229 0714 1516 


1-551 


3583 1942 


1-402 


I 0426 2S79 


1-452 


1 0161 6346 


1-502 


6 8434 5791 


1-552 


-2118 2390 2093 


1-403 


•2458 582S 1628 


1-453 


-2338 6763 1741 


1-503 


4 6177 2750 


1-553 


6 1218 4067 


I •404 


6 1254 6234 


1-454 


6 338S 1004 


I 504 


2 3942 2171 


1-554 


4 0067 7654 


I •405 


3 6705 6453 


1-455 


4 0036 3901 


1-505 


1729 3832 


'■555 


1 8938 2641 


I "406 


1 21S1 2039 


1-456 


I 6708 0199 


1-506 


-2217 9538 7510 


1-556 


-2109 7829 8818 


1-407 


•2448 76S1 2747 


1-457 


-2329 3402 9663 


1-507 


5 7370 2983 


1-557 


7 6742 5973 


1-408 


6 3205 8332 


1-458 


7 0121 2062 


1-508 


3 5224 0030 


1-558 


5 5676 3896 


1-409 


3 8754 8549 


1-459 


4 6862 7161 


1-509 


I 3099 8429 


1-559 


3 4631 2375 


I-4I0 


I 43-^8 3153 


1-460 


2 3627 4730 


1-510 


-2209 0997 7959 


1-560 


I 3607 1201 


1-4II 


•2438 9926 1901 


I -461 


0415 4535 


1511 


6 8917 8399 


1-561 


-2099 2604 0163 


I -41 2 


6 5548 454S 


1-462 


-2317 7226 6343 


1-512 


4 6859 9529 


1-562 


7 1621 9051 


I 413 


4 119s 0850 


1-463 


5 4060 9925 


1-513 


2 4824 1127 


1-563 


5 0660 7655 


1-414 


I 6866 0565 


1-464 


3 0918 5046 


1-514 


2810 2973 


1-564 


2 9720 5765 


; i"4is 


-2429 2561 3448 


1-465 


7799 1477 


1-515 


-2198 0818 4847 


1-565 


8801 3173 


; 1-416 


6 8280 9257 


1-466 


-2308 4702 8986 


1-516 


5 8848 6530 


1-566 


-20SS 7902 9669 


1-417 


4 4024 7749 


1-467 


6 1629 7342 


1-517 


3 6900 7801 


1-567 


6 7025 5044 


1-418 


1 9792 86S1 


1-468 


3 8579 6315 


1-518 


I 4974 8441 


1-568 


4 6168 9090 


1-419 


-2419 5585 1811 


1-469 


1 5552 5673 


1-519 


-2189 3070 8231 


1-569 


2 5333 1597 


1-420 


7 1401 6897 


1-470 


-2299 2548 5187 


1-520 


7 1188 6952 


1-570 


4518 2357 


I -42 1 


4 7242 3697 


I -47 1 


6 9567 4626 


1-521 


4 9328 4384 


1-571 


•2078 3724 1 163 


1-422 


2 3107 1969 


1-472 


4 6609 3761 


1-522 


2 7490 0310 


1-572 


6 2950 7805 


i'423 


•2409 8996 1472 


1-473 


2 3674 2362 


1-523 


56?3 4511 


1-573 


4 2198 2078 


1-424 


7 4909 1966 


1-474 


0762 0200 


1-524 


-217S 3878 6768 


1-574 


2 1466 3772 


I "425 


S 0846 3208 


1-475 


-2287 7872 7045 


1-525 


6 2105 6865 


1-575 


075s 2681 


1-426 


2 6807 4959 


1-476 


5 5006 2670 


1-526 


4 0354 4582 


1-576 


•2068 0064 8598 


1-427 


2792 6978 


1-477 


3 2162 6844 


1-527 


1 8624 9703 


1-577 


5 9395 1315 


1-428 


•2397 8801 9025 


1-478 


9341 9340 


1-528 


-2169 6917 2010 


1-578 


3 8746 0626 


1-429 


5 4835 0860 


1-479 


-2278 6543 9929 


1-529 


7 5231 1287 


1-579 


I 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 


1-581 


7 6922 6060 


1-432 


-2388 3078 2698 


1-482 


I 8286 7979 


1-532 


1 0302 8764 


1-582 


5 635s 9684 


1-433 


5 9207 1291 


1-483 


•2269 5579 8665 


1-533 


-2158 8703 3751 


1-583 


3 5809 8872 


1-434 


3 5359 8476 


1-484 


7 2895 6306 


1-534 


6 7125 4625 


1-584 


1 5284 3418 


1-435 


1 1536 4014 


1-485 


5 0234 0676 


1-535 


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 9200 


1-487 


4978 8698 


1-537 


2521 0412 


1-587 


5 3830 7153 


1-438 


4 0208 8371 


1-488 


-2258 2385 1896 


1-538 


-2148 1029 2678 


1-588 


3 3387 1081 


1-439 


I 6480 4944 


1-489 


5 9814 0919 


1-539 


5 955S 9755 


1-589 


1 2963 9343 


1-440 


-2369 2775 8682 


1-490 


3 7265 5539 


1-540 


3 8110 1427 


1-590 


-2039 2561 1734 


1-441 


6 9°94 9347 


1-491 


I 4739 5532 


I -54 1 


1 6682 7481 


1-591 


7 2178 8051 


1-442 


4 5437 6704 


1-492 


-2249 2236 0673 


1-542 


•2139 5276 7701 


1-592 


5 1816 8090 


1-443 


2 1804 0515 


1-493 


6 9755 0736 


1-543 


7 3892 1874 


1-593 


3 1475 1647 


1-444 


-2359 8194 0545 


1-494 


4 7296 5497 


1-544 


5 2528 9786 


1-594 


1 1153 8519 


1-445 


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 


1-546 


9866 5972 


1-596 


7 0572 1395 


1-447 


2 7505 5581 


1-497 


-2238 ooss 5719 


1-547 


-2128 8567 3820 


1-597 


5 0311 6993 


1-448 


3989 8123 


1-498 


5 7686 7026 


1-548 


6 7289 4554 


1-598 


3 0071 5094 


1-449 


-2348 0497 5706 


1-499 


3 5340 1911 


1-549 


4 6032 7960 


1-599 


9851 5495 



[i-6oo — 1799] 



OF THE DESCENDING EXPONENTIAL. 



159 



X 


e-x 


X 


Q-X 


X 


c-^ 


X 


g-a; 


I -600 


-2018 9651 7995 


1-650 


-1920 4990 8621 


1-700 


-1826 8352 4053 


1-75° 


•1737 7394 345° 


I -60 1 


6 9472 2392 


I -65 1 


■1918 5795 4705 


1-701 


5 0093 1840 


1-751 


6 0025 6365 


I -602 


4 9312 8483 


1-652 


6 6619 2647 


1-702 


3 1852 2128 


1-752 


4 2674 2879 


1-603 


2 9173 6067 


1-653 


4 7462 2256 


1-703 


I 3629 4735 


I-7S3 


2 5340 2821 


1-604 


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 


I -606 


6 8876 5767 


1-656 


-1909 0105 9164 


1-706 


5 9070 4645 


1-756 


7 3442 1474 


1-607 


4 8817 7312 


1-657 


7 1025 3523 


1-707 


4 0920 4706 


1-757 


5 6177 3391 


1-608 


2 8778 9345 


1-658 


5 1963 8593 


1-708 


2 2788 6175 


1-758 


3 8929 7870 


1-609 


8760 1666 


1-659 


3 2921 4183 


1-709 


4674 8872 


1-759 


2 1699 4738 


I-6I0 


-1998 8761 4075 


1-660 


I 3898 OIOI 


1-710 


■1808 6579 2617 


1-760 


4486 3823 


I-6II 


6 87S2 6371 


I -66 1 


•1899 4893 6159 


1-711 


6 8501 7227 


1-761 


•1718 7290 4953 


I-6I2 


4 8823 8356 


1-662 


7 5908 2166 


1-712 


5 0442 2522 


1-762 


7 oiii 7956 


I-6I3 


2 8884 9828 


1-663 


5 6941 7932 


1-713 


3 2400 8322 


1-763 


5 2950 2660 


1-614 


8966 0590 


1-664 


3 7994 3267 


1-714 


I 4377 4446 


1-764 


3 5805 8893 


I-6I5 


-1988 9067 0441 


1-665 


I 9065 7982 


1-715 


•1799 6372 0713 


1-765 


I 8678 6485 


I-6I6 


6 9187 9182 


1-666 


0156 1888 


1-716 


7 8384 6944 


1-766 


1568 5263 


1-617 


4 932S 6616 


1-667 


-1888 1265 4795 


1-717 


6 0415 2959 


1-767 


■1708 4475 5057 


I-6I8 


2 9489 2543 


1-668 


6 2393 6516 


1-718 


4 2463 8578 


1-768 


6 7399 5696 


I-6I9 


9669 6765 


1-669 


4 3540 6860 


1-719 


2 4530 3622 


1-769 


5 0340 7009 


1-620 


-1978 9869 9083 


1-670 


2 4706 5639 


1-720 


6614 7911 


1-770 


3 3298 8825 


1-621 


7 0089 9301 


1-671 


5891 2666 


1-721 


-17S8 8717 1267 


1-771 


1 6274 0975 


1-622 


5 0329 7219 


1-672 


-1878 7094 7751 


1-722 


7 0837 3509 


1-772 


-1699 9266 3287 


1-623 


3 0589 2640 


1-673 


6 8317 0708 


1-723 


5 2975 4460 


1-773 


8 2275 5591 


1-624 


I 0868 5368 


1-674 


4 9558 1347 


1-724 


3 5131 3941 


1-774 


6 53°i 7719 


1-625 


-1969 1167 5204 


1-675 


3 0817 9482 


1725 


I 7305 1773 


1-775 


4 8344 9499 


1-626 


7 i486 1952 


1-676 


I 2096 4925 


1-726 


•1779 9496 7778 


1-776 


3 1405 0763 


1-627 


5 1824 5414 


1-677 


•1869 3393 749° 


1-727 


8 1706 1778 


1-777 


1 4482 1341 


1-628 


3 2182 5395 


1-678 


7 4709 698S 


1-728 


6 3933 3595 


1-778 


-1689 7576 1064 


1-629 


I 2560 1698 


1-679 


S 6044 3233 


1-729 


4 6178 3052 


1-779 


8 0686 9763 


1-630 


■1959 2957 4127 


1-680 


3 7397 6039 


1-730 


2 8440 9970 


1-780 


6 3814 7269 


1-631 


7 3374 2485 


1-681 


I 8769 5219 


1-731 


I 0721 4173 


1-781 


4 6959 3413 


1-632 


5 3810 6576 


1-682 


0160 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 


I 634 


I 4742 1179 


1-684 


6 2996 9141 


1-734 


5 7668 8716 


1-784 


-1679 6494 1988 


1-635 


■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 0215 


1-736 


2 2388 8257 


1-786 


6 2934 7810 


1-637 


5 6285 6201 


1-687 


7391 3734 


1-737 


4775 2451 


1-787 


4 6180 2249 


1-638 


3 6839 0594 


1-688 


-1848 8893 2326 


1-73S 


-1758 7179 2693 


1-788 


2 9442 4150 


1-639 


I 7411 9355 


1-689 


7 0413 5807 


1-739 


6 9600 8807 


1-789 


I 2721 3345 


1-640 


-1939 8004 2291 


1-690 


5 1952 3993 


1-740 


5 2040 0617 


1-790 


-1669 6016 9667 


1-641 


7 8615 9206 


1-691 


3 35°9 6698 


1-741 


3 4496 7947 


1-791 


7 9329 2949 


1-642 


5 9246 9908 


1-692 


I 5085 3738 


1-742 


I 6971 0623 


1-792 


6 2658 3025 


1-643 


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 


1-645 


1256 2794 


1-695 


5 9922 9028 


1-745 


6 4498 8967 


1-795 


I 2745 2346 


1-646 


■1928 1964 6705 


1-696 


4 1572 1568 


1-746 


4 7043 1271 


1-796 


•1659 6140 7930 


1-647 


6 2692 3436 


1-697 


2 3239 7524 


1-747 


2 9604 8046 


1-797 


7 9552 9475 


1-648 


4 3439 2794 


1-698 


4925 6712 


1-748 


I 2183 9117 


1-798 


6 2981 6816 


1-649 


2 4205 4586 


1-699 


•1828 6629 8949 


1-749 


•1739 4780 4310 


1-799 


4 6426 9788 



160 






MR F. W. NEWMAN'S TABLE 




[1-800 — 1-999 


X 


.- 


X 


C-* 


X 


Q-X 


X 


Q-X 


i-Soo 


•1652 988S 8221 


1-850 


-1572 3716 6314 


1-900 


-1495 6S61 9223 


1-95° 


-1422 7407 1586 


i-8oi 


I 3367 1955 


I -85 1 


8000 7740 


1-901 


4 1912 5363 


1-951 


I 3186 8628 


I -802 


•1649 6862 0822 


1-852 


•1569 2300 6246 


1-902 


2 6978 0922 


1-952 


•I4I9 8980 7802 


1-803 


8 0373 4659 


1-853 


7 6616 1675 


1-903 


I 2058 5751 


1-953 


8 4788 8965 


I 804 


6 3901 3298 


1-854 


6 0947 3871 


1-904 


-1489 7153 9701 


I 954 


7 0611 1977 


1-805 


4 7445 6577 


1-855 


4 5294 2675 


1-905 


8 2264 2622 


1-955 


5 6447 6694 


i-8o6 


3 1006 4331 


1-856 


2 9656 7933 


1-906 


6 7389 4366 


1-956 


4 2298 2976 


1-807 


I 4583 6394 


1-857 


I 4034 94S7 


1-907 


5 2529 4784 


1-957 


2 8163 06S1 


I -808 


-1639 8177 2603 


1-858 


-1559 8428 7182 


1-908 


3 76S4 3727 


1-958 


I 4041 9668 


1-809 


8 1787 2794 


1-859 


8 2838 0861 


1-909 


2 2854 1047 


1-959 


-1409 9934 9795 


i-Sio 


6 5413 6803 


1-860 


6 7263 0368 


1-910 


8038 6595 


1-960 


8 5842 0921 


i-Sii 


4 9056 4466 


I-86I 


5 1703 5549 


1-911 


-1479 3238 0224 


1-961 


7 1763 2906 


I-8l2 


3 2715 5620 


1-862 


3 6159 6246 


1-912 


7 8452 1786 


1-962 


5 7698 5609 


1-813 


1 6391 oiox 


1-863 


2 0631 2304 


1-913 


6 3681 1132 


1-963 


4 3647 8888 


1-814 


0082 7745 


1-864 


5118 3569 


1-914 


4 8924 8114 


1-964 


2 961 1 2604 


1-815 


-1628 3790 8390 


1-865 


•1548 9620 9885 


i'9i5 


3 4183 2586 


1-965 


1 5588 6616 


i-8i6 


6 7515 1874 


1-866 


7 4139 1097 


1-916 


1 9456 4400 


1-966 


1580 0784 


1-817 


5 1255 8032 


1-867 


5 8672 7051 


1-917 


4744 3408 


1-967 


-1398 7585 4968 


1-818 


3 5°i2 6704 


1-868 


4 3221 7592 


1-918 


■1469 0046 9464 


1-968 


7 3604 9027 


1-819 


I 8785 7725 


1-869 


2 7786 2565 


1-919 


7 5364 2420 


1-969 


5 9638 2823 


1820 


2575 °934 


1-870 


I 2366 1815 


1-920 


6 0696 2130 


1-970 


4 5685 6215 


1-82 1 


•1618 6380 6169 


1-871 


-1539 6961 5190 


1-921 


4 6042 8447 


1-971 


3 1746 9064 


1-822 


7 0202 3268 


1-872 


8 1572 2534 


1-922 


3 1404 1225 


1-972 


I 7822 1230 


1-823 


5 4040 2069 


1-873 


6 6198 3693 


1-923 


1 6780 0316 


1973 


° 3911 2575 


1-824 


3 7894 2410 


1-874 


5 0S39 8515 


1-924 


2170 5575 


1-974 


-1389 0014 2959 


1825 


2 1764 4130 


I-S75 


3 5496 6845 


1-925 


-1458 7575 6856 


i'975 


7 6131 2243 


1-826 


5650 7068 


1-876 


2 0168 8530 


1-926 


7 2995 4013 


1-976 


6 2262 0288 


1-827 


•1608 9553 1062 


1-877 


4856 3417 


1-927 


5 8429 6900 


1-977 


4 8406 6956 


1-828 


7 3471 5952 


1-878 


-1528 9559 1352 


1-928 


4 3878 5371 


1-978 


3 4565 2108 


1-829 


5 7406 1577 


1-879 


7 4277 2183 


1-929 


2 9341 9281 


1-979 


2 0737 5606 


1-830 


4 1356 7775 


i-88o 


5 9°io 5757 


1-93° 


1 4819 8484 


1-980 


6923 7311 


1-831 


2 5323 4388 


i-88i 


4 3759 1921 


1-931 


0312 2835 


1-981 


-1379 3123 7085 


1-832 


9306 1253 


1-882 


2 8523 0522 


1-932 


■1448 5819 2190 


1-982 


7 9337 4791 


^■^3Z 


-1599 3304 8212 


1-883 


I 3302 1409 


I '933 


7 1340 6403 


1983 


6 5565 0290 


1-834 


7 7319 5i°3 


1-884 


-15 19 8096 4429 


1-934 


5 6S76 5329 


1-984 


5 1806 3444 


1-835 


6 1350 1768 


1-885 


8 2905 9429 


I '935 


4 2426 8824 


1-985 


3 8061 4117 


1-836 


4 5396 8046 


1-886 


6 7730 6259 


1-936 


2 7991 6743 


1-986 


2 4330 2170 


1-837 


2 9459 3779 


1-887 


5 2570 4766 


1937 


I 3570 8942 


1-987 


1 0612 7467 


1-838 


1 3537 8806 


1-888 


3 7425 4799 


1-938 


-1439 9164 5277 


1-988 


■1369 6908 9870 


1-839 


•1589 7632 2968 


1-889 


2 2295 6206 


1939 


8 4772 5^04 


1989 


8 3218 9241 


1840 


8 1742 6107 


1-890 


7180 8836 


1-940 


7 0394 9778 


1-990 


6 9542 5445 


1-841 


6 5868 8063 


1-891 


•1509 2081 2538 


1-941 


5 6031 7656 


I -99 1 


5 5879 8345 


1-842 


5 0010 8678 


1-892 


7 6996 7161 


1-942 


4 1682 9095 


1-992 


4 2230 7803 


1-843 


3 4168 7793 


1-893 


6 1927 2554 


1-943 


2 7348 395° 


1993 


2 8595 3684 


1844 


I 8342 5250 


1-894 


4 6872 8566 


1-944 


1 3028 2079 


1-994 


I 4973 5850 


1-845 


2532 0890 


1-895 


3 1833 5046 


1-945 


-1429 8722 3338 


1-995 


1365 4167 


1-846 


-1578 6737 4555 


1-896 


1 6809 1845 


1-946 


8 4430 7584 


1-996 


•1358 7770 8497 


1-847 


7 0958 6088 


1-897 


1799 8813 


1-947 


7 0153 4675 


1-997 


7 4189 8705 


1-848 


5 5195 5330 


1-898 


-1498 6805 5798 


1-948 


5 589° 4467 


1-998 


6 0622 4654 


1-849 


3 9448 2125 


1-899 


7 1826 2651 


1949 


4 1641 6819 


1-999 


4 7068 6210 



[2 00O 


-2-t99] 


OF 


THE DESCENDING EXPONENTIAL. 




161 


X 


(,~X 


X 


C-x 


X 
2-100 


g~X 


1 X 

1 


e-* 


2'000 


■1353 3528 3237 


2-050 


•12S7 3490 35S8 


•1224 5642 8253 


2-150 


-1164 8415 7773 


2-OOI 


2 0001 5599 


2-051 


6 0623 3030 


2-IOI 


3 3403 3032 


1 2-151 


3 6773 1838 


2'002 


6488 3161 


2-052 


4 7769 1079 


2-102 


2 1176 0146 


1 2-152 


2 5142 2271 


2-003 


■1349 2988 57S8 |i 2-053 


3 4927 7605 


2-103 


8960 9471 


:i 2-153 


I 3522 8955 


2-004 


7 9502 3344 


2-054 


2 2099 2481 


2-104 


•1219 6758 0886 


|;2-x54 

2-155 


1915 1774 


2-005 


6 6029 5696 


2-055 


9283 5578 


2-105 


8 4567 4269 


•1159 0319 0613 


2 -006 


5 2570 2708 


2-056 


■1279 6480 6767 


2-106 


7 2388 9497 


1 2-156 


7 8734 5354 


2-007 


3 9124 4246 


2-057 


8 3690 5921 


2-107 


6 0222 6449 


2-157 


6 7161 5883 


2 -008 


• 2 5692 0175 


2-058 


7 0913 2913 


2-108 


4 8068 5004 


12-158 


5 5600 2084 


2'oog 


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-IIO 


2 3796 6433 


1 

2-i6o 


3 2512 1038 


2-OII 


8 5475 2967 


2-061 


3 2657 9631 


2-III 


I 1678 9066 


2-i6i 


2 0985 3560 


2-012 


7 2096 5119 


2-062 


I 9931 6693 


2-112 


-1209 9573 2815 


2-162 


9470 1292 


2-013 


5 8731 0992 


j 2-063 


7218 0955 


2-113 


8 7479 7560 


1 2-163 


•1149 7966 4119 


2-014 


4 5379 0452 


2-064 


•1269 4517 2289 


2-II4 


7 5398 3180 


1 2-164 


8 6474 1926 


2-015 


3 2040 3366 


2-065 


8 1829 0568 


2-II5 


6 3328 9553 


2-165 


7 4993 4597 


2-oi6 


I 8714 9601 


2-066 


6 9153 5665 


2-Il6 


5 1271 6560 


2-i66 


6 3524 2018 


2-017 


5402 9023 


2-067 


5 6490 7454 


2-II7 


3 9226 4080 


2-167 


5 2066 4074 


2-018 


•1329 2104 1499 


2-06S 


4 3840 5808 


2-Il8 


2 7193 1992 


2-168 


4 0620 0652 


2-019 


7 8818 6895 


2-069 


3 1203 q6oi 


2-II9 


I 5172 0176 


2-169 


2 9185 1635 


2 -020 


6 5546 5080 


2-070 


I 8578 1705 


2-120 


3162 8511 


2-170 


r 7761 6910 


2-021 


5 2287 5921 


2-071 


5965 899s 


2-I2I 


•1199 1165 6879 


2-171 


6349 6363 


2 -02 2 


3 9041 9285 


2-072 


-1259 3366 2345 


2-122 


7 9180 5158 


2-172 


-1139 4948 9879 


2-023 


2 5809 S°39 


2-073 


8 0779 1629 


2-123 


6 7207 3229 


2-173 


8 3559 7345 


2024 


I 2590 3050 


2-074 


6 8204 6720 


2-124 


5 5246 0972 


2-174 


7 2181 8647 


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 3011 9313 


2-077 


3 0556 5584 


2-127 


I 9434 1037 


2-177 


3 8116 4428 


2-028 


5 9845 5037 


2-078 


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 


1 5462 8710 


2-030 


3 3552 1149 


2-080 


•1249 3021 2199 


2-130 


8 3729 3852 


2-180 


4153 0640 


2031 


2 0425 1273 


2 -08 1 


8 0534 4431 


2-131 


7 1851 5957 


2181 


•1129 2854 5611 


2 032 


7311 2602 


2-082 


6 8060 1468 


2-132 


5 9985 6781 


2-182 


8 1567 35" 


2-033 


•1309 4210 5005 


2-083 


5 5598 3186 


2-133 


4 8131 6204 


2-183 


7 0291 4226 


2 '034 


8 1122 8349 


2-084 


4 3148 9460 


2-134 


3 6289 4109 


2-184 


5 9026 7645 


2'°3S 


6 8048 2504 


2-085 


3 0712 0166 


2-135 


2 4459 0376 


2-185 


4 7773 3654 


2-036 


5 4986 7340 


2-o86 


I 8287 5178 


2-136 


I 2640 4889 


2-186 


3 6531 2140 


2-037 


4 1938 2726 


2-087 


5875 4374 


2-137 


0S33 7527 


2-187 


2 5300 2992 


2-038 


2 8902 8531 


2-088 


■1239 3475 7628 


2-138 


•1178 9038 8174 


2-188 


I 4080 6097 


2-039 


1 58S0 4625 


2-089 


8 1088 4S17 


2-139 


7 7255 6712 


2-189 


2872 1343 


2-040 


2871 0878 


2-090 


6 8713 5817 


2-140 


6 5484 3022 


2-190 


-1119 1674 8617 


2-041 j 


-1298 9874 7160 


2-091 


5 6351 0504 


2-I4I 


5 3724 6987 


2-191 


8 04S8 7808 


2-042 


7 6891 3341 


2-092 


4 4000 S755 


2-142 


4 1976 8489 


2-192 


6 9313 8804 


2-043 


6 3920 9290 


2-093 


3 1663 0446 


2-143 


3 0240 7411 


2-193 


5 8150 1493 


2-044 


5 0963 4879 


2-094 


I 9337 5453 


2-144 


1 S516 3635 ^ 


2-194 


4 6997 5764 


2 -04s 


3 8oi8 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 


I 2168 8185 


2-097 


8 2434 9143 


2-147 


8 3413 4950 


2-197 


I 3606 6951 


2-048 


•1289 9263 1036 


2-098 


7 0158 6186 


2-148 


7 1735 9213 


2-198 


2498 6433 


2-049 


8 6370 28S0 

1 


2099 


5 7894 5930 ; 2-149 


6 0070 0193 


2-199 


■1109 1401 6941 



Vol. XIII. Part III. 



22 



lG-2 






MR F. W. NEWMAN'S TABLE 




[2-200—2-399: 


X 


C-* 


X 


(,-X 


X 


g-X 


1 

' X 


g-X 


2 '200 


-iioS 0315 8362 


2-250 


-1053 9922 4562 


2-300 


-1002 5S84 3723 


2-350 


953 6916 2215 


2'20I 


6 9241 0587 


2-251 


2 9387 8020 


2-301 


I 5863 4992 


2-351 


2 7384 0721 


2-202 


5 8177 3504 


2-252 


1 8863 6771 


2-302 


5852 6419 


2-352 


I 7861 4502 


2 -203 


4 7124 7003 


2-253 


8350 0711 


2-303 


999 5851 7906 


2-353 


8348 3461 


2-204 


3 60S3 0973 


2-254 


-1049 7846 9734 


2-304 


8 5860 9350 


2-354 


949 8844 7503 


2-205 


2 5052 5304 


2-255 


8 7354 3736 


2-305 


7 58S0 0654 


2-355 


8 9350 6534 


2 206 


I 4032 9886 


2-256 


7 6872 2612 


, 2-306 


6 5909 1716 


2-356 


7 9866 0458 


2-207 


3024 4608 


2-257 


6 6400 6256 


2-307 


5 5948 2437 


2-357 


7 0390 9181 


2-20S 


-1099 2026 9360 


2-258 


5 5939 4565 


2-308 


4 5997 2718 


2-358 


6 0925 2609 


2-209 


8 1040 4032 


2-259 


4 5488 7432 


2-309 


3 6056 2458 


2-359 


5 1469 0645 


2-2IO 


7 0064 8515 


1 2-260 


3 5048 4755 


2-310 


2 6125 1560 


2-360 


4 2022 3196 


2-2II 


5 9100 2698 


2-261 


2 4618 6428 


2-311 


I 6203 9922 


i 2-361 


3 2585 0167 


2-212 


4 8146 6473 


1 2-262 


I 4199 2347 


2-312 


6292 7447 


2-362 


2 3157 1464 


2-213 


3 7203 9729 


2-263 


3790 2409 


2-313 


989 6391 4034 


2-363 


I 3738 6993 


2-214 


2 6272 2357 


2-264 


•1039 3391 6508 


2-314 


8 6499 9586 


2-364 


4329 6659 


2-215 


I 5351 4248 


2-265 


8 3003 4541 


2-315 


7 6618 4002 


2-365 


939 4930 0368 


2-2 1 6 


4441 5292 


2-266 


7 2625 6404 


2-316 


6 6746 7185 


2-366 


8 5539 8027 


2-217 


•1089 3542 5381 


2-267 


6 2258 1994 


2-317 


5 6SS4 903s 


2-367 


7 6158 9541 


2-218 


8 2654 4405 


2-268 


5 1901 1206 


2-318 


4 7032 9454 


2-368 


6 6787 4816 


2219 


7 1777 2256 


2-269 


4 1554 3937 


2-319 


3 7190 8343 


2-369 


5 7425 3760 


2-220 


6 0910 SS25 


2-270 


3 1 2 18 00S3 


2-320 


2 7358 5604 


2-370 


4 8072 6278 


2-221 


5 °o55 4003 


2-271 


2 0891 9542 


2-321 


I 7536 1139 


2-371 


3 8729 2276 


2-222 


3 9210 7681 


2-272 


I 0576 2210 


2-322 


7723 4849 


2-372 


2 9395 1662 


2-223 


2 8376 9751 


2-273 


0270 7983 


2-323 


979 7920 6636 


2-373 


2 0070 4342 


2-224 


I 7554 0105 


2-274 


-1028 9975 6759 


2-324 


8 8127 6403 


2-374 


1 0755 0222 


2-225 


6741 8635 


2-275 


7 969° 8435 


2-325 


7 8344 4051 


2-375 


144S 9210 


2-226 


-1079 5940 5232 


2-276 


6 9416 2908 


2-326 


6 8570 9482 


2-376 


929 2152 1212 


2-227 


8 5149 9789 


2-277 


5 9152 o°75 


2-327 


5 8S07 2599 


2-377 


8 2864 6136 


2-228 


7 4370 2197 


2-278 


4 8897 9834 


2-328 


4 9053 3304 


2-378 


7 3586 3889 


2-229 


6 3601 2348 


2-279 


3 8654 2082 


2329 


3 9309 1500 


2-379 


6 4317 4378 


2-230 


5 2843 0136 


2-280 


2 8420 ^716 


2-330 


2 9574 7089 


2-380 


5 5057 7510 


2-231 


4 2095 5452 


2-281 


I 8107 3634 


2-331 


I 9849 9974 


2381 


4 5807 3192 


2-232 


3 1358 8189 


2-282 


7984 2734 


2-332 


I 0135 0057 


2-382 


3 6566 1332 


2'233 


2 0632 8240 


2-283 


■1019 7781 3914 


2-333 


0429 7241 


2-383 


2 7334 1838 


2-234 


99'7 5497 


2-284 


8 7588 7072 1 


2-334 


969 0734 1430 


2-384 


I 8iii 4618 


2-235 


•1069 9212 9853 


2-285 


7 7406 2106 


2-335 


8 1048 2526 


2-385 


8897 9579 


2-236 


8 8519 I20I 


2-286 


6 7233 8914 


2-336 


7 1372 0432 


2-386 


919 9693 6628 


2-237 


7 7835 9435 


2-287 


5 7071 7394 


2-337 


6 1705 5053 


2-387 


9 0498 5675 


2-238 


6 7163 4447 1 


2-288 


4 6919 7445 


2-338 


5 2048 6290 


2-388 


8 1312 6626 


2-239 


5 6501 6130 


2-289 


3 6777 8966 


2-339 


4 2401 4048 j 


2-389 


7 2135 9391 


2-240 


4 5850 4379 


2-290 


2 6646 1854 


2-340 


3 2763 8230 [ 


2-390 


6 2968 3877 


2-241 


3 5209 9086 


2-291 


I 6524 6008 


2-341 


2 3135 8740 


2-391 


5 3809 9993 


2-242 


2 4580 0145 


2-292 


6413 1328 


2-342 


I 3517 5480 


2-392 


4 4660 7646 


2-243 


I 3960 7450 


2-293 


-1009 6311 7712 


2-343 


3908 8356 


2-393 


3 5520 6747 1 


2-244 


3352 089s 


2-294 


8 6220 5059 


2-344 


959 4309 7272 


2-394 


2 6389 7203 


2-245 


-1059 2754 0373 


2-295 


7 6139 3268 


2-345 


8 4720 2130 


2-395 


I 7267 8922 


2*246 


8 2166 5779 


2-296 


6 6068 2239 


2-346 


7 5140 2835 


2-396 


8155 1814 


2-247 


7 1589 7006 ! 


2-297 


5 6007 1870 


2-347 


6 .^569 9292 


2-397 


909 9051 5788 


2-248 


6 1023 3950 i 


2-298 


4 5956 2062 , 


2-348 


5 6009 1405 


2-398 


8 9957 0752 


2-249 


5 0467 6503 

1 


2-299 


3 591S 2713 i 


2-349 


4 6457 9078 


2-399 


8 0871 6616 

1 



2-400— 2-599] 



OF THE DESCENDING EXPONENTIAL. 



163 



X 


Q-X 


X 


Q-X 


1 
X 


^-x 


X 


f>~X 


2 -400 


9°7 179s 3289 


2-450 


862 


■ 

9358 6499 


2-500 


820 8499 8624 


2-550 


780 8166 6001 


2-401 


6 2728 0679 


2-451 


2 


0733 6045 


2-501 


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 


S59 


4910 1582 


2-504 


7 5731 4435 


2-554 


7 6996 3158 


2-405 


2 6549 5609 


2-455 


8 


6319 5441 


2-505 


6 7559 7985 


2-555 


6 9223 2067 


2-406 


I 7527 5231 


2-456 


7 


7737 5163 


2-506 


5 9396 3212 


2-556 


6 1457 8669 


2-407 


8514 5029 


2-457 


6 


9164 0662 


2-507 


5 1241 0032 


2-557 


5 3700 2884 


2 -40 8 


S99 9510 491 I 


2-458 


6 


0599 1853 


2-508 


4 3093 8364 


2-558 


4 5950 4637 


2-409 


9 0515 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 6823 9241 


2-560 


3 0474 0443 


2-41 1 


7 2552 4170 


2-461 


3 


4955 8719 


2-51 1 


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 


S09 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 6019 


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 8656 


2-468 


7 


5419 8002 


2-518 


6 2068 6990 


2-568 


6 S876 9688 


2-419 


° 1058 3552 


2-469 


6 


6948 6167 


2-519 


5 4010 6600 


2-569 


6 1211 9250 


2-420 


8S9 2161 7459 


2-470 


5 


8485 9001 


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 062S 2854 


2-424 


5 6664 1415 


2-474 


2 


4719 5343 


2-524 


I 3841 1142 


2-574 


2 3001 4712 


2-425 


4 781 1 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 


7770 7040 


2-427 


3 0133 9642 


2-477 


839 


9483 2490 


2-527 


8 9835 6171 


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 3871 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 9S26 7894 


2-432 


8 6093 4873 


2-482 


5 


7590 6516 


2-532 


4 99S6 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 S53S 8608 


2-484 


4 


0892 1743 


2-534 


3 4102 0628 


2-584 


4 7151 3392 


2-435 


5 9774 7048 


2-485 


3 


255s 4512 


2-535 


2 6171 9265 


2-585 


3 9607 9601 


2-436 


5 1019 3085 


2-486 


2 


4227 0606 


i 2-536 


I 8249 7163 


2-586 


3 2072 1207 


2-437 


4 2272 6632 


2-487 


I 


5906 9943 


1 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 


S29 


9291 8011 


2-539 


789 4530 5637 


2-589 


9509 7648 


2-440 


I 6085 1462 


2-490 


9 


0996 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 9105 


2-544 


5 5156 4282 


2-594 


7 2055 9286 


2-445 


7 2613 4902 


2-495 


4 


964s 1392 


2-545 


4 7305 1981 


2-595 


6 4587 6074 


2-446 


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 


2-598 


4 2227 4017 


2-449 


3 7992 3247 


2-499 


I 


6712 4679 


2549 


I 5978 6721 


2-599 


3 47S8 8942 



22- 



IG4 






MR F. W. NEWMAN'S TABLE 




[2-600 — 2-799] 


X 


(,-X 


X 


Q-X 


X 


1 

f>-X 


X 


(,-X 


2 -600 


742 7357 8214 


2-650 


706 5 12 1 3060 


2-700 


672 0551 2740 


2-750 


639 2786 1207 


2 -60 1 


I 9934 1760 


2-651 


5 8059 7161 


2-701 


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 


5i°9 1376 


2-653 


4 3957 7034 


2-703 


0419 8324 


2-753 


7 3636 5011 


2-604 


739 7707 7298 


2-654 


3 6917 2665 


2-704 


669 3722 7616 


2-754 


6 7266 0504 


2-605 


9 0313 7197 


2-655 


2 9S83 8665 


2-705 


8 7032 3846 


2-755 


6 0901 9669 


2 -606 


8 2927 0999 


2-656 


2 2S57 49C4 


2-706 


8 0348 6947 
7 3671 6850 


2-756 


5 4544 2443 


2-607 


7 5547 8631 


2-657 


1 5838 1492 


2-707 


2-757 


4 8192 8763 


2-6oS 


6 8176 0017 


2-658 


8825 817S 


2-708 


6 7001 3491 


2-758 


4 1847 8564 


2-609 


6 081 I 5086 


2-659 


1820 4952 


2-709 


6 0387 6801 


2-759 


3 5509 1784 


2-610 


5 3454 3763 


2-660 


690 4822 1745 


2-710 


5 36S0 6715 


2-760 


2 9176 8360 


2-6II 


4 6104 5974 


2-661 


8 7S30 8485 


2-711 


4 7030 3166 


2-761 


2 2850 8227 


2-6l2 


3 8762 1647 


2-662 


8 0846 5105 


2-712 


4 0386 6086 1 


2-762 


I 6531 1323 


2-613 


3 1427 07°7 


2-663 


7 3869 1532 


2-713 


3 3749 541 I 


2-763 


I 0217 7583 


2-614 


2 4099 3°8i 


2-664 


6 6898 7698 


2-714 


2 7119 1074 


2-764 


3910 6946 


2-615 


I 6778 8696 


2-665 


S 9935 3533 


2-715 


2 0495 3007 


2-765 


629 7609 9348 


2-616 


^0 9465 7479 


2-666 


5 2978 8968 


2-716 


I 3878 1146 


2-766 


9 1315 4727 


2617 


2159 9356 


2-667 


4 6029 3932 


2-717 


7267 5423 


2-767 


8 5027 3018 


2-6i8 


729 4861 4256 


2-668 


3 9=86 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 2S27 


2-670 


2 5-2 5309 


2-720 


8 7475 4426 


2-770 


6200 4742 


2-621 


7 3009 6354 i 2"67i 


I 8300 7698 


2-721 


8 0891 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 15-5 


2-673 


4477 9957 


2-723 


6 7742 6303 


2-773 


4 7430 0425 


2-624 


5 1223 3023 


2-674 


6S9 7576 9688 


2-724 


6 1178 1705 


2-774 


4 1185 7352 


2-625 


4 3975 7034 


2-675 


9 0682 8394 


2-725 


5 4620 2718 


2-775 


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 


I 6270 8552 


2-629 


I 5057 6752 


2-679 


6 3175 1601 


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 


1 2-631 


0641 9804 


2-68i 


4 9462 5270 


2-731 


I S4IO 2977 


2-7S1 


619 7649 9879 


12-632 


719 3444 9375 


2-682 


4 2616 4881 


2-732 


8898 1441 


2-782 


9 1455 4357 


2'633 


8 625s 0881 


2-683 


3 5777 2918 


2-733 


2392 4993 


2-783 


8 5267 0750 


2-634 


7 9°72 425° 


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 290S 9031 


2-636 


6 4728 6286 


I 2-686 


I 5300 6901 


2-736 


8 2914 5533 


2-786 


6 6739 0796 


2-637 


5 7567 4812 


1 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 


12-639 


4 3266 6518 


2-689 


679 4885 4263 


2739 


6 3494 9536 


2-789 


4 8266 5850 


2-640 


3 6126 9556 


2-690 


8 8093 9372 


I 2-740 


5 7034 6S93 


I 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 


2643 


I 4750 6552 


2-693 


6 7760 1712 


2-743 


3 7692 6129 


1 2-793 


2 3722 6392 


2-644 


7639 4607 


2-694 


6 099s 7938 


2-744 


3 1258 13S0 


1 2-794 


I 7601 9774 


2-645 


0535 3739 


2-695 


5 4238 1774 


2-745 


2 4830 0944 y 2-795 


I 1487 4332 


2-646 


709 3438 387(5 


2-696 


4 7487 3152 


2-746 


I 8408 4757 1 2-796 


5379 0005 


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 


2-698 


3 4005 8266 


2-748 


5584 4870 2-798 


9 3180 4451 


2-649 


7 2189 9611 


i 2-699 


2 7275 1866 


2-749 


639 9182 1042 1 2-799 


8 7090 3103 



[2 800 2-999] 



OF THE DESCENDING EXPONENTIAL. 



165 



X 


e-x 


X 


(,-X 


X 

2-900 
2-901 
2-902 
2-903 
2-904 




C-x 


X 


e-^ 


2-800 

2-801 
2-802 
2-803 
2-804 


608 

7 

6 
6 

5 


1006 2625 
4928 2957 
8856 4038 
2790 5809 
6730 8207 


2-850 
2-851 
2-852 

; 2-853 
2-854 


578 
7 
7 
6 
6 


4432 0875 
8650 5467 
2874 7845 
7104 7952 
1340 5730 


55° 

549 

9 

8 

8 


2322 0056 
6822 4338 
1328 3589 
5S39 7753 
0356 6775 


2-950 

2-951 
2-952 
2-953 
2-954 


523 

2 

2 

I 
I 


3970 5948 
8739 2403 
3513 1146 
8292 2124 
3076 5284 


2-805 

2 -806 
2-807 
2-808 
2-809 


5 
4 
3 

3 




0677 1172 
4629 4644 
8587 8563 
2552 2867 
6522 7497 


2-855 
2-856 

2-857 
2-858 
2-859 


5 
4 
4 
3 
3 


5582 1121 
9829 4068 
4082 4513 
8341 2399 
2605 7669 


2-905 
2-906 
2-907 
2-908 
2-909 


7 
6 

6 
5 
5 


4879 0601 
9406 9176 

3940 2445 

8479 0353 
3023 2846 


2-955 
2-956 

2-957 
2-958 
2-959 






519 

9 

8 


7866 0575 
2660 7945 
7460 7342 
2265 8713 
7076 2007 


2-8io 

2-8II 
2-812 
2-813 
2-814 


2 
I 


599 


0499 2392 

4481 7492 
8470 2737 
2464 8067 
6465 3421 


2 -860 
2 -861 
2-862 
2-863 
2-864 


2 
2 

I 




6876 0265 
1152 0130 
5433 7206 
9721 1436 
4014 2764 


2-910 
2-91 1 
2-912 

2-913 
2-914 


4 
4 
3 
3 
2 


7572 9869 
2128 1368 
6688 7288 

1254 7575 
5826 2175 


2-960 
2-961 
2-962 
2-963 
2-964 


8 

7 
7 
6 
6 


1891 7172 
6712 4156 
1538 2907 

6369 3373 
1205 5502 


2-815 
2-816 
2-817 

2-8i8 
2-819 


9 
S 

7 
7 
6 


0471 8740 
44S4 3964 
8502 9032 
2527 38S6 
6557 8464 


2-865 

2-866 
2-867 
2-868 
2-869 


5G9 

9 
8 
8 
7 


8313 1132 
2017 6483 
6927 8760 
1243 7906 
5565 3865 


2-915 
2-916 

3-917 
2-918 
2-919 


2 
I 


539 


0403 i°33 
4985 4095 
9573 1307 
4166 2614 
8764 7963 


2-965 
2-966 
2-967 
2-968 
2-969 


5 
5 
4 
4 
3 


6046 9244 
0893 4547 

5745 1358 
0601 9627 
5463 9302 


2-820 
2-821 
2-822 
2-823 
2-824 


6 

5 
4 
4 
3 


0594 2709 
4636 6559 
8684 9956 
2739 2839 
6799 5150 


2-870 
2-871 
2-872 
2-873 
2-874 


6 
6 

S 
5 
4 


9892 6580 
4225 5993 
8564 2049 
2908 4690 
7258 3861 


2-920 
2-921 
2-922 
2-923 
2-924 


9 

8 
8 
7 
7 


3368 7300 
7978 0571 
2592 7721 
7212 8697 
1838 3446 


2-970 
2-971 
2-972 
2-973 
2-974 


3 

2 

2 
I 



0331 0331 

5203 2664 
0080 6249 

4963 i°35 
9S50 6970 


2-825 
2-826 
2-827 
2-82S 
2-829 


3 

I 
I 



0865 6829 

4937 7817 
9015 8054 
3099 7481 
7189 6039 


2-875 
2-876 
2-877 
2-878 
2-879 


4 
3 
3 

2 

I 


1613 9504 

5975 1563 
0341 99S2 

4714 4704 
9092 5674 


2-925 
2-926 

2-927 
2-928 
2-929 


6 

6 
5 
5 
4 


6469 igi2 

I 105 4044 
5746 9786 
0393 9086 
5046 1890 


2-975 
2-976 

2-977 
2-978 

2-979 



5°9 
9 
8 
S 


4743 4004 
9641 2085 
4544 1 163 
9452 1186 
4365 2104 


2-830 
2-831 
2-832 
2-833 
2-834 




589 
S 
8 
7 


1285 3669 
5387 0312 
9494 5909 
3608 0401 
7727 3728 


2-88o 
2-881 
2-882 
2-883 
2-884 


I 


559 
9 


3476 2834 
7865 6129 
2260 5503 
6661 0900 
1067 2263 


2-930 
2-931 
2-932 
2-933 
2-934 







2 

2 

1 


9703 8145 
4366 7797 
9035 0792 
3708 7077 
S387 6600 


2-980 
2-981 
2-982 
2-983 
2-984 


7 
7 
6 
6 
5 


9283 3S65 
4206 6419 

9134 9715 
4068 3703 
9006 8331 


2-835 
2-836 
2-837 

2-838 
2-839 


7 
6 
6 

5 
4 


1852 5833 
59S3 6657 
0120 6141 
4263 4225 
8412 0853 


2-885 
2-886 
2-887 
2-888 
2-889 


8 
7 
7 
6 
6 


547S 9536 
9896 2665 

4319 1593 
8747 6263 
3181 6621 


2-935 
2-936 

2-937 
2938 

2-939 


I 





529 

9 


3071 9306 

7761 5143 
2456 4058 

7156 599S 
1862 0909 


2-985 
2-986 
2-987 
2-988 
2-989 


5 
4 
4 
3 
3 


3950 3549 
8898 9307 

3852 5554 
8811 2240 

3774 9313 


2-840 
2-841 
2-842 

2-843 
2-844 


4 
3 
3 

2 

I 


2566 5964 
6726 9502 
0893 1406 
5065 1619 
9243 0083 


2-890 
2-891 
2-892 
2-893 
2-894 


S 
5 
4 
4 
3 


7621 2611 
2c66 4178 
6517 1265 
0973 3817 
5435 1779 


2-940 
2-941 
2-942 
2-943 
2-944 


8 

8 

7 

,7 

- 6 


6572 8738 
1288 9434 
6010 2942 
0736 9210 
5468 8186 


2-990 

2-991 

2-992 

2-993 
2-994 




I 
I 



8743 6724 
3717 4422 
8696 2358 
3680 0481 
8668 8741 


2-845 
2-846 

2-847 
2-848 

2-849 


1 


579 
9 


3426 6740 
7616 1531 
1811 4398 
6012 5283 
0219 4128 


2-895 
2-896 
2-897 
2-898 
2-899 


2 
2 

1 
I 



9902 5095 
4375 3710 
8853 7568 
3337 6616 
7S27 0797 


2-945 
2-946 

2-947 
2-948 
2-949 


6 

5 
4 
4 
3 


0205 9816 
494S 4048 
9696 0830 
4449 Olio 
9207 1833 


2-995 
2-996 
2-997 
2-998 
2-999 



499 
9 
8 
8 


3662 7087 
8661 5470 
3665 3839 
8674 2146 
36SS 0339 



16G 



MR F. W. NEWMAN'S TABLE 



[3-000—3-199] 



X 


e-« 


X 


f,-X 


X 


Q-X 


X 


C-x 


3-000 
3001 
3-002 
3003 
3004 


497 8706 8368 
7 3730 6185 

6 8759 3739 

6 3793 0981 
5 8831 7860 


3-050 
3-051 
3-052 
3-053 
3-054 


473 
3 

2 

2 
I 


5892 4391 
1158 9138 
6430 1 197 
1706 0520 
69S6 7060 


3-100 
3-101 
3-102 
3-103 
3-104 


450 


449 
9 
8 


4920 2393 
0417 5708 
5919 4027 
1425 7305 
6936 5497 


3-150 
3-151 
3152 
3-153 
3-154 


42S 
8 
7 
7 
6 


5212 6867 
0929 6159 
6650 8260 
2376 3128 
8io6 0720 


3005 
3-006 
3-007 
3008 
3-009 


5 3875 4328 
4 89 2 4 0335 

4 3977 5831 
3 9036 0767 
3 4099 5093 


3-055 
3-056 
3-057 
3-058 
3-059 






469 

9 


2272 0770 
7562 1603 
2856 9511 
S156 4448 
3460 6367 


3-105 
3-106 

3-107 
3-108 
3-109 


8 

7 
7 
6 
6 


2451 8559 

7971 6445 

3495 9111 
9024 6512 

4557 8603 


3-155 
3-156 
3-157 
3-158 
3-159 


6 

5 
5 
5 
4 


3S40 0992 

9578 3903 
5320 9410 
1067 7470 
6S18 8041 


3010 
3-011 
3-012 
3013 

3"oi4 


2 9167 8760 
2 4241 1719 
I 9319 3920 
I 4402 5315 
9490 5853 


3-060 
3-061 
3-062 
3-063 
3-064 


8 

8 
7 
7 
7 


8769 5220 
40S3 0961 
9401 3542 
4724 291S 
0051 904 X 


3-110 
3-III 

3-II2 

3-113 
3-1x4 


6 

5 
5 
4 
4 


0095 5340 
5637 6678 
1184 2572 

6735 2978 
2290 7851 


3-160 
3-161 
3-162 

3-163 
3-164 


4 
3 
3 

2 


2574 1080 
8333 6545 
4097 4393 
9865 4582 
5637 7069 


3-015 
3-016 

3017 
3-0x8 
3-019 


4583 5487 
489 9681 4166 
9 47S4 1842 
8 9891 8466 
8 5004 3989 


3-065 
3-066 
3-067 
3-068 
3-069 


6 
6 

5 
5 
4 


5384 1864 
0721 1342 
6062 7426 
1409 0071 
6759 9230 


3-115 
3-116 

3-117 
3-118 
3-119 


3 
3 
2 
2 
2 


7850 7147 
3415 0822 
8983 8S31 
4557 1130 
0134 7674 


3-165 
3-166 

3-167 
3-168 
3-169 


2 

I 
I 




1414 18x3 
7194 8772 
2979 7902 
8768 9162 
4562 2510 


3-020 
3-021 
3022 
3-023 
3-024 


8 0121 8362 
7 5244 1536 
7 0371 3463 
6 S5°3 4093 
6 0640 3378 


3-070 
3-071 
3-072 
3-°73 

3-074 


4 
3 
3 
2 

n 


2115 4857 
7475 6905 
2840 5328 
8210 0079 

3584 1X12 


3-120 
3-121 
3-122 

3-123 
3-124 


X 

I 





439 


5716 8420 
1303 3322 
6S94 2338 

2489 5423 
80S9 2533 


3-170 
3-171 
3-172 
3-173 
3-174 




419 

9 
8 
8 


0359 7903 
6161 5300 
1967 4658 
7777 5937 
3591 9093 


3-025 
3-026 
3-027 
3028 
3-029 


5 5782 1270 
5 0928 7720 
4 6080 2679 
4 1236 6098 
3 6397 7930 


3-075 
3-076 

3-077 
3-078 
3-079 


I 
I 





8962 838X 
4346 1840 

9734 1442 
5126 7142 
0523 8893 


3-125 
3-126 
3-127 
3-128 
3-129 


9 
8 
8 

8 

7 


3693 3623 
9301 8651 

4914 7571 
0532 0341 
6153 6916 


3-175 
3-176 

3-177 
3-178 
3-179 


7 
7 
7 
6 
6 


9410 4084 

5233 0870 
1059 9409 
6890 9657 
2726 1575 


3-030 
3031 
3-032 
3-033 
3-034 


3 1563 8126 
2 6734 6638 
2 1910 3417 
I 7090 8415 
I 2276 1584 


3-080 
3-081 
3-082 
3-083 
3-084 


459 
9 
8 
8 

7 


5925 6649 
1332 0364 

6742 9993 
2158 5489 
7578 6807 


3-130 
3-131 
3-132 
3-133 
3-134 


7 
6 
6 
5 
5 


1779 7253 
7410 1307 
3044 9035 
8684 0394 
4327 5340 


3-180 
3-181 
3-182 
3-183 
3-184 


5 
5 
5 
4 
4 


8565 512 1 
4409 0251 
0256 6926 
6108 5104 
1964 4742 


3-035 
3036 

3-037 
3-038 
3-039 


7466 2875 
2661 2242 
479 7860 9635 
9 3065 5007 
8 8274 8309 


3-085 
3-086 
3-087 
3-088 
3-089 


7 
6 
6 
5 
5 


3003 3900 
8432 6724 
3866 5231 
9304 9378 
4747 9118 


3-135 
3-136 
3-137 
3-138 
3-139 


4 
4 
4 
3 
3 


9975 3829 
5627 5S18 
1284 1263 
6945 0121 
2610 2348 


3-185 
3-186 

3-187 
3-188 

3-189 


3 
3 
2 
2 
2 


7824 580X 
3688 8238 
9557 2011 
5429 7079 
1306 3402 


3-040 

3041 
3-042 

3-043 
3-044 


8 3488 9494 
7 8707 8514 

7 3931 5321 
6 9159 9867 
6 4393 2x05 


3-090 
3-091 
3-092 
3-093 
3-094 


5 
4 
4 
3 
3 


0195 4405 

5647 5194 
1 104 1439 

6565 3096 
2031 0118 


3-140 
3-141 
3-142 
3-143 
3-144 


2 

1 
I 

X 


8279 7902 

3953 6738 
9631 8814 
5314 4086 
looi 2511 


3-190 
3-191 
3-192 
3-193 
3-194 


I 
I 





7187 0939 
3071 9647 
8960 9486 
4854 0414 
0751 2391 


3-045 
3-046 

3-047 
3-048 

3-049 


5 963X 1987 
5 4873 9465 
5 01 2 1 4493 

4 5373 702X 1 
4 0630 7003 j 


3-095 
3-096 

3-097 
3-098 
3-099 


2 
2 

I 

I 



7501 2460 
2976 0078 
8455 2925 
3939 0957 
9427 4128 


3-145 
3-146 

3-147 
3-148 
3149 




429 
9 
8 


6692 4047 
2387 8649 
8087 6275 
3791 6882 
9500 0427 


3-195 
3-196 

3-197 
3-198 
3-199 


409 

9 
8 
8 
8 


6652 5376 

2557 9327 
8467 4204 
43S0 9965 
0298 6570 



[3-200— 3-399] 



OF THE DESCENDING EXPONENTIAL. 



167 



X 




Q-X 


X 


(,-X 


X 


Q-X 


X 


(,-X 

1 


3-200 


407 


6220 3978 


3-250 


387 7420 7832 


3-3°° 


368 8316 


7401 


3-350 


350 8435 


4101 


3-20I 


7 


2146 2149 


3-251 


7 3545 3005 


3-301 


8 4630 


2669 


3-351 


4928 


7284 


3-202 


6 


8076 1041 


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 0781 


3-304 


7 3592 


9404 


3-354 


9 4429 


6986 


3-205 


s 


5890 1638 


3-255 


5 8082 0663 


3-305 


6 9921 


1836 


3-355 


9 0937 


0155 


3-206 


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 


3-210 


3 


5661 3272 


3-260 


3 8839 8017 


3-310 


5 1617 


3754 


3-360 


7 3525 


8945 


3-2II 


3 


1627 6831 


3-261 


3 5002 8807 


3-311 


4 7967 5832 


3-361 


7 0054 


1048 


3-212 


2 


7598 0705 


3-262 


3 1169 7947 


3-312 


4 4321 


4390 


3-362 


6 6585 


7851 


3'2i3 


2 


3572 4856 


3-263 


2 7340 5398 


3-313 


4 0678 


9391 


3-363 


6 3120 


9320 


3-214 


I 


9550 9242 


3-264 


2 3515 1123 


3-314 


3 7040 


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 


3-217 





7510 3413 


3-267 


I 2061 7556 


3-317 


2 6145 


3100 


3-367 


4 9296 


1 164 


3-218 





3504 8341 


3-268 


° 8251 5993 


3-318 


2 2520 


9771 


3-368 


4 5848 


5444 


3-219 


399 


9503 3304 


3-269 


4445 2512 


3-319 


I 8900 


2668 


3-369 


4 2404 


4182 


3-220 


9 


5505 8261 


3-270 


0642 7075 


3-320 


I 5283 


1754 


3-37* 


3 8963 


7343 


3-221 


9 


1512 3174 


3-271 


379 6843 9645 


3-321 


I 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 


<S 


3537 2704 


3-273 


8 9257 8652 


3-323 


4453 


5784 


3-373 


2 8662 


3030 


3-224 


7 


9555 7242 


3-274 


8 5470 5013 


3-324 


0850 


9264 


3-374 


2 5235 


3545 


3-225 


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 


I 8391 


7297 


3-227 


6 


7634 9472 


3-277 


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 


I 1561 


7783 


3-229 


5 


9707 6073 


3-279 


6 6590 3SS4 


3-329 


8 2S91 


6075 


3-379 


8151 


921S 


3-230 


5 


5749 8788 


3-280 


6 2825 6S07 


3-330 


7 93'o 


5067 


3-380 


4745 


4734 


3-231 


5 


1796 1062 


3-281 


5 9064 7358 


3-331 


7 5732 


9853 


3-381 


1342 


4298 


3-232 


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 


3-234 


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 


3-239 


2 


0307 8583 


3-289 


2 9112 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 


3-241 


I 


2475 0779 


3-291 


2 1661 4168 


3-341 


4 0153 


8476 


3-391 


6 7498 


5071 


3-242 





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 


3S9 6856 4357 


3-295 


6S04 5047 


3-345 


2 6021 


5158 


3-395 


5 4055 


4171 


3-246 


9 


2961 5271 


3-296 


3099 5530 


3-346 


2 2497 


2567 


3-396 


5 0703 


0382 


3-247 


8 


9070 5114 


3-297 


369 9398 3044 


3-347 


I 8976 


5201 


3-397 


4 7354 


0100 


3-24S 


8 


5183 3847 


3-298 


9 5700 7552 


3-348 


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 



lt>S 



MR F. W. NEWMAN'S TABLE 



[3-400— 3-599] 



X 


1 


•Ay 


f,-X 


3-500 


C-x 


X 


C-x 


3-400 


333 7326 9960 1 


3-450 


317 4563 6378 


301 9738 3422 


3-550 


287 2463 9654 


3 "401 


3 3991 3371 


3-451 


7 1390 6609 


3-501 


I 6720 1133 


3-551 


6 9592 9372 


3-402 


3 0659 0122 


3-452 


6 S220 S554 


3-502 


1 3704 9010 


3-552 


6 6724 7785 


3-403 


2 7330 oiSo 


3-453 


6 5054 2182 


3-503 


I 0692 7025 


3-553 


6 3859 4866 


3404 


2 4004 3510 


3-454 


6 1890 7459 i 


3-504 


7683 5146 


3-554 


6 0997 0586 


3-405 


2 06S2 0081 


3-455 


5 8730 4356 


3-505 


4677 3344 


3-555 


5 8137 4916 


3-406 


I 7362 9859 || 3-456 


5 5573 2840 


3-506 


1674 1589 


3-556 


5 5280 7827 


3-407 


I 4047 281 1 1 3-457 


5 2419 2880 


3-507 


299 8673 9851 


3-557 


5 2426 9291 


3-40S 


I 0734 8903 i 3-458 


4 9268 4444 


3-508 


9 5676 8100 


3-558 


4 9575 9279 


3-409 


7425 S102 


3-459 


4 6120 7500 


3-509 


9 2682 6305 


3-559 


4 6727 7762 


3-410 


4120 0376 


3-460 


4 2976 2018 


3-510 


8 9691 4437 


3-560 


4 3882 4714 


3"4ii 


0817 5690 


3-461 


3 9834 7966 


3-511 


8 6703 2466 


3-561 


4 1040 0104 


3-4I2 


529 751S 4013 


3-462 


3 6696 5312 


3-512 


8 3718 0362 


3-562 


3 8200 3904 


3-413 


9 4222 5311 


3-463 


3 3561 4025 


3-513 


8 0735 809s 


3-563 


3 5363 6086 


3414 


9 0929 9552 


3-464 


3 0429 4073 


3-514 


7 7756 5636 


3-564 


3 2529 6622 


3-415 


8 7640 6701 


3-465 


2 7300 5426 


3-515 


7 4780 2954 


3-565 


2 9698 54S4 


3'4i<' 


8 4354 6727 1 


3-466 


2 4174 8052 


3-516 


7 1807 0020 


3-566 


2 6S70 2642 


3-417 


S 1071 9597 


3-467 


2 1052 1920 


3-517 


6 S836 6804 


3-567 


2 4044 S06.9 


3-418 


7 7792 5277 


3-468 


I 7932 6998 


3-518 


6 5869 3277 


3-568 


2 1222 1737 


3-419 


7 4516 3735 


3-469 


I 4816 3256 


3-519 


6 2904 9408 


3-569 


I S402 3616 


3-420 


7 1243 4939 


3-470 


I 1703 0661 


3-520 


5 9943 5168 


3-570 


I 5585 3680 


.3-421 


6 7973 8854 3-471 


8592 9184 


3-521 


5 6985 0528 


3-571 


I 2771 1899 


3-422 


6 4707 5450 1 3-472 


5485 8792 


3-522 


5 4029 5457 


3-572 


9959 8247 


3423 


6 1444 4693 '! 3-473 


23S1 9456 


3-523 


5 1076 9927 


3-573 


7151 2694 


3-424 


5 8184 6550 


3-474 


309 9281 H43 


3-524 


4 S127 3908 


3-574 


4345 5212 


3-425 


325 4928 0989 


3-475 


9 6183 3823 


3-525 


294 5180 7369 


3-575 


1542 5774 


3-426 


5 1674 7977 


3-476 


9 30S8 7465 


3-526 


4 2237 0283 


3-576 


279 8742 4351 


1 3427 


4 8424 7482 


3-477 


8 9997 2038 


3-527 


3 9296 2619 


3-577 


9 5945 0916 


1 3428 


4 5177 9471 , 


3-478 


8 6908 7511 


3-528 


3 6358 4348 


3-578 


9 3150 5440 


j 3-429 


4 1934 3912 il 3-479 


8 3823 3853 


3-529 


3 3423 5440 


3-579 


9 0358 7896 


3-430 


3 8694 0773 j; 3-480 


8 0741 1033 


3-530 


3 0491 5867 


3-580 


8 7569 8255 


3431 


3 5457 0020 3-481 


7 7661 9020 


3-531 


2 7562 5599 


3-581 


8 4783 6490 


3432 


3 2223 1622 3-482 


7 4585 7785 


3-532 


2 4636 4607 


3-582 


8 2000 2573 


3433 


2 8992 5546 1 3-483 


7 1512 7295 


; 3-533 


2 1713 2860 


3-583 


7 9219 6476 


3-434 


2 5765 1760 :! 3-484 


6 8442 7520 


3-534 


I S793 0331 


3-584 


7 6441 8171 


3-435 


2 2541 0232 3-485 


6 5375 8429 


3-535 


I 587s 6990 


3-585 


7 3666 7630 


3-436 


I 9320 0929 ' 3-486 


6 231 1 9993 


3-536 


I 2961 2807 


3-586 


7 0894 4826 


3-437 


I 6102 3819 


3-487 


5 9251 2179 


13-537 


I 0049 7754 


3-587 


6 8124 9731 


3-438 


I 2887 8871 


3-488 


5 6193 4958 


3-538 


7 141 3802 


3-588 


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 


3-540 


1332 7082 


3-590 


5 9833 0423 


3-441 


3263 6670 


3-491 


4 7038 6544 


3-541 


2S9 8432 8257 


3-591 


5 7074 5887 


3-442 


0062 0044 


3-492 


4 3993 1388 


3-542 


9 5535 8416 


3-592 


5 4318 8922 


3-443 


319 6863 5419 


3-493 


4 0950 6671 


3-543 


9 2641 7530 


3'593 


5 1565 9500 


3444 


, 9 3668 2762 


3-494 


3 7911 2364 


3-544 


8 9750 5571 


3-594 

j 


4 8815 7594 


3-445 


j 9 0476 2043 


3'495 


3 4874 8436 


3-545 


8 6862 2500 


3-595 


4 6068 3176 


3-446 


8 7287 3228 


3-496 


3 1841 4857 


3-546 


8 3976 8316 


j 3-596 


4 3323 6218 


3'447 


1 8 4101 6286 


! 3-497 


2 8811 1597 


3-547 


8 1094 2963 


3-597 


4 0581 6694 


3-448 


1 8 0919 1185 


3-498 


2 5783 8624 


3-548 


7 8214 6421 


3-598 


3 7842 4576 


3 449 


\ 7 7739 7893 


1 3-499 


2 2759 5909 


3-549 


7 5337 8661 


3-599 

1 


3 5105 9836 



[3"6oo— 3799] 



OF THE DESCENDING EXPONENTIAL. 



1G9 



X 


(,-X 


X 


C-x 


X 


(>-X 


X 


Q-X 


3-600 


273 


2372 2447 


3-650 


259 9112 8779 


3-700 


247 2352 


6470 


3-750 


235 


1774 5856 


3-601 


2 


9641 2382 


3-651 


9 6515 0642 


3-701 


6 9881 


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 


3'6o3 


2 


4187 4114 


3-653 


9 1327 2236 


3-703 


6 4946 


7036 


3-753 


4 


4729 8342 


3-604 





1464 5S56 


3-654 


8 8737 1916 


3-704 


6 2482 


9889 


3-754 


4 


2386 2764 


3-605 


I 


8744 4813 


3-655 


8 6149 7483 


3-705 


6 0021 


7368 


3-755 


4 


0045 0609 


3-606 


1 


6027 0957 


3-656 


8 3564 8912 


3-706 


5 7562 


9446 


3-756 


3 


7706 1855 


3-607 


I 


3312 4262 


3-657 


8 09S2 6177 


3-707 


5 5106 


6ior 


3-757 


3 


5369 6478 


3-608 


I 


0600 4700 


3-658 


7 8402 9252 


3-708 


5 2652 


7306 


3-758 


3 


303s 4454 


3-609 





7891 2244 


3-659 


7 5825 8110 


3-709 


5 020I 


3038 


3-759 


3 


0703 5761 


3-610 





5184 6866 


3-660 


7 3251 2726 


3-71° 


4 7752 


3272 


3-760 


2 


8374 0375 


3-611 





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 


3'6i3 


9 


7081 2937 


3-663 


6 5543 0869 


3-713 


4 0420 


0741 


3763 


2 


1399 3826 


3-614 


9 


4385 5605 


3-664 


6 2978 S261 


3-714 


3 79S0 


8738 


3-764 


I 


9079 1435 


3-615 


9 


1692 5217 


3-665 


6 0417 1284 


3-715 


3 5544 


1115 


3-765 


r 


676J 2235 


3-616 


8 


900* 1746 


3-666 


5 7857 9910 


3-716 


3 3109 


7848 


3-766 


I 


4445 6203 


3-617 


8 


6314 5165 


3-667 


5 5301 4115 


3-717 


3 0677 


8912 


3-767 


I 


2132 3315 


3-618 


8 


3629 5447 


3-668 


5 2747 3873 


3-718 


2 8248 


4282 


3-768 





9S21 3549 


3-619 


8 


0947 2565 


3-669 


5 0195 9159 


3-719 


2 5821 


3935 


3-769 





7512 6881 


3-620 


7 


8267 6493 


3-670 


4 7646 9947 


3-720 


2 3396 


7846 


3-770 





5206 3287 


3-621 


7 


559° 7203 


3-671 


4 5100 621 1 


3-721 


2 0974 


5991 


3-771 





2902 2746 


3-622 


7 


2916 4669 


3-672 


4 2556 7926 


3-722 


I 8554 


8346 


3-772 





0600 5234 


3-623 


7 


0244 8865 


3-673 


4 0015 5066 


3-723 


I 6137 


4887 


3-773 


229 


8301 0728 


3-624 


6 


7575 9763 


3-674 


3 7476 7607 


3-724 


I 3722 


5588 


3-774 


9 


6003 9205 


3-625 


266 


4909 7336 


3-675 


253 4940 5523 


3-725 


I 1310 


0427 


3-775 


9 


3709 0642 


3-626 


6 


2246 1559 


3-676 


3 2406 8787 


3-726 


8899 


9380 


3-776 


9 


1416 5016 


3-627 


5 


9585 2405 


3-677 


2 9875 7377 


3-727 


6492 


2421 


3-777 


8 


9126 2305 


3-628 


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 


8 


4552 5532 


3-630 


5 


1 6 18 4409 


3-680 


2 2297 4835 


3-730 


239 9283 5837 


3-780 


8 


2269 1426 


3-631 


4 


8968 1479 


3-681 


I 9776 4467 


3-731 


9 6885 


4994 


3-781 


7 


9988 0142 


3-632 


4 


6320 5037 


3-682 


I 7257 9298 


3-732 


9 4489 


8119 


3-782 


7 


7709 1658 


3-633 


4 


3675 5060 


3-683 


1 4741 93°o 


3-733 


9 2096 


5190 


3-783 


7 


5432 5951 


3634 


4 


1033 1519 


3-684 


1 2228 4451 


3-734 


8 9705 


6181 


3-784 


7 


315S 299S 


3-635 


3 


8393 4388 


3-685 


9717 4723 


3-735 


8 7317 


1069 


3-785 


7 


0SS6 2777 


3-636 


3 


5756 3641 


3-686 


7209 0093 


3-736 


8 4930 


9831 


3-786 


6 


S616 5265 


3-637 


3 


3121 9252 


3-687 


4703 °S34 


3-737 


8 2547 


2441 


3-787 


6 


6349 0439 


3-638 


3 


0490 1 194 


3-688 


2199 6023 


3-738 


8 0165 


8878 


3-788 


6 


4083 S276 


3-639 


2 


7860 9441 


3-689 


249 9698 6534 


3-739 


7 7786 


9116 


3-789 


6 


1820 8755 


3-640 


2 


5234 3966 


3-690 


9 7200 2042 


3-740 


7 5410 


3131 


3-790 


5 


9560 1851 


3-641 


2 


2610 4744 


3-691 


9 4704 2522 


3-741 


7 3036 


0901 


3-791 


5 


7301 7544 


3-642 


I 


99S9 1748 


3-692 


9 2210 7949 


3-742 


7 0664 


2402 


3-792 


5 


5045 5809 


3-643 


I 


7370 4952 


3-693 


8 9719 8298 


3-743 


6 8294 


7609 


3-793 


5 


2791 6624 


3-644 


1 


4754 4329 


3-694 


8 7231 3544 


3-744 


6 5927 


6499 


3-794 


5 


0539 9968 


3-645 


I 


2140 9854 


3-695 


8 4745 3662 


3-745 


6 3562 


9048 


3-795 


4 


S290 5817 


3-646 





9530 1501 


3-696 


8 2261 8628 


3-746 


6 1200 


5233 


3-796 


4 


6043 4149 


3-647 





6921 9243 


3-697 


7 9780 8417 


3-747 


5 8S40 


5030 


3-797 


4 


3798 4941 


3-648 





4316 3°S4 


3-698 


7 7302 3003 


3-748 


5 6482 


8415 


3-798 


4 


1555 S171 


3-649 





1713 2908 


3-699 


7 4826 2363 


3-749 


5 4127 


5365 


3-799 


3 


9315 3817 



Vol. XIII. Part III. 



23 



170 



MR F. W. NEWMAN'S TABLE 



[3-800—3-999] 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


(,-X 


3-800 


223 


7077 


1856 


3-S50 


212 


7973 6438 


' 3-900 


202 


4191 1446 


■ 3-950 


192 5470 1775 


3-801 


3 


4841 


2266 


3-851 


2 


5846 7338 


3-901 


2 


2167 9652 


3-951 


2 3545 6698 


3-802 


3 


2607 


5024 


! 3-852 


2 


3721 9496 


3-902 


2 


0146 8080 


3-952 


2 1623 0855 


3803 


3 


0376 


0109 


13-853 


2 


1599 2892 


3-903 




8127 6709 


3-953 


I 9702 4229 


3804 


2 


8146 


7497 


3-854 




9478 7504 


3-904 




6110 5519 


3-954 


I 7783 68oi 


3-805 




5919 


7166 


3-855 




7360 3310 


3-905 




4095 4491 


3-955 


I 5866 8549 


3-806 




3694 


909s 


y^i(> 




5244 0290 


3-906 




2082 3604 


3-956 


I 3951 9457 


3-807 




1472 


3261 


3857 




3129 S422 


3-907 




0071 2837 


3-957 


I 2038 9504 


3-808 




9251 


9641 


3-858 




1017 7686 


3-908 





8062 2172 


3-958 


I 0127 S672 


3-809 


I 


7033 


8214 


3-859 





8907 8060 


3-909 





6055 1586 


3-959 


8218 6940 


3-810 




4817 


S957 


3-860 





6799 9523 


3-910 





4050 1062 


3-960 


6311 4291 


3-811 




2604 


1849 


3-861 





4694 2054 


3-911 





2047 0578 


3-961 


4406 0705 


3-812 




0392 


6866 


3-862 





2590 5632 


3-912 





0046 0114 


3-962 


2502 6164 


3-813 





81S3 


3987 


3-863 





0489 0236 


3-913 


199 


8046 9651 


3-963 


0601 0647 


3-814 





5976 


3190 


3-864 


209 


8389 5845 


3-914 


9 


6049 916S 


3-964 


189 S701 4136 


3-815 





3771 


4453 


3-865 


9 


6292 2437 


3-915 


9 


4054 8645 


3-965 


9 6803 6612 


3-816 





156S 


7754 


3-866 


9 


4196 9993 


3-916 


9 


2o6i 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 


3-818 


9 


7170 


0381 


3-868 


9 


0012 7909 


3-918 


8 


8081 6642 


3-968 


9 1121 7774 


3-819 


9 4973 


9662 


3-869 


8 


7923 8227 


3-919 


8 


6094 5763 


3-969 


8 9231 6008 


3-820 


9 


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 


3-921 


8 


2126 3567 


3-971 


8 5456 9136 


3-822 


8 


839S 


9119 


3-872 


8 


1669 4375 


3922 


8 


0145 2211 


3-972 


8 3572 3991 


3823 


8 


6211 


6068 


3-873 




9588 8085 


3-923 


7 


8166 0655 


3-973 


8 1689 7682 


3-824 


8 


4026 


4879 


3-874 




7510 2592 


3-924 


7 


6188 8882 


3-974 


7 9S09 0189 


3-825 


218 


1843 


5531 


3-875 


207 


5433 7873 


3-925 


197 4213 6871 


3-975 


1S7 7930 1495 


3-826 


7 


9662 


8001 


3-876 




3359 39°9 


3-926 


7 


2240 4602 


3-976 


7 6053 1580 


3-827 


7 


7484 


2268 


3-877 




1287 0679 


3-927 


7 


0269 2055 


3-977 


7 4178 0425 


3-828 


7 


5307 


8309 


3-878 


6 


9216 8161 


3-928 


6 


8299 9211 


3-978 


7 2304 8013 


3-829 


7 


3133 


6104 


3-879 


6 


7148 6336 


3-929 


6 


6332 6050 


3-979 


7 0433 4323 


3-830 


7 


0961 


5630 


3-880 


6 


5082 5182 


3-930 


6 


4367 2553 


3-980 


6 8563 9338 


3-831 


6 


8791 


6865 


3-881 


6 


3018 4678 


3-931 


6 


2403 8699 


3-981 


6 6696 3038 


3-832 


6 


6623 


9789 


3-882 


6 


0956 4805 


3-932 


6 


0442 4469 


3-982 


6 4830 5405 


3-833 


6 


4458 


4378 


3883 


S 


8896 5542 


3-933 


5 


8482 9843 


3-983 


6 2966 6421 


3-834 


6 


2295 


0613 


3-884 


5 


6838 6868 


3-934 


5" 


6525 4803 


3-984 


6 1 104 6066 


3-835 


6 


0133 


8470 


3-885 


5 


4782 8762 


3-935 


5 


4569 9327 


3-985 


5 9244 4323 


3-836 


5 


7974 


7929 


3-886 


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 


S 


0664 6994 


3-987 


5 5529 6594 


3-838 


5 


3663 


1563 


3-888 


4 


8627 7648 


3-938 


4 


8715 0097 


3-988 


5 3675 0572 


3-839 


5 


1510 


5696 


3-889 


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 


3-990 


4 9971 4120 


3-841 


4 


7211 


8486 


3-891 


4 


2491 0911 


3-941 


4 


2877 6251 


3-991 


4 8122 3653 


3-842 


4 


5065 


7100 


3-892 


4 


0449 6209 


3-942 


4 


0935 7186 


3-992 


4 6275 1667 


3-843 


4 


2921 


7165 


3-893 


3 


8410 1912 


3-943 


3 


8995 7530 


3-993 


4 4429 8143 


3-844 


4 


0779 


8659 


3-894 


3 


6372 7999 


3-944 


3 


7057 7265 


3-994 


4 2586 3064 


3-845 


3 


8640 


1561 


3-895 


3 


4337 4449 


3-945 


3 


5 121 6369 


3-995 


4 0744 6411 


3-846 


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 S308 


3-848 


3 


2233 


8499 


3-898 


2 


8243 5780 


3-948 


2 


9324 9714 


3-998 


3 5230 6822 


3849 

1 


3 


0102 


6818 


3-899 


2 


6216 3482 


3-949 


2 


7396 6108 


3-999 


3 3396 3689 



[4'ooo- 


-4-199] 


OF 


THE DESCENDING EXPONENTIAL. 




17] 


X 


Q-X 


1 X 


Q-X 


X 


Q-X 


X 


Q-X 


4'ooo 


183 1563 8889 


4-050 


174 2237 4639 


4-100 


165 7267 5402 


4-150 


157 6441 6485 


4-001 


2 9733 2405 


4-051 


4 0496 0973 


4-101 


5 5611 1010 


4-15' 


7 4865 9948 


4'002 


2 7904 4218 


4-052 


3 8756 4712 


4-102 


5 3956 3174 


4-152 


7 3291 9160 


4-003 


2 6077 4310 


4-053 


Z 7018 5838 


4-103 


5 2303 1878 


4-153 


7 1719 4104 


4-004 


2 4252 2663 


4-054 


3 5282 4334 


4-104 


5 0651 7105 


4-154 


7 0148 4766 


4-005 


2 2428 9259 


4-055 


3 3548 0183 


4-105 


4 9001 8838 


4-155 


6 8579 1130 


4-006 


2 0607 4079 


4-056 


3 1815 3368 


4-106 


4 7353 7062 


4-156 


6 7011 3179 


4-007 


I 8787 7105 


4-057 


3 0084 3871 


4-107 


4 5707 1759 


4-157 


6 5445 0898 


4-008 


I 6969 8319 


4-058 


2 8355 1674 


4-108 


4 4062 2913 


4-158 


6 3880 4272 


4*009 


I 5153 7702 


4-059 


2 6627 6762 


4-109 


4 2419 0507 


4-159 


6 2317 3284 


4-010 


I 3339 5237 


4-060 


2 4901 9115 


4-110 


4 0777 4526 


4-160 


6 0755 7920 


4-0 1 1 


I 1527 0905 


4-061 


2 3177 8718 


4-111 


3 9137 4953 


4-161 


5 9195 8163 


4-012 


9716 4690 


4-062 


2 1455 5552 


4-112 


3 7499 1771 


4-162 


5 7637 3998 


4'°i3 


7907 6570 


4-063 


I 9734 9601 


4-113 


3 5862 4964 


4-163 


5 60S0 5410 


4-014 


6100 6530 


4-064 


I 8016 0847 


4-114 


3 4227 4515 


4-164 


5 4525 2382 


4-015 


4295 4551 


4-065 


r 6298 9273 


4-115 


3 2594 0409 


4-165 


5 2971 4900 


4-016 


2492 0615 


4-066 


I 4583 4863 


4-116 


3 0962 2629 


4-166 


5 1419 2947 


4-017 


0690 4704 


4-067 


I 2S69 7598 


4-117 


2 9332 1159 


4-167 


4 9868 6509 


4-018 


179 8890 6799 


4-068 


I 1157 7462 


4-118 


2 7703 5981 


4-168 


4 8319 5569 


4-019 


9 7092 6884 


4-069 


9447 4437 


4-119 


2 6076 7081 


4-169 


4 6772 0113 


4-020 


9 5296 4939 


4-070 


7738 8507 


4-120 


2 4451 4442 


4-170 


4 5226 0124 


4-021 


9 3502 0948 


4-071 


6031 9655 


4-121 


2 2827 8047 


4-171 


4 36S1 5587 


4-022 


9 1709 4891 


4-072 


4326 7862 


4-122 


2 1205 7880 


4-172 


4 2138 6488 


4-023 


8 9918 6752 


4-073 


2623 3113 


4-123 


I 9585 3926 


4-173 


4 0597 2809 


4-024 


8 8129 6512 


4-074 


0921 5390 


4-124 


1 7966 6167 


4-174 


3 9057 4537 


4-025 


178 6342 4153 


4-075 


169 9221 4677 


4-125 


161 6349 4588 


4-175 


153 7519 1655 


4-026 


8 4556 965S 


4-076 


9 7523 0956 


4-126 


I 4733 9173 


4-176 


3 5982 4148 


4-027 


8 2773 300S 


4-077 


9 5S26 4209 


4-127 


I 3119 9904 


4-177 


3 4447 2002 


4-028 


8 0991 4186 


4-078 


9 4131 4421 


4-128 


I 1507 6767 


4-178 


3 2913 5199 


4-029 


7 9211 3174 


4-079 


9 2438 1575 


4-129 


9S96 9745 


4-179 


3 13S1 3726 


4-030 


7 7432 9954 


4-0S0 


9 0746 5653 


4-130 


8287 8823 


4-180 


2 985° 7567 


4-031 


7 5656 4508 


4-081 


8 9056 6638 


4-131 


6680 3982 


4-181 


2 8321 6706 


4-032 


7 3S81 6819 


4-082 


8 7368 4513 


4-132 


5074 5209 


4-182 


2 6794 1128 


4-033 


7 2108 6868 


4-083 


8 56S1 9263 


4-133 


3470 2487 


4-183 


2 526S 0818 


4-034 


7 0337 4639 


4-084 


8 3997 0869 


4-134 


1867 5799 


4-184 


2 3743 5761 


4-035 


6 8568 0113 


4-085 


8 2313 9315 


4-135 


0266 5130 


4-185 


2 2220 5942 


4-036 


6 6800 3273 


4-086 


8 0632 4585 


4-136 


159 8667 0463 


4-186 


2 0699 1344 


4-037 


6 5034 4101 


4-087 


7 8952 6661 


4-137 


9 7069 1784 


4-187 


I 9179 1954 


4-03S 


6 3270 2579 


4-088 


7 7274 5526 


4-138 


9 5472 9074 


4-18S 


I 7660 7755 


4-039 


6 1507 8690 


4-089 


7 5598 1164 


4-139 


9 3878 2320 


4-189 


I 6143 8733 


4-040 


5 9747 2416 


4-090 


7 3923 3558 


4-140 


9 2285 1504 


4-190 


I 4628 4873 


4-041 


5 79S8 3739 


4-091 


7 2250 2691 


4-141 


9 0693 6612 


4-191 


I 3114 6159 


4-042 


5 6231 2642 


4-092 


7 0578 8547 


4-142 


8 9103 7626 


4-192 


I 1602 2576 


4-043 


5 4475 9108 


4-093 


6 8909 I 109 


4-143 


8 7515 4531 


4-193 


I 0091 4109 


4-044 


5 2722 3118 


4-094 


6 7241 0360 


4-144 


8 5928 7312 


4-194 


8582 0742 


4-045 


5 0970 4656 


4-095 


6 5574 6283 


4-145 


8 4343 5951 


4-195 


7074 2462 


4-046 


4 9220 3703 


4-096 


6 3909 8862 


4-146 


8 2760 0434 


4-196 


5567 9252 


4-047 


4 7472 0243 


4-097 


6 2246 8080 


4-147 


8 1178 0745 


4-197 


4063 logS 


4-048 


4 5725 4257 ' 


4-098 


6 0585 3920 


4-148 


7 9597 6868 


4-198 


° 2559 7985 


4-049 


4 3980 5728 i 


4-099 


5 8925 6366 


4-149 


7 S018 8786 


4-199 


1057 989S 






17 



MR F. W. NEWMAN'S TABLE 



[4-00— 4-399] 



X 


C-x 


X 


Q-X 


X 


Q-X 


X 


Q-X 


4-200 


'I49 


9557 6820 


4-250 


♦ 142 


6423 


3909 


4-300 


135 6855 9012 


4-350 


129 o68i 2580 


4-201 


. 1-9 


8058 87 39 


4-251 


•^ 


4997 


6805 


4-3°! 


5 5499 7235 


4-351 


8 9391 2219 


4-202 


' 9 


6561 5638 


4-252 


2 


3573 


3951 


4-302 


5 4144 901-3 


4-352 


8 8102 4752 


4-203 


9 


5065 7502 


4-253 


2 


2150 


5332 


4-303 


5 2791 4333 


4-353 


8 6815 0165 


4-204 


9 


3571 4318 


4-254 


n 


0729 


°935 


4-304 


5 1439 3180 


4-354 


8 5528 8447 


4"205 


9 


207S 6069 


4-255 




9309 


0745 


4-305 


5 0088 5542 


4-355 


8 4243 9584 


4-206 


9 


0587 2741 


4-256 




7890 


4749 


4-306 


4 8739 1404 


4-356 


8 2960 3564 


4-207 


8 


9097 4319 


4-257 




6473 


2931 


4-307 


4 7391 0754 


4-357 


8 1678 0373 


4-20S 


8 


7609 0787 


4-258 




5057 


5278 


4-308 


4 6044 3578 


4-358 


8 0396 9999 


4-209 


8 


6122 2132 


4-259 




3643 


1776 


4-309 


4 4698 9863 


4-359 


7 91x7 2428 


4-210 


8 


4636 8338 


4-260 




2230 


2410 


4-310 


4 3354 9594 


4-360 


7 7838 7649 


4-211 


8 


3152 9390 


4-261 




oSiS 


7166 


4-3 1 1 


4 2012 2759 


4-361 


7 6561 5649 


4-212 


8 


1670 5274 


4-262 





9408 


6031 


4-312 


4 0670 9344 


4-362 


7 5285 6414 


4"2i3 


8 


01S9 5975 


4-263 





7999 


S990 


4-313 


3 9330 9336 


4-363 


7 4010 9932 


4-214 


7 


8710 1477 


4-264 





6592 


602S 


4-314 


3 7992 2721 


4-364 


7 2737 6190 


4"2i5 




7232 1767 


4-265 





5186 


7133 


4-315 


3 6654 9486 


4-365 


7 1465 5175 


4-216 




5755 6829 


4-266 





3782 


2289 


4-316 


3 5318 9618 


4-366 


7 0194 6875 


4-217 




4280 6648 


4-267 





2379 


1484 


4-317 


3 3984 3102 


4-367 


6 S925 1277 


4-218 




2S07 1211 


4-268 





0977 


4702 


4-318 


3 2650 9927 


4-368 


6 7656 8368 


4-219 




1335 0501 


4-269 


139 


9577 


1930 


4-319 


3 1319 0078 


4-369 


6 6389 8136 


4-220 


6 


9S64 4505 


4-270 


9 


8178 


3153 


4-320 


2 9988 3542 


4-370 


6 5124 0568 


4-221 


6 


S395 3207 


4-271 


9 


6780 


S359 ' 


4-321 


2 8659 0307 


4-371 


6 3859 5651 


4-222 


6 


6927 6593 


4-272 


9 


5384 


7532 


4-322 


2 7331 0357 


4-372 


6 2596 3372 


4-223 


6 


5461 4649 


4-273 


9 


3990 


0659 


4-323 


2 6004 36S1 


4-373 


6 1334 3720 


4-224 


6 


3996 7359 


4-274 


9 


2596 


7726 


4-324 


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 7556 0392 


4-227 


5 


9611 3271 


4-277 


8 


8425 


2427 


4-327 


2 0710 9446 


4-377 


5 6299 H17 


4-228 


5 


8-152 4454 


4-278 


8 


7037 


5114 


4-328 


I 9390 8938 


4-378 


5 5043 4406 


4-229 


5 


6(595 °2I7 


4-279 


8 


5651 


1672 


4-329 


I 8072 1623 


4-379 


5 37S9 0244 


4"23o 


5 


5239 0548 


4-280 


8 


4266 


2086 


4-330 


I 6754 7490 


4-380 


5 2535 8621 


4'23i 


5 


3784 5432 


4-281 


8 


2S82 


•6343 


4-331 


I 5438 6524 


4-381 


5 1283 9523 


4-232 


5 


2331 4853 


4-282 


8 


1500 


4429 


4-332 


I 4123 8713 


4-382 


5 0033 2938 


4'233 


5 


0S79 8797 


4-283 


8 


0119 


6330 


4-333 


I 2810 4042 


4-383 


4 8783 8S53 


4'234 


4 


9429 7250 


4-284 


7 


8740 


2032 


4-334 


I 1498 2500 


4-384 


4'7535 7256 


4'235 


4 


7981 0198 


4-285 


7 


7362 


1521 


4-335 


I 0187 4073 


4-385 


4 6288 8134 


4236 


4 


(>S7,Z 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 


4-239 


4 


2200 6641 


4-289 


7 


1863 


7°77 


4-339 


4957 1252 


4389 


4 1313 6152 


4-240 


4 


0759 1843 


4-290 


7 


0492 


5297 


4-340 


3652 8203 


4-390 


4 0072 9220 


4-241 


3 


9319 1453 


'4-291 


6 


9122 


7222 


4-341 


2349 8191 


4-391 


3 8833 4689 


4-242 


3 


7880 5455 


4-292 


6 


7754 


2838 


4-342 


1048 1203 


4-392 


3 7595 2547 


4-243 


3 


6443 3837 


4-293 


6 


6387 


2132 


4-343 


129 9747 7224 


4-393 


3 6358 2780 


4-244 


3 


5007 6583 


4-294 


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 51 II 


4-296 


6 


2294 


1941 


4-346 


9 5854 3223 


4-396 


3 2654 7612 


4-247 


3 


0709 0S64 


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 0191 9154 


4-249 


2 


7850 5277 


4-299 


5 


8213 


4358 


4-349 


9 1972 5849 


4-399 


2 8962 3384 



[4-400- 


-4-599] 


OF 


THE DESCENDING EXPONENTIAL. 




173 


X 


Q-X 


X 


1 c-=« 


X 


Q-X 


X 


™ 

Q-X 


4-400 


122 7733 9903 


\ 4-45° 


116 7856 6970 


4-500 


III 0899 6538 


4-550 


105 6720 4384 


4-4° I 


2 6506 8700 


4-451 


6 6689 4241 


4-5°! 


97S9 3094 


j 4-551 


5 5664 2461 


4-402 


2 52S0 9761 


4-452 


6 5523 3178 


4-502 


8680 0748 


4-552 


5 4609 1095 


4-403 


2 4056 3076 


4-453 


6 4358 3770 


4-503 


7571 9489 


4-553 


5 3555 0275 


4-404 


2 2S32 8631 


4-454 


6 3194 6006 


4-504 


6464 9306 


4-554 


5 2501 9991 


4-405 


2 1610 6415 


4-455 


6 2031 9874 


4-505 


5359 0187 


4-555 


5 1450 0232 


4'4o6 


2 0389 6414 


4-456 


6 0870 5363 


4-506 


4254 2122 


4-556 


5 0399 0987 


4-407 


I 9169 8618 


4-457 


5 9710 2460 


4-507 


3150 5099 


4-557 


4 9349 2247 


4-408 


I 7951 3013 


4-458 


5 8551 1154 


4-508 


2047 9108 


4-558 


4 8300 3999 


4-409 


I 6733 9588 


4-459 


5 7393 1434 


4-509 


0946 4137 


4-559 


4 7252 6235 


4-410 


I 5517 8330 


4-460 


5 6236 3287 


4-510 


109 9846 0176 


4-560 


4 6205 8943 


4-411 


I 4302 9227 


4-461 


5 5080 6703 


4-5 1 1 


9 S746 7213 


4-561 


4 5160 2114 


4-412 


1 3089 2267 


4-462 


5 3926 1670 


4-512 


9 7648 5238 


4-562 


4 4115 5736 


4"4i3 


I 1876 7438 


4-463 


5 2772 8176 


4-513 


9 6551 4239 


4-563 


4 3071 9799 


4-414 


I 0665 4728 


4-464 


5 1620 6210 


4-514 


9 5455 4205 


4-564 


4 2029 4293 


4-415 


9455 4125 


4-465 


5 0469 5760 


4-515 


9 4360 5127 


4-565 


4 0987 9207 


4-416 


8246 5616 


4-466 


4 9319 6815 


4-516 


9 3266 6992 


4-566 


3 9947 4531 


4-417 


7038 9190 


4-467 


4 S170 9362 


4-517 


9 2173 9789 


4-567 


3 8go8 0254 


4-418 


5832 4834 


4-468 


4 7023 3392 


4-51S 


9 10S2 3508 


4-568 


3 7869 6367 


4-419 


4627 2536 


4-469 


4 5876 8892 


4-519 


8 9991 8138 


4-569 


3 6832 2858 


4-420 


3423 2285 


4-470 


4 4731 585° 


4-520 


8 8902 3668 


4-570 


3 5795 9718 


4-421 


2220 4067 


4-471 


4 3587 4256 


4-521 


8 7814 0087 


4-571 


3 4760 6935 


4-422 


1018 7872 


4-472 


4 2444 409S 


4-522 


8 6726 7385 


4-572 


3 3726 4500 


4-423 


119 9818 3688 


4-473 


4 1302 5364 


4-523 


8 5640 5549 


4-573 


3 2693 2403 


4-424 


9 8619 1501 


4-474 


4 0161 S044 


4-524 


8 4555 4570 


4-574 


3 i66i 0632 


4-425 


9 7421 1301 


4-475 


113 9022 2124 


4-525 


108 3471 4436 


4-575 


103 0629 9178 


4-426 


9 6224 3075 


4-476 


3 7883 7595 ,' 


4-526 


8 2388 5137 


4-576 


2 9599 8030 


4-427 


9 5028 6811 


4-477 


3 6746 4445 


4-527 


8 1306 6662 


4-577 


2 8570 7178 


4-428 


9 3834 2497 


4-478 


3 5610 2663 


4-528 


8 0225 9001 


4-578 


2 7542 6612 


4-429 


9 2641 0122 


4-479 


3 4475 2236 1 


4-529 


7 9146 2141 


4-579 


2 6515 6322 


4-430 


9 1448 9673 


4-480 


3 3341 3155 ' 


4-530 


7 8067 6073 


4-580 


2 5489 6296 


4-431 


9 0258 1138 


4-481 


3 2208 5406 


4-531 


7 6990 0785 


4-581 


2 4464 6526 


'4-432 


S 9068 4506 


4-482 


3 1076 8980 


4-532 


7 5913 6267 


4-582 


2 3440 7000 


4-433 


8 7879 9765 


4-483 


2 9946 3864 1 


4-533 


7 4838 2509 


4-583 


2 2417 7708 


4-434 


8 6692 6903 


4-484 


2 8817 0048 


4-534 


7 3763 9499 


4-584 


2 1395 8641 


4-435 


8 5506 5907 


4-485 


2 768S 7521 ' 


4-535 


7 2690 7226 


4-585 


2 0374 9788 


4-436 


8 4321 6767 


4-486 


2 6561 6270 


4-536 


7 1618 5681 


4-586 


r 9355 1138 


4-437 


8 3137 9470 


4-487 


2 5435 6284 


4-537 


7 0547 4851 


4-587 


I 8336 2682 


4-438 


8 1955 4004 1 


4-488 


2 4310 7553 1 


4-538 


6 9477 4728 


4-588 


I 7318 4409 


4-439 


8 0774 0358 


4-489 


2 3187 0065 1 

i 


4-539 


6 8408 5298' 


4-589 


I 6301 6310 


4-440 


7^9593 8520 1 


4-490 


2 2064 3809 


4-540 


6 7340 6553 


4-590 


I 5285 8373 


4-441 


7 8414 8477 


4-491 


2 0942 8774 


4-541 


6 6273 84S2 


4-591 


I 4271 0590 


4-442 


7 7237 0219 1 


4-492 


I 9822 4948 


4-542 


6 5208 1073 


4-592 


I 3257 2949 


4-443 


7 6060 3733 I 


4-493 


I 8703 2320 


4-543 


6 4143 4316 


4-593 


I 2244 5440 


4-444 


7 4884 9007 


4-494 


I 7585 0880 


4-544 


6 3079 8201 


4-594 


I 1232 8054 


4-445 


7 3710 6031 


4-495 


I 6468 0615 


4-545 


6 2017 2716 


4-595 


I 0222 0781 


4-446 


7 2537 4791 


4-496 


I 5352 151S 


4-546 


6 0955 7852 


4-596 


9212 3610 ! 


4-447 


7 1365 5277 


4-497 


I 4237 3568 


4-547 


5 9S95 3597 


4-597 


8203 6530 1 


4-448 


7 0194 7477 


4-498 


I 3123 6764 


4-548 


5 8835 9941 


4-598 


7195 9533 


4-449 


6 9025 1379 


4-499 


I 2011 1091 


4-549 


5 7777 6873 


4-599 


6189 2608 

J 



174 






MR F. ^V. NEWMAN'S TABLE 




[4-600—4-799] 


X 


Q-X 


X 
4-650 


Q-X 


X 

4-700 


(,-X 


X 


Q-X 


4'6oo 


0100 51S3 5745 


95 6160 1930 


90 9527 7102 


4-750 


86 5169 5203 


4-601 


-:17s 8933 


4-651 


5 5204 5108 


4-701 


8618 6371 


4-751 


6 4304 7832 


4'6o2 


3175 2163 


4-652 


5 4249 7837 


4-702 


7710 4726 


4-752 


6 3440 9104 


4-603 


2172 5425 


4-653 


5 3296 0109 


4-703 


6803 2158 


4-753 


6 2577 9011 


4-604 


1170 8709 


4-654 


5 2343 1914 


4-704 


5896 8658 


4-754 


6 1715 7544 


4"6o5 


0170 2005 


4-655 


5 1391 3242 


4-705 


4991 4218 


4-755 


6 0854 4693 


4-606 


.--199 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 


4-707 


3183 2477 


4-757 


5 9134 4809 


4-60S 


9 7174 1861 


4-658 


4 8541 4272 


4-708 


22S0 5159 


4-758 


5 8275 7759 


4-609 


9 6177 5104 


4-659 


4 7593 3599 


4-709 


1378 6864 


4-759 


5 7417 9291 


4-610 


a^9 5iS| 830S 
9 418V 1464 


4-660 


4 6646 2402 


4-710 


0477 7582 


4-760 


5 6560 9397 


4-6 1 1 


4-661 


4 5700 0671 


4-711 


89 9577 7306 


4-761 


5 5704 8069 


4-612 


9 3193 4562 


4-662 


4 4754 8397 


4-712 


9 8678 6025 


4-762 


5 4849 5298 


4-613 


9 2200 7591 


4-663 


4 3810 5571 


4-713 


9 7780 3731 


4-763 


5 3995 1076 


4-614 


9 1209 0543 


4-664 


4 2S67 2183 


4-714 


9 6883 0414 


4-764 


5 3141 5393 


4"6i5 


9 0218 3407 


4-665 


4 1924 8223 


4-715 


9 59S6 6067 


4-765 


5 22S8 8242 


4-616 


8 9228 6173 


4-666 


4 0983 3683 


4-716 


9 5091 0679 


4-766 


5 1436 9614 


4-617 


8 8239 8831 


4-667 


4 0042 8553 


4-717 


9 4196 4242 


4-767 


5 05S5 9500 


4-618 


8 7252 1372 


4-668 


3 9103 2823 


4-718 


9 3302 6748 


4-768 


4 9735 7892 


4-619 


8 6265 3785 


4-669 


3 8164 6484 


4-719 


9 2409 8186 


4-769 


4 8886 4782 


4-620 


<50?8 5279 6061 


4-670 


3 7226 9527 


4-720 


9 1517 8548 


4-770 


4 8038 0160 


4-621 


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 


4623 


8 2328 1966 


4-673 


3 4419 4851 


4-723 


8 8847 3091 


4-773 


4 5497 7143 


4-624 


8 1346 3594 


4-674 


3 3485 5327 


4-724 


8 7958 9061 


4-774 


4 4652 6392 


4-625 


98 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 


4-628 


7 7428 8143 


4-678 


2 9759 0485 


4-728 


8 4414 1646 


4-778 


4 1280 7768 


4-629 


7 6451 8740 


4-679 


2 8829 7542 


4-729 


8 3530 1925 


4-779 


4 0439 9166 


4-630 


O0I 7 5475 9i°2 


4-680 


2 7901 3887 


4-730 


8 2647 1040 


4-780 


3 9599 8967 


4"63i 


7 4500 9219 


4-681 


2 6973 9511 


4-731 


8 1764 8980 


4-781 


3 8760 7165 


4-632 


7 3526 9081 


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 


i 4-786 


3 4577 3800 


4'637 


6 8671 4223 


4-687 


2 1428 7596 


4-737 


7 6490 1487 


14-787 


3 3743 2197 


4-638 


6 7703 2351 


4-688 


2 0507 7914 


4-738 


7 5614 0967 


4-788 


3 2909 8933 


4-639 


6 6736 0156 


4-689 


I 9587 7437 


4-739 


7 4738 9202 


4-789 


3 2077 3997 


4-640 


00^6 5769 7627 


4-690 


I 8668 6156 


4-740 


7 3864 6185 


4-790 


3 1245 7382 


4-641 


6 4804 4757 


4-691 


I 7750 4062 


4-741 


7 2991 1907 


4-791 


3 0414 9079 


4-642 


6 3840 1535 


4692 


I 6833 1 145 


4-742 


7 2 1 18 6359 


4-792 


2 9584 9081 


4'643 


6 2876 7951 


4-693 


1 5916 7397 


4-743 


7 1246 9531 


i 4-793 


2 8755 7378 


4-644 


6 1914 3995 


4-694 


I 5001 2807 


4"744 


7 0376 1417 


4-794 


2 7927 3963 


4*645 


6 0952 9660 


■ 4-695 


I 4086 7368 


4-745 


6 9506 2006 


4-795 


2 7099 8828 


4-646 


5 9992 4933 


4-696 


I 3173 1069 


4-746 


6 S637 1290 


,4-796 


2 6273 1963 


4-647 


5 9°32 9807 


4-697 


I 2260 3903 


4-747 


6 7768 9260 


4-797 


2 5447 3361 


4-648 


5 8074 4270 


4-698 


I 1348 5859 


4-748 


6 6901 5908 


4-798 


2 4622 3013 


4-649 


5 7116 8315 


4-699 


I 0437 6928 


4-749 


6 6035 1225 


U-799 


2 3798 0912 



[4-800—4-999] 



OF THE DESCENDING EXPONENTIAL. 



175 



X 


(,-X 


X 


g-X 


X 


C-x 


, X 


g-a: 


4-800 


82 2974 7049 


4-850 


78 


2837 7549 


4-900 


74 4658 3071 


4-950 


70 


8340 8929 


4-801 


2 2152 1415 


: 4-851 


8 


2055 3084 


4-901 


4 3914 0210 


4-951 





7632 9061 


4-802 


2 1330 4003 


4-852 


8 


1273 6440 


4-902 


4 3170 478S 


4-952 





6925 6268 


4-803 


2 0509 4805 


4-853 


8 


0492 7609 


4-903 


'4 2427 6798 


4-953 





6219 0546 


4-804 


I 9689 381 1 


4-854 


7 


9712 6583 


4-904 


4 1685 6232 


4-954 





5513 1885 


4-805 


I 8870 1014 


4-855 




8933 3353 


4-905 


4 0944 3083 


4-955 





4808 0280 


4-806 


I 8051 6406 


4-856 




8154 7913 


1 4-906 


4 0203 7343 


4-956 





4103 5722 


4-807 


I 7233 9979 


4-857 




7377 0255 


4-907 


3 9463 9006 


4-957 





3399 8206 


4-808 


I 6417 1724 


4-858 




6600 0370 


4-908 


3 8724 8063 


4-958 





2696 7723 


4-809 


I 5601 1633 


4-859 




5823 8251 


4-909 


3 7986 4507 


4-959 





1994 4268 


4-810 


I 4785 9698 


4-860 




5048 3891 


4-910 


3 7248 8331 


4-960 





1292 7833 


4-8 1 1 


I 3971 5910 


4-861 




4273 7281 


4-91 1 


3 6511 9528 


4-961 





0591 8410 


4-812 


I 3158 0263 


4-862 




3499 8414 


4-912 


3 5775 8090 


4-962 


69 9891 5993 1 


4-813 


I 2345 2747 


4-863 




2726 7282 


4-913 


3 5040 4009 


4-963 


9 


9192 0576 


4-814 


1 1533 3355 


4-864 




1954 3877 


4-914 


3 4305 7279 


4-964 


9 


8493 2150 


4-815 


I 0722 2078 


4-865 




1182 8191 


4-915 


3 3571 7892 


4-965 


9 


7795 0709 


4-816 


991 1 8908 


4-866 




0412 0218 


4-916 


3 2838 5841 


4-966 


9 


7097 6246 


4-817 


9102 3837 


4-867 


6 


9641 9948 


! 4-917 


3 2106 1118 


4-967 


9 


6400 875.1 
5704 8226 


4-818 


8293 6858 


4-868 


6 


8872 7375 


j 4-918 


3 1374 3716 


4-968 


9 


4-819 


7485 7961 


4-869 


6 


8104 2491 


: 4-919 


3 0643 3628 


4-969 


9 


5009 4655 


4-820 


6678 7139 


4-870 


6 


7336 5288 


4-920 


2 9913 0847 


4-970 


9 


4314 8035 


4-821 


5872 4384 


4-871 


6 


6569 5758 


4-921 


2 9183 5364 


4-971 


9 


3620 8357 


4-822 


5066 9688 


4-872 


6 


5803 3894 


4-922 


2 8454 7174 


4-972 


9 


2927 5616 


4-823 


4262 3042 


4-873 


6 


5037 9688 


4-923 


2 7726 6267 


4-973 


9 


2234 9804 


4-824 


3458 4439 


4-874 


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 


4-826 


1853 1328 


4-876 


6 


2746 2941 


4-926 


2 5546 7184 


4-976 


9 


0161 3874 


4-827 


1051 6805 


4-S77 


6 


1983 9290 


4-927 


2 4821 5343 


4-977 


8 


9471 5709 


4-828 


0251 0292 


4-878 


6 


1222 3260 


4-928 


2 4097 0750 


4-978 


8 


878-2 4440 


4-829 


79 9451 1782 


4-879 


6 


0461 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 


4-831 


9 7853 S737 


4-881 


5 


8942 081 I 


4-931 


2 1928 0390 


4-981 


8 


6719 1931 


4-832 


9 7056 4186 


4-8S2 


5 


81S3 5183 


4-932 


2 1206 4718 


4-982 


8 


6032 8171 


4-833 


9 6259 7605 


4-883 


S 


7425 7138 


4-933 


2 0485 6258 


4-983 


8 


5347 1272 


4-834 


9 5463 8988 


4-884 


5 


6668 6667 


4-934 


I 9765 5003 


4-984 


8 


4662 1226 


4-835 


9 4668 8325 


4-885 


5 


5912 3762 


4-935 


I 9046 0946 


4-985 


8 


3977 8027 


4-836 


9 3874 5609 


4-886 


5 


5156 8417 


4-936 


I 8327 4079 


4-986 


8 


3294 1668 


4-837 


9 3081 0831 


4-887 


S 


4402 0623 


4-937 


I 7609 4395 


4-987 


8 


2611 2142 


4-838 


9 2288 3984 


4-888 


5 


3648 0373 


4-938 


I 6892 1888 


4-988 


8 


1928 9442 


4-839 


9 1496 5060 


4-8S9 


5 


2894 7659 


4-939 


I 6175 6549 


4-989 


8 


1247 3561 


4-840 


9 °7°5 4052 


4-890 


5 


2142 2475 


4-940 


I 5459 8372 


4-990 


8 


0566 4492 


4-841 


8 9915 0950 


4-891 


5 


1390 4812 


4-941 


I 4744 7350 


4-991 


7 


9886 2229 


4-842 


8 9125 5747 


4-892 


5 


0639 4663 


4-942 


I 4030 3475 


4-992 


7 


9206 6765 


4-843 


8 8336 8436 


4-S93 


4 


9889 2020 


4-943 


I 3316 6741 


4-993 


7 


8527 8094 


4-844 


8 7548 9°o7 i 


4-894 


4 


9139 6876 


4-944 


I 2603 7139 


4-994 


7 


7S49 6207 


4-845 


8 6761 7455 


4-895 


4 


8390 9224 


4-945 


I 1891 4664 


4-995 


7 


7172 1099 


4-846 


8 5975 3770 


4-896 


4 


7642 9055 


4-946 


I 1179 9308 


4-996 


7 


6495 2763 


4-847 


8 5189 7945 


4-897 


4 


6S95 6363 


4-947 


I 0469 1063 i 


4997 


7 


5819 1191 


4-848 


8 4404 9971 


4-898 


4 


6149 I I 40 


4-948 


9758 9923 


4-998 


7 


5143 6378 


4-849 


8 3620 9842 


4-899 


4 


54°3 3379 


4-949 


9049 5881 


4-999 


7 


4468 8316 



176 






MR F. W. NEWMAN'S 


TABLE 




[5-000— 5-1 99j 


X 


C-x 


X 


Q-X 


i ^ 


Q-X 


X 


Q-X 


5"ooo 


67 3794 6999 


5-°50 


64 0933 3446 


5-100 


60 9674 6565 


s-is° 


57 9940 4727 


5-001 


7 3121 24^b 


5-051 


4 0292 7316 


5-101 


9065 2866 


5'iSi 


7 9360 8221 


5-002 


7 24-lS 4572 


5-052 


3 9652 75S9 


5-102 


8456 525S 


|5'i52 


7 8781 7508 


5-003 


7 1776 344S 


5-053 


3 9013 4259 


5-103 


7S4S 3734 


5-153 


7 8203 2584 


5-004 


7 I 104 9043 


5-054 


3 8374 7319 


5-104 


7240 8288 


5-154 


7 7625 3441 


' 5-005 


7 0434 1348 


5-055 


3 7736 6762 


5'io5 


6633 8915 


5-155 


7 704S 0075 


5-006 


6 9764 035S 


5-056 


3 7099 2583 


5-100 


6027 5608 


15-156 


7 6471 2479 


5 "007 


6 9094 6065 


5-057 


3 6462 4775 


5-107 


5421 S362 


5-157 


7 5895 0648 


5-008 


6 S425 S463 


5-058 


3 5826 3331 


j 5-108 


4S16 7170 


5-158 


7 5319 4576 


5009 


6 7757 7546 


5-059 


3 5190 8246 


5'io9 


4212 2025 


5-159 


7 4744 4257 


5-010 


6 7090 3306 


5-060 


3 4555 9513 


'5-110 


360S 2924 


5-160 


7 4169 9686 


5 on 


6 6423 5737 


5-061 


3 3921 7125 


5-III 


3004 9S58 


5-161 


7 3596 0856 


15-012 


6 5757 4832 


5-062 


3 32SS 1076 


5-112 


2402 2S22 


5-162 


7 3022 7762 


I50I3 


6 5092 0585 


5-063 


3 2655 1361 


5'ii3 


1800 1810 


5-163 


7 245° 0398 


1 5'oi4 


6 4427 2989 


5-064 


3 2022 7972 


5-114 


1198 6816 


5-164 


7 1877 8759 


!5-°i5 


6 3763 2037 


5-065 


3 139 1 0903 


5-115 


0597 7834 


5-165 


7 1306 2839 


5016 


6 3099 7723 


5-066 


3 0760 0148 


'5-116 


59 9997 4S58 


,5-166 


7 0735 2632 


ls;oi7 


6 2437 0039 


5-067 


3 0129 5700 


5-117 


9 9397 7883 


5-167 


7 0164 8132 


1 5-018 


6 1774 89S0 


5-068 


2 9499 7554 


5-118 


9 8798 6901 


5-168 


6 9594 9333 


j 5-019 


6 1 113 4539 


5-069 


2 8870 5703 


5-119 


9 S200 1907 


5-169 


6 9025 6231 


1 5-020 


6 0452 6709 


5-070 


2 8242 0141 


5-120 


9 7602 2895 


5-170 


6 8456 8819 


5-021 


5 9792 54S4 


5-071 


2 7614 0861 


S-121 


9 7004 9S59 


5-171 


6 7888 7092 


5 02 2 


5 9133 0856 


5-072 


2 6986 7857 


5-122 


9 6408 2793 


5-172 


6 7321 1043 


5-023 


5 8474 2820 


5-073 


2 6360 1 1 23 


5'i23 


9 5812 1691 


5-173 


6 6754 0668 


5 '024 


5 7816 1368 


5-074 


2 5734 0653 


5-124 


9 5216 6548 


5-174 


6 6187 5960 


5-025 


65 7158 6495 


5-075 


62 5108 6440 


5-125 


59 4621 7356 


5-175 


56 5621 6914 


5-026 


5 6501 8193 


5-076 


2 4483 8478 


5-126 


9 4027 4111 


5-176 


6 5056 3524 


5 '02 7 


5 5845 6456 


5-077 


2 3859 6761 


5-127 


9 3433 6806 


5-177 


6 4491 5785 


5028 


5 5190 1278 


5-078 


2 3236 1282 


5-128 


9 2840 5436 


5-178 


6 3927 3691 


5-029 


5 4535 2652 


5-079 


2 2613 2036 


5-129 


9 2247 9993 


5-179 


6 3363 7236 


5-030 


5 3881 0570 


5-080 


2 1990 9016 


5-130 


9 1656 0474 


5-180 


6 2800 6414 


5031 


5 3227 5028 


5-081 


2 1369 2216 


5-131 


9 1064 6870 


5-181 


6 2238 1221 


5-032 


5 2574 6018 


5-082 


2 0748 1629 


5-132 


9 0473 9178 


5-182 


6 1676 1650 


5-033 


5 1922 3534 I 


5-083 


2 0127 7250 


5-133 


8 9883 7390 


5-183 


6 1 1 14 7696 


5-034 


5 1270 7569 , 


5-084 


1 9507 9073 


5-134 


8 9294 1501 


5-184 


6 0553 9353 


5-035 


5 0619 8117 


5-085 


I 8888 7090 


5-135 


8 8705 1505 


5-185 


5 9993 6615 


5036 


4 99<59 5171 


5-086 


I 8270 1297 


5-136 


8 8116 7396 


5-186 


5 9433 9478 


5-037 


4 9319 8724 


5-087 


I 7652 1686 


5-I37 


8 7528 9168 


5-187 


5 8874 7934 


S-038 


4 8670 8771 


5-088 


I 7034 8251 


5-138 


8 6941 6816 


5-188 


5 8316 1980 


5-039 


4 8022 5304 


5-089 


I 6418 0987 


5-139 


8 6355 0333 


5-189 


5 7758 1609 


5-040 


4 7374 8318 


5090 


1 5801 9887 


5-140 


8 5768 9713 


5-19° 


5 7200 6815 


5-041 


4 6727 7806 


5-091 


I 5186 4945 


5-141 


8 5183 4951 


5-191 


5 6643 7593 


5-042 


4 6o8i 3760 : 


5-092 


I 4571 6155 


5-142 


8 4598 6041 


5-192 


5 6087 3938 


5043 


4 5435 6176 


5-093 


I 3957 3511 


5-143 


8 4014 2977 


5-193 


5 5531 5843 


5044 


4 4790 5046 


5-094 


I 3343 7006 


5-144 


8 3430 5753 


5-194 


5 4976 3304 


5-045 


4 4146 0364 


5-095 


I 2730 6635 


5-145 


8 2847 4364 ! 


5-195 


5 4421 6315 


5046 


4 35°2 2123 


5-096 


I 2118 2391 


5-146 


8 2264 8803 ; 


5-196 


5 3867 4870 


5-047 


4 2859 0317 


5-097 


I 1506 4268 


5-147 


8 1682 9064 


5-197 


5 3313 8963 


5048 


4 2216 4940 


5-098 


I 0895 2260 


5-148 


8 iioi 5143 


5-198 


5 2760 8590 


5049 


4 1574 5985 


5099 


I 0284*6361 


5-149 


8 0520 7032 


5'i99 


5 2208 3744 



[5 -200— 5-399] 



OF THE DESCENDING EXPONENTIAL. 



177 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


5-200 


55 1656 4421 


5-250 


52 4751 8399 


5-300 


49 9159 3907 


5-350 


47 4815 0999 


5-201 


5 1105 0614 


S-251 


2 4227 3503 


5-301 


9 8660 4808 


5-351 


7 4340 5222 


5-202 


5 0554 2318 


5-252 


2 3703 3850 


5-302 


9 8162 0696 


5-352 


7 3866 4187 


5'203 


5 0003 9527 


5-253 


2 3179 9434 


5-303 


9 7664 1565 


5-353 


7 3392 7892 


5 '204 


4 9454 2237 


5-254 


2 2657 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 


5-256 


2 1612 7555 


5-306 


9 6173 4013 


5-356 


7 1974 7389 


5-207 


4 7808 3311 


5-257 


2 1091 4035 


5-307 


9 5677 4759 


5-357 


7 1503 0001 


5-208 


4 7260 7966 


5-258 


2 0570 5726 


5-308 


9 5182 0461 


S-358 


7 1031 7328 


5-209 


4 6713 8093 


S-259 


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 671S 


5-360 


7 0090 6107 


5-211 


4 5621 4744 


5-261 


I 9011 2011 


5-311 


9 3698 7261 


5-361 


6 9620 7551 


5-212 


4 5076 1256 


5-262 


I 8492 4493 


5-312 


9 3205 2741 


5-362 


6 9151 3691 


5-213 


4 4531 3219 


5-263 


I 7974 2160 


5-313 


9 2712 3154 


5-363 


6 8682 4522 


5-214 


4 3987 0628 


5-264 


I 7456 5007 


5-314 


9 2219 8494 


5-364 


6 8214 0040 


5-215 


4 3443 3476 


5-265 


1 6939 3028 


5-315 


9 1727 8755 


5-365 


6 7746 0240 


5-216 


4 2900 1759 


5-266 


I 6422 6219 


5-316 


9 1236 3934 


5-366 


6 7278 5118 


5-217 


4 2357 5471 


5-267 


I 59°6 4574 


5-317 


9 0745 4026 


5-367 


6 681 1 4669 


5-218 


4 1815 4606 


5-268 


I 5390 8088 


5-318 


9 0254 9025 


5-368 


6 6344 8887 


5-219 


4 1273 9160 


5-269 


I 4875 6756 


5-319 


8 9764 8926 


5-369 


6 5878 7769 


5-220 


4 0732 9126 


5-270 


I 4361 0573 


5-320 


8 927s 3725 


5-370 


6 5413 1310 


5-221 


4 0192 4500 


5-271 


I 3846 9533 


5-321 


8 8786 3417 


5-371 


6 4947 9505 


5-222 


3 9652 5276 


5-272 


I 3333 3632 


5-322 


8 8297 7997 


5-372 


6 4483 2350 


5-223 


3 9"3 1448 


5-273 


I 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 


S3 8035 9960 


5-275 


51 1795 6708 


5-325 


48 6835 1014 


5-375 


46 3091 8733 


5-226 


3 7498 2289 


5-276 


I 1284 1310 


5-326 


8 6348 5096 


5-376 


6 2629 0129 


5-227 


3 6960 9993 


5-277 


I 0773 1024 


5-327 


8 5862 4042 


5-377 


6 2-166 6152 


5-228 


3 6424 3067 


5-278 


r 0262 5846 


5-328 


8 5376 7847 


5-378 


6 1704 6796 


5-229 


3 5888 1505 


5-279 


9752 5770 


! 5-329 


8 4891 6505 


5-379 


6 1243 2056 


5-230 


3 5352 5303 


5-280 


9243 0793 


5-330 


8 4407 0012 


5-380 


6 0782 1930 


5-231 


3 4817 4453 


5-281 


8734 0907 


5-331 


8 3922 8363 


5-381 


6 0321 6411 


S-232 


3 4282 8952 


5-282 


8225 6109 


5-332 


8 3439 1554 


5-382 


5 9861 5496 


5-233 


3 3748 8793 


5-283 


7717 6393 


15-333 


8 2955 9579 


5-383 


5 9401 9179 


5-234 


3 3215 3972 


5-284 


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 0112 


5-385 


5 8484 0322 


S-236 


3 2150 0322 


5-286 


6196 7688 


5-336 


8 1509 2611 


5-386 


5 S025 7773 


5-237 


3 1618 1481 


5-287 


5690 8251 


; 5-337 


8 I027 9925 


5-3S7 


5 7567 9805 


5-238 


3 1086 7957 


5-288 


5185 3870 


5-338 


8 0547 2050 


5-388 


5 7110 6412 


5-239 


3 0555 9744 


5-289 


4680 4542 


5-339 


8 0066 8980 


5-389 


5 6653 7591 


5-240 


3 0025 6836 


5-290 


4176 0260 


5-340 


7 9587 0710 


5-390 


5 6197 3336 


5-241 


2 9495 9228 


5-291 


3672 1019 


iS-341 


7 9107 7237 


5-391 


5 5741 3643 


5-242 


2 8966 6915 


5-292 


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 


55-394 


5 4376 1889 


5-245 


2 7382 1694 


5-295 


1661 4375 


5-345 


7 7195 1205 


5-395 


5 3922 0399 


5-246 


2 6855 0509 


5-296 


1 160 0268 


5-346 


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 


S-299 

1 


49 9658 7997 


5-349 


7 5290 1525 


15-399 


5 2109 9783 



Vol. XIII. Part III. 



24 



178 



MR F. W. XKWMAN'S TABLE 



[5-400— 5-599] 



i ^ 




Q-X 




. X 


Q-X 


X 


Q-X 


X 


Q-X 


1 

5-400 


45 


165S 


0943 


5-450 


42 9630 4691 


5-500 


40 8677 1438 


5-550 


38 8745 7243 


5-401 


5 


1206 


6619 


5-451 


2 9201 0533 


5-501 


8268 6710 


5-551 


8 


8357 1729 j 


5-402 


5 


0755 


680S 


5-452 


2 8772 0668 


5-502 


7860 6064 


5-552 


8 


7969 0099 


5-403 


5 


0305 


1504 


5 453 


2 S343 5091 


5 ■503 


7452 9496 


5-553 


8 


7581 2348 1 


5-404 


4 


9S55 


0703 


5-454 


2 7915 3797 


5-504 


70+5 7003 


5-554 


8 


7193 8473 


5-405 


4 


9405 


4401 


5-455 


2 7487 6782 


5-505 


6638 8581 


5-555 


8 


6806 8470 


5-406 


4 


8956 


2593 


5-456 


2 7060 4042 


5-506 


6232 4225 


5-556 


8 


6420 2335 


5-407 


4 


8507 


5274 


5 -45 7 


2 6633 5572 


5-507 


5826 3931 


5-557 


8 


6034 0064 


5-408 


4 


S059 


2441 


5-458 


2 6207 1369 


5-508 


5420 7695 . 


5-558 


8 


564S 1653 


5-409 


4 


7611 


40S8 


5-459 


2 5781 1428 


5-509 


5015 5514 


5-559 


8 


5262 7099 


5-410 


4 


7164 


021I 


5-460 


2 5355 5745' 


5-510 


4610 7383 


5-560 


8 


4877 6398 


5"4n 


4 


6717 


0806 


5-461 


2 4930 4315 


'5-511 


4206 3298 


5-561 


8 


4492 9545 


5-412 


4 


6270 


5868 


5-462 


2 4505 7135 


5-512 


3802 3255 


5-562 


8 


4108 6537 


5-413 


4 


5S24 


5393 


5-463 


2 4081 4199 


5-513 


3398 7250 


5-563 


8 


3724 7371 


5-414 


4 


5378 


9376 


5-464 


2 3657 5505 


5-514 


2995 5279 


5-564 


8 


3341 2041 


S-415 


4 


4933 


7S13 


5-465 


2 3234 1047 


5-515 


2592 7338 


5-565 


8 


2958 0545 


5-416 


4 


4489 


0699 


5-466 


2 2811 0821 


5-516 


2190 3423 


5-566 


8 


2575 2879 


5-417 


4 


4044 


8030 


5-467 


2 2388 4824 


5-517 


1788 3530 


5-567 


8 


2192 9038 


5-418 


4 


3600 


9801 


5-468 


2 1966 3050 


5-518 


1386 7655 


5-568 


8 


1810 9020 


5419 


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 8110 


5-522 


39 9784 4252 


5-572 


8 


0286 7088 


5-423 


4 


1388 


5110 


5-473 


I 9861 7393 


5-523 


9 9384 8406 


5-573 




9906 6121 


5-424 


4 


0947 


3431 


5-474 


I 9442 0874 


5-524 


9 8985 6554 


5-574 




9526 8954 


5-425 


44 0506 


6162 


5-475 


41 9022 8550 


5-525 


39 85S6 8692 


5-575 


37 


9147 5582 


5-42^ 


4 


0066 


3297 


5-476 


I 8604 0416 


5-526 


9 8188 4816 


5-576 




8768 6002 


5-427 


3 


9626 


4834 


5-477 


I 8185 6468 


5-527 


9 7790 4921 


5-577 




8390 0209 


5-428 





9187 


0766 


5-478 


I 7767 6701 


5-528 


9 7392 9004 


5-578 




801 I 8200 


5429 


3 


8748 


1091 


5-479 


I 7350 1113 


5-529 


9 6995 7062 


5-579 




7633 9971 


5-430 


3 


8309 5803 i 


5-480 


I 6932 9698 


5-530 


9 6598 9089 


5-5S0 




7256 5519 


5-43» 


3 


7871 


4898 , 


5-481 


I 6516 2452 


5-531 


9 6202 50S2 


5-581 




6879 4839 


5-432 




7433 


8371 • 


5-482 


I 6099 9372 


5-532 


9 5806 5038 


5-582 




6502 7928 


5-433 


3 


6996 


6219 


5-483 


I 5684 0452 


5-533 


9 5410 8951 


5-583 




6126 4782 


5-434 


3 


6559 


8437 


5-484 


I 5268 5689 


5-534 


9 5015 6818 


5-584 




5750 5397 


5-435 


3 


6123 


5021 


5-485 


I 4853 5079 


5-535 


9 4620 8636 


5-585 




5374 9770 


5-436 


3 


5687 


5966 


5-486 


I 4438 8618 


5-536 


9 4226 4400 


5-586 




4999 7896 


5-437 


3 


5252 


1268 


5-487 


I 4024 6301 


5-537 


9 3832 4106 


5-587 




4624 9773 


5-438 


3 


4S17 


0922 


5-488 


I 3610 8124 


5-538 


9 3438 7750 


5-588 




4250 5395 


5 439 


3 


4382 


4924 


5-489 


r 3197 4083 


5-539 


9 3045 5329 


5-589 




3876 4761 


5-440 


3 


3948 


3271 


5-490 


I 2784 4174 


5-540 


9 2652 6838 


5-590 




3502 7865 


5-441 


3 


3514 


5956 


5-491 


1 2371 8393 


5-541 


9 2260 2274 


5-591 




3129 4704 


5-442 


3 


3081 


2977 


5-492 


I 1959 6736 


5-542 


9 1868 1632 


5-592 




2756 5274 


5-443 


3 


2648 


4329 


5-493 


I 1547 9198 


5-543 


9 1476 4909 


5-593 




2383 9572 


5-444 


3 


2216 


0007 


5-494 


1 1136 5777 


5 -544 


9 1085 2101 


5-594 




2011 7594 


5-445 


3 


1784 


0008 


5-495 


I 0725 6465 


5-545 


9 0694 3204 


5-595 




1639 9335 


5-446 


3 


1352 


4326 


5-496 


I 0315 1262 


5-546 


9 0303 8214 


5-596 




1268 4794 


5-447 


3 


0921 


2957 


5-497 


9905 0162 


5-547 


8 9913 7126 


5-597 




0897 3965 


5-448 


3 


0490 


5898 


5-498 


9495 3160 


5-548 


8 9523 9938 


5-598 




0526 6845 


5-449 


3 


0060 


3144 , 


5-499 


9086 0254 


5-549 


8 9134 6645 


5-599 




0156 3430 



[5 6oo- 


-5-799] 


OF 


THE DESCENDING 


EXPONENTIAL. 




17 


X 


c-^ 


X 


Q-X 


X 


Q-X 


X 


Q-X 


5-600 


36 9786 3716 


5-650 


35 1751 6775 


'5-700 


33 4596 5457 


SlSo 


31 8278 0796 


5-601 


6 9416 7701 


5-651 


5 1400 1016 


,5-701 


3 4262 1164 


\ 5-751 


I 7959 9606 


5-602 


6 9047 5380 


5-652 


5 104S 8772 


1 5-702 


3 3928 0214 


S-752 


I 7642 1596 


5-603 


6 8678 6749 


5-653 


5 0698 0038 


1 5-703 


3 3594 2603 


5-753 


1 7324 6762 


5-604 


6 8310 1805 


: 5-654 


5 0347 4810 


5-704 


3 3260 8328 


5-754 


I 7007 5102 


5-605 


6 7942 0544 


5-655 


4 9997 3087 


5-705 


3 2927 7385 


; 5-755 


I 6690 6611 


5-606 


6 7574 2963 


5-656 


4 9647 4863 


5-706 


3 2594 9772 


5-756 


I 6374 1287 


5-607 


6 7206 9057 


5-657 


4 929S 0136 


5-707 


3 2262 5484 


: 5-757 


1 6057 9127 


5-608 


6 6839 8823 


5-658 


4 8948 8902 


5-708 


3 1930 4520 


;s-758 


I 5742 0128 


5-609 


6 6473 2258 


5-659 


4 S600 I 15 7 


5-709 


3 1598 6874 


; 5-759 


I 5426 4286 


5-610 


6 6106 9358 


5-660 


4 8251 6898 


5-710 


3 1267 2545 


5-760 


I 5111 1598 


5-611 


6 5741 0118 


5-661 


4 79°3 6122 


5-711 


3 0936 1528 


: 5-761 


I 4796 2062 


5-612 


6 5375 4536 


5-662 


4 7555 8825 


5-712 


3 0605 3821 


5-762 


I 4481 5673 


5-613 


6 5010 2608 


5-663 


4 7208 5003 


5-713 


3 0274 9419 


5-763 


I 4167 2429 


5-614 


6 4645 4330 


5-664 


4 6861 4654 


5-714 


2 9944 8321 


5-764 


I 3853 2327 


5-615 


6 4280 9698 


' 5-665 


4 6514 7773 


5-715 


2 9615 0522 


5-765 


I 3539 5364 


5-616 


6 3916 8709 


5-666 


4 6168 4357 


5-716 


2 9285 6019 


5-766 


I 3226 1536 


5-617 


6 3553 1359 


5-667 


4 5822 4403 


5-717 


2 8956 480S 


5-767 


I 2913 0840 


5-618 


6 3189 7645 


5-668 


4 5476 7907 


5-718 


2 8627 6888 


5-768 


I 2600 3273 


5-619 


6 2826 7563 


5-669 


4 5131 4866 


5-719 


2 8299 2254 


5-769 


I 2287 8S32 


5-620 


6 2464 I 109 


5-670 


4 4786 5276 


5-720 


2 7971 0902 


5-770 


I 1975 7514 


5-621 


6 2101 8279 


5-671 


4 4441 9134 


5-721 


2 7643 2831 


5-771 


I 1663 9316 


5-622 


6 1739 9071 


5-672 


4 4097 6437 


5-722 


2 7315 8036 


5-772 


I 1352 4234 


5-623 


6 1378 34S0 


5-673 


4 3753 7180 


5-723 


2 69S8 6514 


5-773 


I 1041 2266 


5624 


6 1017 1503 


5-674 


4 3410 1361 


5-724 


2 6661 8261 


5-774 


I 0730 3409 


5-625 


36 0656 3136 


5-675 


34 3066 8976 


5-725 


32 ^iis 3276 


5-775 


31 0419 7659 


5-626 


6 0295 8375 


5-676 


4 2724 0022 


5-726 


2 6009 1554 


5-776 


I 0109 5013 


5-627 


5 9935 7218 


5-677 


4 2381 4495 


5-727 


2 5683 3092 


5-777 


9799 5468 


5-628 


5 9575 9660 


5-678 


4 2039 2392 1 


5-728 


2 5357 7887 


5-778 


9489 9021 


5-629 


5 9216 5697 


5-679 


4 1697 3709 


5-729 


2 5032 5935 


5779 


9180 5668 


5-630 


5 8857 5327 


5-680 


4 1355 8443 


5-730 


2 4707 7234 


5-780 


8S71 5408 


5-631 


5 8498 8546 


5-681 


4 1014 6591 


5-731 


2 4383 1779 


5-781 


8562 S237 


5632 


5 8140 5349 


5-682 


4 0673 8149 


5-732 


2 4058 9569 


5-782 


8254 4151 


5-633 


5 7782 5734 


5-683 


4 0333 3114 


5-733 


2 3735 0599 


5-783 


7946 3147 


5-634 


5 7424 9696 


5-684 


3 9993 1482 


5-734 


2 341 1 4867 


5-784 


7638 5223 


5-635 


5 7067 7233 


5-685 


3 9653 3250 


5-735 


2 3088 2368 


5-785 


7331 0376 


5-636 


5 6710 8341 


5-686 


3 9313 8414 


5-736 


2 2765 3101 


5-786 


7023 8602 


5-637 


5 6354 3015 


5-687 


3 8974 6972 


5-737 


2 2442 7061 


5-787 


6716 9S98 


5-638 


5 599S 1253 


5-688 


3 8635 8919 , 


5-738 


2 2120 4246 


5-788 


6410 4261 


5-639 


5 5642 3052 


5-689 


3 8297 4253 


5-739 


2 1798 4651 


5-789 


6104 1688 


5-640 


5 5286 8406 


5-690 


3 7959 2969 


5-740 


2 1476 8275 


5-790 


5798 2176 


5-641 


5 4931 7314 


5-691 


3 7621 5066 


5-741 


2 1155 5114 


5-791 


5492 5723 


5-642 


5 4576 9770 


5-692 


3 7284 0538 


5-742 


2 0S34 5164 


5-792 


5187 2324 


5-643 


5 4222 5773 


5693 


3 6946 9383 


5-743 


2 0513 8422 


5-793 


4882 1977 


5-644 


5 3868 5318 


5-694 


3 6610 1598 : 

i 


5-744 


2 0193 4886 


5-794 


4577 4679 


5-645 


5 3514 8401 


s-695 


3 6273 7179 


5-745 


I 9873 4552 


5-795 


4273 0427 


5-646 


5 3161 5020 


5-696 


3 5937 6123 


5-746 


I 9553 7416 


5-796 


3968 9217 


5-647 


5 2808 5170 


5-697 


3 5601 8426 1 


5-747 


1 9234 3476 


5-797 


3665 1047 


5-648 


5 2455 8848 


5-698 


3 5266 4085 


5-748 


I 8915 2728 


5-798 


3361 5914, 


5-649 


5 2103 6051 


5-699 


3 4931 3°97 


5-749 


I 8596 5169 


5799 


3058 3814 



24- 



ISO 



MR F. W. NEWMAN'S TABLE 



[5-800—5-999] 



X 


e-x 


X 


(,-X 


X 


e-x 


X 


C-x 


5-800 
5-801 
5-802 

s-803 

5-804 


2° 






2755 4745 
2452 8704 
2150 56S7 

1545 5691 

1546 8714 


5-850 
5-851 
5-852 
5-853 
5-854 


28 
8 
8 
8 
8 


7989 9158 
7702 0698 
7414 5116 
7127 2407 
6840 2570 


5-900 

5-901 
5-902 

5-903 
5-904 


27 


3944 
3670 

3397 
3123 
2850 


4819 
6743 
1404 
8799 
8926 


5-950 
5-951 
5-952 
5-953 
5-954 


26 
6 
6 

5 
5 


0584 0518 
0323 5980 
0063 4046 
9803 471 1 

9543 7975 


5-805 
5-806 

5807 
5-808 
5-809 









1245 4753 
0944 3804 
0643 5864 

0343 °93i 
0042 9002 


5-855 
5-856 
5-857 
5-858 
5-859 


8 
8 
8 
8 
8 


6553 5601 
6267 1498 
5981 0257 
5695 1S76 
5409 6352 


5-905 
5-906 

5-907 
5-908 

5-909 




257S 
2305 
2033 
1761 

1490 


17S1 
7361 
5665 
6689 

0431 


5-955 
5-956 
5 -95 7 
5-958 
5-959 


5 
5 
5 
5 
5 


9284 3835 
9025 2287 
8766 3329 
8507 6959 
8249 3174 


5-810 
5-811 
5-812 
5-813 
5-814 


29 
9 
9 
9 
9 


9743 o°72 
9443 4140 
9144 1203 

884s 1257 
8546 4299 


5-860 
5-861 
5-862 
5-863 
5-864 


8 
8 
8 
8 

8 


5124 3683 
4839 3864 
4554 6894 
4270 2769 
3986 1487 


5-910 
5'9ii 
5'9i2 
5-913 
5-914 




1218 

0947 
0676 
0406 
0135 


6S87 
6056 
7934 
2519 
9808 


5-960 
5-961 
5-962 
5-963 
5-964 


5 
5 
5 
5 
5 


7991 1972 
7733 3350 
7475 7304 
7218 3834 
6961 2936 


5-8IS 
5-816 

5-817 
5-SI8 

5-819 


9 
9 
9 
9 
9 


8 248 0327 

7949 9338 
7652 1328 

7354 6294 
7057 4234 


S-865 
5-866 

S-867 
5-868 
5-869 


8 
8 
8 
8 
8 


3702 3045 
341S 7440 

3135 4670 
2852 4730 
2569 7619 


5-915 
5-916 

5-917 
5-918 
5-919 


6 
6 
6 
6 
6 


9865 9799 
9596 2488 
9326 7873 
9°S7 5951 
8788 6720 


5-965 
5-966 

5-967 
5-968 

5-969 


5 
5 
5 
5 
5 


6704 4607 
6447 8846 
6191 5649 
5935 5014 
5679 6938 


5-820 
5-821 
5-822 

5-823 
5-824 


9 
9 
9 
9 
9 


6760 5145 
6463 9023 
6167 5866 
5871 5670 
5575 8433 


5-870 
5-871 
5-872 
5-873 
5-874 


8 
8 
8 
8 
8 


2287 3334 
2005 1872 
1723 3229 
1441 7404 
1 160 4394 


5-920 
5-921 
5-922 

5-923 
5-924 


6 
6 
6 
6 
6 


8520 
8251 
7983 
7715 
7448 


0177 
6319 

5143 
6648 
0829 


5-970 
5-971 
S-972 
5-973 
5-974 


5 
5 
5 
5 
5 


5424 1419 
5168 S454 
4913 8041 
4659 0177 
4404 4S60 


5-825 

5-826 
5-827 
5-828 
5-829 


29 
9 
9 
9 
9 


5280 4152 
4985 2824 
4690 4446 

4395 9°i4 
4101 6527 


5-875 
S-876 
5-877 
5-878 

5-879 


28 
8 
8 

8 
7 


0879 4194 
0598 6804 
0318 2220 
0038 0439 
9758 1458 


5-925 
5-926 

5-927 
5-928 

5-929 


26 
6 
6 
6 
6 


7180 

6913 
6646 

6380 
6114 


76S5 
7213 
9410 

4273 
iSoo 


5-975 
5-976 
5-977 
5-978 
5-979 


25 
5 
5 
5 
5 


4150 20S7 
3896 1855 
3642 4162 
3388 9006 
3135 63S3 


5-830 
5-831 
5-832 
5-833 
5-834 


9 

I 

9 
9 


3807 6980 

3514 0372 
3220 6698 

2927 5957 
2634 8146 


5-880 
5-881 
5-882 
5-883 
5-884 


7 
7 
7 
7 
7 


9478 5275 
9199 1887 
8920 1290 
8641 3483 
8362 8462 


5-930 
5-931 
5-932 
5-933 
5-934 


6 
6 
6 
6 
6 


5848 
5582 
5317 
5051 
4786 


1989 
4836 
033S 
8494 
9300 


5-980 

5-981 

5-982 

5-983 
5-984 


5 
5 
5 
5 
5 


2882 6292 
2629 8730 

2377 3694 
2125 1182 
1873 1191 


5-835 
5-836 
5-837 
5-838 
5-839 


9 
9 
9 
9 
9 


2342 3260 
2050 1298 
1758 2257 
1466 6133 
1175 2923 


5-885 
5-886 
5-887 
5-888 

5-889 


7 
7 
7 
7 
7 


8084 6225 
7806 6769 
7529 0091 
7251 6188 
6974 5058 


5-935 
5-936 
5-937 
5-938 
5-939 


6 
6 
6 
6 
6 


4522 
4257 
3993 
37^9 
3466 


2754 
8854 
7596 
8978 
2997 


5-985 
5-986 

5-987 
5-988 

5 989 


5 
5 
5 
5 
5 


1621 3719 
1369 8762 
1118 6320 
0867 6389 
0616 8966 


5-840 
5-841 
5-842 

5 -843 
5-844 


9 
9 
9 
9 
8 


0884 2626 
0593 5237 
0303 °754 
0012 9175 

9723 0495 


5-890 
5-891 
5-892 
5-893 
5-894 


7 
7 
7 
7 
7 


6697 6697 
6421 I 103 
6144 8274 
5868 8206 
5593 0896 


5-940 
5-941 
5-942 
5-943 
5-944 


6 
6 
6 
6 
6 


3202 

2939 
2677 
2414 
2152 


9651 
8937 
0852 

5394 
2561 


5-990 
5-991 
5-992 
5-993 
5-994 


5 
5 
4 
4 
4 


0366 4050 
0116 1637 
9866 1726 
9616 4313 
9366 9397 


5-845 
5-846 

5-847 
5-848 

5-849 


8 
8 
8 
8 
8 


9433 4713 
9144 1825 
8855 1828 
8566 4720 
8278 0498 


5-895 
5-896 
5-897 
5-898 

5-899 


7 
7 
7 
7 
7 


5317 6343 
5042 4543 
4767 5493 
4492 9191 
4218 5634 


5-945 
5-946 
5-947 
5-948 
5 949 


6 
6 
6 
6 
6 


1890 2348 
1628 4755 
1366 9778 
1105 7415 
0844 7662 


5-995 
5-996 
5-997 
5-998 
5-999 


4 
4 
4 
4 
4 


9117 6974 
8868 7042 
8619 9599 
8371 4642 
8123 2169 



[6-OO0 


—6-199] 


OF 


THE DESCENDING EXPONENTIAL. 




181 


X 


Q-X 


X 


Q-X 


X 

i 


Q-X 


1 
X 


Q-X 


6'ooo 


24 787s 2177 


6-050 


23 5786 2006 


6-IOO 


22 4286 7719 


6-150 


21 3348 1770 


6001 


4 7627 4663 


6 05 1 


3 5550 5322 


6-IOI 


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 


6-053 


3 5079 9020 


6-103 


2 3614 9199 


6-153 


I 2709 0916 


6-004 


4 6885 6971 


6-054 


3 4844 9396 


6-104 


2 3391 4168 


6-154 


I 2496 4888 


6-005 


4 6638 9349 


6-055 


3 4610 2121 


6-105 


2 316S 1370 


6-155 


I 2284 0985 


6-006 


4 6392 4192 


6-056 


3 4375 7191 


6-106 


2 2945 0804 


6-156 


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 


6-<S68 


3 3907 4361 


6-108 


2 2499 6358 


6-158 


I 1648 2005 


6-009 


4 5654 3496 


6-059 


3 3673 6456 


6-109 


2 2277 2474 


6-159 


I 1436 6581 


6010 


4 5408 8180 


6-060 


3 3440 08S7 


6-IIO 


2 2055 0813 


6-160 


I 1225 3272 


6011 


4 5163 5319 


6-061 


3 3206 7653 


6-111 


2 1833 1372 


6-161 


I 1014 2074 


6-OI2 


4 4918 4909 6-062 


3 2973 6751 


6-112 


2 1611 4149 


6-162 


I 0803 2987 


6-013 


4 4673 6948 


6-063 


3 2740 8179 


6-113 


2 1389 9143 


6-163 


I 0592 6007 


6-014 


4 4429 1434 


6-064 


3 2508 1934 


6-114 


2 ii68 6350 


6-164 


1 0382 1134 


6-015 


4 4184 8364 


6-065 


3 2275 8014 


6-115 


2 0947 5769 


6-165 


I 0171 8364 


6-oi6 


4 3940 7737 


6-066 


3 2043 6417 


6-ii6 


2 0726 7398 


6-i66 


9961 7697 


6-017 


4 3696 9548 


6-067 


3 1811 7141 


6-117 


2 0506 1234 


6-167 


9751 9128 


6-018 


4 3453 3797 


6-068 


3 1580 0182 


6-ii8 


2 0285 7275 


6-168 


9542 2658 


6-oig 


4 3210 0480 


6-069 


3 1348 5540 


6-119 


2 0065 5518 


6-169 


9332 8282 


6-020 


4 2966 9595 


6-070 


3 1117 3211 


6-I20 


I 9845 5963 


6-170 


9123 6000 


6-021 


4 2724 1140 


6-071 


3 0886 3192 


6-I2I 


I 9625 8606 


6-171 


8914 5809 


6-022 


4 2481 5112 


6-072 


3 0655 5483 


6-122 


1 9406 3445 


6-172 


8705 7708 


6-023 


4 2239 1509 


6-073 


3 0425 0081 


6-123 


1 9187 0478 


6-173 


8497 1694 


6024 


4 1997 0328 


6-074 


3 0194 6982 


6-124 


1 8967 9703 


6-174 


8288 7764 


6-025 


24 1755 1567 


6-075 


22 9964 6186 


6-125 


21 8749 iu8 


6-175 


20 80S0 5917 


6-026 


4 1513 5224 


6-076 


2 9734 7689 


6-126 


I 8530 4720 


6-176 


7872 6151 


6-027 


4 1272 1296 


6-077 


2 9S°5 1490 


6-127 


I 8312 0508 


6-177 


7664 8464 


6-028 


4 1030 9781 


6-078 


2 9275 7586 , 


6-128 


1 8093 S479 


6-178 


7457 2854 


6-029 


4 0790 0676 


6-079 


2 9046 5974 


6-129 


I 7875 8630 


6-179 


7249 9318 


6-030 


4 0549 3979 


6-080 


2 8817 6653 


6-130 


I 7658 0961 


6-180 


7042 7855 


6031 


4 0308 9687 


6-081 


2 8588 9620 


6-131 


I 7440 5468 


6-181 


6835 8462 


6032 


4 0068 7798 


6-082 


2 8360 4873 


6-132 


I 7223 2149 


6-182 


6629 1137 


6-033 


3 9828 8310 


6-083 


2 8132 2409 


6-133 


I 7006 1003 


6-183 


6422 5879 


6-034 


3 9589 1221 


6-084 


2 7904 2227 


6-134 


I 6789 2026 


6-184 


6216 2684 


6-035 


3 9349 6527 


6-085 


2 7676 4324 


6-135 


I 6572 5218 


6-185 


6010 1553 


6-036 


3 9110 4227 


6-086 


2 7448 8698 


6-136 


I 6356 0575 


6-i86 


5804 2481 


6-037 


3 8871 4318 


6-087 


2 7221 5346 


6-137 


I 6139 8096 


6-187 


5598 5467 


6-038 


3 8632 6798 


6-088 


2 6994 4266 


6-138 


I 5923 7778 


6-i88 


5393 0509 


6-039 


3 8394 1664 


6-089 


2 6767 5457 


6-139 


1 5707 9620 


6-189 


5187 7605 


6-040 


3 8155 8914 


6-090 


2 6540 8915 


6-140 


I 5492 3618 


6-190 


4982 6753 


6-041 


3 7917 8545 


6-091 


2 6314 4638 


6-141 


I 5276 9772 


6-191 


4777 7951 


6-042 


3 7680 0556 


6-092 


2 6088 2625 


6-142 


I 5061 8078 


6-192 


4573 1197 


6-043 


3 7442 4943 


6-093 


2 5862 2872 


6-143 


I 4846 8535 


6-193 


4368 6488 


6-044 


3 7205 1705 


6-094 


2 5636 5378 


6-144 


I 4632 I 140 


6-194 


4164 3823 


6-045 


3 6968 0839 


6-095 


2 5411 0141 


6-145 


I 4417 5892 


6-195 


3960 3200 


6 046 


3 6731 2342 


6-096 


2 5185 7157 


6-146 


I 4203 2788 


6-196 


3756 4616 


6-047 


3 6494 6213 


6-097 


2 4960 6426 


6-147 


I 3989 1825 


6-197 


3552 8070 


6-048 


3 6258 2449 


6-098 


2 4735 7944 


6-148 


I 3775 3003 


6-198 


3349 3559 


6-049 


3 6022 1048 


6-099 


2 4511 1709 


6-149 


I 3561 6319 


6-199 


3146 1082 



IS: 



T^IR F. W. NEWMAN'S TABLE 



[6'2oo — 6"399] 



X 


Q-X 


X 


Q-X 


X 


Q-X 1 


X 


f>-X 


6-200 


20 2943 0636 


6-250 


19 3045 4136 


6-300 


iS 3630 4777 


6-350 


17 4674 7136 


6-201 


2740 2220 


6-251 


9 2 85 2 4647 


6-301 


8 3446 9390 


6-351 


7 4500 1262 


6-202 


2537 5831 


6-252 


9 2659 7086 1 


6-302 


8 3263 5838 


6-352 


7 4325 7133 


6-203 


2335 1468 


6-253 


9 2467 1452 


6-303 


8 3080 4118 


6-353 


7 4151 4747 


6204 


2132 9127 


6-254 


9 2274 7743 1 


6-304 


8 2897 4229 


6-354 


7 3977 4103 


6-205 


1930 S809 


6-255 


9 2082 5956 


6-305 


8 2714 6169 


6-355 


7 3S03 5199 


6-206 


1729 0509 


6256 


9 1890 6090 


6-306 


8 2531 9936 


6-356 


7 3629 8032 


6-207 


1527 4227 


6-257 


9 1698 8143 1 


6-307 


8 2349 5528 


6-357 


7 3456 2602 


6-20S 


1325 9960 


6-258 


9 1507 2113 ^ 


6-308 


8 2167 2945 


6-358 


7 3282 8906 


6-209 


II24 7706 


6259 


9 1315 7998 j 


6-309 


8 19S5 2182 1 


6-359 


7 3109 6944 


6-210 


0923 7464 


6-260 


9 1124 5797 


6-310 


8 1803 3239 


6-360 


7 2936 6712 


6-21 I 


0722 9231 


6-261 


9 0933 5506 


6-311 


8 1621 6115 


6-361 


7 2763 8209 


6-2 12 


0522 3005 


6-262 


9 0742 7125 1 


6-312 


8 1440 0806 


6-362 


7 2591 1435 


6-213 


0321 S784 


6-263 


9 0552 0651 


6-313 


8 1258 7312 


6-363 


7 2418 6386 


6-214 


0121 6567 


6-264 


9 0361 6083 


6-314 


8 1077 5631 


6-364 


7 2246 3061 


6-215 


19 9921 6350 


6-265 


9 0171 3418 


6-315 


8 0896 5761 


6-365 


7 2074 1459 


6-216 


9 9721 S133 


6266 


8 9981 2655 


6-316 


8 0715 7699 


6-366 


7 1902 1578 


6-217 


9 9522 1913 


6-267 


8 9791 3792 


6-317 


8 0535 1445 


6-367 


7 173° 3416 


6-218 


9 9322 7689 


6-268 


8 9601 6827 


6-318 


8 0354 6995 


6-368 


7 1558 6970 


6-219 


9 9123 5457 


6-269 


8 9412 1758 


6-319 


8 0174 4350 


6-369 


7 1387 2241 


6-220 


9 8924 5217 


6-270 


8 9222 8583 


6-320 


7 9994 3506 


6-370 


7 1215 9225 


6-221 


9 8725 6966 


6-271 


8 9033 7300 


6321 


7 9S14 4462 


6-371 


7 1044 7922 


6-222 


9 8527 0703 


6272 


8 8S44 790S 


6-322 


7 9634 7217 


6-372 


7 0873 8329 


6223 


9 8328 6424 


6-273 


8 8656 0404 


6-323 


7 9455 1767 


6-373 


7 0703 0445 


6224 


9 8130 4129 


6-274 


8 8467 47S6 


6-324 


7 9275 8113 


6-374 


7 0532 4268 


6-225 


19 7932 3815. 


6-275 


18 8279 1054 


6-325 


17 9096 6250 


6-375 


17 0361 9796 


6-226 


9 7734 5481 


6276 


8 8090 9204 


6-326 


7 8917 6179 


6-376 


7 oigi 7027 


6-227 


9 7536 9124 


6-277 


8 79°2 9235 


6-327 


7 8738 7898 


6-377 


7 0021 5961 


6-228 


9 7339 4742 


6-278 


8 7715 "45 


6-328 


7 8560 1403 


6-378 


6 9851 6595 


6-229 


9 7142 2334 


6-279 


8 7527 4932 


6-329 


7 8381 6694 


6-379 


6 9681 8927 


6-230 


9 6945 1897 


6-280 


8 7340 0594 


6-330 


7 8203 3769 


6-380 


6 9512 2957 


6-231 


9 6748 3429 


6-281 


8 7152 8130 


6-331 


7 8025 2626 


6-381 


6 9342 8681 


6-232 


9 6551 6929 


6-282 


8 6965 7537 


6332 


7 7847 3263 


6-382 


6 9173 6099 


6233 


9 6355 2395 


6-283 


8 6778 8814 


6-333 


7 7669 5679 


6-383 


6 9004 5208 


6-234 


9 6158 9824 


6-284 


8 6592 1959 


6-334 


7 7491 9871 


6-384 


6 8835 6008 


6235 


9 5962 9214 


6-285 


8 6405 6970 


6-335 


7 7314 5839 


6-385 


6 8666 8496 


6-236 


9 5767 0564 


6-286 


8 6219 3844 


6-336 


7 7137 3579 


6-386 


6 8498 2670 


6-237 


9 5571 3872 


6-287 


8 6033 2581 


6-337 


7 6960 3091 


6-387 


6 8329 8530 


6-238 


9 5375 9136 


6-288 


8 5847 3179 


6-33S 


7 6783 4372 


6-388 


6 8161 6073 


6-239 


9 5180 6354 


6-289 


8 5661 5634 


6-339 


7 6606 7421 


6-389 


6 7993 5297 


6240 


9 4985 5523 


6-290 


8 5475 9947 


6-340 


7 6430 2237 


6-390 


6 7S25 6201 


6-241 


9 4790 6642 


6-291 


8 5290 6114 


6341 


7 6253 8816 


6-391 


6 7657 8784 


6-242 


9 4595 9709 


6-292 


8 5i°5 4134 


6342 


7 6077 7159 


6392 


6 7490 3043 


6243 


9 4401 4722 


6-293 


8 4920 4005 


6-343 


7 5901 7262 


6-393 


6 7322 8977 


6-244 


9 4207 1679 


6-294 


8 4735 5725 


6-344 


7 5725 9123 


6-394 


6 71SS 6585 


6-245 


9 4013 0578 


6-295 


8 455° 9293 


6-345 


7 5550 2743 


6-395 


6 69S8 5864 


6246 


9 3819 1417 


6296 


8 4366 4706 


6346 


7 5374 8117 


6396 


6 6821 6812 


6-247 


9 3625 4194 


6-297 


8 4182 1963 


6-347 


7 5'99 5246 


6-397 


6 6654 9429 


6-248 


9. 3431 8908 


6-298 


8 3998 1062 


6-348 


7 5024 4126 


6-398 


6 6488 3713 


6249 


9 3238 5556 


6-299 


8 3814 2000 


6-349 


7 4849 4757 


6-399 


6 6321 9661 



[6-400— 6-599] 



OF THE DESCENDING EXPONENTIAL. 



183 



X 


C-x 


X 


(,-X 


X 


Q-X 


X 


e-x 


6-400 


16 


6155 7273 


6-450 


15 8052 2169 


6-500 


15 0343 9193 


6-550 


14 3011 5598 


6-401 


6 


5989 6546 


6-451 


5 7894 2437 


6-501 


5 0193 6505 


6-551 


4 2868 6197 


6-402 


6 


5823 7479 


6-452 


5 7736 4283 


6-502 


5 0043 5319 


6-552 


4 2725 8225 


6-403 


6 


5658 0071 


6-453 


5 7578 7707 


6-503 


4 9893 5634 


6-553 


4 2583 1681 


6-404 


6 


5492 4319 


6-454 


5 7421 2707 


6-504 


4 9743 7448 


6-554 


4 2440 6561 


6-405 


^ 6 


5327 0222 


6-455 


5 7263 9282 


'6-505 


4 9594 0759 


6-555 


4 2298 2867 


6-406 


6 


5161 7778 


6-456 


5 7106 7428 


6-506 


4 9444 5566 


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 


6-458 


5 6792 8434 


6-508 


4 9145 9661 


[6-558 


4 1872 0315 


6-409 


6 


4667 0349 


6-459 


5 6636 1289 


6-509 


4 8996 8947 


: 6-559 


4 1730 2304 


6-410 


6 


4502 4502 


6-460 


5 6479 5710 


6-510 


4 8S47 9723 


i 6-560 


4 1588 5710 


6-41 1 


6 


4338 0300 


6-461 


5 6323 1697 


6-511 


4 8699 1987 


6-561 


4 1447 0532 


6-412 


6 


4173 7741 


6-462 


5 6166 9246 


6-512 


4 8550 5738 


6-562 


4 1305 6769 


6-413 


6 


4009 6S24 


6-463 


5 6010 8358 


6-513 


4 8402 0975 


6-563 


4 1164 4418 


6-414 


6 


3845 7547 


6-464 


5 5854 9029 


6-514 


4 8253 7696 


6-564 


4 1023 3479 


6-415 


6 


3681 9908 


6-465 


5 5699 1259 


6-515 


4 8105 5899 


6-565 


4 0882 3951 


6-416 


6 


3518 3906 


6-466 


5 5543 5046 


6-516 


4 7957 5584 


6-566 


4 0741 5831 


6-417 


6 


3354 954° 


6-467 


5 5388 0389 


6-517 


4 7809 6748 


6-567 


4 0600 9119 


6-418 


6 


3191 6807 


6-468 


5 5232 7285 


6-518 


4 7661 9390 


6-568 


4 0460 3812 


6-419 


6 


3028 5706 


6-469 


5 5°77 5734 


6-519 


4 7514 3508 


6-569 


4 0319 9910 


6-420 


6 


2865 6235 


6-470 


5 4922 5733 


6-520 


4 7366 9102 


6-570 


4 0179 7412 


6-421 


6 


2702 8393 


6-471 


5 4767 7282 


6-521 


4 7219 6170 


6-571 


4 0039 6315 


6-422 


6 


2540 2177 


6-472 


5 4613 0378 


6-522 


4 7072 4709 


6-572 


3 9899 6619 


6-423 


6 


2377 7588 


6-473 


5 4458 5020 


6-523 


4 6925 4720 


6-573 


3 9759 8322 


6-424 


6 


2215 4622 


6-474 


5 4304 1207 


6-524 


4 6778 6200 


6-574 


3 9620 1422 


6-425 


16 


2053 3278 


6-475 


15 4149 8937 


6-525 


14 6631 9147 


6-575 


13 9480 5918 


6-426 


6 


1891 3555 


6-476 


5 3995 8209 


6-526 


4 6485 3561 


6-576 


3 9341 1810 


6-427 


6 


1729 545° 


6-477 


5 3841 9020 


6-527 


4 6338 9439 


6-577 


3 9201 9094 


6-428 


6 


1567 8963 


6-478 


5 3688 1370 


6-528 


4 6192 6781 


6-578 


3 9062 7771 


6-429 


6 


1406 4092 


6-479 


5 3534 5257 


6-529 


4 6046 5585 


6-579 


3 8923 7838 


6-430 


6 


1245 0834 


6-480 


5 3381 0679 


6-530 


4 5900 5850 


6-580 


3 8784 929s 


6-431 


6 


1083 9190 


6-481 


5 3227 7635 


6-531 


4 5754 7573 


6-581 


3 8646 2139 


6-432 


6 


0922 9156 


6-482 


5 3074 6123 


6-532 


4 5609 0754 


6-582 


3 8507 6370 


6-433 


6 


0762 0731 


6-483 


5 2921 6142 


6-533 


4 5463 5391 


6-583 


3 8369 1986 


6-434 


6 


0601 3914 


6-484 


5 2768 7691 


6-534 


4 5318 1483 


6-584 


3 8230 8986 


6-435 


6 


0440 8702 


6-485 


5 2616 0767 


6-535 


4 5172 9028 


6-585 


3 8092 736S 


6-436 


6 


C280 5096 


6-486 


5 2463 5369 


6-536 


4 5027 8024 


6-586 


3 7954 7130 


6-437 


6 


0120 3092 


6-487 


5 2311 1495 


6-537 


4 4882 8471 


6-587 


3 7816 S273 


6-438 


5 


9960 2689 


6-488 


5 2158 9145 


6-538 


4 4738 0367 


6-588 


3 7679 0793 


6-439 


5 


9S00 3886 


6-489 


5 2006 8317 


6-539 


4 4593 3710 


6-589 


3 7541 4691 


6-440 


S 


9640 6681 


6-490 


5 1854 9008 


6-540 


4 4448 8499 


6-590 


3 7403 9964 


6-441 


5 


9481 1072 


6-491 


5 1703 1218 


6-541 


4 4304 4732 


6-591 


3 7266 66io 


6-442 


5 


9321 7058 


6-492 


5 1551 4945 


6-542 


4 4160 2409 


6-592 


3 7129 4630 


6-443 


5 


9162 4637 


6-493 


5 1400 0188 


6-543 


4 4016 1527 


6-593 


3 6992 4021 


6-444 


5 


9003 3808 


6-494 


5 1248 6944 


6-544 


4 3872 2085 


6-594 


3 6855 4781 


6-445 


5 


8844 4569 


6-495 


5 i°97 5213 


6-545 


4 3728 4083 


6-595 


-3 6718 6911 


6-446 


5 


8685 6918 


6-496 


5 0946 4993 


6-546 


4 3584 7517 


6-596 


3 6582 0407 


6-447 


5 


8527 0855 


6-497 


5 0795 6283 


6-547 


4 3441 2387 


6-597 


3 6445 5269 


6-448 


5 


8368 6376 


6-498 


5 0644 9080 


6-548 


4 3297 8692 


6-598 


3 6309 1496 


6-449 


5 


8210 3481 


6-499 


5 0494 3384 


6-549 


4 3154 6429 


6-5.99 


3 6172 9086 



1S4 



MR F. \V. NEWMAN'S TABLE 



[6-6oo — 6-799] 



X 


(,-X 


X 


g-X 


X 


Q-X 


X 


Q-X 


6600 


13 


6036 8037 


6-650 


12 9402 2105 


6-700 


12 3091 1903 


6-750 


II 7087 9621 


6 60 1 


3 


5900 8349 


! 6-651 


2 9272 8730 


6-701 


2 2968 1606 


6-751 


I 6970 9326 


6602 


3 


5765 0020 


1 6-652 


2 9143 6647 


6-702 


2 2S45 2539 


6-752 


I 6854 0202 


6-603 


3 


5629 3049 


6-653 


2 9014 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 


1 6620 5457 


6-605 


3 


5358 3174 


6-655 


2 8756 8143 


6-705 


2 2477 2704 


6-755 


I 6503 9834 


6-606 


3 


5223 0267 


6-656 


2 88^8 1219 


6-706 


2 2354 8543 


6-756 


I 6387 5377 


6607 


3 


50S7 S713 


6-657 


2 8499 55S0 


6-707 


2 2232 5606 


6-757 


I 6271 2083 


6-6oS 


3 


4952 8509 


6-658 


2 S371 1227 


6-70S 


2 2110 3S92 


6-758 


I 6154 9952 


6609 


3 


4817 9655 


6-659 


2 8242 S157 


6709 


2 1988 339S 


6-759 


I 6038 8983 


6-6io 


3 


4683 2149 


6-66o 


2 81 14 6370 


6-710 


2 1866 4124 


6-760 


1 5922 9174 


6-6II 


3 


4548 599° 


J 6-661 


2 7986 5864 


6-711 


2 1744 6070 


6-761 


I 5807 0524 


6-6 1 2 


3 


4414 1177 


6-662 


2 7858 6638 


6-712 


2 1622 9232 


6-762 


1-5691 3032 


6-613 


3 


4279 770S 


6-663 


2 7730 8691 


6-713 


2 1501 3611 


6-763 


I 5575 6698 


6614 


3 


4145 5581 


6-664 


2 7603 2020 


6-714 


2 1379 9204 


6-764 


I 5460 1519 


6-615 


3 


4011 4796 


6-665 


2 7475 6626 


6-715 


2 1258 6012 


6-765 


I 5344 7494 


6-616 


3 


3877 5351 


6-666 


2 7348 2507 


6-716 


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 5 114 2905 


6-618 


3 


3610 0476 


6-668 


2 7093 8087 


6-718 


2 0895 3705 


6-768 


I 4999 2337 


6-619 


3 


3476 5°43 


6-669 


2 6966 7784 


6-719 


2 0774 5356 


6-769 


I 4SS4 2920 


6-620 


3 


3343 0946 


6-670 


2 6839 8751 


6-720 


2 0653 8214 


6-770 


I 4769 4651 


6621 


3 


3209 8181 


6-671 


2 6713 0986 


6-721 


2 0533 3279 


6-771 


I 4654 7530 


6-622 


3 


3076 6749 


6-672 


2 65S6 4489 


6-722 


2 0412 7549 


6-772 


I 454° 1555 


6623 


3 


2943 6647 


6-673 


2 6459 9257 


6-723 


2 0292 4023 


6-773 


I 4425 6726 


6624 


3 


2S10 7S75 


6-674 

( 


- 6333 5290 


6-724 


2 0172 1701 


6-774 


I 43 II 3042 


6-625 


13 


2678 0431 


6-675 


12 6207 2586 


6-725 


12 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 9100 


6-627 


3 


2412 9522 


6-677 


2 5955 0963 


6727 


I 9S12 1938 


6-777 


I 3968 8S41 


6-628 


3 


2280 6054 


6-678 


2 5829 2042 


6-728 


I 9692 4415 


6-778 


I 3854 9722 


6-629 


3 


2148 3909 


6-679 


2 5703 4379 


6-729 


I 9572 8089 


6779 


I 3741 1742 


6630 


3 


2016 3086 


6-68o 


2 5577 7973 


6-730 


I 9453 2958 


6-780 


I 3627 4898 


6631 


3 


1884 3583 


6-68i 


2 5452 2823 


6-731 


I 9333 9022 


6-781 


I 3513 9191 


6632 


3 


1752 5398 


6-682 


2 5326 8927 


6-732 


I 9214 6280 


6-782 


I 3400 4620 


6-633 


3 


1620 8532 


6-683 


2 5201 6284 


6-733 


I 9095 4729 


6-783 


I 3287 1182 


6-634 


3 


1489 2981 


6-684 


2 5076 4894 


6-734 


I 8976 4370 


6-784 


I 3173 8877 


6635 


3 


1357 874s 


6-685 


2 4951 4754 


6-735 


I 8857 5200 


6-785 


I 3060 7704 


6636 


3 


1226 5823 


6-686 


2 4826 5864 


6-736 


I 8738 7219 


6786 


I 2947 7661 


6-637 


3 


1095 4213 


6-687 


2 4701 8222 


6-737 


I 8620 0425 


6-787 


I 2834 8748 


6-638 


3 


0964 3914 


6-688 


2 4577 1827 


6-738 


I 8501 4818 


6-788 


I 2722 0963 


6639 


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 


I 2496 8774 


6'64i 


3 


0572 0870 


6*69 1 


2 4204 0112 


6-741 


I 8146 5101 


6-791 


1 2384 4368 


6642 


3 


0441 5802 


6692 


2 4079 8693 


6-742 


I S028 4226 


6-792 


I 2272 1085 


6643 


3 


03 1 1 2038 


6-693 


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 


I 2047 7887 


6645 


3 


0050 8418 


1 6-695 


2 3708 1875 


6-745 


I 7674 8679 


6-795 


I 1935 7969 


6-646 


2 


9920 8560 


6-696 


2 3584 5411 


6-746 


I 7557 2519 


6-796 


I 1823 9171 


6647 


2 


9791 0001 


6-697 


2 3461 0184 


6-747 


I 7439 7534 


6797 


I 1712 1490 


6648 


2 


9661 2739 


6-698 


2 3337 6190 


6-748 


I 7322 3723 


6-798 


I 1600 4927 


6*649 


2 


9531 677s 


6-699 


2 3214 3431 


6-749 


I 7?b5 1086 


6799 


I 1488 9480 



[6-8oo — 6-999] 



OF THE DESCENDING EXPONENTIAL. 



X 


Q-X 


X 


(,-X 


X 


(.-X 


X 


Q-X 


6 -800 


II 1377 5148 


6-850 


10 5945 5693 


6-goo 


10 0778 5429 


6-950 


9 5863 5154 


6-8oi 


I 1266 1929 


'6-S51 


° 5839 6767 


6-901 


0677 8147 


6-951 


9 5767 6998 


6-8o2 


I I 154 9824 


'6-852 


5733 8S99 


6-902 


0577 1872 


6-952 


9 5671 9799 


6-803 


I 1043 8829 


6-853 


5628 2089 


6-903 


0476 6603 


6-953 


9 5576 3558 


6-804 


I 0932 8945 


6-854 


5522 6334 


6-904 


0376 2339 


6-954 


9 5480 8272 


6-805 


I 0822 OI7I 


6-855 


5417 1636 


6-905 


0275 9078 


6-955 


9 5385 3941 


6-8o6 


I O7II 2505 


6-856 


5311 7991 


6-906 


0175 6820 


6-956 


9 5290 0564 


6-807 


I 0600 5946 


I 6-857 


° 5206 5399 


6-907 


0075 5564 


6-957 


9 5194 8140 


6 -808 


1 0490 0493 


6-858 


5101 3860 


6-908 


09 9975 5309 


6-958 


9 5099 6667 


6-809 


I 0379 6144 


6-859 


4996 3371 


6-909 


9 9875 6053 


6959 


9 5004 6146 


6-810 


I 0269 2900 


' 6-86o 


4891 3933 


6-910 


9 9775 7796 


6-960 


9 4909 6575 


6-811 


I 0159 0758 


6-S61 


4786 5543 


6-911 


9 9676 0537 


6-961 


9 4S14 7952 


6-812 


I 0048 9718 


6-862 


4681 8201 


6-gi2 


9 9576 427s 


6-962 


9 4720 0278 


6-813 


9938 9778 


6-863 


4577 1906 


6-913 


9 9476 9008 


6-963 


9 4625 3552 


6-814 


9829 0938 


6-864 


4472 6657 


6-914 


9 9377 4737 


6-964 


9 4530 7771 


6-815 


9719 3196 


6-865 


4368 2453 


6-915 


9 9278 1459 


6-965 


9 4436 2936 


6-816 


9609 6551 


6-866 


4263 9292 


6-916 


9 9178 9173 


6-966 


9 4341 9045 


6-817 


9500 1003 


6-S67 


° 4159 7174 ' 


6-917 


9 9079 7880 


6-967 


9 4247 6097 


6-818 


9390 6549 


6-868 


C 4055 6097 ; 


6-918 


9 8980 7577 


6-968 


9 4153 4092 


6-819 


9281 3189 


6869 


3951 6061 


6-919 


9 8S81 8264 


6-969 


9 4059 3029 


6-820 


9172 0922 


6-870 


3S47 7064 


6-920 


9 8782 9940 


6-970 


9 3965 2906 


6-821 


9062 9747 


6-871 


3743 9i°7 


6-921 


9 8684 2604 


6-971 


9 3871 3723 


6-822 


8953 9662 


6-872 


3640 2186 


6-922 


9 8585 6255 


6-972 


9 3777 5478 


6-823 


S845 0667 


6-873 


° 3536 6302 


6-923 


9 8487 0891 


6-973 


9 36S3 8171 


6-824 


8736 2761 


6-874 


° 3433 1453 


6-924 


9 8388 6513 


6-974 


9 3590 1801 


6-825 


10 8627 5941 


6-875 


10 3329 7639 


6-925 


9 8290 3118 


6-975 


9 3496 6367 


6-826 


8519 0208 


6-876 


3226 4857 


6-926 


9 8192 0706 


6-976 


9 3403 1868 


6-827 


8410 5561 


6-877 


3123 3109 


6-927 


9 8093 9276 


6-977 


9 3309 8303 


6-828 


8302 1997 


6-878 


3020 2391 1 


6-928 


9 7995 8827 


6-978 


9 3216 5671 


6-829 


8193 9516 


6-879 


2917 2703 


6-929 


9 7897 9358 


6-979 


9 3123 3972 


6-830 


8085 8118 


6-880 


2814 4045 


6-930 


9 7800 0868 


6-980 


9 3030 3203 


6-831 


7977 7800 


6-881 


2711 6415 


6-931 


9 7702 3356 


6-981 


9 2937 3365 


6-832 


7S69 8562 


6-882 


2608 9S12 


6-932 


9 7604 6821 


6-982 


9 2844 4456 


6833 


7762 0402 


6883 


2506 4235 


6-933 


9 7507 1262 


6-983 


9 2751 6476 


6-834 


7654 3321 


6-884 


2403 9683 1 


6934 


9 7409 6679 


6984 


9 2658 g423 


6-835 


7546 7315 


6-885 


2301 6155 


6-935 


9 7312 3069 


6-985 


9 2566 3297 


6-836 


7439 2386 


6-886 


2199 3651 


6-936 


9 7215 0432 


6-986 


9 2473 Sog6 


6-837 


7331 8530 


6-8S7 


2097 2168 


6-937 


9 7117 8768 


6-987 


9 2381 3S20 


6838 


7224 5748 


6-888 


1995 1706 


6-938 


9 7020 8075 


6-988 


9 2289 0467 


6-839 


7117 4038 


6-889 


1893 2264 


6-939 


9 6923 8351 


6-989 


9 2196 8039 


6-840 


7010 3400 


6-890 


1791 3841 t 


6-940 


9 6826 9597 


6-ggo 


9 2104 6532 


6-841 


6903 3831 


6-891 


1689 6436 


6-941 


9 6730 1812 


6991 


9 2012 5946 


6-842 


6796 5332 


6-892 


1588 0048 


6-942 


9 6633 4993 


6-992 


9 1920 62S0 


6-843 


6689 7900 


6-893 


i486 4676 ' 


6-943 


9 6536 9141 


6-993 


9 182S 7533 


6-844 


6583 1536 


6-894 


1385 0318 


6-944 


9 6440 4255' 


6 994 


9 1736 9704 


6-845 


6476 6237 


6-895 


1283 6975 


6-945 


9 6344 0332 


6-995 


9 1645 2793 


6-846 


6370 2002 


6-896 


1182 4644 


6-946 


9 6247 7374 


6-996 


9 1553 6798 


6-847 


6263 8833 


6-897 


io8r 3325 


6-947 


9 6151 5377 


6-997 


9 1462 1719 


6-848 


6157 6725 


6-898 


09S0 3017 


6-948 


9 6055 4343 


6-998 


9 1370 7555 


6-849 


6051 5679 


6899 


0879 3719 


6-949 


9 5959 4268 


6-999 


9 1279 4304 



Vol. XIII. Part 111. 



25 



1S6 



MR F. W. NEWMAN'S TABLE 



[yooo— 7-199] 



X 


g-X 


X 


c- 


X 


e-* 


X 


Q-X 


7'ooo 


9 118S 1965 


7-050 


8 6740 S957 


7-100 


8 


2510 4923 


7-150 


7 8486 4081 


7'ooi 


9 i°97 0539 


7051 


8 6654 1982 


7-101 


8 


2428 0231 


7-151 


7 8407 9610 


7002 


9 1006 0024 ! 


7-052 


8 6567 5873 


7-102 


8 


2345 6362 


7-152 


7 8329 5922 


7-003 


9 0915 0419 


7053 


8 6481 0630 


7-103 


8 


2263 3318 


7-153 


7 8251 3017 


7-004 


9 0S24 1723 


7-054 


8 6394 6251 


7-104 


8 


2181 1096 


7-154 


7 8173 0896 


7-005 


9 0733 3935 


7-055 


8 6308 2737 


7-105 


8 


2098 9695 


7-155 


7 8094 9555 


7-006 


9 0642 7055 


7-056 


8 6222 0086 1 


7-106 


8 


2016 9116 


7-156 


7 8016 8996 


7-007 


9 0552 1081 


7-057 


8 6135 8296 ' 


7-107 


8 


1934 9357 


7-157 


7 7938 9217 


7-008 


9 0461 6012 


7-058 


8 6049 7369 


7-108 


8 


1853 0417 


7-158 


7 7861 0217 


7-009 


9 0371 1848 


7-059 


8 5963 7301 


7-109 


8 


1771 2296 


7-159 


7 7783 1996 


7-010 


9 0280 8588 


7-060 


8 5877 8094 


7-110 


8 


1689 4992 


7-160 


7 7705 4553 


, 7"oii 


9 0190 6231 


7 06 1 


8 5791 9745 


7-111 


8 


1607 8505 


7-161 


7 7627 7887 


7-012 


9 0100 4776 


7-062 


8 5706 2254 


7-112 


8 


1526 283s 


7-162 


7 7550 1997 


7'oi3 


9 0010 4221 


7063 


8 5620 5620 


7-113 


8 


1444 7979 


7163 


7 7472 6883 


7014 


8 9920 4567 


7064 


8 5534 9843 


7-114 


8 


1363 3938 


7-164 


7 7395 2543 


7"oi5 


8 9830 5812 


7-065 


8 5449 4920 


7-115 


8 


1282 0711 


7-165 


7 7317 8977 


7-016 


8 9740 7955 


7-066 


8 5364 0852 


7-116 


8 


1200 8297 


7-166 


7 7240 6185 


7-017 


8 9651 0995 


7067 


8 5278 7638 


7-117 


8 


I I 19 6694 


7-167 


7 7163 4165 


7-018 


8 9561 4932 


7-068 


8 5193 5277 


7-118 


8 


1038 5903 


7-168 


7 7086 2916 


7-019 


8 9471 9765 


7-069 


S 5108 3767 


7-119 


8 


0957 5922 


7-169 


7 7009 2439 


7 -020 


8 9382 5493 


7-070 


8 5023 3109 


7-120 


8 


0876 6751 


7-170 


7 6932 2731 


17021 


8 9293 2114 


7-071 


8 4938 3301 


7-121 


8 


0795 8389 


7-171 


7 6855 3793 


7 02 2 


8 9203 9628 


7-072 


8 4S53 4342 


7-122 


8 


0715 0834 


7-172 


7 6778 5623 


7-023 


8 91 14 8034 


7-073 


8 4768 6232 


7-123 


8 


0634 4087 


7-173 


7 6701 8221 


1 7-024 


8 9025 7332 


1 7-074 


8 4683 8969 


7-124 


8 


0553 8146 


7-174 


7 6625 1587 


,7-025 


8 8936 7519 


7-075 


8 4599 2554 


7-125 


8 


0473 3010 


7-175 


7 6548 5718 


7-02(5 


8 8847 8596 


7076 


8 4514 6984 


7-126 


8 


0392 8679 


7-176 


7 6472 0615 


7-027 


8 8759 0562 


7-077 


8 4430 2259 


7-127 


8 


0312 5152 


7-177 


7 6395 6277 


'7-028 


8 8670 3415 


7-078 


8 4345 8379 


7-128 


8 


0232 2429 


7-178 


7 6319 2702 


1 7-029 


8 8581 7155 


7-079 


8 4261 5342 


7-129 


8 


0152 0507 


7-179 


7 6242 9891 


7-030 


8 8493 1780 


7 080 


8 4177 3148 


7-130 


8 


0071 9387 


7-180 


7 6166 7842 


7 031 


8 8404 7291 


7-081 


8 4093 1796 


7-131 




9991 9068 


I7-181 


7 6090 655s 


7032 


8 8316 3685 


7082 


8 4009 1284 


7-132 




9911 9549 


7-182 


7 6014 6029 


7-033 


8 8228 0963 


7-083 


8 3925 1613 


7-133 




9832 0829 


7-183 


7 5938 6263 


1 7-034 


8 8139 9123 


7084 


8 3841 2781 


7-134 




9752 2907 


,7-184 


7 5862 7256 


7-03^ 


8 8051 8165 


7-085 


8 3757 4787 


1 7-135 




9672 5783 


7-185 


7 5786 9008 


7036 


8 7963 8087 


7-086 


8 3673 7631 


7-136 




9592 9455 


1 7-186 


7 5711 1518 


7-037 


8 7875 8888 


7-087 


8 3590 1312 


7137 




9513 3924 


7-187 


7 5635 4785 


7038 


8 7788 0569 


7088 


8 3506 5828 


7-138 




9433 9187 


7-188 


7 5559 8808 


7039 


8 7700 3127 


7-089 


8 3423 1180 


7-139 




9354 5245 


,7-189 


7 5484 3587 


7-040 


8 7612 6562 


7-090 


8 3339 7365 


7-140 




9275 2096 


' 7-190 


7 5408 9120 


7-041 


8 7525 0873 


7-091 


8 3256 4385 


7-141 




9195 9741 


7-191 


7 5333 5408 


7-042 


8 7437 6060 


7-092 


8 3173 2236 


7-142 




91 16 8177 


7-192 


7 5258 2449 


J7-043 


8 7350 2121 


7-093 


8 3090 0920 


7-143 




9037 7404 


7-193 


7 5183 0243 


7-044 


8 7262 9055 


7-094 


8 3007 0434 


7-144 

1 




8958 7422 


7-194 


7 5107 8789 


7-04?; 


8 717s 6863 


7095 


8 2924 0779 


7-145 




8879 8229 


7-195 


7 5032 8085 


7-046 


8 7088 5541 


7-096 


8 2841 1952 


1 7-146 




8800 9825 


7-196 


7 4957 8132 


7-047 


8 7001 5091 


7-097 


8 2758 3955 


7-147 




8722 2209 


7-197 


7 4882 8929 


7048 


8 6914 5511 


7-098 


8 2675 6784 


7-148 




8643 5380 


7-198 


7 4808 0474 


7-049 


8 6827 (>^°o 


7099 


8 2593 0441 


7149 




8564 9338 


1 7-199 


7 4733 2768 



[7- 



-7-399I 



OF THE DESCENDING EXPONENTIAL. 



187 



X 


Q-X 


X 


(,~X 


1 X 


Q-X 


X 


1 


7-200 


7 4658 5808 


7-250 


7 1017 4389 


7-300 


6 7553 8775 


7-350 


6 


4259 2360 


1 7'20I 


7 4583 9596 


7-251 


7 0946 4569 


7-301 


6 7486 3574 


7-351 


6 


4x95 0089 


7'202 


7 4509 4129 


7-252 


7 0875 5459 


7 '302 


6 7418 9048 


7-352 


6 


4130 8460 


7-203 


7 4434 9407 


7-253 


7 0804 705S 


7-303 


6 7351 5196 


7-353 


6 


4066 7472 


7-204 


7 4360 5430 


7-254 


7 0733 9365 


7-304 


6 7284 2017 


7-354 


6 


4002 7125 


17-205 


7 4286 2196 


7-255 


7 0663 2379 


7-305 


6 7216 9511 


7-355 


6 


3938 7417 


1 7-206 


7 4211 9705 


7-256 


7 0592 6100 


7-306 


6 7149 7678 


7-356 


6 


3874 8350 


i 7-207 


7 4137 7956 


7-257 


7 0522 0527 


7-307 


6 70S2 6516 


7-357 


6 


3810 9921 


i 7-208 


7 4063 6949 


7-258 


7 0451 5659 


7-308 


6 7015 6025 


7-358 


6 


3747 2130 


7-209 


7 3989 6682 


7-259 


7 0381 1495 


7-309 


6 6948 6204 


7-359 


6 


3683 4976 


7-210 


7 3915 7155 


7-260 


7 0310 8036 


7-310 


6 6881 7052 


7-360 


6 


3619 8459 


7-211 


7 3S41 8368 


7-261 


7 0240 5279 


7-311 


6 6814 8569 


7-361 


6 


3556 2579 


7-212 


7 3768 0318 


7-262 


7 0170 3225 


7-312 


6 6748 0755 


7-362 


6 


3492 7334 


7-213 


7 3694 3007 


7-263 


7 0100 1872 


7-313 


6 6681 3607 


7-363 


6 


3429 2724 


7-214 


7 3620 6432 


7-264 


7 0030 1221 


7-314 


6 6614 7127 


7-364 


6 


3365 8748 


7-215 


7 3547 0594 


7-265 


6 9960 1270 


7-315 


6 6548 1313 


7-365 


6 


3302 5406 


7-216 


7 3473 5491 


7-266 


6 9890 2018 


7-316 


6 6481 6164 


7-366 


6 


3239 2697 


7-217 


7 3400 1 123 


7-267 


6 9820 3465 


7317 


6 6415 1680 


7-367 


6 


3176 0621 


7-218 


7 3326 7488 


7-268 


6 9750 5611 


7-318 


6 6348 7861 


7-368 


6 


3112 9176 


7-219 


7 3253 4587 


7-269 


6 9680 8454 


7-319 


6 6282 4705 


7-369 


6 


3049 8362 


7"220 


7 3180 2419 


7-270 


6 9611 1994 


7-320 


6 6216 2211 


7-370 


6 


2986 8179 


7-221 


7 3107 0982 


7-271 


6 9541 6230 


7-321 


6 6150 0380 


7-371 


6 


2923 8626 


7-222 


7 3034 0277 


7-272 


6 9472 1161 


7-322 


6 60S3 9210 


7-372 


6 


2860 9702 


7-223 


7 2961 0301 


7.-273 


6 9402 6787 


7-323 


6 6017 8701 


7-373 


6 


2798 1406 


7-224 


7 2888 1056 


7-274 


6 9333 3107 


7-324 


6 5951 8853 


7-374 


6 


2735 3739 


7-225 


7 2815 2539 


7-275 


6 9264 0121 


7-325 


6 5885 9663 


7-375 


6 


2672 6698 


7-226 


7 2742 4751 


7-276 


6 9194 7S27 


7-326 


6 5820 1133 


7-376 


6 


2610 0285 


7-227 


7 2669 7689 


7-277 


6 9125 6225 


7-327 


6 5754 3261 


7-377 


6 


2547 4498 


7-228 


7 2597 1355 


7-278 


6 9056 5314 


7-328 


6 56S8 6046 


7-378 


6 


2484 9336 


7-229 


7 2524 5746 


7-279 


6 8987 5094 


7-329 


6 5622 9489 


7-379 


6 


2422 4799 


7-230 


7 2452 0863 


7-280 


6 8918 5564 


7-330 


6 5557 3587 


7-380 


6 


2360 0886 


7-231 


7 2379 6704 


7-281 


6 8849 6723 


7-331 


6 5491 8341 


7-381 


6 


2297 7597 


7-232 


7 2307 3270 


7-282 


6 8780 8570 


7-332 


6 5426 3750 


7-382 


6 


2235 4931 


7-233 


7 2235 055S 


7-283 


6 8712 1105 


7-333 


6 5360 9813 


7-383 


6 


2173 2887 


7-234 


7 2162 8568 


7-284 


6 8643 4328 


7-334 


6 529s 6530 


7-384 


6 


2111 1465 


7-235 


7 2090 7300 


7-285 


6 8574 8236 


7-335 


6 5230 3900 


7-385 


6 


2049 0664 


7-236 


7 2018 6753 


7-286 


6 8506 2831 


7-336 


6 5165 1922 


7-386 


6 


1987 0483 


7-237 


7 1946 6927 


7-287 


6 8437 8110 


7-337 


6 5100 0596 


7-387 


6 


1925 0922 


7-238 


7 1874 7819 


7-288 


6 8369 4074 


7-338 


6 5034 9921 


7-388 


6 


1863 19S1 


7-239 


7 1802 9431 


7-289 


6 8301 0722 


7-339 


6 4969 9896 


7-389 


6 


1801 3658 


7-240 


7 1731 1760 


7-290 


6 8232 8053 


7-340 


6 4905 0521 


7-390 


6 


1739 5953 


7-241 


7 1659 4807 


7-291 


6 8164 6066 


7-341 


6 4840 1795 


7-391 


6 


1677 8866 


7-242 


7 1587 8570 j 


7-292 


6 8096 4760 


7-342 


6 4775 3717 


7-392 


6 


1616 2396 


7-243 


7 1516 3049 


7-293 


6 8028 4136 


7-343 


6 4710 6287 


7-393 


6 


1554 6541 


7-244 


7 1444 8244 


7-294 


6 7960 4192 


7-344 


6 4645 9504 


7-394 


6 


1493 1302 


7-245 


7 1373 4153 


7-295 


6 7892 4927 


7-345 


6 4581 3368 


7-395 


6 


1431 6678 


7-246 


7 1302 0775 


7-296 


6 7824 6342 


7-346 


6 4516 7877 


7-396 


6 


1370 2669 


7-247 


7 I2j-' 8lII 


7-297 


6 7756 8434 


7-347 


6 4452 3032 


7-397 


6 


1308 9273 


7-248 


7 "59 (5i59 


7-298 


6 7689 1205 


7-348 


6 4387 8831 


7-398 


6 


1247 6490 


7-249 


7 10S8 491,8 


7-299 


6 7621 4652 


7-349 


6 4323 5274 


7-399 


6 


11S6 4320 



25—2 



188 



MR F. \y. NEWMAN'S TABLE 



[7-400— 7-599] 



X 


C-x 


X 


.- 


X 


C-x 


X 

7-550 


g-X 


7-400 


6 1125 2761 


7-450 


5 8144 


1612 


7-500 


5 5308 4370 


5 


2611 0127 


7-401 


6 1064 1S14 


7-451 


5 8086 


0461 


7'5°i 


5 5253 1562 


7-551 


5 


2558 4280 


7-402 


6 1003 1477 


7-452 


5 8027 


9891 


7-502 


5 5197 9307 


7-552 


5 


2505 8958 


7 •403 


6 0942 1 75 1 


7-453 


5 7969 


9901 


7-5°3 


5 5142 7603 


7-553 


5 


2453 4162 


7-404 


6 0S81 2634 


7-454 


5 7912 


0491 


7-504 


5 5087 6451 


7-554 


5 


2400 9890 


7-405 


6 0S20 4125 


7-455 


5 7S54 


1660 


7-505 


5 5032 5850 


7-555 


5 


2348 6142 


7-406 


6 0759 6225 


7-456 


5 7796 


3407 


7-506 


5 4977 5799 


7-556 


5 


2296 2917 


7-407 


6 0698 8933 


7-457 


5 7738 


5733 


7-507 


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 


7-511 


5 4703 3781 


7-561 


5 


2035 4629 


7-412 


6 0396 1563 


7-462 


5 7450 


6010 


7-512 


5 4648 7021 


7-562 


5 


1983 4534 


7-413 


6 0335 7903 


7-463 


5 7393 


1791 


7-513 


5 4594 0S07 


7563 


5 


1931 4960 


7-414 


6 0275 4847 


7-464 


5 7335 


8146 


7-514 


5 4539 5139 


7-564 


5 


1879 5904 


7-415 


6 0215 2393 


7-465 


5 7278 


5074 


7-515 


5 4485 0017 


7-565 


5 


1827 7368 


7-416 


6 0155 0542 


7-466 


5 7221 


2575 


7-5'6 


5 -1430 5439 


7-566 


5 


1775 9349 


7-417 


6 0094 9292 


7-467 


5 7164 


0649 


7-5I7 


5 4376 1406 


7-567 


5 


1724 1849 


7-418 


6 0034 8643 


7-468 


5 7106 


9294 


7-518 


5 4321 7916 


7-568 


5 


1672 4865 


7-419 


5 9974 S594 


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 


7-421 


5 985s 0296 


7'47i 


5 6935 


8653 


7-521 


5 4159 0704 


7-571 


5 


1517 7014 


7-422 


5 9795 2045 


7-472 


5 6878 


9579 


7-522 


5 4104 9384 


7-572 


5 


1466 2094 


7-423 


5 9735 4391 


7-473 


5 6822 


1074 


7-523 


5 4050 8605 


7-573 


5 


1414 7689 


7424 


5 9675 7336 


7-474 


5 6765 


3137 


7-524 


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 


7576 


5 


1260 7558 


7-427 


5 9496 9746 


7-477 


5 6595 


2729 


7-527 


5 383s 0889 


7-577 


5 


1209 5207 


7-428 


5 9437 5074 


7-478 


5 6538 


7060 


7-528 


5 3781 2807 


7-578, 


5 


1158 3367 


7-429 


5 9378 0996 


7-479 


5 64S2 


1955 


7-529 


5 3727 5263 


7-579 


5 


1 107 2040 


7-430 


5 9318 7512 


7-480 


5 6425 


7415 


7-530 


5 3673 8257 


7-580 


5 


1056 1223 


7-431 


5 9259 4621 


7-481 


5 6369 


3440 


7-531 


5 3620 1787 


7-581 


5 


1005 0917 


7-432 


5 9200 2322 


7-482 


5 6313 


0028 


7-532 


5 3566 5853 


7-582 


5 


0954 1121 


7 433 


5 9141 0616 


7-483 


5 6256 


7180 


7-533 


5 3513 0455 


7-583 


5 


0903 1835 


7-434 


5 9081 9501 


7-484 


5 6200 


4894 


7-534 


5 3459 5592 


7-584 


5 


0S52 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 


7-486 


5 6088 


2007 


7-536 


5 3352 7469 


7-586 


5 


0750 7027 


7-437 


5 8904 9698 


- 7-487 


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 


s 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 


17-541 


5 3086 6490 


7-591 


5 


0497 5826 


7-442 


5 8611 1801 


7-492 


5 5752 


6791 


[7-542 


5 3033 5889 


7-592 


S 


0447 1 102 


7 443 


5 8552 5982 


7 '493 


5 5696 


9543 


i 7-543 


5 2980 5818 


7-593 


5 


0396 6883 


7 444 


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 


0295 9957 


' 7 446 


5 8377 2036 


7-496 


5 5530 


1138 


7-546 


5 28218782 


7-596 


5 


.0245 7248 


7-447 


5 8318 8556 


7-497 


5 5474 


6115 


i 7-547 


5 2769 0827 


7-597 


■ 5 


0195 5042 


7-448 


5 8260 5659 


7-498 


5 5419 


1646 


7-548 


5 2716 3400 


7-59S 


5 


0145 3338 


7 449 


5 8202 3344 


7-499 


5 5363 


7731 


1 7-549 


5 2663 6500 


7:j99 


5 


0095 2135 



[7 -600—7 799] 



OF THE DESCENDING EXPONENTIAL. 



189 



X 


Q-X 


X 


c-«= 


X 


e-* 


;?? 


Q-X 


7-600 


S 


0045 1433 


7-650 


4 7604 4129 


7-700 


4 52S2 7183 


7-750 


4 


3074 2541 


7-601 


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 


7-603 


4 


9S95 2329 


7-653 


4 7461 8137 


7-7°3 


4 5147 0737 


7-753 


4 


2945 2249 


7-604 


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 8698 


7-755 


4 


2859 4203 


7-606 


4 


9745 7715 


7-656 


4 7319 6416 


7-706 


4 5011 8354 


7-756 


4 


2816 5823 


7-607 


4 


9696 0506 


7-657 


4 7272 3456 


7-707 


4 4966 8461 


; 7-757 


4 


2773 7S71 


7-608 


4 


9646 3794 


7-658 


4 7225 0969 


7-708 


4 4921 9017 


7-758 


4 


2731 0347 


7-609 


4 


9596 757S 


7-659 


4 7177 8954 


7-709 


4 4877 0023 


17-759 


4 


2688 3251 


7-610 


4 


9547 1858 


7-660 


4 7130 7411 


7-710 


4 4832 1477 


7-760 


4 


2645 6581 


7-611 


4 


9497 6634 


7-661 


4 7083 6339 


7-711 


4 4787 3380 


[7-761 


4 


2603 0337 


7-612 


4 


9448 1905 


7-662 


4 7036 5738 


7-712 


4 4742 5730 


7-762 


4 


2560 4520 


7-613 


4 


9398 7670 


7-663 


4 69S9 5607 


7713 


4 4697 8528 


7-763 


4 


2517 9128 


7-614 


4 


9349 3929 


7-664 


4 6942 5946 


7-714 


4 4653 1773 


, 7-764 


4 


2475 4162 


7'6iS 


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 


7-617 


4 


9201 5666 


7-667 


4 6801 9779 


7-717 


4 4519 4185 


7-767 


4 


2348 1809 


7-618 


4 


9152 3896 


7-668 


4 6755 1993 


7-718 


4 4474 9213 


7-768 


4 


2305 8538 


7-619 


4 


9103 2618 


7-669 


4 6708 4675 


7-719 


4 4430 4686 


7-769 


4 


2263 5691 


7-620 


4 


9054 1831 


7-670 


4 6661 7824 


7-720 


4 4386 0604 


7-770 


4 


2221 3267 


7-621 


4 


9005 1534 


7-671 


4 6615 1439 


7-721 


4 4341 6965 


7-771 


4 


2179 1265 


7-622 


4 


8956 1728 


7-672 


4 6568 5521 


7-722 


4 4297 3770 


7-772 


4 


2136 9684 


7-623 


4 


8907 2411 


7-673 


4 6522 0068 


7-723 


4 4253 1017 


7-773 


4 


2094 8525 


7-624 


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 


7-626 


4 


8760 7392 


7-676 


4 6382 6499 


7-726 


4 4120 5414 


7-776 


4 


1968 7572 


7-627 


4 


8712 0028 


7-677 


4 6336 2904 


7-727 


4 4076 4429 


7-777 


4 


1926 8094 


7-628 


4 


S663 3152 


7-678 


4 6289 9773 


7-728 


4 4032 3885 


7-778 


4 


1884 9036 


7-629 


4 


8614 6762 


7-679 


4 6243 7105 


7-729 


4 3988 3781 


7-779 


4 


1843 0396 


7-630 


4 


8566 0858 


7-680 


4 6197 4899 


7-730 


4 3944 4117 


7-780 


4 


180I 2175 


7-631 


4 


8517 5440 


7-681 


4 6151 3155 


7-731 


4 3900 4893 


7-781 


4 


1759 4371 


7-632 


4 


8469 0507 


7-6S2 


4 6105 1872 


7732 


4 3856 6107 


7-782 


4 


I717 69S6 


7-633 


4 


8420 6059 


7-683 


4 6059 1051 


7-733 


4 3812 7760 


7-783 


4 


1676 0017 


7-634 


4 


8372 2095 


7-684 


4 6013 0690 


7-734. 


4 3768 9851 


7-784 


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 


I55I 161I 


7-637 


4 


8227 3103 


7-687 


4 5875 2366 


7-737 


4 3637 8750 


7787 


4 


1509 6307 


7-638 


4 


8179 1071 


7-688 


4 5829 3843 


7-738 


4 3594 2589 


7-788 


4 


1468 I418 


7-639 


4 


8130 9521 


7-689 


4 5783 5778 


7739 


4 3550 6864 


7-789 


4 


1426 6944 


7-640 


4 


8082 8452 


7-690 


4 5737 8172 


7-740 


4 3507 1575 


7-790 


4 


1385 2884 


7-641 


4 


8034 7864 


7-691 


4 5692 1022 


7-741 


4 3463 6721 


7-791 


4 


1343 9238 


7-642 


4 


7986 7756 1 


7-692 


4 5646 4329 


7-742 


4 3420 2302 


7-792 


4 


1302 6005 


7-643 


4 


7938 8128 


7-693 


4 5600 8093 


7-743 


4 3376 8316 


7-793 


4 


I261 3186 


7-644 


4 


7890 89S0 


7-694 


4 5555 2313 


7-744 


4 3333 4765 


7-794 


4 


1220 0779 


7-645 


4 


7S43 031° 


7-695 


4 5509 6988 


7-745 


4 3290 1647 


7-795 


4 


I178 8784 


7-646 


4 


7795 2119 


7-696 


4 5464 2119 


7-746 


4 3246 8961 


7-796 


4 


II37 7201 


7-647 


4 


7747 4406 


7-697 


4 5418 77°4 


7-747 


4 3203 6709 


7-797 


4 


1096 6030 


7-648 


4 


7699 717° 


7-698 


4 5373 3743 


7-748 


4 3160 4888 


7-798 


4 


1055 5269 


7-649 


4 


7652 0411 


7-699 


4 5328 0236 


7-749 


4 3117 3499 


7-799 


4 


IOI4 4919 



IPO 



MR F. W. NFAVMAN'S TABLE 



[7-Soo— 7-999] 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


7 800 
7-801 
7-802 
7-803 
7-804 


4 

4 
4 
4 
4 


0973 4979 
0932 5449 
0891 6328 
0S50 7616 
0809 9312 


7-850 
7-851 
7-852 
7-853 
7-854 


3 
3 
3 
3 
3 


8975 1968 
8936 24H 

8897 3243 
8S58 4464 
8819 6074 


\ 7-9°° 
7-901 
7-902 

7-903 
7-904 


3 
3 
3 
3 
3 


7074 3540 
7037 2982 
7000 2794 
6963 2976 
6926 3528 


7-950 
7-951 

7-952 
17-953 
i 7-954 


3 5266 2165 
3 5230 9679 
3 5195 7545 
3 5160 5763 

3 5125 4333 


7-805 
1 7-806 
7-807 
7-808 
7-809 


4 

: 

4 
4 


0769 1417 
0728 3929 
06S7 6849 
0647 0176 
0606 3909 


7-855 
7-856 

7-857 
7-858 

7-859 


3 

3 
3 
3 


S780 8072 
8742 0458 

8703 3231 
8664 6391 
8625 9938 


7-905 

: 7-906 

7-907 
7-908 

1 7-909 


3 
3 
3 
3 
3 


6889 4449 
6852 5739 
6815 7398 
6778 9424 
6742 1819 


7-955 
7-956 
7-957 
7-958 
7-959 


3 5090 325s 
3 5055 2527 
3 5020 2149 
3 4985 2122 
3 4950 2445 


7-810 
7-8H 
7-812 

7-813 
7-814 


4 
4 
4 
4 
4 


0565 8048 
0525 259- 
0484 7542 
0444 2S97 
0403 8656 


7-860 
7-861 
7-862 
7-863 
7-864 


3 
3 
3 
3 
3 


8587 3871 
8548 8190 
8510 2895 
8471 7984 
8433 3459 


f 7-910 

1 7-911 
, 7-912 

7-913 
7-914 


3 
3 
3 
3 
3 


6705 4580 
6668 7709 
6632 1205 
6595 5067 
6558 9295 


7-960 
7-961 
7-962 
7-963 
7-964 


3 4915 3117 
3 4880 4139 

3 4845 5509 
3 4S10 7228 

3 4775 9294 


7-815 
7S16 

7-817 
7-818 
7-819 


4 
4 
4 
4 
4 


0363 4820 

0323 1386 
0282 8357 

0242 5730 
0202 3505 


7-865 
7 -866 
7-867 
7-868 
7-869 


3 
3 
3 
3 
3 


8394 9317 
8356 5560 
8318 2186 
8279 9195 
8241 6588 


7-915 
7-916 
7-917 
7-918 
7-919 


3 
3 
3 
3 
3 


6522 3888 
6485 8847 
6449 4170 
6412 9858 
6376 5910 


7-965 
7-966 

7-967 
7-968 

7-969 


3 4741 1709 
3 4706 4471 
• 3 4671 7580 
3 4637 1036 
3 4602 4838 


7-820 
7-821 
7-822 
7-823 
7-824 


4 
4 

4 
4 
4 


0162 1683 
0122 0262 
0081 9242 
0041 8623 

0001 8405 


7-870 
7-871 
7-872 
7-873 
7-874 


3 
3 
3 
3 
3 


8203 4362 
8165 2519 
8127 1057 
808S 9977 
8050 9277 


7-920 
7-921 
7-922 

7-923 
7-924 


3 
3 
3 
3 
3 


6340 2326 
6303 9106 
6267 6248 

6231 3753 
6195 1620 


7-970 
7-971 
7-972 

7-973 
7-974 


3 4567 8986 
3 4533 3480 
3 4498 8319 
3 4464 3503 
3 4429 9032 


7-825 
7-826 
7-827 
7-828 
7-829 


3 
3 
3 
3 
3 


9961 8586 
9921 9167 
9882 0148 
9842 1527 
9802 3304 


7-875 
7-876 

7-877 
7-878 

7-879 


3 
3 
3 
3 
3 


8012 8958 

7974 9019 
7936 9460 
7899 0280 
7861 1479 


7-925 
7-926 
7-927 
7-928 
7-929 


3 
3 
3 
3 
3 


6158 9850 
6122 8441 
6086 7393 
6050 6706 
6014 6379 


7-975 
7-976 

7-977 
7-978 

7-979 


3 4395 4905 
3 4361 1122 
3 4326 76S2 
3 4292 4586 
3 4258 1833 


7-830 

7-831 
7-832 
7-833 
7-834 


3 
3 
3 
3 
3 


9762 5480 
9722 8053 
9683 1024 

9643 4391 
9603 8155 


7-880 
7-881 
7-882 
7-883 
7-884 


3 
3 

\ 

3 


7823 3057 

7785 5°i3 
7747 7347 
7710 0058 
7672 3146 


7-930 
7-931 
7-932 
7-933 
7-934 


3 
3 
3 
3 
3 


5978 6413 
5942 6806 
5906 7559 
5870 8671 
5835 0142 


7-980 
7-981 
7-982 

7-983 
7-984 


3 4223 9422 
3 4189 7354 
3 4155 5628 
3 4i2t 4243 
3 4087 3199 


7-835 
7-836 
7-837 
7-838 

7-839 


3 
3 
3 
3 
3 


9564 2315 
9524 6870 
9485 1821 
9445 7166 
9406 2906 


7-885 
7-886 
7-887 
7-888 
7-889 


3 
3 
3 
3 
3 


7634 6612 
7597 0453 
7559 4671 
7521 9264 
7484 4232 


7-935 
7-936 
7-937 
7-938 
7-939 


3 
3 
3 
3 
3 


5799 1971 
5763 4158 
5727 6702 
5691 9604 
5656 2863 


7-985 
7-986 

7-987 
7-988 
7-989 


3 4053 2496 
3 4019 2134 
3 3985 2112 
3 3951 2429 

3 3917 3087 


7-840 
7-841 
7-842 
7843 
7-844 


3 
3 
3 
3 
3 


9366 9040 
9327 5568 
9288 2489 
9248 9803 
9209 7509 


7-890 
7-891 
7-892 

7-893 
7-894 


3 
3 
3 
3 
3 


7446 9575 
7409 5293 
7372 1384 
7334 7850 
7297 4688 


7-940 

7-941 
7-942 

7-943 
7-944 


3 

3 
3 
3 
3 


5620 6478 
5585 0450 
5549 4777 
5513 9460 
5478 4498 


7-990 
7-991 
7-992 
7-993 
7-994 


3 3883 4083 
3 3849 5418 
3 3815 7092 
3 3781 9104 
3 3748 1454 


7845 
7-846 

7-847 
7-848 

7-849 


3 
3 
3 
3 
3 


9170 5608 
9131 4098 
9092 2980 
9053 2252 
9014 1915 


7-895 
7-896 

7-897 
7-898 
7-899 


3 
3 
3 
3 
3 


7260 1900 
7222 9484 
7185 7441 
7148 5769 
7111 4469 


7-945 
7-946 

7-947 
7-948 
7-949 


3 
3 
3 
3 
3 


5442 9891 
5407 5638 
5372 1740 
5336 8195 
5301 5003 


7-995 
7-996 

7-997 
7-998 
7-999 


3 3714 4141 
3 3680 7166 
3 3647 0527 
3 3613 4224 
3 3579 8258 



[S003 S199] 



OF THE DESCENDING EXPONENTIAL. 



191 



X 

8-000 


e-x 


X 


Q-X 


X 


(,-X 


X 

8-150 


e-x 


5360 


3 


3546 


2628 


8-050 


3 


1910 1922 


8-100 


3 0353 


9138 


2 8873 


8-OOI 


3 


3512 


7333 


8-051 


3 


1878 2980 


8-IOI 


3 0323 


5751 


8-151 


2 8844 


6769 


8-002 


3 


3479 


2373 


8-052 


3 


1846 4356 


8-102 


3 0293 


2666 


8-152 


2 8815 


8466 


8-003 


3 


3445 


7748 


8-053 


3 


1814 6051 


8-103 


3 0262 


9885 


8-153 


2 S787 


0451 


8'oo4 


3 


3412 


3457 


8-054 


3 


1782 8064 


8-104 


3 0232 


7407 


8-154 


2 8758 


2725 


8-005 


3 


3378 


9501 


8-055 


3 


1751 0395 


8-105 


3 0202 


5230 


8-155 


2 8729 


5286 


8-006 


3 


3345 


5878 


8-056 


3 


1719 3043 


8-106 


3 0172 


3356 


8-156 


2 8700 


8134 


8-007 


3 


3312 


2589 


8-057 


3 


1687 6009 


8-107 


3 0142 


1783 


8-157 


2 8672 


1270 


8 -008 


3 


3278 


9633 


8-058 


3 


1655 9291 


8-108 


3 OH2 


0512 


8-158 


2 8643 4692 ] 


8-009 


3 


3245 


7010 


8-059 


3 


1624 2890 


8-109 


3 0081 


9542 j 


8-159 


2 8614 


8400 


8-010 


3 


3212 


4719 


8-060 


3 


1592 6805 


8-IIO 


3 0051 


8873 : 


8-i6o 


2 8586 


2395 


8-OII 


3 


3179 


2760 


8-o6i 


3 


1561 1036 


8-111 


3 0021 


8505 


8-161 


2 8557 


6675 


8-012 


3 


3146 


1133 


8-062 


3 


1529 5583 


8-112 


2 9991 


8436 


8-162 


2 8529 


1241 


8013 


3 


3112 


9838 


8-063 


3 


1498 0445 


8-113 


2 9961 


8668 


8-163 


2 8500 


6093 


8014 


3 


3079 


8874 


8-064 


3 


1466 5622 


8-114 


2 9931 


9199 


8-164 


2 8472 


1229 


8-015 


3 


3046 


8240 


8-065 


3 


1435 i"4 


8-115 


2 9902 


0029 ! 


8-165 


2 8443 


6650 


8016 


3 


3013 


7937 


8-066 


3 


1403 6920 


8-n6 


2 9872 


1158 


8-166 


2 8415 


2356 


8-017 


3 


2980 


7964 


8-067 


3 


1372 3040 


8-117 


2 9842 


2587 


8-167 


2 8386 


8345 


8-oi8 


3 


2947 


8321 


8-068 


3 


1340 9474 


8-118 


2 9812 


4313 


8-168 


2 8358 


4619 


8019 


3 


2914 


9007 


8069 


3 


1309 6221 


8-119 


2 9782 


6338 


8-169 


2 8330 


1176 


8-020 


3 


2S82 


0023 


8-070 


3 


1278 3281 


8-120 


2 9752 


8660 


8-170 


2 8301 


8016 


8 -02 1 


3 


2849 


1367 


8-071 


3 


1247 0654 


8-121 


2 9723 


1280 


8-171 


2 8273 


5140 


8-022 


3 


2816 


3040 


8-072 


3 


1 2 15 8340 


8-122 


2 9693 


4198 


8-172 


2 8245 


2546 


8023 


3 


2783 


5041 


8-073 


3 


1184 6337 


8-123 


2 9663 


7412 


8-173 


2 8217 


0235 


8-024 


3 


2750 


7370 


8-074 


3 


1153 4647 


8-124 


2 9634 


0923 


8-174 


2 818S 


8205 


8-025 


3 


2718 


0026 


8-075 


3 


1122 3268 


8-125 


2 9604 


4730 


8-I7S 


2 8160 


6458 


8-026 


3 


2685 


3010 


8-076 


3 


1091 2200 


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 


8-177 


2 8104 


3808 


8-028 


3 


2619 


9957 


8-078 


3 


1029 0997 


8-128 


2 9515 


7927 


8-178 


2 8076 


2905 


8029 


3 


2587 


3920 


8-079 


3 


0998 0861 


8-129 


2 9486 


2916 


8-179 


2 8048 


22S2 


8030 


3 


2554 


8209 


8-080 


3 


0967 1035 


8-130 


2 9456 


8201 


8-180 


2 S020 


1940 


8031 


3 


2522 


2823 


8-081 


3 


0936 1519 


8-131 


2 9427 


3780 


8-i8i 


2 7992 


1878 


8-032 


3 


2489 7763 


8-082 


3 


0905 2312 


8-132 


2 9397 


9653 


8-182 


2 7964 


2096 


8-033 


3 


2457 


3028 


8-083 


3 


0874 3414 


8-133 


2 9368 


5820 


8-183 


2 7936 


2594 


8-034 


3 


2424 


8617 


8-084 


3 


0843 4825 


8-134 


2 9339 


2281 


8-184 


2 790S 


3371 


8-03S 


3 


2392 


4530 


8-085 


3 


0812 6545 


8-I3S 


2 9309 


9036 i 


8-185 


2 7SS0 


4427 


8-036 


3 


2360 


0768 


8-086 


3 


0781 8572 


8-136 


2 9280 


6083 


8-186 


2 7852 


5762 


8-037 


3 


2327 


7329 


8-087 


3 


0751 0907 


8-137 


2 9251 


3423 


8-187 


2 -S24 


7375 


8-038 


3 


2295 


4213 


8-088 


3 


0720 3550 


8-138 


2 9222 


1056 


8-188 


2 7796 


9267 


8-039 


3 


2263 


1420 


8-089 


3 


0689 6500 


8-139 


2 9192 


8981 


8-189 


2 7769 


1437 


8-040 


3 


2230 


8950 


8-090 


3 


0658 9757 


8-140 


2 9163 


7198 


8-190 


2 7741 


3884 


8-041 


3 


2198 


6802 


8-091 


3 


0628 3321 


8-141 


2 9134 


5707 


8-191 


2 7713 


6609 


8-042 


3 


2166 


4976 


8-092 


3 


0597 7190 


8-142 


2 9105 


45°7 


8-192 


2 7685 


9611 


8-043 


3 


2134 


3472 


8-093 


3 


0567 1366 


8-143 


2 9076 3598 


8-193 


2 765S 


2890 


8-044 


3 


2102 


2289 


8-094 


3 


0536 5848 


8-144 


2 9047 


2979 . 


8-194 


2 7630 


6445 


8-045 


3 


2070 


1427 


8-095 


3 


0506 0634 


8-145 


2 9018 


2652 


8-195 


2 7603 


0277 


8-046 


3 


2038 


0886 


8096 


3 


0475 5726 


8-146 


2 8989 


2614 


8-196 


2- 7575 


4384 


8-047 


3 


2006 


0666 


8-097 


3 


0445 1 123 


8-147 


2 8960 


2866 


8-197 


2 7547 


876S 


8-048 


3 


1974 


0765 


8-098 


3 


0414 6824 


8-148 


2 8931 


3408 


8-198 


2 7520 


3427 


8-049 


3 


1942 


1184 


8099 


3 


0384 2829 


8-149 


2 8902 


4239 


8-199 


2 7492 


8361 



102 



l^IR F. W. NEWMAN'S TABLE 



[S-200— 8399] 



X 

8-200 

8-201 

S-202 

■ 8203 

8-204 

j 8-205 
j 8206 

8-207 
' 8-208 

8-209 

8-210 
8-2 1 1 
8-212 

j 8-213 

,8-214 

! 8-215 

I 8-216 

8-217 

8-2i8 

8-219 

8-220 

I 8-221 

I 8-222 

8-223 

8-224 



25 
26 



8-22 
8-22 
8-227 
8-2 

8-2 



18 
29 



8 
8 
8 
8 



8-230 
•231 
•232 

•234 

8-235 
8236 

8237 
8238 
8-239 
8-240 
8-241 I 
8-242 I 

8-243 
8-244 

8-245 
8246 

8-247 
8-248 
8-249 . 



7465 
7437 
7410 

7383 

7355 

73^8 
7301 
7273 
7246 
7219 

7192 
7164 

7137 
7110 

7083 

7056 
7029 

7002 

6975 
694S 

6921 
6S94 
6867 
6S40 
6814 

6787 
6760 

6733 
6706 
6680 

6653 
6626 
6600 
6573 
6547 

6520 
6494 
6467 
6441 
6414 

6388 
6362 

633s 
6309 

6283 

6256 
6230 
6204 
6178 
6151 





X 


3570 


8-250 


9°54 


8-251 


4S12 


8-252 


0S44 


8-253 


7150 


8-254 


3729 


8-255 


05S2 


8-256 


7708 


8-257 


5i°7 


8-258 


2778 


8-259 


0721 


8-260 


8936 


8-261 


7423 


8-262 


6182 


8-263 


5211 


8-264 


45" 


8-265 


40S2 


8-266 


3923 


8-267 


4034 


8-268 


4415 


8-269 


5065 


8-270 


5984 


8271 


7173 


8-272 


8630 


8-273 


0356 


8-274 


2349 


8-275 


4611 


8-276 


7140 


8-277 


9936 


8-278 


3000 


8-279 


633° 


8-2S0 


9927 


8-281 


3790 


8-282 


7919 


8-283 


?3i4 


8-284 


6975 


8-285 


1900 


8-286 


7091 


8-287 


2546 


8-288 


8266 


8-289 


4249 


8-290 


0497 


8-291 


7008 


8-292 


3783 


8-293 


0821 


8294 


8121 


8-295 


5684 


8-296 


3S'o 


8-297 


1597 


8-298 


9946 


8-299 



2 6125 S557 

2 6099 7429 

2 6073 6562 

2 6047 5956 

2 6021 5610 



X 



5995 
5969 
5943 
5917 
5891 

5865 
5840 
5814 
5788 
5762 

5736 
5711 
5685 
5659 
5634 

5608 
5582 
5557 
5531 
5506 

5480 
5455 
5429 
5404 
5379 



2 5353 

2 5328 

2 53°3 

2 5277 

2 5252 

2 5227 
2 5202 

2 S'76 

2 5151 
2 5126 

2 5i°i 
2 5076 

5051 
5026 
5001 

4976 

495' 
4926 
4901 
4876 



5525 
5699 
6133 
6S27 
7780 

8991 
0461 
2190 

4177 
6422 

8924 
1684 
4701 
7974 

i5°5 

5291 
9334 
3632 
8187 
2996 

806 1 
3380 

8954 
4782 
0864 

7200 

3790 
06-52 
7728 
5077 
2678 
0531 
8637 
6994 

5603 

4463 
3574 

2935 
2548 
2410 

2523 
2885 

3497 
4358 
5468 



8-300 
8-301 

'I ^■^°- 

; 8-303 
; S-304 

i 8-305 
8-306 

8-307 
8-308 
8309 

8-310 
8-311 
8-312 

8-313 
8-314 

8-315 
8-316 

8317 
8-318 

8-319 

8-320 
8-321 
8322 
8-323 
8-324 

8-325 
8-326 
8-327 
8-328 



8-330 
8-331 
8-332 
8-333 
8-334 

8-335 
8-336 
8-337 
8-338 
8-339 
8-340 
8-341 
8-342 
8-343 
8-344 

8-345 
8-346 

8-347 
8-348 

8-349 



e- 



4851 
4S26 
4802 
4777 
4752 

4727 
4703 
4678 

4653 
4629 

4604 
4579 

4555 
4530 
4506 

4481 

4457 
4432 
4408 

43S3 

4359 
4335 
4310 
4286 
4262 

423S 
4213 
4189 

4165 
4141 



6827 

8434 
0290 

2394 

4745 

7344 
0190 

3284 
6624 
0210 

4043 
8122 

2447 
7017 

1S33 

6S93 
2199 
7749 
3543 
9582 

5864 
2390 

9159 
6172 

3427 
0925 
8665 
6647 
4872 
3337 



2 4117 2045 

2 4093 0993 

2 4069 01S3 

2 4044 9613 

2 4020 9283 

2 3996 9194 

2 3972 9345 

2 3948 9735 

2 3925 0365 

2 3901 1235 

2 3877 2343 

2 3853 3690 

2 3829 5275 

2 3805 7099 

2 3781 9161 

2 3758 1461 

2 3734 3998 

2 3710 6773 

2 3686 9784 

2 3663 3033 



X 



8-350 

8-351 
8-352 
8-353 
8-354 

8-355 
8-356 
8-357 
8-358 
8-359 
8-360 
8-361 
S-^62 

8-363' 
8-364 

8-365 
8-366 

8-367 
8-368 
8-369 

8-370 

8-371 
8-372 

8-373 
8-374 

8-375 
8-376 
8-377 
8-378 
8-379 
8-380 
8-i8i 
8-382 

8-383 
8-384 

8-385 
8-386 

8-387 
8-388 
8-389 

8-390 

8-391 
8-392 

8-393 
S-394 

8-395 
8-396 
8-397 
8-398 
8-399 



2 3639 
2 3616 

2 3592 
2 3568 
2 3545 

2 3521 
2 3498 
2 3474 
2 3451 

2 3427 

2 3404 
2 3381 
2 3357 
2 3334 
2 3311 

2 3287 

2 3264 

2 3241 

2 3217 

2 3194 



3171 

3148 

3125 
3102 

3079 

3055 
3032 
3009 
2986 
2963 

2940 
2918 

2895 
2872 
2849 

2826 
2803 
2780 
2758 

2735 



2 2712 

2 2690 

2 2667 

2 2644 

2 2622 

2 2599 

2 2576 

2 2554 

2 2531 

2 2509 



6518 

0240 
4198 

8391 
2821 

7486 
2386 

7521 
2891 

8495 

4334 
0406 

6713 
3253 
0026 

7033 
4272 

1744 
9448 
7385 

5554 
3954 
2586 
1449 
0543 

9867 
9423 
9209 

9224 
9470 

9945 
0650 

1584 
2747 
4138 

5758 
7607 
9683 
1987 

4519 

7278 
0264 
3478 
6917 

0584 
4476 
8595 
2939 
7509 
2304 



[8-400— 8-599] 



OF THE DESCENDING EXPONENTIAL. 



193 



X 


Q-X 


0? 


Q~X 


X 


Q-X 


X 


Q-X 


8-400 


2 


2486 7324 


8-450 


2 1390 0415 


8-500 


2 0346 8369 


8-550 


I 9354 5099 


8-401 


2 


2464 2569 


8-451 


2 1368 6622 


8-501 


2 0326 5002 


8-551 


I 9335 1651 


8-402 


2 


2441 8039 


8-452 


2 1347 3042 


8-502 


2 0306 1839 


8-552 


I 9315 8396 


8-403 


2 


2419 3733 


8-453 


2 1325 9676 


8-503 


2 0285 8879 


8-553 


I 9296 5334 


8-404 


2 


2396 9651 


8-454 


2 1304 6523 


8-504 


2 0265 612I 


8-554 


I 9277 2465 


8-405 


'J 


2374 5794 


8-455 


2 1283 3583 


8-505 


2 0245 3566 


8-555 


I 9257 9789 


8-406 


'T 


2352 2160 


8-456 


2 1262 0855 


8-506 


2 0225 I214 


8-556 


I 9238 7306 


8-407 


2 


2329 8749 


8-457 


2 1240 8341 


8-507 


2 0204 9064 


8-557 


I 9219 5015 


8-408 


2 


2307 5562 


8-458 


2 I2I9 6039 


8-508 


2 0184 7116 


8-558 


I 9200 2916 


8-409 


2 


2285 2598 


8-459 


2 I 198 3949 


8-509 


2 0164 5369 


8-559 


I 9181 1009 


8-410 


2 


2262 9857 


8-460 


2 1177 2071 


8-510 


2 0144 3825 


8-560 


I 9161 9294 


8-4II 


2 


2240 7338 


8-461 


2 I 156 0404 


8-511 


2 0124 2482 


8-561 


I 9142 7770 


8-412 


2 


2218 5042 


8-462 


2 II34 8950 


8-512 


2 0104 1340 


8-562 


I 9123 6438 


8-413 


2 


2196 2968 


8-463 


2 III3 7706 


S-5I3 


2 0084 0399 


8-563 


I 9104 5297 


8-414 


2 


2174 1116 


8-464 


2 1092 6674 


8-514 


2 0063 9659 


8-564 


I 9085 4347 


8-4IS 


2 


2151 9486 


8-465 


2 I07I 5853 


8-515 


2 0043 9120 


8-565 


I 9066 3588 


8-416 


2 


2129 8077 


8-466 


2 1050 5243 


8-516 


2 0023 8781 


8-566 


I 9047 3020 


8-417 


2 


2107 6890 


8-467 


2 1029 4843 


8-517 


2 0003 8642 


8-567 


I 9028 2642 


8-418 


2 


2085 5923 


8-468 


2 1008 4653 


8-518 


I 9983 8703 


8-568 


I 9009 2455 


8-419 


2 


2063 5178 


8-469 


2 0987 4673 


8-519 


I 9963 8965 


8-569 


I 8990 2457 


8-420 


2 


2041 4653 


8-470 


2 0966 4903 


8-520 


I 9943 9425 


8-570 


I 8971 2650 


8-421 


2 


2019 4348 


8-471 


2 0945 5343 


8-521 


I 9924 0086 


8-571 


I 8952 3032 


8-422 


2 


1997 4264 


8-472 


2 0924 5993 


8-522 


I 9904 0945 


8-572 


I 8933 3604 


8-423 


2 


1975 4400 


8-473 


2 0903 6851 


8-523 


I 9884 2004 


8-573 


I 8914 4365 


8-424 


2 


1953 4755 


8-474 


2 0882 7919 


8-524 


I 9864 3261 


8-574 


I 8895 5315 


8-425 


2 


1931 5330 


8-475 


2 0861 9195 


8-525 


I 9844 4717 


8-575 


I 8876 6454 


8-426 


2 


1909 6124 


8-476 


2 0841 0680 


8-526 


I 9824 6372 


8-576 


I 8857 7782 


8-427 


2 


1887 7138 


8-477 


2 0820 2374 


8-527 


I 9804 8224 


8-577 


I 8838 9298 


8-428 


2 


1865 8370 


8-478 


2 0799 4276 


8-528 


I 9785 0275 


8-578 


I 8820 1003 


8-429 


2 


1843 9821 


8-479 


2 0778 6385 


8-529 


I 9765 2524 


8-579 


I 8801 2896 


8-430 


2 


1822 1490 


8-480 


2 0757 8703 


8-530 


I 9745 4970 


8-580 


I 8782 4977 


8-431 


2 


iSoo 3378 


8-481 


2 0737 1228 


8-531 


I 9725 7614 


8-581 


I 8763 7246 


8-432 


2 


1778 5483 


8-482 


2 0716 3960 


8-532 


I 9706 045s 


8-582 


I 8744 9703 


8-433 


2 


1756 7807 


8-483 


2 0695 6900 


8-533 


I 9686 3493 


8-583 


1 8726 2347 


8-434 


2 


1735 °348 


8-484 


2 0675 0046 


8-534 


I 9666 6728 


8-584 


I 8707 5178 


8-435 


2 


1713 3106 


8-485 


2 0654 3400 


8-535 


I 9647 0159 


8-585 


I 8688 8196 


8-436 


2 


1691 6081 


8-486 


2 0633 6960 


8-536 


I 9627 3787 


8-586 


I 8670 1402 


8-437 


2 


1669 9274 


8-487 


2 0613 0726 


8-537 


I 9607 7612 


8-587 


I 8651 4793 


8-438 


2 


1648 2683 


8-488 


2 0592 4698 


8-538 


I 9588 1632 


8-588 


I 8632 8372 


8-439 


2 


1626 6308 


8-489 


2 0571 8876 


8-539 


I 9568 5848 


8-589 


I 8614 2137 


8-440 


2 


1605 0150 


8-490 


2 0551 3260 


8-540 


I 9549 0260 


8-590 


I 8595 6088 


8-441 


2 


1583 4208 


8-491 


2 0530 7850 


8-541 


I 9529 4868 


8-591 


I 8577 0224 


8-442 


2 


1561 8482 


8-492 


2 0510 2644 


8-542 


I 9509 9670 


8-592 


1 8558 4547 


8-443 


2 


1540 2971 


8-493 


2 0489 7644 


8-543 


I 9490 4668 


8-593 


I 8539 9055 


8-444 


2 


1518 7676 


8-494 


2 0469 2849 


8-544 


I 9470 9861 


8-594 


I 8521 3749 


?-445 


2 


1497 2595 


8-495 


2 0448 8258 


8-545 


I 9451 5248 


8-595 


I 8502 8628 


8-446 


2 


1475 7730 


8-496 


2 0428 3S72 


8-546 


I 9432 0830 


8-596 


I 8484 3692 


8-447 


2 


1454 3080 


I 8-497 


2 0407 9691 


8-547 


I 9412 6607 


8-597 


I 8465 8940 


8-448 


2 


1432 8644 


8-498 


2 0387 5713 


8-548 


I 9393 2577 


8-598 


I 8447 4374 


8-449 


2 


1411 4423 


8-499 


2 0367 1939 


1 8-549 


I 9373 8741 


8-599 


I 8428 9991 



Vol. XIIT. Part HI. 



26 



194 



MR F. W. NEWMAN'S TAHLE 



[8-600—8-799] 



X 


e-* 


! X 


Q-X 


1 
X 


(,~X 


X 

8-750 


Q-X 


s-600 


I S410 5794 


8-650 


I 7512 


6S48 


8-700 


I 665S 5811 


I 5S46 1325 


S-6oi 


I S392 17S0 


8-651 


I 7495 


1809 


8-701 


I 6641 9308 


8-751 


I 5830 2943 


8-6o2 


I S373 795° 


8-652 


1 7477 


6944 


18-702 


I 6625 2972 


8-752 


I 5S14 4719 


8603 


I S355 4304 


8-653 


I 7460 


225s 


1 8-703 


I 660S 6S02 


8-753 


I 5798 6653 


8604 


I 8337 0841 


8-654 


I 7442 


7740 


8-704 


I 6592 0799 


8-754 


I 5782 8746 


8-605 


I 8318 7562 


8-655 


I 74^5 


3399 


8-705 


I 6575 4961 


8-755 


I 5767 0996 


8-6o6 


I S300 4466 


8-656 


1 7407 


9233 


8-706 


I 655S 9289 


8-756 


I 5751 3404 


8-607 


I 8282 1553 


8-657 


I 7390 


5241 


8-707 


I 6542 3782 


8-757 


I 5735 5969 


8-6oS 


I 8263 8S23 


8-658 


I 7373 


1422 


8-708 


I 6525 8441 


8-758 


I 5719 8692 


8-609 


I 8245 6275 


8-659 


I 7355 


7778 


8-709 


I 6509 3265 


8-759 


I 5704 1571 


8-610 


I 8227 3910 


8-660 


I 7338 


4307 


8-710 


I 6492 8254 


8-760 


I 56SS 4608 


8-6 1 1 


1 8209 1728 


8-66i 


I 7321 


1009 


8-711 


I 6476 3409 


8-761 


I 5672 7S02 


8612 


1 8190 9727 


8-662 


1 7303 


7885 


8-712 


I 6459 8727 


8-762 


I 5657 "53 


8-613 


I 8172 790S 


8-663 


I 72S6 


4933 


8-713 


I 6443 4211 


8-763 


I 5641 4660 


8-614 


I S154 6271 


8-664 


I 7269 


2155 


8-714 


I 6426 9859 


8-764 


I 5625 8323 


8-615 


I 8136 4815 


8-665 


I 7251 


9549 


8-71S 


I 6410 5671 


8-765 


I 5610 2143 


8-6i6 


I 8118 3541 


8-666 


I 7234 


7116 


8716 


I 6394 1648 


8-766 


I 5594 6119 


8-617 


I 8100 2448 


8-667 


I 7217 


4855 


8717 


I 6377 7788 


8-767 


I 5579 0251 


8-6i8 


I 8082 1536 


8-668 


I 7200 


2766 


8-718 


I 6361 4092 


8-768 


I 5563 4539 


8-619 


I 8064 0805 


8-669 


I 7183 


0849 


8-719 


I 6345 0560 


8-769 


I 5547 8982 


8-620 


I 8046 0255 


8-670 


I 7165 


9104 


8-720 


1 6328 7191 


8-770 


I 5532 3581 


8-621 


I 8027 9885 


S-671 


I 714S 


7531 


8-721 


I 6312 3985 


8-771 


I 5516 8335 


8-622 


1 8009 9695 


8-672 


I 7131 


6129 


8-722 


I 6296 0943 


8-772 


I 5S°i 3244 


8-623 


I 7991 9685 


8-673 


I 7114 


4898 


8-723 


I 6279 8063 


8-773 


I 54S5 8308 


8-624 


I 7973 9855 


8-674 


I 7097 


3839 


8-724 


1 6263 5347 


8-774 


1 5470 3527 


8-625 


I 7956 0205 


8-675 


I 70S0 


2951 


8-725 


I 6247 2793 


8-775 


I 5454 8901 


8-626 


I 7938 0735 


8-676 


I 7063 


2233 


8-726 


I 6231 0401 


8-776 


I 5439 4429 


8-627 


1 7920 1444 


8-677 


I 7046 


16S6 


8-727 


I 6214 8172 


8-777 


I 5424 0112 


8-628 


I 7902 2332 


8-678 


I 7029 


1310 


8-728 


I 6198 6104 


8-77S 


I 5408 5949 


8629 


I 7884 3399 


8-679 


I 7012 


1 104 


8-729 


I 6182 4199 


8-779 


I 5393 1940 


8-630 


I 7866 4645 


8-680 


I 6995 


1067 


8-730 


1 6166 2456 


8-780 


I 5377 8085 


8-631 


I 7848 6070 


8-68i 


I 6978 


1201 


8-731 


I 6150 0874 


8-781 


I 5362 4384 


8-632 


I 7830 7673 


8-682 


I 6961 


1505 


8-732 


I 6133 9454 


8-782 


I 5347 0836 


8-633. 


I 7S12 9454 


8-683 


I 6944 


1978 


8-733 


I 6117 8195 


8-783 


I 5331 7442 


8-634 


• 7795 1414 


8-684 


I 6927 


2621 


8-734 


I 6101 7098 


8-784 


I 5316 4201 


8-635 


' 7777 3551 


S-685 


I 6910 


3433 


8-735 


I 6085 6161 


8-785- 


I 5301 1114 


8-636 


I 7759 5867 


8-686 


I 6893 


4414 


8-736 


I 6069 5385 


8-786 


I 5285 8179 


8-637 


I 7741 8360 


8-687 


I 6S76 


5564 


8-737 


1 6053 4770 


8-787 


I 5270 5397 


8-638 


1 7724 1030 


8-688 


I 6859 6883 


8-738 


I 6037 4315 


8-788 


I 5255 2768 


8639 


I 7706 3878 


8-689 


I 6842 


8370 


8-739 


I 6021 4022 


8-789 


I 5240 0292 


8-640 


I 7688 6902 


8-690 


I 6826 


0026 


8-740 


I 6005 3888 


8-790 


I 5224 7968 


8-641 


I 7671 0104 


8-691 


I 6809 


1850 


8-741 


I 5989 3914 


8-791 


I 5209 5796 


8-642 


I 7653 3482 


8-692 


I 6792 


3842 


8-742 


I 5973 4100 


8-792 


I 5194 3776 


8-643 


I 7635 7037 


8-693 


I 6775 


6002 


8-743 


I 5957 4446 


8-793 


I 5179 1908 


8-644 


I 7618 0768 


8-694 


I 6758 


833° 


8-744 


I 5941 4951 


8-794 


I 5164 0192 


8-645 


I 7600 4675 


8-695 


1 6742 


0826 


8-745 


I 5925 5616 


8-795 


I 5148 8628 


8646 


1 7582 8758 


8696 


I 6725 


3489 


8-746 


1 5909 6440 


8-796 


I 5133 7215 


8647 


I 7565 3017 


8-697 


I 6708 6319 


8-747 


I 5893 7423 


8-797 


I 5118 5953 


8-648 


1 7547 7452 


8-698 


1 6691 


9316 


8748 


I 5877 8565 


8-798 


I 5103 4843 


|8-649 


I 7530 2063 


i 8-699 


1 6675 


2480 


8-749 


I 5861 9866 


8-799 


I 5088 3883 



[8 Soo— 8-999] 



OF THE DESCENDING EXPONENTIAL. 



195 



X 


C-x 


X 


(,-X 


X 
8-goo 


e- 


X 


e-x 


8-800 


I 5073 307s 


8-850 


I 4338 1736 


I 3638 8926 


8-950 


I 2973 7160 


8-801 


I 5058 2417 


8-851 


I 4323 8426 


8-901 


I 3625 2606 


8-951 


I 2960 7488 


8-8o2 


I 5043 1910 


8-852 


I 4309 5259 


8-902 


I 3611 6421 


8-952 


I 2947 7945 


8-803 


I 5028 1553 


8-853 


I 4295 2236 


8-903 


I 3598 0373 


8-953 


I 2934 8532 


8-804 


I 5°i3 1347 


8-854 


I 4280 9355 


8-904 


I 3584 4460 


8-954 


I 2921 9248 


8-805 


I 4998 1291 


8-855 


I 4266 6617 


8-905 


I 3570 8684 


8-955 


I 2909 0093 


8-806 


I 4983 13S4 


8-856 


I 4252 4021 


8-906 


I 3557 3°43 


8-956 


I 2896 1068 


8-807 


I 4968 1628 


8-857 


I 4238 1569 


8-907 


I 3543 7538 


8-957 


I 2883 2171 


8-8o8 


I 4953 2021 


8-858 


I 4223 9258 


8-908 


I 3530 2168 


8-958 


I 2870 3403 


8-809 


I 4938 2564 


8-859 


I 4209 7090 


S-gog 


I 3516 6933 


8-959 


I 2857 4764 


8-8io 


I 4923 3256 


8-860 


I 4195 5064 


8-910 


I 3503 1834 


8-960 


I 2844 6254 


8-811 


I 4908 4097 


8-861 


I 4181 3180 


8-911 


I 34S9 6870 


8-961 


I 2831 7872 


8-812 


I 4893 5088 


8-862 


I 4167 1437 


8-912 


I 3476 2040 


8-962 


I 2818 9618 


8-813 


I 4878 6227 


8-863 


I 4152 9837 


8-913 


I 3462 7345 


8-963 


I 2806 1492 


8-814 


I 4863 7515 


8-864 


I 4138 8378 


8-gi4 


I 3449 2785 


8-964 


I 2793 3495 


8-815 


I 4848 8952 


8-865 


I 4124 7060 


8-915 


I 3435 8360 


8-965 


I 2780 5625 


8-816 


I 4834 0537 


8-866 


I 4110 58S4 


8-916 


I 3422 4069 


8-966 


I 2767 7884 


8-817 


I 4819 2271 


8-867 


I 4096 4848 


8-917 


I 3408 9912 


8-967 


I 2755 0270 


8-8i8 


I 4804 4153 


8-868 


I 4082 3954 


8-918 


I 3395 5889 


8-968 


I 2742 2783 


8-819 


I 47S9 6182 


8-869 


1 4068 3200 


8-919 


I 3382 2000 


8-969 


I 2729 5424 


8-820 


I 4774 8360 


8-870 


I 4054 2588 


8-920 


I 3368 8245 


8-970 


I 2716 8192 


8-821 


I 4760 0686 


8-871 


I 4040 2115 


8-921 


I 3355 4623 


8-971 


I 2704 1087 


8-822 


I 4745 3159 


8-872 


I 4026 1783 


8-922 


I 3342 1135 


8-972 


1 2691 4110 


8-823 


I 4730 5779 


8-873 


I 4012 1592 


8-923 


I 3328 7781 


8-973 


I 2678 7259 


8-824 


I 4715 8547 


8-874 


I 3998 1540 


8-924 


I 3315 4560 


8-974 


I 2666 0535 


8-825 


I 4701 1462 


8-875 


I 3984 1629 


8-925 


I 3302 1472 


8-975 


I 2653 3938 


8-826 


I 4686 4524 


8-876 


I 3970 1857 


8-926 


I 32S8 8517 


8-976 


I 2640 7467 


8-827 


I 4671 7733 


8-877 


I 3956 2225 


8-g27 


I 3275 5695 


8-977 


■- I 2628 1123 


8-828 


I 4657 1089 


8-878 


I 3942 2732 


8-928 


I 3262 3005 


8-978 


I 2615 4905 


8-829 


I 4642 4591 


8-879 


I 3928 3379 


8-929 


I 3249 0449 


S-979 


I 2602 8813 


8-830 


I 4627 8239 


8-880 


I 3914 4165 


8-930 


I 3235 8024 


8-980 


I 2590 2847 


8-831 


I 4613 2034 


8-881 


I 3900 5091 


8-931 


I 3222 5733 


8-g8i 


I 2577 7008 


8-832 


I 4598 5975 


8-882 


I 3S86 6155 


8-932 


I 3209 3573 


8-g82 


I 2565 1293 


8-833 


I 4584 0062 


8-883 


I 3872 7359 


8-933 


I 3196 1545 


8-983 


1 2552 5705 


8-834 


I 4569 4295 


8-884 


I 3858 8700 


8-934 


I 31S2 9650 


8-984 


I 2540 0242 


8-835 


I 4554 8674 


8-885 


I 3845 0181 


8-935 


I 3169 7886 


8-985 


I 2527 4904 


8-836 


I 4540 3198 


8-886 


I 3S31 1800 


8-936 


I 3156 6254 


8-986 


I 2514 9692 


8-837 


I 4525 7867 


8-887 


I 3817 3557 


8-937 


I 3143 4754 


8-987 


I 2502 4605 


8-838 


I 4511 2682 


8-888 


I 3803 5453 


8-938 


I 3130 3384 


8-988 


I 2489 9643 


8-839 


I 4496 7642 


8-889 


I 3789 7486 


8-939 


I 3117 2147 


8-989 


I 2477 4806 


8-840 


I 4482 2747 


8890 


I 3775 9658 


8-940 


I 3104 1040 


8-990 


I 2465 0093 


8-841 


I 4467 7996 


8-891 


I 3762 1967 


8-941 


I 3091 0065 


8-991 


I 2452 5505 


8-842 


I 4453 3391 


8-892 


I 3748 4414 


8-942 


I 3077 9220 


8-992 


I 2440 1042 


8-843 


I 4438 8929 


8-893 


I 3734 6998 


8943 


I 3064 8506 


8-993 


I 2427 6703 


8-844 


I 4424 4613 


8-894 


I 3720 9720 


8-944 


I 3051 7923 


8-994 


I 2415 2489 


8-845 


I 4410 0440 


8-895 


I 3707 2579 


8-945 


1 3038 7470 


8-995 


r 2402 8398 


8-846 


I 4395 6412 


8-896 


1 3693 5575 


8-g46 


I 3025 7148 


8-996 


I 2390 4432 


8-847 


I 4381 2527 


8-897 


I 3679 8708 


8-947 


I 3012 6956 


8-997 


I 2378 0589 


8-848 


I 4366 8787 


8-898 


I 3666 1977 


S-948 


I 2999 6894 


8-gg8 


I 2365 6871 


8-849 


I 4352 519° 


8-899 


I 3652 5384 


8-g49 


I 2986 6962 


8-999 


I 2353 3276 



2G— 2 



196 



MR F. W. NEWMAN'S TABLE 



[9' 



-9-199] 



1 
X 


Q-X 


i ^ 


e-^ 


X 


p-X 


X 


(,-X 


9-000 


I 2340 9S04 


9-050 


I 1739 1037 


9-100 


I 1166 580S 


9-150 


1 0621 9803 


9-001 


I 232S 6456 


9-051 


I 1727 3704 


9-101 


1 1155 4198 


9-151 


I 06 II 3636 


9-002 


I 2316 3231 


9-052 


I 1715 6489 


9-102 


I 1 144 2700 


9-152 


I 0600 7575 


9-003 


I 2304 0129 


9053 


I 1703 9391 


9-103 


I 1133 1313 


9-153 


1 0590 1621 


9004 


I 2291 7I5I 


j 9-054 


I 1692 2410 


9-104 


I 1122 0037 


9-154 


1 0579 5772 


9005 


I 2279 4295 


9-055 


I 1680 5546 


9-105 


I mo 8873 


9-155 


I 0569 0029 


9-006 


I 2267 1562 


9-056 


I 1668 8799 


9-106 


I 1099 7820 


9156 


1 0558 4392 


9-007 


I 2254 8952 


9057 


I 1657 2169 


9-107 


I 1088 6S77 


9-157 


1 0547 8860 


9008 


I 2242 6464 


9-058 


I 1645 5655 


9-108 


I 1077 6046 


9-158 


1 °S37 3434 


9-009 


I 2230 4099 


9-059 


I 1633 9258 


9-109 


I 1066 5325 


9-159 


1 0526 8113 


9-010 


I 221S 1856 


9-060 


I 1622 2976 


9-110 


I 1055 4715 


9-160 


1 0516 2898 


9-01 1 


I =205 9735 


9-061 


I 1610 6812 


9-111 


1 1044 4216 


9-161 


1 0505 7788 


9012 


I 2193 7736 


9-062 


I 1599 0763 


9-112 


I 1033 3827 


9-162 


I 0495 2782 


9-013 


I 21SI 5860 


9-063 


I 1587 4830 


9'ii3 


I 1022 3548 


9-163 


I 04S4 7882 


9-014 


1 2169 4105 


9-064 


I 1575 9013 


9-114 


1 1011 3379 


9-164 


I 0474 3087 


9015 


I 2157 2471 


9-065 


I 1564 3312 


9-115 


I 1000 3321 


9-165 


1 0463 8396 


9016 


I 2145 0960 


9-066 


I 1552 7726 


9116 


1 0989 3373 


9-166 


I 0453 3810 


9017 


I 2132 9569 


9-067 


I 1541 2256 


9-117 


1 0978 3534 


9-167 


I 0442 9328 


9018 


I 2120 8300 


9068 


I 1529 6902 


9-118 


1 0967 3806 


g-i68 


1 0432 4951 


9-019 


I 2108 7153 


9-069 


I 1518 1663 


9-119 


1 0956 4187 


9-169 


I 0422 0678 


9-020 


I 2096 6126 


9-070 


I 1506 6539 


9-120 


I 0945 4677 


9-170 


I 0411 6510 


9-021 


I 2084 5220 


9-071 


I 1495 1529 


9-121 


1 0934 5277 


9-171 


I 0401 2445 


9-022 


I 2072 4436 


9072 


I 1483 6635 


9-122 


1 0923 5987 


9-172 


I 0390 8485 


9023 


I 2060 3772 


9-073 


I 1472 1856 


9-123 


I 0912 6805 


9-173 


I 03S0 4628 


9024 


I 2048 3228 


9-074 


I 1460 7192 


9-124 


I 0901 7733 


9-174 


I 0370 0875 


9-025 


I 2036 2805 


9-075 


I 1449 2642 


9-125 


1 0890 8770 


9-175 


I 0359 7226 


9026 


I 2024 2502 


9-076 


I 1437 8206 


9-126 


I 0879 9915 


9-176 


I 0349 3681 


9-027 


I 2012 2320 


9-077 


I 1426 38S5 


9-127 


I 0869 1170 


9-177 


I 0339 0239 


9028 


I 2000 2258 


9-078 


I 1414 9679 


9-128 


I 0858 2533 


9-178 


I 0328 6900 


9029 


I 1988 2315 


9-079 


I 1403 5586 


9-129 


I 0847 4005 


9-179 


I 0318 3665 


9030 


1 1976 2493 


9-080 


I 1392 1607 


9-130 


I 0836 5585 


9-180 


I 0308 0533 


9-031 


I 1964 2790 


9-081 


I 1380 7743 


9-131 


I 0825 7274 


9-181 


I 0297 7504 


9032 


I 1952 3207 


9-082 


I 1369 3992 


9-132 


1 0S14 9070 


9182 


1 02S7 4578 


9033 


I 1940 3744 


9-083 


I 1358 0355 


9-133 


I 0804 0975 


9-183 


1 0277 1755 


9034 


1 1928 4400 


9-084 


I 1346 6831 


9-134 


I 0793 2988 


9-184 


I 0266 9034 


9035 


I I916 5175 


9-085 


I 1335 3421 


9-135 


I 0782 5109 


9-185 


1 0256 6417 


9036 


1 1904 6070 


9086 


I 1324 0124 


9136 


I 0771 7338 


9-186 


I 0246 3902 


9037 


I 1892 7083 


9-087 


1 1312 6941 


9-137 


1 0760 9675 


9-187 


I 0236 1489 


9038 


I 1880 8215 


9-088 


I 1301 3870 


9-138 


1 0750 2119 


9-188 


I 0225 9179 


9039 


1 1868 9466 


9-089 


I 1290 0913 


9-139 


I 0739 4670 


9-189 


I 0215 6971 


9-040 


I 1857 0836 


9-090 


I 1278 8068 


9-140 


1 0728 7329 


9-190 


I 0205 4865 


9041 


I 1845 2325 


9-091 


I 1267 5337 


9-141 


I 0718 0096 ' 


9-191 


I 0195 2861 


9042 


» ^^iz 3932 


9-092 


I 1256 2718 


9-142 


I 0707 2969 


9-192 


I 0185 0959 


9"043 


I I82I 5657 


9-093 


I 1245 02 1 1 


9-143 


I 0696 5950 


9-193 


I 0174 9159 


9-044 


I 1809 7500 


9-094 


I 1233 7817 ; 


9-144 


I 0685 9037 


9-194 


I 0164 7461 


9"045 


I 1797 9462 


9-095 


I 1222 5536 


9-145 


I 0675 2232 


9-195 


I 0154 5864 


9-046 


I 1786 I54I 


9-096 


I 1211 3366 


9-146 


I 0664 5533 


9-196 


1 0144 4369 


9047 


I 1774 3739 


9-097 


I 1200 1309 


9-147 


I 0653 8940 


9-197 


I 0134 2975 


9048 


I 1762 6054 


9-098 


I 1 188 9363 


9-148 


1 0643 2455 


9-198 


I 0124 1683 


9049 


I 1750 8487 


, 9-099 


I "77 753° 


9-149 


I 0632 6076 


9-199 


I 0114 0492 



[9-200— 9-399] 



OF THE DESCENDING EXPONENTIAL. 



197 



X 


Q-X 


X 


c-^ 


X 


Q-X 


X 


Q-X 


9'200 


I 0103 9402 


9-250 


9611 1652 


9-300 


9142 4231 


9-350 


8696 5419 


9-201 


I 0093 8413 


9-251 


9601 5588 


9-301 


9133 2853 


9-351 


8687 8497 


g-202 


I 0083 7526 


9-252 


. 9591 9621 


9-302 


9124 1566 


9-352 


8679 1662 


9-203 


I 0073 6738 


9-253 


9582 3749 


9-303 


9115 0370 


9-353 


8670 4914 


9-204 


I 0063 6051 


9-254 


9572 7973 


9-304 


9105 9265 


9*354 


8661 8252 


9-205 


I 0053 5466 


9-255 


9563 2293 


9-305 


9096 8251 


9-355 


8653 1677 


9-206 


I 0043 4980 


9-256 


9553 6709 


9-306 


9087 7328 


9-356 


8644 5189 


9-207 


I °°32, 4596 


9-257 


9544 1220 


9-307 


9078 6496 


9-357 


8635 8787 


9-208 


I 0023 431 1 


9-258 


9534 5826 


9-308 


9069 5755 


9-358 


8627 2471 


9-209 


I 0013 4127 


9"259 


9525 0528 


9-309 


9060 5105 


9-359 


8618 6242 


9-210 


1 0003 4043 


9-260 


9515 5325 


9-310 


9051 4545 


9-360 


8610 0099 


9-211 


9993 4059 


9-261 


9506 0217 


9'3ii 


9042 4076 


9-361 


8601 4042 


9212 


9983 4175 


9262 


9496 5205 


9-312 


9033 3697 


9-362 


8592 8071 


9-213 


9973 4390 


9-263 


9487 0287 


9-313 


9024 3408 


9-363 


85S4 2185 


9-214 


9963 4706 


9-264 


9477 5464 


9-314 


9015 3210 


9-364 


8575 6386 


9-215 


9953 5121 


9-265 


9468 0736 


9-315 


9006 3102 


9-365 


8567 0673 


9-216 


9943 5636 


9-266 


9458 6103 


9-316 


8997 3084 


9-366 


8558 5045 


9-217 


9933 6250 


9-267 


9449 1564 


9-317 


8988 3156 


9-367 


8549 9502 


9-218 


9923 6963 


9-268 


9439 7119 


9-318 


8979 3318 


9-368 


8541 4046 


9-219 


9913 7776 


9-269 


9430 2769 


9-319 


8970 3569 


9-369 


8532 8674 


9-220 


9903 8688 \ 


9-270 


9420 8514 


9-320 


8961 3910 


9-370 


8524 3388 


9-221 


9893 9698 


9-271 


9411 4352 


9-321 


8952 4341 


9-371 


8515 8188 


9-222 


9884 0808 


9-272 


9402 0285 


9-322 


8943 4862 


9-372 


8507 3072 


9-223 


9874 2017 


9-273 


9392 6312 


9-323 


8934 5472 


9-373 


8498 8041 


9-224 


9864 3324 


9-274 


9383 2432 


9-324 


8925 6171 


9-374 


8490 3096 


9-225 


9854 4730 


9*275 


9373 8647 


9-325 


8916 6959 


9-375 


8481 8235 


9-226 


9844 6235 


9-276 


9364 4955 


9-326 


8907 7837 


9-376 


8473 3459 


9-227 


9834 7838 


9-277 


9355 1357 


9-327 


8898 8803 


9-377 


8464 8768 


9-228 


9S24 9539 


9-278 


9345 7852 


9-328 


8889 9859 


9-378 


8456 4162 


9-229 


9815 1338 


9-279 


9336 4441 


9-329 


888i 1004 


9*379 


8447 9640 


9-230 


9805 3236 


9-280 


9327 1123 


9*330 


8872 2237 


9-380 


8439 5202 


9-231 


9795 5232 


9-281 


9317 7S99 


9-331 


8863 3559 


9-381 


8431 0849 


9-232 


9785 7326 


9-282 


9308 4768 


9-332 


8854 4970 


9-382 


8422 6581 


9-233 


9775 9517 


9-283 


9299 1729 


9-333 


8845 6469 


9-383 


8414 2396 


9*234 


9766 1807 


9-284 


9289 8784 


9-334 


8836 8057 


9*384 


8405 8296 


9"23S 


9756 4194 


9-285 


9280 5932 


9-335 


8827 9733 


9-385 


8397 4280 


9-236 


9746 6678 


9-286 


9271 3172 


9-336 


8819 1497 


9386 


8389 0347 


9-237 


9736 9260 


9-287 


9262 0505 


9-337 


8810 3350 


9-387 


8380 6499 


9-238 


9727 1940 


9288 


9252 7931 


9*338 


8801 5291 


9-388 


8372 2734 


9"239 


9717 4716 


9-289 


9243 545° 


9339 


8792 7319 


9-389 


8363 9053 


9-240 


97°7 759° 


9-290 


9234 3060 


9-340 


8783 9436 


9-390 


8355 5456 


9-241 


9698 0561 


9-291 


9225 0763 


9-341 


8775 1640 


9-391 


8347 1942 


9-242 


9688 3629 


9-292 


9215 8559 


9-342 


8766 3933 


9-392 


8338 8512 


9'243 


9678 6794 


9-293 


9206 6446 


9-343 


8757 6313 


9-393 


8330 5165 


9-244 


9669 0055 


9-294 


9197 4426 


9*344 


8748 87S0 


9-394 


8322 1902 


9*245 


9659 3414 


9'295 


9188 2497 


9-345 


8740 1335 


9-395 


8313 8722 


9-246 


9649 6869 


9-296 


9179 0661 


9-346 


8731 39V7 


9*396 


8305 5624 


9-247 


9640 0420 


9-297 


9169 8916 


9*347 


8722 6707 


9-397 


8297 2610 


9-248 


9630 4068 


9-298 


9160 7263 


9-348 


8713 9524 


9-398 


8288 9679 


9-249 


9620 7812 


9-299 


9151 5701 


9-349 


8705 2428 


9*399 


8280 6831 



19S 



MR F. W. NEWMAN'S TABLE 



[9-400— 9-599] 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


9-400 


8272 4065 


9-450 


7S68 9565 


9-500 


7485 1830 


9-550 


7120 1263 


9-401 


8264 13S3 


9-451 


7861 0915 


9-50I 


7477 7015 


9-551 


7113 0097 


9-402 


8255 8783 


9-452 


7853 2343 


9-502 


7470 2276 


9-552 


7105 9003 


9 "403 


8247 6265 


9-453 


7845 3850 


9503 


7462 7611 


9-553 


7098 7979 


9-404 


8239 3830 


9-454 


7837 5436 


9-504 


7455 3020 


9-554 


7091 7027 


9 '405 


8231 1477 


9-455 


7829 7099 


9-505 


■7447' 8505 


9-555 


7084 6145 


9-406 


8222 9207 


9-456 


7821 8841 


9-506 


7440 4063 


9-556 


7077 5334 


9407 


8214 7019 


9-457 


7814 0662 


9-507 


7432 9697 


9-557 


7070 4595 


9-408 


8206 4913 


9-458 


7S06 2560 


9-508 


74-^5 5404 


9-558 


7063 3925 


9-409 


8198 2889 


9-459 


7798 4537 


9-509 


7418 1186 


9-559 


7056 3327 


9-410 


8190 0947 


9-460 


7790 6591 


9-510 


7410 7042 


9-560 


7049 2799 


9-411 


81S1 90S7 


9-461 


7782 8723 


9-511 


7403 2972 


9-561 


7042 2341 


9-412 


8173 7309 


9-462 


7775 0933 


9'5i2 


7395 8976 


9-562 


703s 1954 


9-413 


8165 5613 


9-463 


7767 3221 


9-513 


738S 5054 


9-563 


7028 1637 


9-414 


8157 3998 


9-464 


7759 5587 


9-514 


7381 1206 


9-564 


7021 1391 


9415 


8149 2465 


9-465 


7751 8030 


9-515 


7373 7431 


9-565 


7014 1214 


9-416 


8141 1013 


9-466 


7744 0551 


9-516 


7366 3731 


9-566 


7007 1 108 


9-4I7 


8132 9643 


9-467 


7736 3149 


9-517 


7359 0104 


9-567 


7000 1072 


9-418 


8124 8354 


9-468 


7728 5825 


9-518 


7351 6551 


9-568 


6993 U06 


9-419 


8116 7146 


9-469 


7720 8577 


9-519 


7344 3071 


9-569 


6986 1210 


9-420 


8108 6019 


9-470 


7713 1407 


9-520 


7336 9664 


9-570 


6979 1384 


9-421 


8100 4974 


9-471 


7705 4315 


9-521 


7329 6331 


9-571 


6972 1627 


9-422 


8092 4009 


9-472 


7697 7300 


9-522 


7322 3072 


9-572 


6965 1940 


9423 


8084 3126 


9-473 


7690 0360 


9-523 


7314 9885 


9-573 


6958 2323 


9-424 


8076 2323 


9-474 


7682 3498 


9-524 


7307 6772 


9-574 


6951 2776 


9-425 


8068 1601 


9-475 


7674 6713 


9-525 


7300 3732 


9-575 


6944 3298 


9-426 


8060 0960 


9-476 


7667 0005 


9-526 


7293 0764 


9-576 


6937 3889 


9-427 


8052 0399 


9-477 


7659 3373 


9-527 


7285 7870 


9-577 


6930 4550 


9-428 


8043 9919 


9-478 


7651 6818 


9-528 


7278 5049 


9-578 


6923 5280 


9429 


8035 9519 


9-479 


7644 0339 


9-529 


7271 2300 


9-579 


6916 6079 


9-430 


8027 9200 


9-480 


7636 3937 


9-530 


7263 9624 


9-580 


6909 6948 


9-431 


8019 8961 


9-481 


7628 7611 


9-531 


7256 7021 


9-581 


6902 7885 


9-432 


Son 8802 


9-482 


7621 1362 


9-532 


7249 4490 


9-582 


6895 8892 


9-433 


8003 8723 


9-483 


7613 5189 


9-533 


7242 2032 


9-583 


6888 9967 


9-434 


799S 8724 


9484 


7605 9091 


9-534 


7234 9646 


9-584 


6882 1112 


9-435 


7987 8806 


9-485 


7598 3070 


9-535 


7227 7332 


9-585 


687s 2325 


9-436 


7979 8967 


9-486 


7590 7125 


9-536 


7220 5091 


9-586 


6868 3607 


9-437 


7971 9208 


9-487 


7583 1256 


9-537 


7213 2922 


9-587 


6861 4958 


9-438 


7963 9528 


9-488 


7575 5463 


9-538 


7206 0825 


9-588 


6854 6377 


9-439 


7955 9929 


9-489 


7567 9745 


9-539 


7198 8S00 


9589 


6847 7865 


9-440 


7948 9408 


9-490 


7560 4103 


9-540 


7191 6848 


9-590 


6840 9422 


9-441 


7940 0968 


9-491 


7552 8537 


9-541 


7184 4967 


9-591 


6834 1046 


9-442 


7932 1606 


9-492 


7545 3046 


9-542 


7177 3158 


9-592 


6827 2739 


9-443 


7924 2325 


9-493 


7537 7631 


9-543 


7170 1420 


9 593 


6820 4501 


9-444 


7916 3122 


9-494 


7530 2291 


9-544 


7162 9755 


9-594 


6813 6330 


9-445 


7908 3998 


9-495 


7522 7026 


9-545 


7155 8161 


9-595 


6806 8228 


9-446 


7900 4954 


9-496 


75151837 


9-546 


7148 6638 


9-596 


6800 0194 


9-447 


7892 5988 


9-497 


7507 6722 


9-547 


7141 5188 


9-597 


6793 2228 


9-448 


7884 7102 


9-498 


7500 1683 


9-548 


7134 3808 


9-598 


6786 4329 


9 449 


7876 8294 


9-499 


7492 6719 


9-549 


7127 2500 


9 '5 99 


6779 6499 



[9-600—9799] 



OF THE DESCENDING EXPONENTIAL. 



199 



X 


(,-X 


X 


Q~X 


X 


Q-X 


X 


Q-X 


9'6oo 


6772 8736 


9-650 


6442 5567 


9-700 


6128 3495 


9-750 


5829 4664 


9"6oi 


6766 1042 


9-651 


6436 II73 


9-701 


6122 2242 


9-751 


5823 6398 


9'6o2 


6759 3414 


9-652 


6429 6S44 


9-702 


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 01 7 1 


9-706 


6091 6895 


9-756 


5794 5943 


9-607 


6725 6291 


9-657 


6397 6163 


9-707 


60S5 6008 


9-757 


5788 8026 


9-608 


6718 9068 


9-658 


6391 2219 


9-708 


6079 5183 


9758 


5783 0167 


9-609 


6712 1913 


9-659 


6384 8339 


9-709 


6073 4418 


9-759 


5777 2365 


9-610 


6705 4824 


9-660 


6378 4522 


9-710 


6067 3714 


9-760 


5771 4622 


9-611 


6698 7803 


9-661 


6372 0770 


97 1 1 


6061 3071 


9-761 


5765 6936 


9-612 


6692 0849 


9-662 


6365 7081 


9-712 


6055 2488 


9-762 


5759 9308 


9-613 


6685 3961 


9-663 


6359 3455 


9-713 


6049 1966 


9-763 


5754 1737 


9-614 


6678 7141 


9-664 


6352 9894 


9-714 


6043 1504 


9-764 


5748 4225 


9-615 


6672 0387 


9-665 


6346 6396 


9715 


6037 1103 


9-765 


5742 6769 


9-616 


6665 3700 


9-666 


6340 2961 


9-716 


6031 0762 


9-766 


5736 9371 


9-617 


6658 7079 


9-667 


6333 9590 


9-717 


6025 0481 


9-767 


5731 2030 


9-618 


6652 0526 


9-668 


6327 6282 


9-718 


6019 0261 


9-768 


5725 4747 


9-619 


6645 4038 


9-669 


6321 3°37 


9-719 


6013 OIOI 


9-769 


5719 7521 


9-620 


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 9981 


9-772 


5702 6185 


9-623 


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 


9-774 


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 9350 


9-727 


5965 0979 


9-777 


5674 1766 


9628 


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 S396 


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 


S94I 2851 


9-781 


5651 5252 


9-632 


6559 5727 


9-682 


6239 6586 


9-732 


5935 3468 


9-782 


5645 8765 


9'633 


6553 0164 


9-683 


6233 4220 


9'733 


5929 4144 


9783 


5640 2335 


9'634 


6546 4667 


9-684 


6227 1917 


9-734 


5923 4880 


9-784 


5634 5961 


9-635 


6539 9235 


9-685 


6220 9676 


9735 


5917 5675 


9-785 


5628 9643 


9-636 


6533 3868 


9-686 


6214 7498 


9-736 


5911 6529 


9-786 


5623 3381 


9'637 


6526 8567 


9-687 


6208 5381 


9-737 


5905 7442 


9-787 


5617 7176 


9-638 


6520 3331 


9-688 


6202 3327 ^ 


9-738 


5S99 8414 


9-788 


5612 1027 


9-639 


6513 8160 


9-689 


6196 1335 


9739 


5893 9445 


9-789 


5606 4934 


9-640 


6507 3055 


9-690 


6189 9404 


9-740 


5888 0535 


9-790 


5600 8897 


9-641 


6500 8014 


9-691 


6183 7536 


9-741 


5882 1684 


9-791 


5595 2916 


9-642 


6494 3°39 


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 32S2 


9-694 


6165 2301 


9'744 


5S64 5483 


9-794 


5578 5309 


9-645 


6474 8501 


9-695 


6159 0680 


9-745 


5858 6867 


9795 


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 


9798 


5556 2614 


9-649 


6449 0025- 


9-699 


6134 4809 


9-749 


5835 2987 


9-799 


5550 7079 



200 



MR F. W. NEWMAN'S TABLE 



[9-800— 9-999] 



1 


C-x 


X 


C-x 


X 


C-x 


1 
X 


c-x 


9-800 


5545 1599 


9-850 


5274 7193 


9-900 


5017 4682 


|9-9S° 


4772 7634 


9-801 


5539 6175 


9-851 


5269 4472 


9-901 


5°i2 4532 


9951 


4767 993° 


9-802 


5534 0S07 


9-852 


5264 1804 


9-902 


5037 4433 


; 9-952 


4763 2274 


9-803 


552S 5494 


9-853 


5258 918S 


9-903 


5032 4383 


: 9-953 


4758 4666 


9 804 


55-3 °^3(> 


9-854 


5253 6625 


9-904 


4997 4384 


9-954 


4753 7105 


9-805 


5517 5033 


9-855 


5248 4115 


9-905 


4992 4435 


9-955 


474S 9591 


9806 


55 1 1 9886 


9-856 


5243 1657 


9-906 


4987 4535 


9-956 


4744 2126 


9-807 


5506 4794 


9-857 


5237 9252 


9-907 


49S2 4686 


9-957 


4739 4707 


9-808 


55°° 9756 


9-858 


5232 6899 


9-908 


4977 48S6 


9-958 


4734 7336 


9809 


5495 4774 


9-859 


5227 4598 


9-909 


4972 5136 


9-959 


4730 0012 


9-810 


5489 9847 


9-860 


5222 2350 


9-910 


4967 5436 


9-960 


4725 2736 


9-811 


5484 4974 


9-861 


5217 0154 


9-911 


4962 5785 


9-961 


4720 5507 


9-812 


5479 0157 


9-862 


521 1 8010 


9-912 


4957 6184 


9-962 


4715 8325 


9-813 


5473 5394 


9-863 


5206 5918 


9-913 


4952 6633 


9-963 


4711 1190 


9-814 


5468 0686 


9-864 


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 


9-816 


5457 1434 


9-866 


5190 9954 


9-916 


4937 8275 


9-966 


4697 0068 


9-817 


5451 6890 


9-867 


5185 8070 


9-917 


4932 8922 


9-967 


4692 3122 


9818 


5446 2400 


9-868 


5180 6238 


9-918 


4927 9618 


9-968 


4687 6222 


9819 


5440 7965 


9869 


5175 4457 


9-919 


4923 0363 


9-969 


4682 9369 


9-820 


5435 3584 


9-870 


5170 2729 


9-920 


4918 1157 


9-970 


4678 2563 


9-821 


5429 9258 


9-871 


5165 1052 


9-921 


4913 2000 


9-971 


4673 5804 


9822 


5424 4986 


9-872 


5159 9427 


9-922 


4908 2893 


9-972 


4668 9092 


9-823 


5419 0768 


9873 


5154 7853 


9-923 


4903 3834 


9-973 


4664 2426 


9-824 


5413 6604 


9-874 


5149 6331 


9-924 


4898 4825 


9-974 


4659 5807 


9-825 


5408 2495 


9-875 


5144 4860 


9-925 


4893 5865 


9-975 


4654 9234 


9826 


5402 8439 


9-876 


5139 3441 


9-926 


48S8 6953 


9-976 


4650 2708 


9-827 


5397 4438 


9-877 


5134 2073 


9-927 


4883 8091 


9-977 


4645 6229 


9-828 


5392 0490 


9-878 


5129 0757 


9-928 


4878 9277 


9-978 


4640 9796 


9-829 


5386 6597 


9-879 


5123 9492 


9-929 


4874 0512 


9-979 


4636 3409 


9-830 


5381 2757 


9-880 


5118 8278 


9-93° 


4869 1796 


9-980 


4631 7069 


9-831 


5375 8971 


9-881 


5113 711S 


9-931 


4864 3129 


9-981 


4627 0775 


9-832 


537° 5239 1 


9-882 


5108 6004 


9-932 


4859 4510 


9982 


4622 4528 


9833 


5365 1560 


9-883 


5103 4943 


9933 


4854 594° 


9-983 


4617 8326 


9-834 


5359 7936 


9-884 


5098 3934 


9934 


4849 7418 


9-984 


4613 2171 


9835 


5354 4365 


9-885 


5°93 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 7383 


9-887 


5083 121 I 


9-937 


4835 2144 


9-987 


4599 3982 


9-838 


5338 3972 


9-888 


5078 0405 


9-938 


4830 3816 


9-988 


4594 Son 


9-839 


5333 0615 


9-889 


5072 9650 


9-939 


4825 5536 


9-989 


4590 2086 


9-840 


5327 73" 


9-890 


5067 8946 


9-940 


4820 7305 


9-990 


4585 6207 


9-841 


5322 4060 


9-891 


5062 8292 


9-941 


4815 9121 


9-991 1 


4581 0373 


9842 


5317 0863 


9-892 


5057 768J 


9-942 


481 1 0986 


9-992 1 


4576 4586 


9-843 


53^1 7719 


9-893 


5052 7137 


9943 


4806 2899 


9-993 


4571 8844 


9844 


5306 4627 


9-894 


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 


9846 


5295 8604 


9-896 


5°37 5783 


9-946 


4791 8927 


9-996 


4558 1893 


9-847 


5290 5672 


9-897 


5°32 5432 


9-947 


4787 1032 


9-997 


4553 6334 


9848 


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 



f 



[lO'' 



10-199] 



OF THE DESCENDING EXPONENTIAL. 



201 



X 

/ 


Q-X 


X 


Q-X 


X 


(,-X 


X 


e-" 


lO'OOO 


4539 9930 


10-050 


4318 


5749 


lo-ioo 


4107 9555 


10-150 


3937 6082 


lO'OOI 


4535 4552 


10 051 


4314 


2585 


lO'IOI 


4103 8496 


10151 


3933 7025 


I0"002 


4530 9221 


10-052 


4309 


9464 


10-102 


4099 7478 


10-152 


3899 8008 


io'oo3 


4526 3934 


10-053 


4305 


6386 


10-103 


4095 6501 


10-153 


3895 9029 


1 0-004 


4521 8693 


10-054 


4J0I 


335' 


10-104 


4091 5565 


10-154 


3892 0089 


io'oo5 


4517 3497 


10-055 


4297 


0359 


10-105 


4087 4670 


10-155 


3888 1 189 


10-006 


4512 8346 


10-056 


4292 


7410 


io-io6 


4083 3816 


10-156 


3884 2327 


10-007 


4508 3240 


10-057 


4288 


4504 


10-107 


4079 3002 


10-157 


3880 3504 


10-008 


4503 8179 


10-058 


4284 


1641 


10-108 


4075 2230 


10-158 


3876 4720 


10-009 


4499 3163 


10-059 


4279 


8821 


10-109 


4071 1498 


10-159 


3S72 5975 


10-010 


4494 8193 


io-o6o 


4275 


6044 


lo-i 10 


4067 0807 


io-i6o 


3S68 726S 


10-01 1 


4490 3267 


10-061 


4271 


3309 


lO-III 


4063 0156 


10161 


3S64 8600 


IOOI2 


4485 8386 ! 


10-062 


4267 


0617 


IO-II2 


405S 9546 


10-162 


3860 9971 


10-013 


4481 3550 


10-063 


4262 


7968 


10-113 


4054 8977 


10-163 


3857 1380 


10-014 


4476 8759 


10-064 


4258 


5361 


10-114 


4050 8448 


10164 


3S53 2828 


10-015 


4472 4013 


10-065 


4254 


2797 


lo-iis 


4046 7960 


10-165 


3849 4315 


10-016 


4467 931 1 


10-066 


4250 


0275 


lo-i 16 


4042 7512 


io-i66 


3845 5839 


10-017 


4463 4654 


10-067 


4245 


7796 


I0-II7 


4038 7105 


10-167 


3841 7403 ; 


10-018 


4459 0042 


10-068 


4241 


5360 


io-ii8 


4034 6738 


10-168 


3S37 9005 


10-019 


4454 5474 


10-069 


4237 


2966 


10-119 


4030 6412 


10-169 


3834 0645 


10-020 


4450 0951 


10-070 


4233 


0614 


10-120 


4026 6125 


10-170 


3830 2323 i 


10-021 


4445 6472 


10-071 


4228 


8305 


10-121 


4022 5879 


10-171 


3826 4040 j 


10-022 


4441 2038 


10-072 


4224 


6037 


10-122 


4018 5674 


10-172 


3822 5795 1 


10-023 


4436 7643 1 


10-073 


4220 


3813 


10-123 


4014 5508 


10-173 


3S18 7589 


10-024 


4432 3303 


10-074 


4216 


1630 


10-124 


4010 5383 


10-174 


3814 9420 


10-025 


4427 gooi 


10-075 


4211 


9489 


10-125 


4006 5297 


10-175 


3S1I 1290 


10-026 


4423 4745 i 


10-076 


4207 


7391 


10-126 


4302 5252 


10-176 


3807 3197 


10-027 


4419 0532 ; 


10-077 


4203 


5334 


10-127 


3998 5247 


10-177 


3S03 5143 


10-028 


4414 6363 \ 


10-078 


4199 


3320 


10-128 


3994 5282 


10-178 


3799 7127 


10-029 


4410 2239 


10-079 


4195 


1348 


10-129 


3990 5356 


10-179 


3795 9149 


10-030 


4405 8159 


10-080 


4190 


9417 


10-130 


3986 5471 


io-i8o 


3792 1209 


10-03 r 


4401 4123 


10-081 


4186 


7529 


10-131 


3,'82 5625 


io-i8i 


3788 3307 


10-032 


4397 0131 


10-082 


4182 


5682 


10-132 


3978 5820 


10-182 


3784 5442 


10-033 


4392 6183 


10-083 


4178 3877 


10-133 


3974 6054 


10-183 


3780 7616 


10-034 


4388 2278 


10-084 


4174 


2114 


10-134 


3970 6327 


10-184 


3776 9827 


10-035 


4383 8418 


10-085 


4170 


0393 


10-135 


3966 6641 


10-185 


3773 2076 


10-036 


4379 4601 


10-086 


4165 8714 


10-136 


3962 6994 


10-186 


3769 4363 


10-037 


4375 0829 


10-087 


4161 


7076 


10-137 


3958 7387 


10-187 


3765 6687 


10-038 


4370 7100 


10-088 


4157 


5479 


10-138 


3954 7819 


io-i88 


3761 9049 


10-039 


4366 3415 


10-089 


4153 


3925 


10-139 


3950 8291 


10-189 


3758 1449 \ 


10-040 


4361 9773 


10-090 


4149 


2412 


10-140 


3946 8803 


10-190 


3754 3887 


10-041 


4357 6175 


10-091 


4145 


0940 


10-141 


3942 9354 


10-191 


3750 6361 


10-042 


4353 2621 


10-092 


4140 


9510 


10-142 


3938 9944 


10-192 


3746 8874 


10-043 


4348 9"° 


10-093 


4136 


8121 


10-143 


3935 0574 


10-193 


3743 1424 


10-044 


4344 5642 


10-094 


4132 


6773 


10-144 


3931 1243 


10-194 


3739 40 I I 


ro-045 


4340 2218 


■ 10-095 


4128 


5467 


10-145 


3927 1951 


' 10-195 


3735 6636 


10-046 


4335 8838 


10-096 


4124 


4202 


10-146 


3923 2699 


10196 


3731 9298 


10-047 


4331 5501 


10-097 


4120 


2979 


10-147 


3919 3486 


10-197 


3728 1997 


10-048 


4327 2207 


10-098 


4116 


1796 


10-148 


3915 4312 


10-198 


3724 4734 


10-049 


4322 8956 


, 10-099 


4112 


0655 


, 10-149 


39»i 5177 


10-199 


3720 7508 



Vol. XITI. Papt J 1 1. 



202 



MR F. W. NEWMAN'S TABLE 



[lO'2O0 10*399] 



r 

X 


C-* 


X 


(,-X 


1 


X 


C-x 




iU 


e-x 


I0'200 


3717 0319 


10-250 


3535 


7501 


10-300 


3363 


309s 


10-350 


3199 2790 


10-20I 


3713 3167 


10-251 


3532 


2161 


10-301 


3359 


9479 


10-351 


3196 0813 


io-2oa 


3709 6052 


10-252 


352S 


6856 


10-302 


3356 


5896 


10-352 


3192 8868 


10-203 


3705 8975 ' 


10-253 


3525 


1587 


10-303 


3353 


2347 


10-353 


3189 6955 


10-204 


3702 1934 


10-254 


3521 


6353 . 


10-304 


3349 


8831 


10-354 


3186 5074 


10-205 


3698 4931 


10-255 


3518 


1154 


10-305 


3346 


5349 


10-355 


3183 3225 


io-2o6 


3694 7964 


10-256 


3514 


5991 


10-306 


3343 


1901 


10-356 


3180 1408 


10-207 


3691 i°35 


10-257 


35 1 1 


0S63 


10-307 


3339 


8486 


10-357 


3176 9622 


10-208 


3687 4142 


10-258 


3507 


5769 


10-308 


333(> 


5104 


10-358 


. 3173 7868 


10-209 


3(>'^3 72S7 


10-259 


3504 


0711 


10-309 


3333 


1755 


io-359 


3170 6146 


I0-2IO 


3683 0468 


10-260 


3500 


5688 


10-310 


3329 


8440 


10-360 


3167 4456 


10-2II 


3676 3686 


10-261 


3497 


0700 


10-311 


3326 


515S 


10-361 


3164 2798 


IO-2I2 


3672 6940 


10-262 


3493 


5746 


10-312 


3323 


1910 


10-362 


3161 1171 


10-213 


3669 0232 


10-263 


349° 


0828 


10-313 


3319 


8695 


10-363 


3157 9575 


lp-214 


3665 3560 


10-264 


3486 


5944 


10-314 


33^*^ 


5512 


10-364 


3154 8oii 


10-215 


3661 6925 


10-265 


3483 


1096 


10-315 


3313 


2364 


10-365 


3151 6479 


IO-2l6 


365S 0326 


10-266 


3479 


6282 


10-316 


3309 


9248 


10-366 


3148 4978 


10-217 


3654 3764 


10-267 


3476 


1504 


10-317 


3306 


6165 


10-367 


3145 3509 


10-218 


3650 7238 ! 


10-268 


3472 


6759 


10-318 


3303 


3115 


10-368 


3142 2071 


10-219 


3647 0749 


10-269 


3469 


2050 


10-319 


3300 


0099 


10-369 


3139 0665 


10-220 


3643 4207 


10-270 


3465 


7375 


10-320 


3296 


7115 


10-370 


3135 9290 


10-221 


3639 7881 


10-271 


3462 


2735 


10-321 


3293 


4165 


10-371 


3132 7946 


10-222 


3636 1501 


10-272 


3458 


8130 


10-322 


3290 


1247 


10-372 


3129 6634 


10-223 


3632 5158 


10-273 


3455 


3559 


10-323 


3286 


8362 


10-373 


3126 5353 


10-224 


3628 8851 


10-274 


3451 


9023 


10-324 


3283 


5510 


10-374 


3123 4103 


10-225 


3625 2580 


10-275 


3448 


4521 


10-325 


32S0 


2691 


10-375 


3120 2885 


10-226 


3621 6346 


10-276 


3445 


0054 


10-326 


3276 


9905 


10-376 


3117 1698 


10-227 


3618 0147 


10-277 


3441 


5621 


10-327 


3273 


7151 


10-377 


3114 0542 


10-228 


3614 3985 


10-278 


3438 


1222 


10-328 


3270 


4431 


10-378 


3110 9417 


10-229 


3610 7859 


10-279 


3434 


6858 


10-329 


3267 


1742 


10-379 


3107 8323 


10-230 


3607 1770 


10-280 


3431 


2529 


10-330 


3263 90S7 


10-380 


3104 7260 


10-231 


3633 5716 


10-281 


3427 


8233 


10-331 


3260 


6464 


10-381 


3101 6228 


10-232 


3599 9698 


10-282 


3424 


3972 


10-332 


3257 


3874 


10-382 


3098 5227 


10-233 


3596 3717 


10-283 


3420 


9745 


10-333 


3254 


1316 


10-383 


3095 4258 


10-234 


3592 7771 


10-284 


3417 


5553 


10-334 


3250 


8791 


10-384 


3092 3319 


10-235 


3589 1861 


10-285 


3414 


1394 


10-335 


3247 


6299 


10-385 


3089 241 I 


10-236 


3585 5987 


10-286 


3410 


7270 


10-336 


3244 


3839 


10-386 


3086 1534 


10-237 


3582 0149 


10-287 


3407 


3180 


10-337 


3241 


1411 


10-387 


3083 0688 


10-238 


3578 4347 


10-288 


3403 


9124 


10-338 


3237 


9016 


10-388 


3079 9873 


10-239 


3574 8580 


10-289 


3400 


5101 


10-339 


3234 


6653 


10-3S9 


3076 9088 


10-240 


3571 2850 


10-290 


3397 


1113 


10-340 


3231 


4323 


10-390 


3073 8334 


10-241 


3567 7155 


10-291 


3393 


7159 


10-341 


3228 


2024 


10-391 


3070 7611 


10-242 


3564 1495 


10-292 


3390 


3239 


10-342 


3224 


9759 


10-392 


3067 6919 


10-243 


3560 5872 


10-293 


3386 


9353 


10-343 


3221 


7525 


10-393 


3064 6258 


10-244 


3557 0283 


10-294 


3383 


5500 


10-344 


3218 


5324 


10-394 


3061 5627 


10-245 


3553 4731 


10-295 


338° 


1682 


10-345 


3215 


3154 


10-395 


3058 5026 


10-246 


3549 9214 


10-296 


3376 7897 


10-346 


3212 


1017 


10-396 


3055 4457 


10-247 


3546 3733 


10-297 


3373 


4146 


10-347 


3208 


8912 


10-397 


3052 3917 


' 10-248 


3542 8287 


10-298 


3370 


0429 


10-348 


3205 


6839 


10-398 


3049 3409 


1 10-249 


3539 2876 


10-299 


3366 6745 


10-349 


3202 


4799 


' 10-399 


3046 2931 



[10-400- 10-599] 



OF THE DESCENDING EXPONENTIAL. 



203 



X 


(,-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


io'4oo 


3043 2483 


10-450 


2894 8273 


10-500 


2753 6449 


' 10-550 


2619 3481 


10-401 


3040 2066 


10-451 


2891 9339 


10-501 


2750 8927 


i 10-551 


2616 7301 


IO-402 


- 3337 1679 


10-452 


2889 0434 


10-502 


2748 1431 


10-552 


2614 II47 


10-403 


3034 1322 


10-453 


2886 1558 


10-503 


2745 3964 


10-553 


261I 5019 


10-404 


3031 0996 


10-454 


2883 2711 


10-504 


2742 6523 


10-554 


2608 8917 


10-405 


3028 0700 


i°'455 


2880 3893 


10-505 


2739 9III 


10-555 


2606 2840. 


10-406 


3025 0435 


10-456 


2877 5103 


10-506 


2737 1725 


10-556 


2603 6790 


10-407 


3022 0199 


10-457 


2874 6342 


10-507 


2734 4367 


10-557 


2601 0766 


10-408 


3018 9994 


10-458 


2871 7611 


10-508 


2731 7036 


10-558 


2598 4768 


10-409 


3015 9819 


io'459 


2868 8907 


10-509 


2728 9733 


10-559 


. 2595 8797 


10-410 


3012 9675 


10-460 


2866 0233 


10-510 


2726 2457 


10-560 


2593 2851 


10-41 1 


3009 9560 


10-461 


2863 1587 


10-511 


2723 5208 


10-561 


2590 6931 


10-412 


3006 9476 


10-462 


2860 2969 


10-512 


2720 7987 


10-562 


2588 1037 


10-413 


3003 9421 


10-463 


2857 4381 


i 10-513 


2718 0792 


10-563 


2585 5169 


10-414 


3000 9397 


10-464 


2854 5821 


10-514 


2715 3625 


10-564 


2582 9327 


10-415 


2997 9402 


10-465 


2S51 7289 


10-515 


2712 6485 


10-565 


2580 3510 


10-416 


2994 9438 


10-466 


2848 8786 


10-516 


2709 9372 


10-566 


2577 7720 


10-417 


2991 9503 


10-467 


2846 0312 


10-517 


2707 2286 


10-567 


2575 1955 


10-418 


29S8 9599 


10-468 


2843 1865 


10-518 


2704 5227 


10-568 


2572 6216 


10-419 


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 1191 


10-570 


2567 4815 


10-421 


2980 0064 


10-471 


2834 6698 


10-521 


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 0061 


10-523 


2691 0339 


10-573 


2559 7906 


10-424 


2971 0798 


10-474 


2826 1785 


10-524 


2688 3442 


10-574 


2557 2321 


10-425 


2968 H02 


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 


2959 2192 


10-478 


2814 8964 


10-528 


2677 6123 


10-578 


2547 0236 


10-429 


2956 2615 


10-479 


2812 0829 


10-529 


2674 9360 


10-579 


2544 4778 


10-430 


2953 3067 


10-480 


2809 2722 


10-530 


2672 2624 


10-580 


2541 9346 


10-431 


2950 3549 


10-481 


2806 4643 


i°-S3i 


2669 5915 


10-581 


2539 3940 


10-432 


2947 4060 


10-482 


2803 6593 


10-532 


2666 9233 


10-582 


2536 8559 


io'433 


2544 4601 


10-483 


2800 8570 


10-533 


2664 2577 


10-583 


2534 3203 


io'434 


2941 5171 


10-484 


2798 0576 


10-534 


2661 5947 


10-584 


2531 7872 


io'435 


2938 5770 


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 


io"439 


2926 8462 


10-489 


2784 1022 


10-539 


2648 3200 


10-589 


2519 1599 


10-440 


2923 920S 


10-490 


2781 3195 


10-540 


2645 6730 


10-590 


2516 6420 


10-441 


2920 9984 


10-491 


2778 5396 


10-541 


2643 0287 


10-591 


2514 1266 


10-442 


2918 0788 


10-492 


2775 7624 j 


10-542 


2640 3870 


10-592 


2511 6137 


io"443 


2915 1622 


10-493 


2772 9881 


10-543 


2637 7479 


10-593 


2509 1034 


10-444 


2912 2484 


10-494 


2770 2165 


10-544 


2635 i"S 


10-594 


2506 5955 


10445 


2909 3377 


10-495 


2767 4476 


10-545 


2632 4777 


10-595 


2504 0902 


10-446 


2906 4298 


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 


10-598 


2496 5892 


10-449 


2897 7236 


10-499 


2756 3999 


10-549 


2621 9688 


10-599 


2494 0938 



27—2 



204 



MR F. W. NEWMAN'S TABLE 



[10600 — 10799] 





f,-X 


X 


Q-X 


X 


(,-X 


X 


1 


! 
io-6oo 


2491 6010 


10-650 


2370 0841 


10-700 


2254 4938 


10-750 


2 144 5408 


io-6oi 


24S9 1106 1 


10-651 


2367 7153 


10-701 


2252 2404 


10-751 


2142 3974 


IO"6o2 


2486 6227 \ 


10-652 


2365 3487 


10-702 


2249 9893 


10-752 


2140 2560 


io'6o3 


24S4 1374 


10-653 


2362 9846 


10-703 


2247 7404 


10-753 


2138 1168 


io'6o4 


24SI 6545 


10-654 


2360 6227 


10-704 


2245 4938 


10-754 


2135 9798 


10-605 ' 


2479 1741 


10-655 


2358 2633 


10-705 


2243 2494 


10-755 


2133 8449 


io-6o6 j 


2476 6961 


10-656 


2355 9062 ; 


10-706 


2241 0073 


10-756 


2131 7121 


10-607 i 


2474 2207 ■ 


10-657 


2353 5515 [ 


10-707 


2238 7674 


10-757 


2129 5815 


io-6o8 j 


2471 7477 1 


10-658 


2351 1991 i 


10-708 


2236 5298 


10-758 


2127 4529 


10-609 


2469 2772 


10-659 


2348 8491 j 


10-709 


2234 2944 


10-759 


2125 3265 


10-610 


2466 8091 ' 


10-660 


2346 5014 


10-710 


2232 0612 


10-760 


2123 2023 


io-6ii 


2464 3435 '■ 


io-66i 


2344 1561 


10-71 1 


2229 8302 


10-761 


2121 0801 


IO-6l2 


2461 S804 


10-662 


2341 8131 


10-712 


2227 6015 


10-762 


2n8 9601 


10-613 j 


2459 4198 


10-663 


2339 4725 


10-713 


2225 3750 


10-763 


2II6 8422 


10-614 i 


2456 9616 


10-664 


2337 1342 


10-714 


2223 I50S 


10-764 


2II4 7264 


10-615 ' 


2454 5059 


10-665 


2334 7982 


10-715 


2220 92S7 


10-765 


2II2 6128 


10-616 


2452 0526 


iD-660 


2332 4646 


10-716 


2218 7089. 


10-766 


2II0 5012 


1 10-617 


2449 6018 1 


10-667 


2330 1333 


10-717 


2216 4913 


10-767 


2108 3918 


10-618 


2447 1534 1 


10-668 


2327 8043 


10-718 


2214 2759 


10-768 


2106 2844 


10-619 


2444 7074 


io'669 


2325 4777 


10-719 


2212 0628 


10-769 


2104 1792 


1 10-620 


2442 2640 


10-670 


2323 1533 


10-720 


2209 8518 


10-770 


2102 0761 


10-621 


2439 8229 i 


10-671 


2320 8314 


10-721 


2207 6431 


10-771 


2099 9750 


10-622 


2437 3843 i 


10-672 


2318 5117 


10-722 


2205 4365 


10-772 


2097 8761 


10623 


2434 9481 


10-673 


2316 1943 


10-723 


2203 2322 


10-773 


2095 7793 


10-624 


2432 5144 


10-674 


2313 8793 


10-724 


220I 0301 


10-774 


2093 6846 


1 10-625 


2430 0S31 


10-675. 


2311 5666 


10-725 


2198 S30I 


10775 


2091 5919 


10-626 


2427 6542 


10-676^ 


2309 2562 


10-726 


2196 6324 


10-776 


2089 5014 


1 10-627 


2425 2278 


10-677 


2306 9481 


10-727 


2194 4369 


10-777 


2087 4129 


10-628 


2422 8038 


10-678 


2304 6423 


10-728 


2192 2435 


10-778 


2085 3265 


10-629 


2420 3S22 


10-679 


2302 3388 


10-729 


2190 0524 


10779 


2083 2423 


' 10-630 


2417 9630 


io-68o 


2300 0376 


10730 


2187 8634 


10-7S0 


2081 1601 


i 10-631 


2415 5463 


10-681 


2297 7387 


10-731 


2185 6767 


10-781 


2079 0799 


10-632 


2413 1319 


; 10-682 


2295 4421 


10-732 


2183 4921 


10-782 


2077 0019 


i 10-633 


2410 7200 


10-683 


2293 1478 


10-733 


2i8i 3097 


10-783 


2074 9259 


10-634 


240S 3105 


10684 


2290 8558 


10-734 


2179 1295 


10-784 


2072 8521 


10-635 


2405 9034 


10-685 


2288 5661 


10-735 


2176 9514 


10-785 


2070 7802 


10-636 


2403 4987 


10-686 


2286 2787 


10-736 


2174 7756 


10-786 


2068 7105 


10-637 


2401 0964 


' 10-687 


2283 9935 


10-737 


2172 6019 


10-787 


2066 642^ 


10-638 


2398 6965 


IO-688 


2281 7107 


10-738 


2170 4303 


10-788 


2064 5772 


1 10-639 


2396 2990 


10-689 


2279 4301 


10739 


2168 2610 


10-789 


2062 5137 


10-640 


2393 9°39 


10-690 


2277 1518 


10-740 


2166 0938 


10-790 


2060 4522 


10-641 


2391 5112 


! 10-691 


2274 8758 


i 10-741 


2x63 9288 


10-791 


2058 3928 


10-643 


2389 1209 


1 10692 


2272 6021 


10742 


2I6I 7660 


10-792 


2056 3354 


10-643 ■ 


2386 7329 


10-693 


2270 3306 


10-743 


2159 6053 


10-793 


2054 2801 


10-644 


2384 3474 


; 10-694 


2268 0614 


10-744 


2157 4468 


10-794 


2052 2268 


10-645 


2381 9642 


■ 10-695 


2265 7945 


10-745 


2tS5 2904 


10-795 


2050 1756 


10-646 


2379 5835 


1 10-696 


2263 5298 


! 10-746 


2153 1362 


10-796 


2048 1265 


10-647 


2377 2051 


1 10-697 


2261 2674 


' 10-747 


2150 9841 


10-797 


2046 0794 


10-648 


2374 8291 


1 10-698 


2259 0073 


10-748 


2148 8342 


10-798 


2044 0343 


10-649 

1 


2372 4554 


! 10-699 


2256 7494 


, 10-749 


2146 6864 


10-799 


2041 9913 



[lo 800 — io'99g] 



OF THE DESCENDING EXPONENTIAL. 



205 



X 


C-x 


X 


(,-X 


X 
10-900 


^-x 


X 


(,-X 


10 '800 


2039 9503 


10-850 


1940 4608 


1845 8234 


10-950 


1755 8015 


io'8oi 


2037 9114 


10-851 


1938 5213 


10-901 


1843 9785 


10-951 


1754 0466 


io-8o2 


2035 8745 


10-852 


1936 5837 


10-902 


1842 1354 


10-952 


1752 2934 


io'8o3 


2033 8397 


10-853 


1934 6481 


10-903 


1840 2942 


10-953 


1750 5420 


10 '804 


2031 8068 


10-854 


1932 7144 


10-904 


1838 4548 


10-954 


1748 7923 


10-805 


2029 7760 


10-855 


1930 7827 


10-905 


1836 6173 


10-955 


1747 0444 


io-8o6 


2027 7473 


10-856 


1928 8528 


10-906 


1834 7816 


10-956 


1745 2983 


10-807 


2025 7205 


10-857 


1926 9250 


10-907 


1832 9477 


10-957 


1743 5538 


io-8o8 


2023 6958 


10-858 


1924 9990 


10-908 


1831 1157 


10-958 


1741 8112 


10-809 


2021 6732 


10-859 


1923 0750 


10-909 


1829 2855 


10-959 


1740 0702 


io-8io 


2019 6525 


10-860 


1921 1528 


10-910 


1827 4571 


10-960 


1738 3310 


10-811 


2017 6338 


10-861 


1919 2327 


10-911 


1S25 6306 


10-961 


1736 5935 


10-812 


2015 6172 


10-862 


1917 3144 


10-912 


1823 8059 


10-962 


1734 8578. 


10-813 


2013 6026 


10-863 


1915 3980 


10-913 


1S21 9830 


10-963 


1733 1238 


10-814 


20H 5900 


10-864 


1913 4836 


10-914 


1820 1619 


10-964 


1731 3916 


10-815 


2009 5794 


10-865 


1911 5711 


10-915 


1818 3427 


10-965 


1729 661Q 


10-816 


2007 5709 


10-866 


1909 6604 


10-916 


1816 5252 


10-966 


1727 9323 


10-817 


2005 5643 


10-867 


1907 7517 


10-917 


1814 7096 


10-967 


1726 2052 


io-8i8 


2003 5597 


10-868 


1905 8449 


10-918 


1812 8958 


10-968 


1724 4798 


10-819 


2001 5572 


10-869 


1903 9400 


10-919 


1811 0838 


10-969 


1722 7562 


10-820 


1999 5566 


10-870 


1902 0371 


10-920 


1809 2736 


10-970 


1721 0343 


10-821 


1997 5581 


t 10-871 


1900 1360 


10-921 


1807 4653 


10-971 


1719 3141 


10-822 


1995 5615 


i 10-872 


1898 2368 


10-922 


1805 6587 


10-972 


1717 5957 


10-823 


1993 5669 


10-873 


1896 3395 


10-923 


1803 8539 


10-973 


1715 8789 


10-824 


1991 5744 


10-874 


1894 4441 


10-924 


1802 0510 


10-974 


1714 1639 


10-825 


1989 5838 


10-875 


1892 5506 


10-925 


1800 249S 


10-975 


1712 4506 


10-826 


19S7 5952 


10-876 


1890 6590 


10-926 


1798 4505 


10-976 


1710 7390 


10-827 


19S5 60S6 


i 10-877 


1S88 7693 


10-927 


1796 6529 


10-977 


1709 0291 


10-828 


19S3 6240 


10-878 


1886 8815 


10-928 


1794 8572 


10-978 


1707 3210 


13-829 


1981 6413 


10-879 


1884 9955 


10-929 


1793 0632 


10-979 


1705 6145 


10-830 


1979 6607 


' 10-880 


1883 1115 


10-930 


1791 2711 


10-980 


1703 9097 


10-831 


1977 6820 


■ IO-88I 


1881 2293 


10-931 


1789 4807 


10-981 


1702 2067 


10-832 


1975 7053 


10-882 


1879 3490 


10-932 


1787 6921 


10-982 


1700 5053 


10-833 


1973 7306 


10-883 


1877 4706 


10-933 


1785 9053 


10-983 


169S 8057 


10-834 


1971 7579 


10-884 


1875 5941 


10-934 


1784 1203 


10-984 


1697 1077 


10-835 


1969 7871 


10-885 


1873 7194 


10-935 


1782 3371 


10-985 


1695 4114 


10-836 


1967 8183 


; 10-886 


1871 8466 


10-936 


1780 5556 


10-986 


1693 7169 


10-837 


1965 8514 


i 10-887 


1869 9757 


10-937 


1778 7760 


10-987 


1692 0240 


10-838 


1963 8866 


IO-8S8 


1868 1067 


10-938 


1776 9981 


1 10-988 


1690 3328 


10-839 


1961 9237 


10-889 


1866 2395 


10-939 


1775 2220 


10-989 


168S 6433 


10-840 


1959 9627 


10-890 


1864 3742 


10-940 


1773 4476 


10-990 


1686 9556 


10-841 


1958 0037 


' 10-891 


1862 5108 


10-941 


1771 6751 


10-991 


1685 2694 


; 10-842 


1956 0467 


10-892 


i860 6492 


10-942 


1769 9043 


10-992 


16S3 5850 


J io'843 


1954 0917 


10-893 


1858 7895 


10-943 


1768 1353 


10-993 


i68i 9023 


10-844 


1952 1385 


10-894 


1856 9316 


10-944 


1766 3680 


10-994 


16S0 2212 


10-845 


1950 1874 


10-895 


1855 0756 


10-945 


1764 6025 


10-995 


1678 5418 


10-846 


1948 2382 


10-896 


1853 2215 


10-946 


1762 838S 


10-996 


1676 8641 1 


10-847 


1946 2909 


10-897 


1851 3692 


10-947 


1761 0768 


10-997 


1675 1881 


10-848 


1944 3456 


10-898 


1849 5187 


10-948 


1759 3166 


10-998 


1673 5138 1 


10-849 


1942 4022 


10-899 


1847 6701 


10-949 


1757 5582 


10-999 


167I 8411 



>06 



MR F. W. NEWMAN'S TABLE 



11-199] 



1 

1 * 


p-X 

^ 1 


X 


f.~X 


X 


e-* 1 


X 


e-x 


1 1 000 


1670 1701 


11 050 


15S8 7149 


IIIOO 


1511 2324 


11-150 


1437 5287 


Il'OOI 


166S 5007 


11 05 1 


15S7 1270 


II-lOl 


1509 7219 


11-151 


1436 0919 


I1-002 


1666 8331 


11052 


1585 5407 


I1-I02 


1508 2129 


11-152 


1434 6565 


11-003 


1665 1671 


"•053 


1583 9559 


IIIO3 


1506 7055 


"■153 


1433 2226 


11-004 


1663 5027 


11054 


1582 3727 


IIIO4 


1505 1995 


"•154 


143 1 7901 


1 1 005 


1661 8401 


"•055 


15S0 7912 


11-105 


1503 6951 


"•155 


143° 3590 


1 1 006 


1660 1791 


11-056 


1579 2112 


11-106 


1502 1921 


11-156 


1428 9293 


11-007 


1658 5197 


11057 


1577 6327 


11-107 


1500 6907 


11-157 


1427 5011 


1 1 -008 


1656 8620 


11 058 


1576 0559 


11-108 


1499 1907 


11-158 


1426 0743 


11-009 


1655 2060 


11-059 


1574 4806 


11-109 


1497 6923 


11-159 


1424 6490 


II'OIO 


1653 5516 


1 1 060 


1572 9069 


II-IIO 


1496 1954 


1 1 -160 


1423 2250 


11 01 1 


1651 8989 


ii-o6i 


1571 334S 


Il-III 


1494 6999 


11-161 


1421 8025 


11012 


1650 2478 


1 1 -062 


1569 7643 


1I-II2 


1493 2060 


11-162 


1420 3814 


11-013 


164S 5984 


1 1 063 


I56S 1953 


IIII3 


1491 7135 


11-163 


1418 9618 


11*014 


1646 9506 


1 1 -064 


1566 6279 


11-114 


1490 2225 


11-164 


1417 5435 


11-015 


1645 3045 


11-065 


1565 0620 


I1-II5 


1488 7331 


11-165 


1416 1267 


11 016 


1643 6600 


II 066 


1563 4977 


11-116 


1487 2451 


11-166 


1414 7"3 


11-017 


1642 0172 


11-067 


1561 9350 


11117 


14S5 7586 


11-167 


1413 2973 


11018 


1640 3760 


1 1 068 


1560 3739 


iiiiS 


1484 2735 


11-168 


1411 8847 


11-019 


1638 7364 i 


1 1 -069 


1558 8143 


11-119 


1482 7900 


H-169 


1410 4735 


I I -020 


1637 0985 


11-070 


1557 2562 


I1-I20 


1481 3080 


11-170 


1409 0637 


I I -02 I 


1635 4622 


11-071 


1555 6998 


III2I 


1479 8274 


11171 


1407 6554 


11 022 


1633 8276 


11072 


1554 1448 


11-122 


1478 3483 , 


11-172 


1406 2484 


11 023 


1632 1946 


11073 


1552 5915 


III23 


1476 8707 


11-173 


1404 8429 


1 1 024 


1630 5632 


11074 


1551 0397 


III24 


1475 3946 


11-174 


1403 4387 


11 025 


1628 9334 


11-075 


1549 4S94 


11x25 


1473 9199 


"•175 


1402 0360 


II 026 


1627 3053 


1 1 076 


1547 9407 


11-126 


1472 4467 


11176 


1400 6347 


11 027 


1625 6788 


11-077 


1546 3935 


III27 


1470 9750 


II-I77 


1399 2347 


11 028 


1624 0539 


11-078 


1544 8479 


11-128 


1469 5048 


11178 


1397 8362 


1 1 029 


1622 4307 


11079 


1543 3038 


II-I29 


1468 0360 


11-179 


1396 4390 


I I 030 


1620 8091 


1 1 -080 


1541 7613 


11-130 


1466 56S7 


11-180 


1395 0433 


I I -03 I 


1619 1891 


11081 


1540 2203 


II-I3I 


1465 1029 


11-181 


1393 649° 


11 032 


1617 5707 


11-082 


1538 6808 


11-132 


1463 63S5 


11-182 


1392 2560 


"•033 


1615 9539 


1 1 083 


1537 1429 


III33 


1462 1756 


11-183 


1390 8644 


11034 


1614 3388 


11-084 


1535 6066 


HI34 


1460 7142 


11184 


1389 4743 


"•035 


1612 7253 


11-085 


1534 0717 


III35 


1459 2542 


11-185 


1388 085s 


1 1 -036 


1611 1133 


1 1 086 


1532 5384 


11-136 


1457 7956 


11-186 


1386 6981 


1 1 037 


1609 5030 


11-087 


1531 0066 


11137 


1456 3386 


11-187 


1385 3121 


11-038 


1607 8963 


11-088 


1529 4764 


11-138 


1454 8830 


11-188 


1383 9275 


11039 


1606 2872 


1 1 089 


1527 9477 


11-139 


1453 4288 


11-189 


1382 5442 


11 040 


1604 68i8 


1 1 090 


1526 4205 


11-140 


1451 9761 


11-190 


1381 1624 


11-041 


1603 0779 


1 1 091 


1524 8949 


11141 


1450 5249 . 


11191 


1379 7819 


1 1 -042 


i6oi 4756 


1 1 092 


1523 3707 


11-142 


1449 0751 


11-192 


1378 4028 


11 043 


1599 8749 


11-093 


1521 8481 


II-I43 


1447 6267 


11193 


1377 0251 


11-044 


1598 2759 


11-094 


1520 3270 


i "-'^-^ 


1446 1798 


11-194 


1375 6488 


11045 


1596 6784 


11-095 


1518 8075 


III4S 


1444 7343 


"-195 


1374 2738 


11-046 


1595 0825 


1 1 -096 


1517 2894 


11-146 


1443 2903 


11-196 


1372 9002 


11047 


1593 4882 


11-097 


1515 7729 


11-147 


1441 8478 


11-197 


1371 5280 


1 1 -048 


1591 8955 


11-098 


iSU 2579 


11148 


1440 4066 


11-198 


1370 1572 


11-049 


1590 3044 


' 11-099 


15 1 2 7444 


11-149 


1438 9670 


11199 


1368 7877 



[II-200— ir399] 



OF THE DESCENDING EXPONENTIAL. 



207 



1 ^ 

1 


Q~X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


1 

I I "200 


1367 4196 


: 11-250 


1300 7298 


11-300 


1237 2924 


11-35° 


1 1 76 9490 


II-20I 


1366 0529 


! 11-251 


1299 4297 


11-301 


1236 0557 


11-351 


"75 7726 


II-202 


1364 6875 


1 11-252 


1298 1309 


11-302 


1234 8203 


11-352 


"74 5974 


II-203 


1363 3235 


11-253 


1296 8334 


11-303 


1233 5861 


11-353 


"73 4234 


H'204 


1361 9608 


11-254 


129s 5372 


11-304 


1232 3531 


"•354 


1172 2506 


1 1 '205 


1360 5996 


1 11-255 


1294 2423 


11-305 


I23I I2I4 


11-355 


1171 0789 


1 1 '206 


1359 2396 


i 11-256 


1292 9487 


11-306 


1229 8909 


11-356 


I 169 9084 


II"207 


1357 8811 


11-257 


1 291 6564 


"■307 


1228 6616 


11-357 


"68 7391 


1 1 '208 


1356 5239 


11-258 


1290 3654 


11-308 


1227 4336 


11-358 


"67 5709 


11-209 


1355 1680 


j "•259 


1289 0757 


11-309 


1226 2067 


11-359 


"66 4039 


II-2IO 


1353 8135 


11-260 


1287 7873 


11-310 


1224 9812 


11-360 


1165 2381 


1 1 -2 I I 


1352 4604 


11-261 


1286 5001 


11-311 


1223 7568 


11-361 


"64 0735 


1 1 -2 I 2 


1351 1086 


11-262 


1285 2143 


11-312 


1222 5336 


11-362 


1162 gioo 


11-213 


1349 7582 


11-263 


1283 9297 


11-313 


I22I 3II7 


11-363 


ii6i 7476 


II-2I4 


1348 4091 


11-264 


1282 6464 


11-314 


1220 0910 


11-364 


1160 5865 


II-2I5 


1347 0614 


11-265 


I28I 3644 


"•315 


1218 8715 


11-365 


"59 4265 


II-2l6 


1345 715° 


11-266 


1280 0837 


11-316 


I217 6533 


11-366 


1158 2676 


11-217 


1344 3699 


11-267 


1278 8042 


11-317 


1216 4362 


11-367 


"57 1099 


II-2I8 


1343 0262 


11-268 


1277 5261 


11-318 


I215 2204 


11-368 


"55 9534 


I I -2 I 9 


1341 6839 


11-269 


1276 2492 


11-319 


I214 0058 


11-369 


"54 7980 


11-220 


1340 3429 


11-270 


1274 9736 


11-320 


1212 7924 


11-370 


"53 6438 


II-22I 


1339 0=32 


I I -2 7 I 


1273 6993 


11-321 


I2II 5802 


11-371 


1152 4907 


11-222 


1337 6649 


11-272 


1272 4262 


11-322 


I2IO 3692 


11-372 


1151 3388 


11-223 


^ii(> 3279 


11-273 


I27I 1544 


11-323 


1209 159s 


11-373 


1150 1881 


11-224 


1334 9922 


11-274 


1269 8839 


11-324 


1207 9509 


11-374 


"49 0384 


II-22S 


1333 6579 


11-275 


1268 6146 


11-325 


1206 7436 


11-375 


I 147 8900 


11-226 


1332 3249 


11-276 


1267 3466 


11-326 


1205 5374 


11-376 


1146 7427 


11-227 


1330 9932 


11-277 


1266 0799 


11-327 


1204 3325 


11-377 


"45 5965 


11-228 


1329 6629 


11-278 


1264 8145 


11-328 


1203 1288 


11-378 


"44 4515 


11-229 


1328 3339 


11-279 


1263 5503 


11-329 


1201 9262 


11-379 


"43 3076 


11-230 


1327 0062 


11-280 


1262 2874 


11-330 


1200 7249 


11-380 


1142 1649 


II-23I 


1325 6799 


11-281 


I26I 0257 


11-331 


1199 5248 


11-381 


1141 0233 


11-232 


1324 3549 


11-282 


1259 7653 


11-332 


1198 3259 


11-382 


1139 8828 


11-233 


1323 0312 


11-283 


1258 5062 


11-333 


1197 1281 


"•383 


1138 7435 


11-234 


1321 7088 


11-284 


1257 2483 


11-334 


"95 9316 


11-384 


"37 6053 


"•23s 


1320 3S78 


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 


"35 3324 


11-237 


1317 7496 


11-287 


1253 4822 


11-337 


1192 3492 


11-387 


"34 1976 


11-238 


1316 4325 


11-288 


1252 2294 


11-338 


1191 1574 


11-388 


1 133 0640 


11-239 


1315 1168 


11-289 


1250 9778 


11-339 


1189 9669 


11-389 


1131 9315 


11-240 


1313 8023 


11-290 


1249 7274 


11 -340 


1188 7775 


11-390 


1130 8001 


II-24I 


1312 4892 


11-291 


1248 4783 


H-341 


1187 5893 1 


11-391 


1129 6699 


11-242 


1311 1773 


11-292 


1247 2305 


11-342 


1186 4023 


n-392 


1128 5408 


11-243 


1309 8668 


11-293 


1245 9839 


11-343 


1185 2165 : 


11-393 


1127 4128 


11-244 


1308 5576 


11-294 


1244 7385 


11-344 


1184 0319 


11-394 


1126 2860 


11-245 


1307 2497 


11-295 


1243 4944 


11-345 


1182 8484 


11-395 


1125 1602 


H-246 


1305 9431 


11-296 


1242 2515 


11-346 


ii8i 6662 


11-396 


1124 0356 


11-247 


1304 6378 


11-297 


I24I 0099 


n-347 


1 180 4851 


11-397 


1122 9122 


H-248 


1303 3338 


11-298 


1239 7695 


11-348 


1179 3052 


11-398 


1121 7898 


11-249 


1302 0311 , 


11-299 


1238 53°3 


11-349 


1178 1265 


11-399 


1 1 20 6686 



20S 



MR F. W. NEWMAN'S TABLE 



[11-400— 11-599] 



X 


e-* 


X 


C-* 


X 


Q-X 


X 


(,-X 

\ 


11-400 


1119 5485 


11-450 


1064 9475 


11-500 


1013 0093 


11 -550 


963 6043 


1 1 -401 


iiiS 4295 


IP-45I 


1063 S830 


11-501 


ion 91)68 


1 1 -55 1 


962 6412 


1 1 -402 


III7 3II6 


"•452 


1062 8197 


11-502 


1010 9854 


11-552 


961 6790 


11-403 


III6 1949 


"•453 


1061 7574 


11-503 


1009 9749 


11553 


960 7178 


11-404 


III5 0792 


"•454 


1060 6962 


11-504 


1008 9654 


"•554 


959 7576 


H-40S 


III3 9647 


11-455 


1059 6360 


11-505 


1007 9569 


11-555 


958 7983 


1 1 -406 


II12 8513 


11-456 


1058 5769 


11-506 


1006 9495 


11-556 


957 8400 


11-407 


IIII 7390 


11-457 


1057 5189 


11-507 


1005 9430 


11-557 


956 8826 


11-408 


1110 6278 


11-458 


1056 4619 


11-508 


1004 9376 


11-558 


955 9262 


11-409 


1109 5177 


11-459 


1055 4059 


n-509 


1003 9332 


11-559 


954 97°8 


1 11-410 


I 108 4088 


11-460 


1054 35" 


11-510 


1002 9297 


11-560 


954 0163 


II-4II 


1107 3009 


11-461 


1053 2972 


11-511 


1001 9273 


i 11-561 


953 0627 


II-4I2 


H06 1942 


11-462 


1052 2445 


11-512 


1000 9259 


11-562 


952 1102 


"■413 


I 105 0885 


11-463 


1051 1927 


11-513 


0999 9255 


11-563 


951 1585 


11-414 


IIO3 9S4O 


11-464 


1050 1421 


11-514 


998 9260 


11-564 


950 2078 


11-415 


I 102 8806 


11-465 


1049 0925 


11-515 


997 9276 


11-565 


949 2581 


II-4I6 


iioi 7782 


11-466 


1048 0439 


11-516 


996 9302 


11-566 


948 3093 


11-417 


iioo 6770 


11-467 


1046 9964 


11-517 


995 9337 


11-567 


947 3615 


11-418 


1099 5769 


11-468 


1045 9499 


11-518 


994 9383 


11-568 


946 4146 


II-4I9 


1098 4779 


11-469 


1044 9045 


11-519 


993 9439 


11-569 


945 4687 


11-420 


1097 3799 


11-470 


1043 8601 


11-520 


992 9504 


11-570 


944 5237 


1 1 -42 1 


1096 2831 


11-471 


1042 8168 


11-521 


991 9580 


11-571 


943 5796 


11-422 


1095 1874 


11-472 


1041 7745 


11-522 


990 9665 


11-572 


942 6365 


11-423 


1094 0927 


"•473 


1040 7332 


"•523 


989 9760 


11-573 


941 6943 


11-424 


1092 9992 


11-474 


1039 6930 


11-524 


988 9866 


11-574 


940 7531 


"•425 


1091 9067 


11-475 


1038 6538 


"■525 


987 9981 


11-575 


939 812S 


11-426 


1090 8154 


11-476 


1037 6157 


11-526 


9S7 0106 


11576 


938 8735 


11-427 


1089 7251 


11-477 


1036 5786 : 


11-527 


986 0240 


11-577 


937 9351 


n-428 


1088 6359 


11-478 


1035 5425 


11-528 


985 0385 


11-57S 


936 9976 


11-429 


1087 5478 


11-479 


1034 5075 


11-529 


9S4 0540 . 


11-579 


936 0611 


11 -430 


1086 4608 


1 1 -480 


1033 4735 


11-530 


9S3 0704 


11-580 


935 1255 


"•431 


1085 3749 


11-481 


1032 4406 


11-531 


982 0878 


11-581 


934 1908 


11-432 


1084 2901 


11-482 


1031 4086 


11-532 


981 1062 


11582 


933 2571 


"•433 


1083 2063 


11-483 


1030 3777 


11-533 


980 1256 


11-583 


932 3243 


I '•434 


1082 1237 


11-484 


1029 3479 


11-534 


979 1460 


11-584 


931 3925 


"•435 


io8i 0421 


11-485 


1028 3190 


11-535 


978 1673 


11-5S5 


930 4615 


11-436 


1079 9616 


11-486 


1027 2912 


11-536 


977 1896 


11-586 


929 5315 


"•437 


1078 8821 


11-487 


1026 2645 


11537 


976 2129 


11587 


928 6025 


11-438 


1077 8038 


ir-488 


1025 2387 


11-538 


975 2372 


11-588 


927 6743 


"•439 


1076 7265 


11-489 


1024 2140 


11539 


974 2625 


11-589 


926 7471 


11-440 


1075 6504 


11-490 


1023 1903 


11-540 


973 2887 


11-590 


925 8208 


11-441 


1074 5752 


11-491 


1022 1676 


11-541 


972 3159 


11-591 


924 8955 


11-442 


1073 5012 


11-492 


1021 1459 


11-542 


971 3441 


11-592 


923 9710 


"•443 


1072 4282 


"■493 


1020 1253 


11-543 


970 3732 


11593 


923 0475 


11-444 


1071 3563 


11-494 


1019 1057 


11-544 


969 4033 


11-594 


922 1250 


"•445 


1070 285s 


11-495 


1018 0S71 


11-545 


968 4344 


11-595 


921 2033 


11-446 


1069 2158 


11-496 


1017 0695 


11-546 


967 4664 


11-596 


920 2825 


11-447 


1068 1471 


11-497 


1016 0529 


11-547 


966 4995 


11-597 


919 3627 


11 -448 


1067 0795 


11-498 


1015 0374 


11-548 


965 5334 


11-598 


918 4438 


11-449 


1066 0129 


11-499 


1014 0229 


11-549 


964 5684 


11599 


917 5258 



[ii'6oo — 11799] 



OF THE DESCENDING EXPONENTIAL. 



209 



X 


(,-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


1 1 -600 


916 6088 


n-650 


871 9052 


11-700 


829 3819 


11-750 


788 9325 


1 1 -601 


915 6926 


1 1 -65 1 


871 0338 


11-701 


828 5529 


11-751 


788 1439 


1 1 -602 


914 7774 


11-652 


870 1632 


11-702 


827 7248 


"■752 


787 3562 


11-603 


913 8631 


"■653 


869 2934 


11703 


826 8975 ' 


"■753 


786 5692 


11-604 


912 9497 


11-654 


868 4246 : 


11-704 


826 0710 


"-754 


785 7830 


11-605 


912 0372 


"•655 


867 5566 


1 1 705 


825 2453 


"-75S 


784 9977. 


11-606 


911 1256 


11-656 


866 6895 , 


11-706 


824 4205 


11756 


784 2131 


11-607 


910 2149 


11-657 


865 8232 


11707 


823 5965 


11-757 


783 4292 


11 -608 


909 3051 


11-658 


864 9578 


11-708 


822 7733 


11758 


782 6462 


11-609 


908 3963 


11-659 


864 0933 


11-709 


821 9510 


11-759 


781 8639 


II-6I0 


907 4884 


11 -660 


863 2296 


11-710 


821 1294 


11-760 


781 0825 


ii-6ii 


906 5813 


11-661 


862 3668 


11-711 


820 3087 


11-761 


780 3018 


II-6l2 


905 6752 


11-662 


861 5049 


11-712 


819 4888 


11762 


779 5219 


11-613 


904 7700 


11-663 


860 6438 


11713 


818 6697 


11-763 


778 7427 


11-614 


903 8657 


11-664 


859 7836 


11-714 


817 8515 


11-764 


777 9644 


11-615 


902 9622 


11-665 


858 9242 


11-71S 


817 0340 


11-765 


777 1868 


11-616 


902 0597 


11-666 


858 0657 


11-716 


816 2174 


11-766 


776 4100 


11-617 


901 15S1 


11-667 


857 2081 


11-717 


815 4016 


11767 


775 6340 


11-618 


900 2574 


11-668 


856 3513 


11-718 


814 5866 


11-768 


774 8587 


11-619 


899 3576 


11-669 


855 4954 


11-719 


813 7724 


11-769 


774 0843 


11-620 


898 4587 


11-670 


854 6403 


11-720 


812 9590 


11-770 


773 3i°6 


11-621 


897 5607 


11-671 


853 7861 


11-721 


812 1465 


11-771 


772 5376 


11-622 


896 6636 


11-672 


852 9328 


11-722 


811 3348 


11-772 


771 7655 


11-623 


895 7674 


11-673 


852 0803 


11723 


810 5238 


"•773 


77° 9941 


11-624 


894 8720 


11-674 


851 2286 


11-724 


809 7137 


11774 


77° 2235 


11-625 


893 9776 


11-675 


850 3778 


11-725 


808 9044 


"•775 


769 4537 


n-626 


893 0841 


11-676 


849 5279 


11-726 


808 0959 


11-776 


768 6846 


11-627 


892 1914 


11-677 


848 6788 


11-727 


807 2882 


11-777 


767 9163 


11-628 


891 2997 


11-678 


847 8305 


11-728 


806 4813 


11-778 


767 1488 


11629 


890 4088 


11-679 


846 9831 


11729 


805 6752 


11-779 


766 3820 


11-630 


889 51S9 


11-680 


846 1365 


11-73° 


804 8700 


11-780 


765 6160 


11-631 


888 6298 


11-681 


845 2908 


"-731 


804 0655 


11-781 


764 8508 


11-632 


887 7416 


11-682 


844 4460 


11732 


803 2618 


11782 


764 0863 


11-633 


886 8543 


11-683 


843 6019 


11-733 


802 4590 


11783 


763 3226 


11-634 


885 9679 


11-684 


842 7587 


"■734 


801 6569 


11784 


762 5596 


11-635 


885 0824 


11-685 


841 9164 


11-735 


800 8557 


11-785 


761 7975 


11-636 


884 1977 


! 11-686 


841 0749 


11736 


800 0552 


11786 


761 0361 


11-637 


883 3140 


11-687 


840 2343 


"•737 


799 2556 


11-787 


760 2754 


11-638 


882 4311 


11-688 


839 3944 


11-738 


798 4567 


11788' 


759 5155 


11-639 


881 5491 


11-689 


838 5555 


"■739 


797 6586 


11789 


758 7564 


11-640 


880 6680 


11 -690 


837 7173 


11-740 


796 8614 


11-790 


757 9980 


1 1 -641 


879 7878 


11-691 


836 8800 


11-741 


796 0649 


11791 


757 2404 


11-642 


878 9084 


11-692 


836 0436 


11-742 


795 2692 


11792 


756 4835 


11-643 


878 0300 


11-693 


835 2080 


"-743 


794 4744 


11-793 


755 7274 


11-644 


877 1524 


11-694 


834 3732 


11-744 


793 6803 


11-794 


754 9721 


11-645 


876 2757 


11-695 


833 5392 


"•745 


792 8870 


"•795 


754 2175 


11-646 


875 3998 


11-696 


832 7061 


11-746 


792 0945 


11-796 


753 4636 


11-647 


874 5249 


11-697 


831 8738 


"■747 


791 3028 


11-797 


752 7105 


11-648 


873 6508 


J 11-698 


831 0423 


11-748 


790 5119 


11-798 


751 9582 


11-649 


872 7776 


1 11-699 


830 2117 


! 11-749 


789 7218 


11-799 


751 2066 



Vol. XIII. Part III. 



28 



210 



MR F. W. NEWMAN'S TABLE 



[ifSoo — 11-999] 



X 


C-.C 


X 


g-X 


X 


Q.-X 


i X 


Q-X 


ii-Soo 


750 455S 


11-850 


713 S556 


11-900 


679 0405 


11-950 


645 9233 


ii-Soi 


749 7057 


11-851 


713 1421 


11-901 


678 361S 


11-951 


645 2777 


11-S02 


74S 9564 


H-SS2 


712 4294 


11-902 


677 6837 


11-952 


644 6327 


11-803 


74S 2078 


"•853 


711 7173 


11-903 


677 0064 


11953 


643 9884 


11-804 


747 4600 


11-854 


7x1 0059 


11-904 


676 3297 


11-954 


643 3447 


11-805 


746 7129 


11-855 


710 2953 


11-905 


675 6537 


11-955 


642 7017 


II 806 


745 9665 


11-856 


709 5853 


11-906 


674 9784 


11-956 


642 0593 


11-807 


745 2209 


11-857 


708 8761 


11-907 


674 3038 


11-957 


641 4176 


11-S08 


744 4761 


, "-858 


708 1676 


11-908 


673 6298 


1 11-958 


640 7765 


1 1 -809 


743 7320 


1 11-859 


707 4598 


11-909 


672 9565 


11-959 


640 1360 


11-810 


742 98S6 


11-860 


706 7526 


11-910 


672 2839 


11-960 


639 4962 


ii-Sn 


742 2460 


n-86i 


706 0463 


11 -9 1 1 


671 6120 


11-961 


63S 8571 


11-812 


741 5°4i 


11-862 


705 3406 


11 -9 1 2 


670 9407 


11-962 


638 2185 


11-813 


740 7630 


11-863 


704 6356 


"■913 


670 2701 


11-963 


637 5806 


11-814 


740 0226 


, 11-864 


703 9313 


j 11-914 


669 6001 


11 -964 


636 9454 


H-815 


739 2829 


11-865 


703 2277 


11-915 


668 9309 


11-965 


636 3067 


11-816 


• 73i5 5440 


11-866 


702 5248 


11-916 


668 2623 


11-966 


635 6707 


11-817 


737 8059 


11-S67 


701 8227 


11-917 


667 5944 


11-967 


635 0354 


11-818 


737 0684 


11-868 


701 1212 


11-918 


666 9271 


n-968 


634 4007 


11-819 


736 3317 


11-S69 


700 4204 


H-919 


666 2605 


11-969 


633 7666 


11-820 


735 5958 


11-870 


699 7203 


11-920 


665 5946 


11-970 


633 1331 


H-821 


734 8605 


11-871 


699 0210 


11-921 


664 9293 


11-971 


632 5003 


11-822 


734 1260 


11-872 


698 3223 


11-922 


■664 2647 


11-972 


631 86Si 


11-823 


733 3923 


"-873 


697 6243 


11-923 


663 600S 


11-973 


631 2366 


11-824 


732 6592 


, 11-874 


696 9271 


11-924 


662 9375 


11-974 


630 6057 


11-825 


731 927° 


11-875 


696 2305 


11-925 


662 2749 


11-975 


629 9754 


11-826 


731 1954 


11-876 


695 5346 


11-926 


661 6130 


11-976 


629 3457 


11-827 


73° 4646 


11-877 


694 8394 


11-927 


660 9517 


11-977 


628 7167 


11-828 


729 7345 


11-878 


694 1449 


11-928 


660 2911 


11-978 


628 08S3 


11-829 


729 0051 


11-879 


693 45 1 1 


11-929 


659 6311 


11-979 


627 4605 


11-830 


728 2765 


ii-SSo 


692 7580 


11-930 


658 9718 


11-980 


626 8334 


11-831 


727 5485 


ii-SSi 


692 0656 


11-931 


65S 3132 


II -98 1 


626 2068 


11-832 


726 8214 


11-882 


691 3739 


11-932 


657 6552 


11-982 


625 5809 


^y»33 


726 0949 


11-883 


690 6829 


11-933 


656 9978 


11-9S3 


624 9557 


"•834 


725 3692 


11-884 


689 9925 


"-934 


656 3412 


11-984 


624 3310 


"■835 


724 6442 


11-885 


689 3029 


11-935 


655 6S52 


11-985 


623 7070 


11-836 


723 9199 


11-886 


688 6139 


11-936 


655 0298 


11-986 


623 0836 


11-837 


723 1963 


11-887 


687 9256 


11-937 


654 3751 


11-987 


622 460S 


11-838 


722 4735 


II -888 


687 2380 


11-938 


653 7211 


11-988 


621 8387 


11-839 


721 7514 


11-889 


686 5512 


11-939 


653 0677 


11-989 


621 2172 


11-840 


721 0300 


11-890 


685 8649 j 


11-940 


652 4149 


11-990 


620 5963 


11-841 


720 3093 


11-891 


685 1794 [ 


1 1 -941 


651 7628 


11-991 


619 9760 


11-842 


719 5894 


11-892 


684 4946 ' 


11-942 


651 1 1 14 


11-992 


619 3563 


11-843 


718 8702 


11893 


683 8104 


11-943 


650 4606 


11-993 


618 7373 


11-844 


718 1516 


11-894 


683 1270 


11-944 


649 8105 


11-994 


61S 1188 


11-845 


717 4339 


11-895 


682 4442 


11-945 


649 1610 


11-995 


617 5010 


11-846 , 


716 7168 


11-896 


681 7621 


11-946 


648 5 121 


11-996 


616 8838 


11-847 


716 0004 


11-897 


681 0807 : 


11-947 


647 8640 


11-997 


6i6 2673 


11-848 


715 2848 


11 -898 


680 3999 ) 


11-948 


647 2164 


11-998 


615 6513 


11-849 


714 5699 


11-899 


679 7199 ' 


11-949 


646 5695 


11-999 


615 0360 



[l2'000 1 2 '199] 



OF THE DESCENDING EXPONENTIAL. 



211 



X 


Q-X 


X 


g-a 


X 


g-X 


X 


Q-X 


I2'0O0 


614 4212 


12-050 


584 4555 


12-100 


555 9513 


12-150 


528 


8372 


I2*OOI 


613 8071 


12-051 


583 8714 


12-101 


555 3956 


12-151 


528 


3087 


12 "002 


613 1936 


12-052 


583 2878 


12-102 


554 8405 


12-152 


527 


7806 


12-003 


612 5807 


12-053 


582 7048 


12-103 


554 2860 


I2-153 


527 


2531 


12-004 


611 9684 


12-054 


582 1224 


12-104 


553 7320 


12-154 


526 


7261 


12-005 


611 3568 


i2'o55 


581 5406 


12-105 


553 1785 


12-155 


526 


1997 


12 -006 


610 7457 


12-056 


580 9593 


12-106 


552 6256 


12-156 


525 


6737 


12-007 


610 1353 


12-057 


580 37S7 


12-107 


552 0732 


12-157 


525 


1483 


12-008 


609 5255 


12-05S 


579 7986 


i2-io8 


551 5214 


12-158 


524 


6234 


12-009 


608 9162 


12-059 


579 2191 


12-109 


550 9702 


12-159 


524 


0991 


12-010 


608 3076 


12-060 


578 6401 


12-110 


55° 4195 


12-160 


523 


5752 


12-011 


607 6996 


12-o6l 


578 0618 


I2-1II 


549 8694 


12-161 


523 


°Si9 


12-012 


607 0922 


12-062 


577 4840 


12-112 


549 3198 


12-162 


522 


5291 


12-013 


6o6 4854 


12-063 


576 9068 


12-113 


548 7707 


12-163 


522 


0069 


12-014 


605 8793 


12-064 


576 3302 


12-114 


548 2222 


12-164 


521 


4851 


12-015 


605 2737 


12-065 


575 7541 


12-115 


547 6743 


12-165 


520 


9639 


1 2 -0 1 6 


604 6687 


i2-o66 


575 1787 


I2-I16 


547 1269 


i2-i66 


520 


4432 


12-017 


604 0643 


12-067 


574 6038 


12-117 


546 5800 


12-167 


519 


9230 


12-018 


603 4606 


12-068 


574 0295 


I2-I18 


546 0337 


12-168 


519 


4033 


12-019 


602 8574 


12-069 


573 4557 


I2TI9 


545 4880 


12-169 


518 


8842 


12-020 


602 2549 


12-070 


572 8825 


12-120 


544 9427 


12-170 


518 


3656 


I2-021 


601 6529 


12-071 


572 3°99 


12-121 


544 39S1 


12-171 


517 


8475 


12-022 


601 0516 


12-072 


571 7379 


12-122 


543 8539 


12-172 


517 


3299 


12-023 


600 4508 


12-073 


571 1665 


12-123 


543 3i°4 


12-173 


516 


8128 


12-024 


S99 8507 


12-074 


570 5956 


12-124 


542 7673 


12-174 


516 


2962 


12-025 


599 2511 


12-075 


570 0253 


12-125 


542 2248 


12-175 


515 


7802 


12-026 


598 6522 


12-076 


569 4555 


12-126 


541 6829 


12-176 


515 


2647 


12-027 


598 °S38 


12-077 


568 8S64 


12-127 


541 1415 


12*177 


514 


7497 


12-028 


597 4561 


12-078 


568 317S 


12-128 


540 6006 


12-178 


514 


2352 


12-029 


596 8589 


12-079 


567 7497 


12-129 


540 0603 


12-179 


513 


7212 


12-030 


596 2623 


1 2 -080 


567 1823 


12-130 


539 5205 


I2'l8o 


513 


2077 


12-031 


595 6664 


i2-o8i 


566 6154 


12-131 


538 9812 


12-181 


512 


6948 


12-032 


595 °7ro 


12-082 


566 0490 


12-132 


538 4425 


12-182 


512 


1824 


12-033 


594 4762 


12-083 


565 4833 


12-133 


537 9043 


12-183 


5" 


6704 


12-034 


593 8821 


12-084 


564 9181 


12-134 


537 3667 


12-184 


511 


159° 


12-035 


593 2885 


12-085 


564 3534 


12-135 


536 8296 , 


12-185 


51° 


6481 


12-036 


592 6955 


12-086 


563 7S94 


12-136 


536 2930 


12-186 


51° 


1377 


12-037 


592 1031 


12-087 


563 2259 


12-137 


535 7570 


12-187 


5°9 


6278 


12-038 


591 5113 


12-088 


562 6629 


12-138 


535 2215 


12-188 


5°9 


1185 


12-039 


590 9201 


12-089 


562 1005 


12-139 


534 6866 


i2-i8g 


508 


6096 


12-040 


590 3294 


12-090 


561 5387 


12-140 


534 1522 


12-190 


508 


1012 


12-041 


589 7394 


12-091 


560 9775 


12-141 


533 6183 


12-191 


5°7 


5934 


12-042 


589 1499 


12-092 


560 4168 


12-142 


533 °849 


12-192 


5°7 


0861 


12-043 


588 5611 


I 2 -093 


559 8566 


12-143 


532 5521 


12-193 


■ 506 


5792 


1 2 -044 


587 9728 


J 2-094 


559 2970 


12-144 


532 0198 


12-194 


506 


0729 


12-045 


587 3851 


12-095 


558 7380 


12-145 


531 4881 


12-195 


5°5 


5671 1 

r I 


12-046 


586 7981 


12-096 


558 1796 


12-146 


53° 9568 


12-196 


505 


0618 


12-047 


586 2115 


12-097 


557 6217 


12-147 


53° 4261 


12-197 


504 


557° 


12-048 


585 6256 


12-098 


557 0643 


12-148 


529 8q6o 


12-198 


5°4 


0526 


12-049 


585 0403 


12-099 


556 507s 


12-149 


529 3664 


12-199 


5°3 


5488 



28—2 



212 



MR F. W. NEWMAN'S TABLE 



[12-200— 12-399] 



X 


g-x 


X 


(,-X 


X 


C-x 


X 


C-x 


I 2 -200 


503 0456 


12-250 


478 5II7 


12-300 


455 1744 


12-350 


432 9753 


I2-20I 


502 542S 


12-251 


47S 0335 


12-301 


454 7195 


12-351 


432 5425 


12202 


502 0405 


12-252 


477 5557 


12302 


454 2650 


12-352 


432 1102 


12203 


s°i 5387 


12-253 


477 0783 


12-303 


453 8110 


12-353 


431 6783 


12-204 


501 0374 


12-254 


476 6015 


12-304 


453 3574 


12-354 


431 2469 


12-205 


500 5366 


12-255 


476 1251 


12-305 


452 9042 


12-355 


430 8158 


12-206 


500 0363 


12-256 


475 6493 


12-306 


452 4516 


12-356 


433 3852 


12-207 


499 5365 


12-257 


475 1738 


12-307 


451 9993 


12-357 


429 9551 


I2-20S 


499 0372 


12-258 


474 6989 


12-308 


451 5476 


12-358 


429 5253 


12-209 


498 5385 


12-259 


474 2244 


12-309 


451 0962 


12-359 


429 0960 


12-210 


498 0402 


12-260 


473 7505 


12-310 


453 6454 


12-360 


428 6671 


12-211 


497 5424 


12-261 


473 2769 


12-311 


453 1950 


12-361 


428 2387 


12-212 


497 0451 


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 


12-214 


496 0520 


12264 


471 S592 


12-314 


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 


12366 


426 1028 


12-217 


494 5661 


12-267 


470 4458 


12-317 


447 5'^i9 


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 


46S 0994 


12-322 


445 2699 


12-372 


423 5539 


12223 


491 6075 


12-273 


467 6316 


12-323 


444 8249 


12-373 


423 1305 


12-224 


491 1162 


12-274 


467 1642 


12-324 


444 3833 


12-374 


422 7076 


12-225 


493 6253 


12-275 


466 6972 


12-325 


443 9361 


12-375 


422 2851 


12226 


493 1349 


12-276 


466 2308 


12-326 


443 4924 


12-376 


421 8630 


12-227 


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 


12-229 


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 9°54 


12-331 


441 2805 


12-381 


419 7590 


12232 


487 2029 


12-282 


463 4418 


12-332 


440 8394 


12-382 


419 3394 


12-233 


486 7160 


12-283 


462 9786 


12333 


440 3988 


12-383 


418 9203 


12234 


486 229s 


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 


4S5 2580 


12-286 


461 5917 


12-336 


439 0796 


12-386 


417 6654 


12-237 


4S4 7730 


12-287 


461 1303 


12-337 


438 6407 


12-387 


417 2480 


12-238 


484 28SJ 


12-288 


460 6694 


12-338 


438 2023 


12-388 


416 8309 


12-239 


483 8044 


12289 


460 2090 


12-339 


437 7643 


12389 


416 4143 


12-240 


4S3 3209 


12-290 


459 749=' 


12-340 


437 3268 


12-390 


415 9981 


12241 


482 8378 


12-291 


459 2895 


12-341 


436 8897 


12-391 


415 5823 


12-242 


482 3552 


12292 


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 


12294 


457 9^37 


12344 


435 5810. 


12-394 


414 3374 


12-245 


4S0 9103 


12-295 


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 


i 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 9905 


12-299 


455 6298 


12-349 


433 4085 


12-399 


412 2709 



[I2-400— 12-599] 



OF THE DESCENDING EXPONENTIAL. 



213 



X 


Q-X 


X 


Q-X 


X 


g-X 


X 


Q-X 


12-400 


411 8589 


12-450 


391 7723 


12-500 


372 6653 


12-550 


354 4902 


12-401 


411 4472 


12-451 


391 3807 


12-501 


372 2928 


12-551 


354 1359 


12-402 


411 0360 


12-452 


390 9895 


12-502 


371 9207 


12-552 


353 7819 


12-403 


410 6251 


12-453 


390 5987 


12-503 


371 549° 


12-553 


353 4283 


12-404 


410 2147 


i2"454 


390 2083 


12-504 


371 1776 


12-554 


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 0174 


12-408 


408 S77I 


12-458 


388 6506 


12-508 


369 6959 


12-558 


351 6656 


12-409 


408 1688 


12-459 


388 2621 


12-509 


369 3264 


12-559 


351 3141 


12-410 


407 7608 


12-460 


3S7 8741 


12-510 


368 9572 


12-560 


350 9630 


I2-4II 


407 3532 


1 2 -,1 6 1 


387 4864 


12-511 


368 5885 


12-561 


350 6122 


12-412 


406 9461 


12-462 


387 0991 


12-512 


368 2200 


12-562 


350 2617 


12-413 


4°6 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 7835 


12-518 


366 0173 


12-568 


348 1665 


12-419 


404 1074 


12-469 


384 3989 


12-519 


365 6515 


12-569 


347 8185 


12-420 


4°3 7035 


12-470 


384 0147 


12-520 


365 2860 


12-570 


347 4708 


12-421 


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 0919 


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 


401 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 0470 


12-428 


400 4868 


12-478 


380 9548 


12-528 


362 3754 


12-578 


344 7022 


12-429 


400 0865 


12-479 


380 5740 


12529 


362 0132 


i2'579 


344 3576 


12-430 


399 6866 


12-480 


380 1936 


12-530 


361 6514 


12-580 


344 0134 


12-431 


399 2871 


12-481 


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 


12-433 


398 4893 


12-483 


379 0548 


12-533 


360 5681 


12-583 


342 9829 


1 2 "434 


398 0910 


12-4S4 


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 48S0 


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 1 


12-439 


396 1056 


12-489 


376 7S73 


12-539 


358 4HI 


12-589 


340 9312 


12-440 


395 7096 


12-490 


376 4107 


12-540 


358 0529 


12-590 


340 5904 1 


12-441 


395 3141 


12-491 


376 0344 


12-541 


357 6950 


12-591 


340 2500 


12-442 


394 9190 


12-492 


375 6586 


12-542 


357 3375 


12-592 


339 9099 


12-443 


394 5243 


12-493 


375 2831 


12-543 


356 9833 


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 


12-546 


355 9110 


12-596 


338 5530 


12-447 


392 9494 


12-497 


373 7850 


12-547 


355 5553 


12-597 


338 2146 


12-448 


392 5566 


12-498 


373 4114 


12-548 


355 1999 


12-598 


337 8766 


12-449 

1 


392 1642 


12499 


373 0382 


12-549 


354 8449 


12-599 


337 5389 



2U 



MR F. W. NEWMAN'S TABLE 



[i2-6oo — 12-799] 



X 


C-x 


X 


Q-X 


X 


g-X 


X 


f>-X 


1 2 -600 


337 2015 


12-650 


320 7560 


12-700 


305 1125 j 


12-750 


290 2323 


12-601 


zz(> 8645 


12-651 


320 4354 


12-701 


304 S076 


12-751 


2S9 9419 


1 2 -602 


336 527S 


12-652 


320 II5I 


12-702 


3°4 5029 


12-752 


2S9 6521 


12-603 


zi(> 1914 


12-653 


319 7952 


12-703 


304 1986 


12-753 


289 3626 


12-604 


335 8554 


12-654 


319 4755 


12-704 


303 8945 


12-754 


289 0734 


12-605 


335 5197 


12-655 


319 1562 


12-705 


303 59°8 


12-755 


288 7845 


I 2 -606 


335 1844 


12-656 


318 8372 


12-706 


303 2874 


12756 


288 4959 


12607 


334 S493 


12-657 


318 5185 


12-707 


302 9842 1 


12-757 


288 2075 


12-608 


334 5147 


12-658 


318 2002 


12-708 


332 6814 \ 


12-758 


287 9194 


12-609 


334 1803 1 


12-659 


317 S821 


12-709 


332 3789 1 


12-759 


287 6317 


12-610 


Zo3, S463 


12 -660 


317 5644 


12-710 


302 0766 


12-760 


287 3442 


12-611 
12-612 


333 1793 


12-661 
12-662 


317 2470 
316 9299 


12-711 
12-712 


301 7747 
3=1 4731 


12-761 
12-762 


287 0570 
2S6 7701 


12-613 


332 8463 


12-663 


316 6132 


12-713 


331 1718 


12-763 


286 4834 


12-614 


zz^ 5136 


12-664 


316 2967 


12-714 


300 8707 


12-764 


286 1971 


12-615 


333 1S12 


12-665 


315 9806 


12-715 


300 5700 


12-765 


285 9110 


12-616 


331 8492 


12-666 


315 6647 


12-716 


300 2696 ; 


12-766 


285 6253 


12-617 


zz^ 5175 


12-66^7 


315 3492 


12-717 


299 9695 


12-767 


285 3398 


i2-6i8 


331 1862 


12-668 


315 0340 


12-7IS 


299 6697 


12-768 


2S5 0546 


12-619 


330 8552 


12-669 


314 7192 


12-719 


299 3701 


12-769 


284 7697 


12-620 


1 

zz° 5245 


12-670 


314 4046 


12-720 


299 0709 


12-770 


284 4851 


12-621 


zz^ 1941 


12-671 


314 0904 


12-721 


298 7720 


12-771 


284 2007 


12-622 


329 8641 


12-672 


z'^z 7764 


12-722 


298 4734 


12-772 


283 9167 


12-623 


329 5344 


12-673 


313 4628 


12-723 


298 1750 


12-773 


2S3 6329 


12-624 


329 2050 ' 


12-674 


313 1495 


12-724 


297 8770 


12-774 


283 3494 


12-625 


32S 8760 


12-675 


312 8365 


12-725 


297 5793 


12-775 


283 0662 


12-626 


328 5473 


12-676 


312 5238 


12-726 


297 2819 


12-776 


282 7833 


12-627 


32S 2189 


12-677 


312 2115 


12-727 


296 9847 


12-777 


282 5006 


12-628 


327 8908 


12-678 


311 8994 


12-728 


296 6879 


12-778 


282 2183 


12-629 


327 5631 


12-679 


3" 5877 


12-729 


296 3913 


12-779 


281 9362 


12-630 


327 2357 


1 2 -680 


311 2762 


12-730 


296 0951 


12-780 


281 6544 


12-631 


326 9086 


12-681 


310 9651 


12-731 


295 7992 


12-781 


281 3729 


12-632 


326 5819 


12-682 


31° 6543 


12-732 


295 5335 


12-782 


281 0916 


"■633 


326 255s 


12-683 


310 3438 


12-733 


295 2082 


12-783 


280 8107 


12-634 


325 9294 


12-684 


310 0336 


; 12-734 


294 9131 


12-784 


280 5300 


12635 


325 6036 


12-685 


309 7237 


12-735 


294 6183 


12-785 


280 2496 


12-636 


325 2782 


12-686 


309 4142 


12-736 


294 3239 


12-786 


279 9695 


12-637 


324 9531 


12-687 


309 1049 


12-737 


294 0297 


12-787 


279 6897 


12-638 


324 6283 


12-688 


308 7960 


12-738 


293 735S 


12-788 


279 4101 


12-639 


324 3038 


12-689 


308 4873 


12-739 


293 4422 


12-789 

1 


279 1309 


12-640 


323 9797 


; 12-690 


308 1790 


12-740 


293 1489 


' 12-790 


278 8519 


12-641 


323 6558 


12-691 


307 8710 


12-741 


292 8559 


12-791 


278 5732 


12-642 


323 3323 


12-692 


3°7 5632 


12-742 


292 5632 


12-792 


278 2947 


12-643 


323 0092 


12-693 


307 2558 


12-743 


292 2708 


12-793 


278 0166 


12-644 


322 6863 


12-694 


306 9487 


12-744 

1 


291 9787 


12-794 


277 7387 


12-645 


322 3638 


; 12-695 


306 6419 


12-745 


291 6868 


12-795 


277 4611 


1 2 -646 


322 0416 


' 12-696 


306 3354 


12-746 


291 3953 


12-796 


277 1838 


12-647 


321 7197 


12-697 


306 0293 


12-747 


291 1040 


12-797 


276 9067 


12-648 


321 3982 


12-698 


305 7234 


12-748 


293 8131 


12-798 


276 6300 


12-649 


321 0769 


12-699 


305 4178 


12-749 


293 5224 


12-799 


276 3535 



[i2-8oo — 12-999] 



OF THE DESCENDING EXPONENTIAL. 



:i5 



1 

X 


Q~X 


X 


Q-X 


X 


Q-X 


X 


(,-X 


12-800 


276 0772 


12-850 


262 6128 


12-900 


249 8050 


12-950 


237 6219 


i2-8oi 


275 8013 


12-851 


262 3503 


12-901 


249 5553 


12-951 


237 3844 


I2-8o2 


275 5256 


12-352 


262 0881 


12-902 


249 3059 


12-952 


237 1471 


12-803 


275 2503 


12-853 


261 8261 


12-903 


249 0567 


12-953 


236 9101 


I2-S04 


274 9751 


12-854 


261 5644 


12-904 


248 8078 


12-954 


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 


12-807 


274 I5I5 


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 


260 2599 


12-909 


247 566S 


12-959 


235 4929 


12-810 


273 33°2 


12-860 


259 9998 


12-910 


247 3194 


12-960 


235 2575 


I2-8II 


273 0570 


12-861 


259 7399 


12-911 


247 0722 


12-961 


235 °224 


12-8x2 


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 552S 


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 


i2-8i6 


271 6951 


12-866 


258 4444 


12-916 


245 8399 


12-966 


233 8502 


12-817 


271 4236 


12-867 


258 1861 


12-917 


245 5942 


12-967 


233 6165 


12-818 


271 1523 


12-868 


257 9281 


12-918 


245 3488 


12-968 


233 383° 


12-819 


270 8813 


12-869 


257 6703 


12-919 


245 i°35 


12-969 


233 1497 


12*820 


270 6105 


12-870 


257 4127 


12-920 


244 8586 


12-970 


232 9167 


12-821 


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 9917 


12-876 


255 8729 


12-926 


243 3938 


12-976 


231 5233 


12-827 


268 7229 


12-877 


255 6171 


12-927 


243 i5°5 


12-977 


231 2919 


12-828 


268 4543 


12-878 


255 3616 


12-928 


242 9075 


12-978 


231 0608 


12-829 


268 i860 


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 


12-8S1 


254 5967 


12-931 


242 1799 


12-981 


230 3686 


12-832 


267 3826 


12-882 


254 3422 


12-932 


241 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 9905 


12-838 


265 7831 


12-888 


252 8207 


12-938 


240 4905 


12-988 


228 7617 


12-839 


26s 5175 


12889 


252 5680 


12-939 


240 2502 


12-989 


228 5330 


12-840 


265 2521 


12-890 


252 3156 


12-940 


240 0100 


12-990 


228 3046 


12-841 


264 9870 


12-891 


252 0634 


12-941 


239 77°i 


12-991 


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 


12-993 


227 6207 


12-844 


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 3°5i 


12-948 


238 0976 


12-998 


226 4854 


12-849 


262 8755 


12-899 


250 0549 


12-949 


237 8596 


12-999 


226 2591 



•21G 




MR F. W. NEWMAN'S 


TABLE 




[13-000—13-199; 


X 


Q-X 


X 


(^-X 


X 


Q-X 


X 


Q-X 


13-000 


226 0329 


13-050 


215 0092 


13-100 


204 5231 


13-150 


194 5483 


I3-OOI 


225 8070 


13-051 


214 7943 


13-101 


204 3186 


13-151 


194 3539 


13"002 


225 5813 


13-052 


214 5796 


13-102 


204 1 144 


13-152 


194 1596 


13-003 


225 3558 


: 13-053 


214 3651 


13-103 


203 9104 


13153 


193 9656 


13-004 


225 1306 


13-054 


214 1508 


13-104 


203 7066 


13-154 


193 7717 


13-005 


224 9056 


13-055 


213 9368 


13-105 


203 5030 


13-155 


193 5780 


13-006 


224 6S08 


13-056 


213 7230 


13-106 


203 2996 


13156 


193 3846 


13-007 


224 4562 


13-057 


213 5094 


13-107 


203 0964 


13-157 


193 1913 


13-ooS 


224 2319 


13-058 


213 2960 


13108 


202 8934 


13-158 


192 99S2 


1 3 009 


224 0078 


13-059 


213 0828 


13-109 


202 6906 


\ 13-159 


192 8053 


13-010 


223 7839 


13-060 


212 8698 


13-110 


202 4880 


13-160 


192 6126 


13-on 


223 5602 


13-061 


212 6570 


I3-III 


202 2856 


13-161 


192 4200 


13012 


223 3367 


13-062 


212 4445 


13-112 


202 0834 


13-162 


192 2277 


13-013 


223 113s 


13-063 


212 2321 


13-113 


201 8815 


13-163 


192 0356 


13-014 


222 8905 


13-064 


212 0200 


13-114 


201 6797 


13-164 


191 8436 


' 13-015 


222 6677 


13-065 


211 8081 


i3'iiS 


201 4781 


13-165 


191 6519 


13016 


222 4452 


1 13-066 


211 5964 


13-116 


201 2767 


13166 


191 4603 


13-017 


222 2228 


13-067 


211 3849 


13-117 


201 0755 


13-167 


191 2690 


13-018 


222 0007 


13-068 


211 1736 


13-118 


200 8746 


13-168 


191 0778 


13-019 


221 7788 


13-069 


210 9626 


13-119 


200 6738 


13-169 


190 8868 


13-020 


221 5572 


13-070 


210 7517 


13-120 


200 4732 


13-170 


190 6960 


13-021 


221 3357 


13-071 


210 54H 


13-121 


203 2729 


13-171 


190 5054 


13-022 


221 1145 


13-072 


210 3306 


I3-I22 


200 0727 


13-172 


190 3150 


13-023 


220 8935 


13-073 


210 1204 


13-123 


199 8727 


13-173 


193 1248 


13-024 


220 6727 


13-074 


209 9104 


13-124 


199 6729 


13-174 


189 9348 


13-025 


220 4521 


13-075 


209 7006 


13-125 


199 4734 


13-175 


189 7449 


13-026 


220 2318 


13-076 


209 4910 


13-126 


199 2740 


13-176 


189 5553 


13-027 


220 OII7 


13-077 


209 2816 


13-127 


199 074S 1 


13-177 


189 3658 


13-028 


219 7918 


13-078 


209 0724 


13-128 


19S 8758 


13-178 


189 1766 


13-029 


219 5721 


13-079 


208 8634 


13-129 


19S 6771 


13-179 


188 9875 


13-030 


219 3526 


13-080 


208 6547 


13-130 


198 4785 


13-180 


188 7986 


13-031 


219 1334 


13-081 


208 4461 


I3-I3I 


198 2801 


13-181 


188 6099 


13-032 


218 9144 


13-082 


208 2378 


13132 


198 0819 


13-182 


188 4214 


13-033 


218 6956 


13-083 


208 0296 


13-133 


197 8839 


13-183 


188 2330 


13-034 


218 4770 


13-084 


207 8217 


13-134 


197 6861 


13-184 


188 0449 


13-035 


218 2586 


13-085 


207 6140 


13-135 


197 4886 


13-185 


187 8569 


13036 


218 0405 


13-086 


207 4065 


13-136 


197 2912 


13-186 


187 6692 


13-037 


217 8225 


13087 


207 1992 


13-137 


197 0940 


13-187 


187 4816 


13-038 


217 6048 


13-088 


206 9921 


13-138 


196 8970 


13-188 


187 2942 


13-039 


217 3873 


13089 


206 7852 


13-139 


196 7002 


13-189 


187 1070 


13-040 


217 1700 


13-090 


206 5785 


13-140 


196 5036 


13-190 


186 9200 


13-041 


216 9530 


13-091 


206 3721 


13-141 


196 3072 


13-191 


186 7332 


13-042 


216 7361 


13-092 


206 1658 


13142 


196 mo 


13-192 


186 5465 


13-043 


216 519s 


'3093 


205 9597 ! 


13-143 


195 9150 


13-193 


186 3601 


13-044 


216 3031 


13094 


205 7539 


13-144 


195 7191 


13-194 


186 1738 


13-045 


216 0869 


13-095 


205 5482 


13-145 


195 5235 


13*195 


185 9877 


13-046 


215 8709 


13096 


205 3428 


13-146 


195 3281 


13-196 


185 8018 


13-047 


215 6552 


13-097 


205 1375 


I3I47 


195 1329 


13-197 


185 6161 


13-048 


215 4396 


13098 


204 9325 


I3I48 


194 9378 


13-198 


185 4306 


13-049 i 


215 2243 ; 


13-099 


204 7277 


13149 


194 7430 


13199 


185 2453 



[i3'2oo— 13-399] 



OF THE DESCENDING EXPONENTIAL. 



217 



1 ^ 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


I3'200 


185 0601 


13-250 


176 0346 


13-300 


167 4493 


13-350 


159 2827 


i I3'20I 


184 8751 


13-251 


175 8587 


13-301 


167 2819 


13-351 


159 1235 


I3"202 


184 6904 


13-252 


175 6829 


13-302 


167 1147 


13-352 


158 9645 


i3'203 


184 5058 


13-253 


17s 5°73 


13-303 


166 9477 


13-353 


158 8056 


1 i3'204 


184 3213 


13-254 


175 3319 


13-304 


166 7809 


13-354 


158 6469 


13-205 


184 1371 


13-255 


175 1566 


13-305 


166 6142 


13-355 


158 4883 


13-206 


183 9531 


13-256 


174 9816 


13-306 


166 4476 


13-356 


158 3299 


13-207 


183 7692 


13-257 


174 8067 


13-307 


166 2813 


13-357 


158 1716 


13-208 


183 585s 


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 


13-210 


183 2187 


13-260 


174 2831 


13-310 


165 7832 


13-360 


157 6978 


I3-2II 


183 0356 


13-261 


174 1089 


13-311 


165 6175 


13-361 


157 5402 


13-212 


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 


13-214 


182 4873 


13-264 


173 5873 


13-314 


165 1214 


13-364 


157 0682 


13-215 


182 3049 


13-265 


173 4138 


13-315 


164 9563 


13-365 


156 9II3 


13-216 


182 1227 


13-266 


173 2405 


13-316 


164 7914 


13-366 


156 7544 


13-217 


181 9407 


13-267 


173 0673 


13-317 


164 6267 


13-367 


156 5977 


13-218 


181 7588 


13-268 


172 8944 


13-318 


164 4622 


13-368 


156 4412 


13-219 


181 5771 


13-269 


172 7216 


13-319 


164 2978 


13-369 


156 2849 


13-220 


181 3957 


13-270 


172 5489 


13-320 


164 1336 


13-370 


156 1287 


13-221 


181 2144 


13-271 


172 3765 


13-321 


163 9695 


13-371 


155 9726 


13-222 


181 0332 


13-272 


172 2042 


13-322 


163 8057 


13-372 


155 8167 


13-223 


180 8523 


13-273 


172 0321 


13-323 


163 6419 


13-373 


155 6610 


13-224 


180 6715 


13-274 


171 8601 


13-324 


163 4784 


13-374 


155 5054 


13-225 


180 4909 


13-275 


171 6883 


13-325 


163 3150 


13-375 


155 3500 


13-226 


180 3105 


13-276 


171 5167 


13-326 


163 1517 


13-376 


155 1947 


13-227 


180 1303 


13-277 


171 3453 


13-327 


162 9887 


13-377 


155 0396 


13-228 


179 95°3 


13-278 


171 1740 


13-328 


162 8258 


13-378 


154 8846 


13-229 


179 7704 


13-279 


171 0030 


13-329 


162 6630 


13-379 


154 7298 


13-230 


179 59°7 


13-280 


170 8320 


13-330 


162 5004 


13-380 


154 5752 


13-231 


179 4112 


13-281 


170 6613 


13-331 


162 3380 


13-381 


154 4207 


13-232 


179 2319 


13-282 


170 4907 


13-332 


162 1758 


13-382 


154 2663 


13-233 


179 0528 


13-283 


170 3203 


^IT:>2, 


162 0137 


13-383 


154 1122 


13-234 


178 8738 


13-284 


170 1501 


13-334 


161 8517 


13-384 


153 9581 


13-235 


178 6950 


13-285 


169 9800 


13-335 


161 6900 


13-385 


153 8042 


13-236 


178 5164 


13-286 


169 81C1 


13-336 


161 5284 


13-386 


153 6505 


13-237 


178 3380 


13-287 


169 6404 


13-337 


161 3669 


13-387 


153 4969 


13-238 


178 1597 


13-288 


169 4708 


13-338 


161 2056 


13-388 


153 3435 


13-239 


177 9817 


13-289 


169 3014 


13-339 


161 0445 


13-389 


153 1903 


13-240 


177 8038 


13-290 


169 1322 


13-340 


160 8835 


13-390 


153 0371 


13-241 


177 6261 


13-291 


168 9632 


13-341 


160 7227 


13-391 


152 8842 


13-242 


177 4485 


13-292 


168 7943 


13-342 


160 5621 


13-392 


152 7314 


13-243 


177 2712 


13-293 


168 6256 


13-343 


160 4016 


i-l'2,9Z 


152 5788 


13-244 


177 0940 


13-294 


168 4570 


13-344 


160 2413 


13-394 


152 4262 


i3"245 


176 9170 


13-295 


168 2887 


13-345 


160 0812 


13-395 


152 2739 


13-246 


176 7402 


13-296 


168 1205 


13-346 


159 9211 


13-396 


152 1217 


13-247 


176 5635 


13-297 


. 167 9524 


13-347 


159 7613 


13-397 


151 9696 


13-248 


176 3870 


13-298 


167 7845 


13-348 


159 6016 


13-398 


151 8177 


13-249 


176 2107 


13-299 


167 6168 


13-349 


159 4421 


13-399 


151 6660 

1 



Vol. XIII. Part III. 



21S 



MR F. W. NEWMAN'S TABLE 



[; 3-400— 13-599] 



X 


e-« 


X 


Q-X 


X 


e-« 


X 


Q-X 


i3"4oo 


151 5144 


13-450 


144 1250 


13-500 


137 0959 


13-550 


130 4097 


13-401 


151 3630 


13-451 


143 9809 


13-501 


136 9589 


13-551 


130 2793 


13402 


151 2117 


13-452 


143 8370 


13-502 


136 8220 


13-552 


130 1491 


13-403 


151 0605 


13-453 


143 6932 


13-503 


136 6852 


13-553 


130 0190 


13-404 


150 9096 


13-454 


143 5496 


, 13-504 


136 5486 


13554 


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 II96 


13-507 


136 1396 


13-557 


129 5000 


13-408 


150 3071 


13-458 


142 9766 


13-508 


136 0035 


13-558 


129 3706 


13-409 


150 1569 


13459 


142 8337 


13-509 


135 8676 


13-559 


129 2413 


13-410 


150 0068 


13-460 


142 6909 


13-510 


135 7318 


13-560 


129 II2I 


13-411 


149 8569 


13-461 


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 


13-413 


149 5575 


13-463 


142 2635 


13-513 


^zs 3252 


13-563 


128 7254 


13-414 


149 4080 ' 


13-464 


142 I2I3 


13-514 


135 1899 


13-564 


128 5967 


13-415 


149 2586 


13-465 


141 9792 


13-515 


135 0548 


13-565 


128 4682 


13-416 


149 1095 


13-466 


141 8373 


13-516 


134 9198 


13-566 


128 3398 


13-417 


148 9604 


13-467 


141 6955 


13-517 


134 7850 


13-567 


128 21x5 


13-418 


148 8115 1 


13-468 


141 5539 


USi^^ 


134 6502 


13-568 


I2S 0833 


13-419 


148 6628 


13-469 


141 4124 


13-519 


134 5157 


13-569 


127 9553 


13-420 


148 5142 


13-470 


141 2711 


13-520 


134 3812 


13-570 


127 8274 


13-421 


148 3658 


13-471 


141 1299 


13-521 


134 2469 


13-571 


127 6996 


13-422 


148 2175 


13-472 


140 9888 


13-522 


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 


13-474 


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 


13-426 


147 6258 


13-476 


140 4260 


13-526 


133 5773 


13-576 


127 0627 


13-427 


147 4782 


13-477 


140 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 


13-429 


147 1836 


13-479 


140 0054 


13-529 


133 1772 


13-579 


126 6821 


13-430 


147 0365 


13-480 


139 8654 


13-530 


133 0441 


13-580 


126 5555 


13-431 


146 8895 


13-481 


139 7256 


13-531 


132 9111 


13-5SI 


126 4290 


13-432 


146 7427 


13-482 


139 5860 


13-532 


132 7783 


13-582 


126 3026 


13-433 


146 5960 


13-483 


139 4465 


T-ym 


132 6456 


13-583 


126 1764 


13434 


146 4495 


13-484 


139 3071 


13-534 


132 5130 


13-584 


126 0503 


13-435 


146 3031 


13-485 


139 1678 


13-535 


132 3S05 


13-5S5 


125 9243 


13-436 


146 1569 


13-486 


139 0288 


13-536 


132 2482 


13-586 


125 7984 


13-437 


146 0108 


13-487 


138 8898 


13-537 


132 1160 


13-587 


125 6727 


13-438 


145 8649 


13-488 


138 7510 


13-538 


131 9840 


13-588 


125 5471 


13439 


145 7191 


13-489 


138 6123 


13-539 


131 8521 


13-589 


125 4216 


13-440 


145 5734 


13-490 


138 4737 


13-540 


131 7203 


13-590 


125 2962 


13-441 


145 4279 


13-491 


138 3353 


13-541 


131 5886 


13-591 


125 1710 


13-442 


145 2826 


13-492 


138 1971 


13-542 


131 4571 


13-592 


125 0459 


13-443 


145 1374 


13-493 


138 0589 


13-543 


131 3257 


13-593 


124 9209 


13-444 


144 9923 


13-494 


137 9209 


13544 


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 


13-546 


130 9323 


13-596 


124 5467 


13-447 


144 5580 


13-497 


137 5078 


13-547 


130 8015 


13-597 


124 4222 


13-448 


144 413s 


13-498 


137 3704 


13-548 


130 6707 


13-598 


124 2978 


13-449 

1 


144 2692 


13-499 


137 2331 


13-549 


130 5401 


13-599 


124 1736 



[i3-6oo— 13-799] 



OF THE DESCENDING EXPONENTIAL. 



219 



X 


(,-X 


X 


Q-X 


X 


C-x 


X 


e-x 


lyGoo 


124 0495 


13-650 


117 9995 


13-700 


1 
112 2446 


13-750 


106 7704 


13-601 


123 925s 


13-651 


117 8816 


13-701 


112 1324 


13-751 


106 6637 


13-602 


123 8016 


13-652 


117 7638 


13-702 


112 0204 


13-752 


106 5571 


1 3 '603 


123 6779 


13-653 


117 6461 1 


13-703 


III 9084 


13-753 


. 106 4506- 


13-604 


123 5543 


13-654 


117 5285 


13-704 


in 7965 


13-754 


106 3442 


13-605 


123 4308 


13-655 


H7 4110 


13-705 


III 6848 


13-755 


106 2379 


13-606 


123 3074 


13-656 


117 2937 


13-706 


III 5732 


13-756 


106 1317 


13-607 


123 1842 


13-657 


117 1764 


13-707 


III 4617 


13-757 


106 0256 


13-608 


123 0611 


13-658 


117 0593 


13-708 


III 3503 


13-758 


105 9196 


13609 


122 9381 


13-659 


116 9423 


13-709 


III 2390 


13759 


105 8138 


13-610 


122 8152 


13-660 


116 8254 


13-710 


III 1278 


13-760 


105 7080 


13-611 


122 6924 


13-661 


116 7087 


13-711 


III 0167 


13-761 


105 6024 


13-612 


122 5698 


13-662 


116 5920 


13-712 


no 9057 


13-762 


105 4968 


13-613 


122 4473 


13-663 


116 4755 


13-713 


no 7949 


13-763 


105 3914 


13-614 


122 3249 


13-664 


116 3590 


13-714 


no 6842 


13-764 


105 2860 


13-615 


122 2026 


13-665 


116 2427 


13-715 


no 5735 


13-765 


105 1808 


13-616 


122 0805 


13-666 


116 1266 


13-716 


no 4630 


13-766 


105 0757 


13-617 


121 9585 


13-667 


116 0105 


13-717 


no 3526 


13-767 


104 9706 


13-618 


121 8366 


13-668 


115 8945 


13-718 


no 2423 


13-768 


104 8657 


13-619 


121 7148 


13-669 


IIS 7787 


13-719 


no 1321 


13-769 


104 7609 


13-620 


121 5932 


13-670 


IIS 6630 


13-720 


no 0220 


13-770 


104 6562 


13-621 


121 4716 


13-671 


"5 5474 


13-721 


109 9121 


13-771 


104 5516 


13-622 


121 3502 


13-672 


115 4319 


13-722 


109 8022 


13-772 


104 4471 


13-623 


121 2289 


13-673 


"5 3165 


13-723 


109 6925 


13-773 


104 3427 


13-624 


121 1078 


13-674 


IIS 2013 


13-724 


109 5828 


13-774 


104 2384 


13-625 


120 9867 


13-675 


I IS 0861 


13-725 


109 4733 


13-775 


104 1342 


13-626 


120 8658 


13-676 


114 97" 


13-726 


109 3639 


13-776 


104 0301 


13-627 


120 7450 


13-677 


114 8562 


13-727 


109 2546 


13-777 


103 9262 


13-628 


120 6243 


13-678 


114 7414 


13-728 


109 1454 


13-778 


103 8223 


13-629 


120 5037 


13-679 


114 6267 


13-729 


109 0363 


13-779 


103 7185 


13-630 


120 3833 


13-680 


114 5121 


13-730 


108 9273 


13-780 


103 6148 


13-631 


120 2630 


13-681 


114 3977 


13-731 


108 8184 


13-781 


103 5113 


13-632 


120 1428 


13-682 


114 2833 


13-732 


108 7097 


13-782 


103 4078 


13-633 


120 0227 


13-683 


114 1691 


13-733 


108 6010 


13783 


103 3045 


13-634 


119 9027 


13-684 


114 0550 


13-734 


108 4925 


13-784 


103 2012 


13-635 


119 7829 


13-685 


113 9410 


13-735 


108 3840 


13-785 


103 0981 


13-636 


119 6631 


13-686 


113 8271 


13-736 


108 2757 


13-786 


102 9950 


13-637 


"9 5435 


13-687 


"3 7133 


13-737 


108 1675 


13-787 


I02 8921 


13-638 


119 4241 


13-688 


113 5997 


13-738 


108 0594 


13-788 


102 7892 


13-639 


119 3047 


13-689 


113 4861 


13-739 


107 9514 


13-789 


102 6865 


13-640 


119 1854 


13-690 


"3 3727 


13-740 


107 843s 


13790 


102 5839 


13-641 


119 0663 


13-691 


113 2594 


13-741 


107 7357 


13-791 


102 4813 


13-642 


118 9473 


13-692 


113 1462 


13-742 


107 6280 


13-792 


102 37S9 


13-643 


118 8284 


13-693 


113 0331 


13-743 


107 5204 


13-793 


102 2766 


13-644 


118 7097 


13-694 


112 9201 


13-744 


107 4129 


13-794 


102 1743 


13-645 


118 S910 


13-695 


112 8073 


13-745 


107 3056 


13-795 


102 0723 


13-646 


118 4725 


13-696 


112 6945 


13-746 


107 1983 


13-796 


loi 9702 


13647 


118 3541 


13-697 


112 5819 


13-747 


107 0912 


13-797 


loi 8683 


13-648 


118 2358 


13-698 


112 4693 


13-748 


106 9842 


13-798 


loi 7665 


13-649 


118 1176 


13-699 


112 3569 


13-749 


106 8772 


13-799 


lOI 6648 



29—2 



220 



MR F. W. NEWMAN'S TABLE 



[13800—13-999] 



X 


e-' 


X 


Q-X 


X 


(.-X 


X 


Q-X 


13-800 


loi 5631 


13-850 


96 6098 


13-900 


91 8981 


13-950 


87 4162 


13-801 


loi 4616 


13-851 


96 5133 


13-901 


91 8063 


13-951 


87 32S8 


13-802 


loi 3602 


13-852 


96 4168 


13-902 


91 7145 


13-952 


87 2415 


13-803 


loi 2589 


13-853 


96 3204 


13-903 


91 6228 


13-953 


87 1543 


13S04 


loi 1577 


13-854 


96 2242 


13-904 


91 5313 


13-954 


87 0672 


13-805 


loi 0566 


13-855 


96 1280 


13-905 


91 4398 


13-955 


86 9802 


13-806 


100 9556 


13-856 


96 0319 


13-906 


91 3484 


13-956 


86 8933 


13-807 


100 8547 


13-857 


95 9359 


13-907 


91 2571 


13-957 


86 8064 


13-808 


100 7539 


13-858 


95 8400 


13-908 


91 1659 


13-958 


86 7197 


I3S09 


100 6532 


13-859 


95 7443 


13-909 


91 0748 


13-959 


86 6330 


13-810 


100 5526 


13-860 


95 6486 


13-910 


90 9837 


13-960 


86 5464 


13-811 


100 4521 


13-861 


95 5530 


13-911 


90 8928 


13-961 


86 4599 


13-812 


100 3517 


13-862 


95 4575 


13-912 


90 8019 


13-962 


86 3735 


13-813 


100 2514 


13-863 


95 3620 


13-913 


90 7112 


13-963 


86 2871 


13-814 


100 1512 


13-864 


95 2667 


13-914 


90 6205 


13-964 


86 2009 


13-815 


100 05 1 1 


13-865 


95 1715 


13-915 


90 5299 


13-965 


86 1147 


13-816 


099 95 II 


13-866 


95 0764 


13-916 


90 4395 


13-966 


86 0287 


13-817 


99 8512 


13-867 


94 9814 


13-917 


90 3491 


13-967 


85 9427 


13-818 


99 7514 


13-868 


94 8864 


13-918 


90 2588 


13-968 


85 8568 


13-819 


99 6517 


13-869 


94 7916 


13-919 


90 1685 


13-969 


85 7710 


13-820 


99 5521 


13-870 


94 6968 


13-920 


90 0784 


13-970 


85 6852 


13-821 


99 4526 


13-871 


94 6022 


13-921 


89 9884 


13-971 


85 5996 


13-822 


99 3531 


13-872 


94 5076 


13-922 


89 8984 


13-972 


85 5140 


13-823 


99 2538 


13-873 


94 4132 


13-923 


89 8086 


13-973 


85 4286 


13-824 


99 1546 


13-874 


94 3188 


13-924 


89 7188 


13-974 


85 3432 


13-825 


99 0555 


13-875 


94 2245 


13-925 


89 6292 


13-975 


85 2579 


13-826 


98 9565 


13-876 


94 1304 


13-926 


89 5396 


13-976 


85 1727 


13-827 


98 8576 


13-877 


94 0363 


13-927 


89 4501 


13-977 


85 0875 


13-828 


98 7588 


13-878 


93 9423 


13-928 


89 3607 


13-978 


85 0025 


13-829 


98 6601 


13-879 


93 8484 


13-929 


89 2714 


13-979 


84 9175 


13-830 


98 5615 


13-880 


93 7546 


13-930 


89 1821 


13-980 


84 8327 


13-831 


98 4630 


13-881 


93 6609 


13-931 


89 0930 


13-981 


84 7479 


13-832 


98 3646 


13-882 


93 5673 


13-932 


89 0039 


13-982 


84 6632 


13-833 


98 2663 


13-883 


93 4737 


13-933 


88 9150 


13-983 


84 5785 


13-834 


98 1680 


13-884 


93 3803 


13-934 


88 8261 


13-9S4 


84 4940 


13-835 


98 0699 


13-885 


93 2870 


13-935 


88 7373 


13-985 


84 4096 


13-836 


97 9719 


13-886 


93 1937 


13-936 


88 6486 


13-986 


84 3252 


13-837 


97 8740 


13-887 


93 1006 


13-937 


88 5600 


13-987 


84 2409 


13838 


97 7761 


13-888 


93 0075 


13-938 


88 4715 


13-988 


84 1567 


13-839 


97 6784 


13-889 


92 9146 


13-939 


88 3831 


13-989 


84 0726 


13-840 


97 5808 


13-890 


92 8217 


13-940 


88 2947 


13-990 


83 9886 


13-841 


97 4833 


13-891 


92 7289 


, 13-941 


88 2065 


13-991 


83 9046 


13-842 


97 3858 


13-892 


92 6362 


13-942 


88 1183 


13-992 


83 8208 


13-843 


97 2885 


13-893 


92 5437 


13-943 


88 0303 


13-993 


83 7370 


13-844 


97 1912 


13-894 


92 4512 


13-944 


87 9423 


13994 


83 6533 


13-845 


97 0941 


13-895 


92 3587 


13-945 


87 8544 


13-995 


83 5697 


13-846 


96 9971 


13-896 


92 2664 


13-946 


87 7666 


13996 


83 4861 


13-847 


96 9001 


13-897 


92 1742 


13-947 


87 6788 


13-997 


83 4027 


13-848 


96 8033 


13-898 


92 0821 


j 13-948 


87 5912 


13-998 


83 3193 


13-849 


96 7065 


13-899 


91 9901 


13-949 


87 5037 


13999 


83 2361 



[t4'ooo — 14T99] 



OF THE DESCENDING EXPONENTIAL. 



221 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


14*000 


83 1529 


14-050 


79 0974 


14-100 


75 2398 


14-150 


71 5703 


14-001 


83 0698 


14-051 


79 0184 


14-101 


75 1646 


14-151 


71 4988 


14-002 


82 9867 


14-052 


78 9394 


14-102 


75 0895 


14-152 


71 4273 


14-003 


82 9038 


14-053 


78 8605 


14-103 


75 0144 


14-153 


71 3559 


14-004 


82 8209 


14-054 


78 7817 


14-104 


74 9395 


14-154 


71 2846 


14*005 


82 7381 


14-055 


78 7029 


14-105 


74 8646 


14-155 


71 2134 


14-006 


82 6554 


14-056 


78 6243 


14-106 


74 7897 


14-156 


71 1422 


14-007 


82 5728 


14-057 


78 5457 


14-107 


74 7150 


14-157 


71 0711 


14-008 


82 4903 


14-058 


78 4672 


14-108 


74 6403 


14-158 


71 0001 


14009 


82 4078 


14-059 


78 3888 


14-109 


74 5657 


14-159 


70 9291 


14-010 


82 3255 


14-060 


78 3104 


14-110 


74 4912 


14-160 


70 8582 


I4-0II 


82 2432 


14-061 


78 2321 


14-in 


74 4167 


14-161 


70 7873 


14-012 


82 161O 


14-062 


78 1540 


14-112 


74 3423 


14-162 


70 7166 


14-013 


82 0789 


14-063 


78 0758 


14-113 


74 2680 


14-163 


70 6459 


14*014 


81 9968 


14-064 


77 9978 


14-114 


74 1938 


14-164 


70 5753 


14-015 


81 9149 


14-065 


77 9198 


14-115 


74 "97 


14-165 


70 5048 


14-016 


81 8330 


14-066 


77 8420 


14-116 


74 0456 


14-166 


70 4343 


14-0x7 


81 7512 


14-067 


77 7642 


14-117 


73 9716 


14-167 


70 3639 


14-018 


81 6695 


14-068 


77 6864 


14-118 


73 8976 


14-168 


70 2936 


14-019 


81 5879 


14-069 


77 6088 


14-119 


73 8238 


14-169 


70 2233 


14*020 


81 5063 


14-070 


77 5312 


14-120 


73 7500 


14-170 


70 1531 


14-021 


8i 4249 


14-071 


77 4537 


14-121 


73 6763 


14-171 


70 0830 


14-022 


81 3435 


14-072 


77 3763 


14-122 


73 6026 


14-172 


70 0130 


14023 


81 2622 


14-073 


77 2990 


14-123 


73 5291 


14-173 


69 9430 


14-024 


81 1810 


14-074 


77 2217 


14-124 


73 4556 


14-174 


69 8731 


14-025 


81 0998 


14-075 


77 1445 


14-125 


73 3821 


14-175 


69 8033 


14-026 


81 0188 


14-076 


77 0674 


14-126 


73 3088 


14-176 


69 7335 


14-027 


80 9378 


14-077 


76 9904 


14-127 


73 2355 


14-177 


69 6638 


14-028 


80 8569 


14-078 


76 9134 


14-128 


73 1623 


14-178 


69 5942 


14-029 


80 7761 


14-079 


76 8366 


14-129 


73 0892 


14-179 


69 5246 


14-030 


80 6953 


14-080 


76 7598 


14-130 


73 0161 


14-180 


69 4551 


14-031 


80 6147 


14-081 


76 6830 


14-131 


72 9432 


14-181 


69 3857 


14-032 


80 5341 


14-082 


76 6064 


14-132 


72 8703 


14-182 


69 3163 


14-033 


80 4536 


14-083 


76 5298 


14-133 


72 7974 


14-183 


69 2471 


14-034 


80 3732 


14-084 


76 4533 


14-134 


72 7247 


14-184 


69 1778 


14-035 


80 2929 


14-085 


76 3769 


14-135 


72 6520 


14-185 


69 1087 


14-036 


80 2126 


14-086 


76 3006 


14136 


72 5794 


14-186 


69 0396 


14-037 


80 1324 


14-087 


76 2243 


14-137 


72 5068 


14-187 


68 9706 


14-038 


80 0524 


14-088 


76 1481 


14138 


72 4343 


14-188 


68 9017 


14039 


79 9724 


14-089 


76 0720 


14-139 


72 3619 


14-189 


68 8328 


14-040 


79 8924 


14-090 


75 9960 


14-140 


72 2896 


14-190 


68 7640 


14-041 


79 8126 


14-091 


75 9200 


14-141 


72 2174 


14-191 


68 6953 


14-042 


79 7328 


14-092 


75 8442 


14-142 


72 1452 


14-192 


68 6266 


14-043 


79 6531 


14-093 


75 7683 


14-143 


72 0731 


14-193 


68 5580 


14-044 


79 5735 


14-094 


75 6926 


14-144 


72 0010 


14-194 


68 4895 


14-045 


79 4940 


14-095 


75 6170 


14-145 


71 9291 


14-195 


68 4211 


14-046 


79 4145 


14-096 


75 5414 


14-146 


71 8572 


14-196 


68 3527 


14-047 


79 3351 


14-097 


75 4659 


14-147 


71 7854 


14-197 


68 2843 


14-048 


79 2558 


14-098 


75 39°5 


14-148 


71 7136 


14-198 


68 2161 


14-049 


79 1764 


14-099 


75 3151 


14-149 


71 6419 


14-199 


68 1479 



222 



MR F. W. NEWMAN'S TABLE 



['4- 



!00- 



14-359] 



X 


f,-X 


X 


Q-X 


X 


c- 


X 


Q-X 


14-200 


63 0798 


14-250 


64 7595 


14-300 


61 6012 


14-350 


58 5968 


14-201 


6S 01 18 1 


14251 


64 6948 


14-301 


61 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 


14-255 


64 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 I7I5 


14-357 


58 1881 


14-208 


67 5373 


14-258 


64 2435 


14-308 


61 II03 


14-358 


58 1299 


14-209 


67 4698 


14-259 


64 1793 


14-309 


61 0492 


14-359 


58 0718 


14-210 


67 4024 


14-260 


64 1151 


14-310 


60 9S82 


14-360 


58 0138 


I4'2II 


67 3350 


14-261 


64 0511 


14-311 


60 9272 


14-361 


57 9558 


1 4-2 1 2 


67 2677 


14-262 


63 9S70 


14-312 


60 8663 


14-362 


57 8979 


14213 


67 2005 


14-263 


63 9231 


14-313 


60 8055 


14-363 


57 8400 


14214 


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 


i4'2i6 


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 S653 


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 


14-221 


66 6650 


14-271 


63 4137 


14-321 


6o 3210 


14-371 


57 3791 


14-222 


66 5984 


14-272 


63 3504 


14-322 


60 2607 


14-372 


57 3218 


14-223 


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 


14-225 


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 


14-227 


66 2662 


14-277 


63 0344 


14-327 


59 9602 


! 14-377 


57 0359 


14-228 


66 2000 


14-278 


64 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 


14-380 


56 8650 


14-231 


66 0017 


14-281 


62 7828 


14-331 


59 7208 


14-381 


56 8082 


14-232 


65 9357 


14-282 


62 7200 


14-332 


59 6611 • 


14-382 


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 


1 14-389 


56 3555 


14-240 


65 4104 


14-290 


62 2203 


14-340 


59 1857 


14-390 


56 2992 


14-241 


65 3450 


14-291 


62 1581 


14-341 


59 1266 


i 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 


14246 


65 0191 


14-296 


61 8480 


14-346 


58 8317 


14-396 


55 9624 


14-247 


64 9541 


1 14-297 


61 7862 


14-347 


58 7729 


; 14-397 


55 9065 


14-248 


64 8892 


1 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 



[i4'4oo— 14-599] 



OF THE DESCENDING EXPONENTIAL. 



223 



X 


C-x 


1 

X 


e-x 


X 


g-X 


X 


Q-X 


i4'4oo 


55 7390 


14-450 


53 0206 


14-500 


5° 4348 


i 

14-550 


47 9750 


14-401 


55 6833 


14-451 


52 9676 


14-501 


50 3843 


14-551 


47 9271 


14-402 


55 6277 


14-452 


52 9147 


14-502 


50 3340 


14-552 


47 8792 


14-403 


55 5721 


14-453 


52 8618 


14-503 


50 2837 


i4'553 


47 8313 


14-404 


55 5165 


14-454 


52 8089 


14-504 


SO 2334 


14-554 


47 7835 


14-405 


55 4610 


M-455 


52 7562 


14-505 


50 1832 


14-555 


47 7358 


14-406 


55 405f' 


14-456 


52 7°34 


14-506 


50 1331 


14-556 


47 6880 


14-407 


55 3502 


14-457 


52 6508 


14-507 


50 0S29 


14-557 


47 6404 


14-408 


55 2949 


14-458 


52 5981 


14-508 


50 0329 


14-558 


47 5928 


14-409 


55 2396 


14-459 


52 5456 


1.4-509 


49 9829 


14-559 


47 5452 


14-410 


55 1844 


14-460 


52 4930 


14-510 


49 9329 


14-560 


47 4977 


14-411 


55 1293 


14-461 


52 4406 


14-511 


49 8830 


14-561 


47 4502 


14-412 


55 0742 


14-462 


52 3882 


14-512 


49 8332 


14-562 


47 4028 


14-413 


55 0191 


14-463 


52 3358 


14-513 


49 7833 


14-563 


47 3554 


14-4x4 


54 9641 


14-464 


52 2835 


14-514 


49 7336 


14-564 


47 3081 


14-415 


54 9092 


14-465 


52 2312 


14-515 


49 6839 


14-565 


47 2608 


14-416 


54 8543 


14-466 


52 1790 


14-516 


49 6342 


14-566 


47 2135 


14-417 


54 7995 


14-467 


52 1269 


14-517 


49 5846 


14-567 


47 1663 


14-418 


54 7447 


14-468 


52 0748 


14-518 


49 5350 


14-568 


47 1192 


14-419 


54 6900 


14-469 


52 0227 


14-519 


49 4855 


14*569 


47 0721 


14-420 


54 6353 


14-470 


51 9707 


14-520 


49 4361 


i4'57o 


47 0251 


14-421 


54 5807 


14-471 


51 9188 


14-521 


49 3867 


14-571 


46 9781 


14-422 


54 5262 


14-472 


51 8669 


14-522 


49 3374 


14-572 


46 9311 


14-423 


54 4717 


14-473 


51 8150 


14-523 


49 2880 


14-573 


46 8S42 


14-424 


54 4172 


14-474 


51 7633 


14-524 


49 2388 


14-574 


46 8373 


14-425 


54 3628 


14-475 


51 7"5 


14-525 


49 1896 


14-575 


46 7905 


14-426 


54 3085 


14-476 


SI 6598 


14526 


49 1404 


14-576 


46 7438 


14-427 


54 2542 


14-477 


51 6082 


14-527 


49 0913 


14-577 


46 6970 


14-428 


54 2000 


14-478 


51 5566 


14-528 


49 0422 


14-578 


46 6504 


14-429 


54 1458 


14-479 


51 5051 


14-529 


48 9932 


14-579 


46 6037 


i4'43o 


54 0917 


14-480 


51 4536 


14-530 


48 9442 


14-580 


46 5572 


14-431 


54 0376 


14-481 


51 4022 


14-531 


48 8953 


14-581 


46 5106 


14-432 


53 9836 


14-482 


SI 3508 


14-532 


48 8464 


14-582 


46 4641 


14-433 


53 9297 


14-483 


51 2995 


14-533 


48 7976 


14-583 


46 4177 


14-434 


53 8757 


14-484 


51 2482 


14-534 


48 7488 


14-584 


46 3713 


14-435 


53 8219 


14-485 


51 1970 


14-535 


48 7001 


14-585 


46 3250 


14-436 


53 7681 


14-486 


SI 1458 


14-536 


■ 48 6514 


14-586 


46 2787 


14-437 


53 7144 


14-487 


51 0947 


14-537 


48 6028 


14-587 


46 2324 


14-438 


53 6607 


14-488 


51 0436 


14-538 


48 5542 : 


14-588 


46 1862 


14-439 


53 6070 


14-489 


50 9926 


14-539 


48 5057 


14-589 


46 1400 


14-440 


53 5535 


14-490 


50 9416 


14-540 


48 4572 


14-590 


46 0939 


14-441 


53 4999 


14-491 


50 8907 


14-541 


48 408S 


14-591 


46 0478 


14-442 


53 4465 


14-492 


50 8399 


14-542 


48 3604 


14-592 


46 0018 


14-443 


53 3930 


14-493 


50 7890 


14-543 


48 3120 


14-593 


45 9558 


14-444 


53 3397 


14-494 


50 7383 


14-544 


48 2638 


14-594 


45 9099 


14-445 


53 2864 


14-495 


so 6876 


14-545 


48 2155 


14-595 


45 8640 


14-446 


53 2331 


14-496 


50 6369 


14-546 


48 1673 


14-596 


45 8182 


14-447 


53 1799 


14-497 


50 5863 


14-547 


48 1192 


14-597 


45 7724 


14-448 


53 1267 


14-498 


SO 5357 


14-548 


48 0711 


14-598 


45 7266 


14-449 


53 0736 


14-499 


50 4852 


14-549 


48 0230 


14-599 


45 6809 



•2 -24 



MR F. W. NE^V^[AN'S TABLE 



[14-600— 14799] 



X 


c-^ 


X 


y> — .C 


X 


e-x 


X 


C-x 


14600 


45 6353 


14-650 


43 4096 


14-700 


41 2925 


14-750 


39 2786 


14-601 


45 5S96 


14-651 


43 3662 


14-701 


41 2512 


14-751 


39 2394 


14-602 


45 5441 


14-652 


43 3229 


14-702 


41 2100 


14-752 


39 2001 


14-603 


45 49*^6 


14-653 


43 2796 


14-703 


41 16SS 


14-753 


39 1610 


14-604 


45 4531 


14-654 


43 2363 


14-704 


41 1276 


14-754 


39 1219 


14-605 


45 4076 


14-655 


43 1931 


14-705 


41 0S65 


14-755 


39 0827 


14-606 


45 3623 


14-656 


43 1499 


14-706 


41 0455 


14-756 


39 0437 


14-607 


45 3169 


14-657 


43 1068 


14-707 


41 0044 


14-757 


39 0046 


I4-60S 


45 2716 


14-658 


43 0637 


14-708 


40 9635 


14-758 


38 9656 


14-609 


45 2264 


14-659 


43 0207 


14-709 


40 9225 


14-759 


38 9267 


14-610 


45 1812 


14-660 


42 9777 


14-710 


40 88x6 


14-760 


38 8878 


14-611 


45 1360 


14-661 


42 9347 


14-711 


40 S408 


14-761 


38 8489 


i4-6i2 


45 0909 


14-662 


42 8918 


14-712 


40 7999 


14-762 


38 8101 


14-613 


45 0458 


14663 


42 8489 


14-713 


40 7592 


14763 


38 7713 


14-614 


45 0008 


14-664 


42 8061 


14-714 


40 7184 


14-764 


38 7325 


14-615 


44 9558 


14-665 


42 7633 


14-715 


40 6777 


14765 


38 6938 


14-616 


44 9109 


14-666 


42 7206 


1 14-716 


40 6371 


14-766 


38 6552 


14-617 


44 8660 


14-667 


42 6779 


1 14717 


40 5964 


14-767 


38 6165 


14-618 


44 8212 


14-668 


42 6352 


1 14-718 


40 5559 


14-768 


38 5779 


14-619 


44 7764 


14-669 


42 5926 


14-719 


40 5153 


14-769 


38 5394 


14-620 


44 7316 


14-670 


42 5500 


14-720 


40 4748 


14-770 


38 5009 


14-621 


44 6S69 


14-671 


42 5075 


14-721 


40 4344 


14-771 


38 4624 


14-622 


44 6422 


14-672 


42 4650 


14-722 


40 3940 


14-772 


38 4239 


14-623 


44 5976 


14-673 


42 4226 


14-723 


40 3536 


14-773 


38 3855 


14-624 


44 553° 


14-674 


42 3802 


14-724 


40 3133 


14-774 


38 3472 


14-625 


44 5°85 


14-675 


42 3378 


14-725 


40 2730 


14-775 


38 3088 


14-626 


44 4640 


14-676 


42 2955 


14-726 


40 2327 


14-776 


38 2705 


14627 


44 4196 


14-677 


42 2532 


14727 


40 1925 


14-777 


38 2323 


14-628 


44 3752 


14-678 


42 2110 


14-728 


40 1523 


14-778 


38 1941 


14-629 


44 3308 


14-679 


42 1688 


14-729 


40 1 1 22 


14-779 


38 1559 


14-630 


44 2865 


14-680 


42 1266 


14-730 


40 0721 


14-7S0 


38 1178 


14-631 


44 2423 


14-681 


42 0845 


14-731 


40 0321 


14-781 


38 0797 


14632 


44 1980 


14-682 


42 0425 


14-732 


39 9920 


14-782 


38 0416 


14-633 


44 1539 


14-683 


42 0004 


14-733 


39 9521 


14-783 


38 0036 


14-634 


44 i°97 


14-684 


41 9585 


14-734 


39 9121 


14-784 


37 9656 


14-635 


44 0656 


14-685 


41 9165 


14-735 


39 8722 


14-785 


37 9276 


14636 


44 0216 


14-686 


41 8746 


14-736 


39 8324 


14-786 


37 8897 


14-637 


43 9776 


14-687 


41 8328 


14-737 


39 7926 


14-787 


37 8519 


14-638 


43 9336 


14-688 


41 7910 


14738 


39 7528 


14-788 


37 8140 


14-639 


43 8897 


14-689 


41 7492 


14-739 


39 7131 


14-789 


37 7762 


14-640 


43 8459 


14-690 


41 707s 


14-740 


39 6734 


14-790 


37 7385 


14-641 


43 8020 


14-691 


41 6658 


14-741 


39 6337 


14-791 


37 7008 


14642 


43 7583 


14-692 


41 6241 


14-742 


39 5941 


14-792 


37 6631 


14-643 


43 7145 


14-693 


41 5825 


14-743 


39 5545 


14-793 


37 6254 


14-644 


43 6708 


14-694 


41 5410 


14-744 


39 5150 


14794 


37 5878 


14-645 


43 6272 


14-695 


41 4995 


14-745 


39 4755 


14795 


37 5503 


14-646 


43 5836 


14-696 


41 4580 


14-746 


39 4361 


14-796 


37 5127 


14-647 


43 5400 


14-697 


41 4166 


14-747 


39 3966 


14797 


37 4752 


14-648 


43 4965 


14-698 


41 3752 


14-748 


39 3573 1 


14-798 


37 4378 


14-649 


43 4530 


14-699 


41 3338 1 


14-749 


39 3179 i 


14799 


37 4004 



]i4'8oo — 


14-999] 


OF THE DESCENDING EX 


PONENTIAL. 




225 


X 


Q-X 


X 


Q-X 


X 


g-x 


X 


Q-X 


14-800 




37 3630 


14-850 


35 5408 


14-900 


33 8074 


14-950 


32 1586 


14-801 


37 3256 


14-851 


35 5052 


14-901 


2,?, 7736 


, 14-951 


32 1265 


14-802 


37 2S83 


14-852 


35 4698 


14-902 


Zo 7399 


i 14-952 


32 0944 


14-803 


37 2511 


14-853 


35 4343 


14-903 


2,3, 7062 


14-953 


32 0623 


14-804 


37 213S 


14-854 


35 3989 


14-904 


zz 6725 


14-954 


32 0302 


14-805 


37 1766 


^4-855 


35 3635 


14-905 


ZZ 6388 


14-955 


31 9982 


14-806 


37 139s 


14-856 


35 3282 


14-906 


ZZ 6052 


14-956 


31 9662 


14-807 


37 1024 


14-857 


35 2929 


14-907 


zz 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 


ZZ 5045 


14-959 


31 8705 


14-810 


36 9912 


14-860 


35 1871 


14-910 


ZZ 4710 


14-960 


31 8386 


14-811 


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 


14-962 


31 7750 


14-813 


36 8804 


14-863 


35 °8i7 


14-913 


33 3708 


14-963 


31 7433 


14814 


36 843s 


14-S64 


35 0467 


14-914 


ZZ 3374 


14-964 


31 7115 


14-815 


36 8067 


14-865 


35 °"6 


14-915 


ZZ 3041 


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 


ZZ 2376 


14-967 


31 6165 


14-818 


36 6965 


14-868 


34 9°68 


14-91S 


ZZ 2043 


14-968 


31 5849 


14-819 


36 6598 


14-869 


34 8719 


14-919 


ZZ 1712 


14-969 


31 5534 


14-820 


36 6231 


14-870 


34 837° 


14-920 


33 1380 


14-970 


31 5218 


14-821 


36 5865 


14-871 


34 8022 


14-921 


ZZ 1049 


14-971 


31 4903 


14-822 


36 5500 


14-872 


34 7674 


14-922 


ZZ 0718 


14-972 


31 45S9 


14-823 


36 5134 


14-873 


34 7327 


14-923 


33 0387 


14-973 


31 4274 


14-824 


36 4769 


14-874 


34 6979 


14-924 


ZZ 0057 


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 


14-926 


32 9398 


14-976 


31 3333 


14-827 


36 3677 


14-877 


34 594° 


14-927 


32 9068 


14-977 


31 3020 


14-828 


36 3313 


14-87S 


34 5594 


14-928 


32 8739 


14-978 


31 2707 


14-829 


36 2950 


14-879 


34 5249 


14-929 


32 8411 


14-979 


31 2394 


14-830 


36 2587 


14-880 


34 49=4 


14-930 


32 8083 


14-980 


31 2082 


14-831 


36 2225 


14-881 


34 4559 


14-931 


32 7755 


14-981 


31 1770 


14-832 


36 1863 


14-882 


34 4215 


14-932 


32 7427 


14-982 


31 1458 


14-833 


36 1501 


14-883 


34 3871 


14-933 


32 7100 


14-983 


31 1147 


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 


31 0215 


14-837 


36 0058 


14-887 


34 2498 


14-937 


32 5794 


14-987 


30 9905 


14-838 


35 9698 


14-888 


34 2156 


14-938 


32 5468 


14-988 


30 9595 


14-839 


35 9339 


14-889 


34 1S14 


14-939 


32 5143 


14-989 


30 9286 


14-840 


35 8980 


14-890 


34 1472 


14-940 


32 4818 


14-990 


30 8977 
30 8668 


14-841 


35 8621 


14-891 


34 1131 


14-941 


32 4493 


14-991 


14-842 


35 8262 


14-892 


34 0793 


14-942 


32 4169 


14-992 


30 8359 


14-843 


35 7904 


14-893 


34 0449 


14-943 


32 3845 


14-993 


30 8051 


14-844 


35 7547 


14-894 


34 0109 


14-944 


32 3521 


14-994 


30 7743 


14-845 


35 7189 


14-895 


33 9769 


14-945 


32 3198 


14-995 


30 7436 
30 7128 
30 6821 
30 6515 
30 6208 


14-846 


35 6832 


14-896 


33 9429 


14-946 


32 2875 


14-996 


14-847 


35 6476 


14-897 


33 9°90 


14-947 


32 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 



Vol. XIII. Part III. 



30 



226 



MR F. W. NEWMAN'S TABLE 



[15-000— 15199] 



X 


Q-X 


X 


(,-X 


X 


f>-X 


X 


Q-X 


15-000 


30 5902 


15-050 


29 0983 


15-100 


27 6792 


15-150 


26 3292 


15-001 


30 5596 


15-051 


29 0692 


15-101 


27 6515 


15-151 


26 3029 


15-002 


30 5291 


15-052 


29 0402 


15-102 


27 6239 


15-152 


26 2766 


'5 -^03 


30 4986 


15-053 


29 0112 


15-103 


27 5063 


15-153 


26 2504 


15-004 


30 4680 


15-054 


28 9822 


15-104 


27 5687 


15-154 


26 2241 


15-005 


30 4376 


15-055 


28 9532 


15-105 


27 5411 


15-155 


26 1979 


15-006 


30 4072 


i 15-056 


28 9243 


15-106 


27 5136 


15-156 


26 I717, 


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 


15-158 


26 II94 


1 5 009 


30 3161 


15-059 


28 8376 


15-109 


27 4312 


15-159 


26 0933 


15-010 


30 285S 


15-060 


28 8088 


15-110 


27 4038 


15-160 


26 0673 


15-011 


30 2556 


15-061 


28 7800 


15-111 


27 3764 


15-161 


26 0412 


15-012 


30 2253 


15 062 


28 7512 


15-112 


27 3490 


15-162 


26 0152 


I5°i3 


30 1951 


15-063 


28 7225 


15113 


27 3217 


15-163 


25 9892 


15-014 


30 1649 


15-064 


28 6938 


15-114 


27 2944 


15-164 


25 9632 


i5-°i5 


30 134S 


15-065 


28 6651 


15-115 


27 2671 


15-165 


25 9373 


15-016 


30 1047 


15-066 


28 6365 


15116 


27 2398 


15-166 


25 9113 


15-017 


30 0746 


15-067 


28 6078 


15-117 


27 2126 


15-167 


25 8854 


15-018 


30 °44S 


15-068 


28 5792 


15-118 


27 1854 


15-168 


25 8596 


15-019 


3° 0145 


15-069 


28 5507 


15-119 


27 1582 


15-169 


25 8337 


15-020 


29 9845 


15-070 


28 5221 


15-120 


27 1311 


15-170 


25 8079 


15-021 


29 9545 


15-071 


28 4936 


15-121 


27 1040 


15-171 


25 7821 


15-022 


29 9246 


15-072 


28 4652 


15-122 


27 0769 


15-172 


25 7563 


1 5 •'323 


29 8947 


15-073 


28 4367 


15-123 


27 0498 


15-173 


25 7306 


15-024 


29 8648 


15-074 


28 4083 


15-124 


27 0228 


15-174 


25 7049 


i5'025 


29 8349 


15-075 


28 3799 


15-125 


26 9958 


15-175 


25 6792 


15-026 


29 8051 


15-076 


''zS 3515 


15-126 


26 9688 


15-176 


25 6535 


15-027 


29 7753 


15-077 


28 3232 


15-127 


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 


15-129 


26 8880 


15-179 


25 5767 


i5'O30 


29 6861 


15-080 


28 23S3 


15-130 


26 8611 


15-180 


25 55" 


15-031 


29 6565 


15-081 


28 2101 


15-131 


26 8343 


15-181 


25 5256 


15032 


29 6268 


15-082 


28 1819 


15-132 


26 8075 


15-182 


25 5000 


i5'033 


29 5972 


15-083 


28 1538 


15-133 


26 7807 


15-183 


25 4746 


r5'°34 


29 5676 


15-084 


28 1256 


15-134 


26 7539 


15-184 


25 4491 


1 5 "035 


29 5381 


15-085 


28 0975 


15-135 


26 7272 


15-185 


25 4237 


15 '036 


29 5086 


15086 


28 0694 


15-136 


26 7004 


15-186 


25 3983 


i5'037 


29 4791 


15-087 


28 0414 


15-137 


26 6738 


15-187 


25 3729 


15038 


29 4496 


15-088 


28 0133 


15-138 


26 6471 


15-188 


25 3475 


1 5 '039 


29 4202 


15-089 


27 9853 


15139 


26 6205 


15-189 


25 3222 


15 040 


29 3908 


1 5 -090 


27 9574 


15-140 


26 5939 


15-190 


25 2969 


15-041 


29 3614 


15-091 


27 9294 


15-141 


26 5673 


15-191 


25 2716 


15-042 


29 3320 


15-092 


27 9015 


15-142 


26 5407 


15-192 


25 2463 


15043 


29 3027 


15-093 


27 8736 


15-143 


26 5142 


15-193 


25 2211 


15-044 


29 2734 


15-094 


27 8458 


15-144 


26 4877 


15-194 


25 1959 


15-045 


29 2442 


15-095 


27 8179 


15-145 


26 4612 


15-195 


25 1707 


15-046 


29 2150 


15-096 


27 7901 


15-146 


26 4348 


15-196 


25 1455 


i5'o47 


29 1858 ; 


15-097 


27 7624 


15-147 


26 4083 


15-197 


25 1204 


15-048 


29 1566 1 


15-098 


27 7346 1 


15-148 


26 3820 


15-198 


25 0955 


15 '049 


29 1274 1 


15-099 


27 7069 || 


15-149 


26 3556 


15-199 


25 0704 



[15-200— 15-349] 



OF THE DESCENDING EXPONENTIAL. 



227 



X 


(,-X 


X 


C-x 


X 


e-x 


X 


e-x 


15-200 


25 0452 


15-240 


24 0631 


15-280 


23 1196 


15-320 


22 


2131 


15-201 


25 0201 


15-241 


24 0391 


15-^81 


23 0965 


15-321 


22 


1909 


15-202 


24 9951 


15-242 


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 


15-204 


24 9452 


15-244 


23 9671 


15-284 


23 0273 


15-324 


22 


1244 


i5"2°S 


24 9202 


15-245 


23 9431 


15-285 


23 0043 


15-325 


22 


1023 


15-206 


24 8953 


15-246 


23 9192 


15-286 


22 9813 


15-326 


22 


0802 


15-207 


24 8704 


15-247 


23 8953 


15-287 


22 9583 


15-327 


22 


0581 


15-208 


24 8456 


15-248 


23 8714 


15-288 


22 9354 


15-328 


22 


0361 


15-209 


24 8208 


15-249 


23 847s 


15-289 


22 9124 


15-329 


22 


0140 


15-210 


24 7960 


15-250 


23 8237 


15-290 


22 8895 


15-330 


21 


9920 


15-211 


24 7712 


15-251 


23 8000 


15-291 


22 8667 


15-331 


21 


9701 


15-212 


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 


9262 


15-214 


24 6970 


15-254 


23 7286 


15-294 


22 7982 


15-334 


21 


9042 


15-215 


24 6723 


15-255 


23 7049 


15-295 


22 7754 


15-335 


21 


8823 


15-216 


24 6476 


15-256 


23 6812 


15-296 


22 7526 


15-336 


21 


8605 


15-217 


24 6229 


15-257 


23 6575 


15-297 


22 7299 


15-337 


21 


8386 


15-218 


24 5984 


15-258 


23 6339 


15-298 


22 7072 


15-338 


21 


8168 


15-219 


24 5738 


15-259 


23 6102 


15-299 


22 6845 


15-339 


21 


7950 


15-220 


24 5492 


15-260 


23 5866 


15-300 


22 6618 


15-340 


21 


7732 


15-221 


24 5247 


15-261 


23 5631 


15-301 


22 6391 


15-341 


21 


7514 


15-222 


24 5002 


15-262 


23 5395 


15-302 


22 6165 


15-342 


21 


7297 


15-223 


24 4757 


15-263 


23 5160 


15-303 


22 5939 


15-343 


21 


7080 


15-224 


24 4512 


15-264 


23 4925 


15-304 


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 


15-306 


22 5262 


15-346 


21 


6430 


15-227 


24 37S0 


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 


15-309 


22 4587 


15-349 


21 


5781 


15-230 


24 3050 


15-270 


23 3519 


15-310 


22 4363 








15-231 


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 2820 


15-313 


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 








15-236 


24 1596 


15-276 


23 2123 


15-316 


22 3021 








15-237 


24 1354 


15-277 


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 






1 



30—2 



228 



MR F. W. NEWMAN'S TABLE 
[Second Part. Fourtan decimal places?[ 



[15-350—15-748] 



1 

X 


Q-X 


X 


Q-X 


X 


e-^ 


X 


Q-X 


15-350 


2155 6572 


15-450 


1950 5193 


15-550 


1764 9028 


15-650 


1596 9501 


15-352 


2151 3502 


15-452 


1946 6222 


15-552 


1761 3765 


15-652 


1593 7594 


15-354 


2147 051S 


15-454 


1942 7328 


15-554 


1757 8573 


15-654 


1590 5751 


15-356 


2142 7620 


15-456 


1938 8512 


15-556 


1754 3451 


15-656 


1587 3972 


15-358 


2138 4808 


15-458 


1934 9774 


15-558 


1750 8399 


15-658 


1584 2256 


15-360 


2134 20S1 


15-460 


1931 i"3 


15-560 


1747 3416 


15-660 


1581 0602 


15-362 


2129 9439 1 


15-462 


1927 2530 


15-562 


1743 8504 


15-662 


1577 9012 


15-364 


2125 6882 


15-464 


1923 4023 


15-564 


1740 3661 


15-664 


1574 7485 


15-366 


2121 4411 ^ 


15-466 


1919 5593 ' 


15-566 


1736 8890 


15666 


1571 6021 


I5-36S 


2117 2025 . 


15-468 


1915 7241 


iS-568 


1733 4187 


15-668 


1568 4620 


15-370 


2112 9723 


15-470 


19x1 8964 


15-570 


1729 9553 


15-670 


1565 3284 


15-372 


2108 7507 


15-472 


1908 0764 


15-572 


1726 4988 


15-672 


1562 2009 


15-374 


2104 5374 


15-474 


1904 2641 


15-574 


1723 0493 


15-674 


1559 0796 


15-376 


2IOD 3325 I 


15-476 


1900 4594 


15-576 


1719 6066 


15-676 


1555 9646 


15-378 


2096 1360 1 


15-478 


1896 6623 


15-578 


1716 1709 


15-678 


1552 8557 


15-380 


2091 9479 


15-480 


1892 8727 


15-580 


1712 7420 


15-680 


1549 7531 


15-382 


20S7 7682 


15-482 


i8Sg 0908 


15-582 


1709 3201 


15-682 


1546 6567 


15-384 


20S3 5969 


15-484 


1885 3163 


15-584 


1705 9°49 


15-684 


1543 5665 


15-386 


2079 433S 


15-486 


18S1 5495 


15-586 


1702 4965 


15-686 


1540 4825 


15-388 


2075 2791 


15-488 


1877 7901 


15-588 


1699 0949 


15-688 


1537 4046 


15-390 


2071 1327 


15-490 


1874 0383 


15-590 


1695 7000 


15-690 


1534 3328 


15-392 


2066 9943 


15-492 


1870 2940 


15-592 


1692 3120 


15-692 


1531 2672 


15-394 


2062 8645 


15-494 


1866 5571 


15-594 


1688 9306 


15-694 


1528 2078 


15-396 


2058 7430 


15-496 


1862 8278 


15-596 


1685 5560 


15-696 


1525 1544 


15-398 


2054 6297 


15-49S 


1859 1058 


15-598 


1682 1883 


15-698 


1522 1071 


15-400 


2050 5246 


15-500 


1855 3914 


15-600 


167S 8275 


15-700 


1519 0660 ■ 


15-432 


2046 4276 ' 


15-502 


1S51 6843 


15-602 


167s 4732 


15-702 


1516 0308 


15-404 


2042 33S9 


15-504 


1847 9846 


15-604 


1672 1256 


15-704 


1513 0018 


15-406 


2038 2583 


15-506 


1844 2923 


15-606 


1668 7847 


15-706 


1509 9788 


15-4=8 


2034 1858 


15-508 


1840 6074 


15-608 


1665 4505 


15-708 


1506 9619 


15-410 


2030 1215 


15-510 


1836 9299 


15-610 


1662 1229 


15-710 


1503 9510 


15-412 


2026 0653 


15-512 


1833 2598 


15-612 


1658 8020 


15-712 


1500 9461 


15-414 


2022 0173 


15-514 


1829 5970 


15-614 


1655 4877 


. 15-714 


1497 9472 


15-416 


2017 9773 


15-516 


1825 9414 


15-616 


1652 1800 


15-716 


1494 9543 


15-418 


2013 9454 


15-518 


1822 2931 


15-618 


1648 8790 


15-718 


1491 9673 


15-420 


2009 9215 


15-520 


i8r8 6521 


15-620 


1645 5846 


15-720 


1488 9864 


15-422 


2005 9056 


15-522 


1815 0185 


15622 


1642 2967 


15-722 


i486 0115 


15-424 


ZDOI 8978 


15-524 


1811 3921 


15-624 


1639 0154 


15-724 


1483 0424 


15-426 


1997 8980 


15-526 


1807 7729 


15-626 


1635 7407 


15-726 


1480 0792 


15-428 


1953 9062 


15-528 


1804 I 6 10 


15-628 


1632 4725 


15-728 


1477 1221 


iS'43o 


1989 9224 


15-530 


1800 5562 


15-630 


1629 2108 


15-730 


1474 1708 


15-432 


1985 9466 


15-532 


1796 9587 


15-632 


1625 9556 


15-732 


1471 2254 


15-434 


1981 9786 


15-534 


1793 3684 


15-634 


1622 7069 


15-734 


1468 2859 


15-436 


1978 0186 


15-536 


1789 7853 


15-636 


1619 4647 


15-736 


1465 3522 


15-438 


1974 0666 


15-538 


1786 2093 


15-638 


1616 2291 


15-738 


1462 4244 


15-440 


1970 1224 


15-540 


1782 6403 


15-640 


1613 0000 


15-740 


1459 5025 


15-442 


1966 1861 


15-542 


1779 0785 


15-642 


1609 7771 


15-742 


1456 5864 


15-444 


1962 2576 


15-544 


1775 5239 


15-644 


1606 5607 


15-744 


1453 6761 


'5-446 


1958 3370 


15-546 


1771 9764 


15-646 


1603 3507 


15-746 


1450 7717 


15-448 


1954 4243 


15-548 


1768 4360 


15648 


1600 1472 


15-748 


1447 8731 



[i5'75o— i6-i48] 



OF THE DESCENDING EXPONENTIAL. 

[^Second Part. Fourteen decimal places^ 



229 



X 


(,-X 


X 


g-a: 


X 


e-^ 


X 


Q-X 


15-750 


1444 9802 


15-850 


1307 4722 


15-950 


1 183 0498 


16-050 


1070 4677 


15-752 


1442 0931 


15-852 


1304 8599 


15-952 


1180 6S61 


16-052 


1068 3289 


15-754 


1439 2117 


15-854 


1302 2528 


15-954 


1178 3270 


16-054 


1066 1944 


15-756 


1436 3362 


15-856 


1299 6509 


15-956 


1175 9727 


16-056 


1064 0641 


15-758 


1433 4664 


15-858 


1297 0542 


15-958 


1173 6231 


16-058 


1061 9381 


15760 


1430 6024 


15-860 


1294 4626 


15-960 


1171 2781 


16-060 


1059 8164 


15-762 


1427 7441 


15-862 


1291 8763 


15-962 


1168 9379 


16-062 


1057 6989 


15-764 


1424 8914 


15-864 


1289 2951 


15-964 


1166 6023 


16-064 


1055 5855 


15766 


1422 0444 


15-866 


1286 7191 


15-966 


1164 2714 


16-066 


1053 4765 


15-768 


1419 2031 


15-868 


1284 1483 


15-968 


1161 9452 


16-068 


1051 3716 


15-770 


1416 367s 


15-870 


1281 5825 


15-970 


1159 6234 


16*070 


1049 2710 


15-772 


1413 5376 


15-872 


1279 0219 


15-972 


1157 3065 


16-072 


1047 1746 


15-774 


1410 7134 


15-874 


1276 4665 


15-974 


1154 9942 


16-074 


1045 0823 


15-776 


1407 8948 


15-876 


1273 9161 


15-976 


1152 6865 


16-076 


1042 9943 


15-778 


1405 0819 


15-878 


1271 3708 


15-978 


1150 3835 


16-078 


1040 9103 


15780 


1402 2746 


15-880 


1268 8306 


15-980 


1148 0851 


16-080 


1038 8306 


15-782 


1399 4728 


15-882 


1266 2955 


15-982 


1145 7913 


16-082 


1036 7550 


I5-7S4 


1396 6767 


15-884 


1263 7654 


15-984 


1143 5019 


16-084 


1034 6836 


15-786 


1393 8862 


15-886 


1261 2404 


15-986 


1141 2172 


16-086 


1032 6164 


15-788 


1391 lOIl 


15-888 


1258 7205 


15-988 


1138 9370 


16-088 


'1030 5532 


15-790 


1388 3217 


15-890 


1256 2055 


15-990 


1136 6615 


16-090 


1028 4941 


15-792 


1385 5478 


15-892 


1253 6956 


15-992 


1134 3905 


16-092 


1026 4391 


15-794 


1382 7795 


15-894 


1251 1908 


15-994 


1132 1240 


16-094 


1024 3883 


15-796 


1380 0167 


15-896 


1248 6909 


15-996 


1129 8620 


16-096 


1022 3417 


15-798 


1377 2595 


15-898 


1246 i960 


15-998 


1127 6045 


16-098 


1020 2990 


15-800 


1374 5077 


15-900 


1243 7°6o 


16-000 


1125 3517 


i6-ioo 


1018 2604 


15-802 


1371 7614 


15-902 


1241 2211 


16-002 


1123 1032 


16-102 


1016 2259 


15-804 


1369 0207 


15-904 


1238 7411 


16-004 


1120 8593 


16-104 


1014 1955 


15-806 


1366 2854 


15-906 


1236 2661 


16-006 


1118 6198 


16-106 


1012 1691 


I5-80S 


1363 5555 


15-908 


1233 7961 


16-008 


1116 3848 


16-108 


loio 1468 


15-810 


1360 8311 


15-910 


1231 3309 


16-010 


1114 1543 


16-110 


1008 1285 


15-812 


1358 1122 


15-912 


1228 8707 


l6-012 


nil 9282 


l6-112 


1006 1143 


15-8x4 


1355 3987 


15-914 


1226 4155 


16014 


1109 7066 


16-114 


1004 I04I 


15-816 


1352 6906 


15-916 


1223 9651 


16-016 


1107 4894 


16116 


1002 0979 


15-818 


1349 9879 


15-918 


1221 5196 


16-018 


1105 2766 


16-118 


1000 0957 


15-820 


1347 2906 


15-920 


1219 0790 


16-020 


1103 0683 


16-120 


998 0975 


15-822 


1344 5987 


15-922 


1216 6433 


16-022 


I 100 8644 


16-122 


996 1033 


15-824 


1341 9122 


15-924 


1214 2124 


16-024 


1098 6647 


16-124 


994 1131 


15-826 


1339 2311 


15-926 


1211 7864 


16-026 


1096 4696 


16-126 


992 1268 


15-828 


1336 5553 


15-928 


1209 3653 


16-028 


1094 2789 


16-128 


990 1446 


15-830 


1333 8849 


15-930 


1206 9490 


16-030 


1092 0926 


16-130 


988 1662 


15-832 


1331 2198 


15-932 


1204 5375 


16-032 


1089 9106 


16-132 


986 1919 


15-834 


1328 5600 


15-934 


1202 1308 


16-034 


1087 7330 


16-134 


984 2215 


15-836 


1325 9055 


15-936 


1199 7289 


16-036 


10S5 5596 


16-136 


982 2551 


15-838 


13^3 2564 


15-938 


1197 3319 


16-038 


1083 3907 


16-138 


980 2925 


15-840 


1320 6125 


15-940 


1194 9396 


16-040 


ic8i 2261 


16-140 


978 3338 


15-842 


1317 9739 


15-942 


1192 5521 


16-042 


1079 0658 


16-142 


976 3791 


15-844 


1315 3406 


15-944 


1190 1694 


16-044 


1076 9097 


16-144 


974 4283 


15-846 


1312 7126 


15-946 


1187 7915 


16-046 


1074 7581 


16-146 


972 4814 


15-848 


1310 0898 


15-948 


1185 4182 


16-048 


1072 6107 


16-148 


970 5383 



230 



MR F. W. NEWMAN'S TABLE 
[Si:cofid Part. Fotaieen dicimal places^ 



[16-150— 16-548] 



X 


C-* 


i X 


Q-X 


X 


C-x 


X 


Q-X 


i6"iso 


968 5992 


16-250 


876 4248 


16-350 


793 0220 


16-450 


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 


78S 2781 


16-456 


713 2635 


16-158 


960 88 I 3 


16-258 


869 4414 


16-358 


786 7032 


16-458 


711 8384 i 


i6-i6o 


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 


16-164 


■ 955 -^IZl 


16-264 


864 2403 


16-364 


781 9971 


16-464 


707 5802 


i6-i66 


953 2249 


16266 


862 5136 


16-366 


780 4346 


1 16-466 


706 1665 


i6-i68 


951 3204 


16-268 


860 7903 


16-368 


77S 8753 


! 16-468 


704 7555 


16-170 


949 4197 


16-270 


859 0704 


16-370 


777 3191 


16-470 


703 3474 


16-172 


947 5228 


16-272 


857 3540 


16-372 


775 7660 


16-472 


701 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 


i 16-476 


699 1400 


16-178 


941 8546 


16278 


852 2253 


16-378 


771 1253 


16-478 


697 7431 


i6-i8o 


939 9728 


16-280 


850 5225 


16-380 


769 5846 


16-480 


696 3490 


16182 


938 0947 


16282 


848 8232 


16-382 


768 0470 


16-482 


694 9577 


16-184 


936 2204 


16-284 


847 1273 


16-384 


766 5124 


16484 


693 5692 


16186 


934 3498 


16-286 


845 4347 


16-386 


764 9S09 


16-486 


692 1834 


16-188 


932 4830 


16288 


843 7455 


16-388 


763 4525 


16-488 


690 8004 


16-190 


930 6199 


16-290 


^42 0597 


16-390 


761 9271 


16-490 


6S9 4202 


16192 


928 7605 


16-292 


840 3773 


16-392 


760 4048 


16-492 


688 0427 


16-194 


926 9049 


16-294 


838 6982 


16-394 


75S 8S55 


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 35°i 


16-39S 


755 8560 


16498 


683 9268 


i6-2oo 


921 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 


16504 


679 8355 


i6-2o6 


915 8485 


16-306 


82S 6940 


16-406 


749 8233 


16-506 


678 4772 


16208 


914 0186 


16-308 


827 0382 


16-408 


748 3352 


16-508 


677 1216 


16-210 


912 1924 


16-310 


825 3858 


16-410 


746 S400 


16-510 


675 7687 


16212 


910 3698 


16-312 


823 7367 


16-412 


745 3478 


16-512 


674 4186 


16-214 


908 5509 


16-314 


822 0909 


16-414 


743 8586 


16-514 


673 0711 


i6-2i6 


906 7356 


16-316 


820 4484 


16-416 


742 3724 


16-516 


671 7263 


16218 


904 9240 


16-318 


818 8091 


16-418 


740 8891 


16-518 


670 3842 


l6-220 


903 "59 


16-320 


817 1731 


16-420 


739 4088 


16-520 


669 0447 


16-222 


901 3115 


16-322 


815 5404 


16-422 


737 9315 


16-522 


667 7080 


16-224 


899 5107 


16-324 


813 9110 


16-424 


736 4571 


16-524 


666 3739 


j 16-226 


897 7135 


16326 


812 2848 


16-426 


734 9857 


16-526 


665 0425 


16-228 


895 9199 1 


16-328 


810 6618 


16-428 


733 5172 


16-528 


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 


888 781 I 


16-336 


804 2024 


16-436 


727 6724 


16-536 


658 4252 


16-238 


887 0053 


16-338 


802 5956 


16-438 


726 2185 


16-538 


657 1097 


16-240 


885 2330 


16-340 


800 9920 


16-440 


724 7675 


16-540 


655 7968 


16242 


883 4644 


16342 


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 


16-548 


650 5713 



[i6-55o— 16-948] 



OF THE DESCENDING EXPONENTIAL. 

\Sccond Part. Fourteen dceimal plaees?^ 



231 



X 


(>-X 


1 

X 


Q-X 


X 


Q-X 


! 
X 


Q-X 


16-550 


649 2715 


16-650 


587 4851 


16-750 


531 5785 


16-850 


480 9921 


16-552 


647 9742 


16-652 


586 3II3 


16-752 


530 5164 


16-852 


480 O3II 


16-554 


646 6796 


i 16-654 


585 1399 


16-754 


529 4564 


16-854 


479 °72o 


16-556 


645 3875 


16-656 


583 9707 


16-756 


528 3986 


16-S56 


478 1149 


16-558 


644 0980 


16658 


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 


16-566 


638 9658 


16-666 


578 1601 


16-766 


523 1410 


16-866 


473 3575 


16-568 


637 6892 


16-668 


577 0050 


16-768 


522 0957 


16-S68 


472 4118 


16570 


636 4150 


16-670 


575 8521 


16-770 


521 0526 


16-870 


47T 4679 


16-572 


63s 1435 


16-672 


574 7016 


16-772 


520 ciis 


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 9008 


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 


16-584 


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 0621 


16-6S8 


565 5795 


16-788 


5" 7575 


16-888 


463 0574 


16-590 


623 8132 


16-690 


564 4495 


16-790 


510 7350 


16-890 


462 1322 


16-592 


622 5668 


16-692 


563 3217 


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 


16896 


459 3677 


16-598 


618 8426 


16-698 


559 9519 


16-798 


506 6654 


16-898 


458 4499 


1 6 -600 


617 6061 


16-700 


558 8331 


16-800 


505 6531 


16-900 


457 5339 


16-602 


616 3722 


16-702 


557 7166 


i6-8o2 


504 6428 


16-902 


456 6197 


16-604 


615 1406 


16-704 


556 6023 


16-804 


503 6346 


16-904 


455 7074 


i6-6o6 


613 9116 


16-706 


555 4902 


i6-8o6 


502 6283 


16-906 


454 7969 


16-608 


612 6850 


16-708 


554 3803 


16-808 


501 6241 


16-908 


453 8882 


i6-6io 


611 4609 


16-710 


553 2726 


i6-8io 


500 6218 


16-910 


452 9813 


16-612 


610 2392 


16-712 


552 1672 


16-812 


499 6215 


16-912 


452 0763 


16-614 


609 0199 


16-714 


551 0640 


16-814 


498 6233 


16-914 


451 1730 


16-616 


607 8031 


16-716 


549 9630 


i6-8i6 


497 6271 


16-916 


450 2716 


16-618 


606 5S87 


16-718 


548 8641 


16818 


496 6328 


16-918 


449 3719 


16-620 


605 3767 


16-720 


547 767s 


16-820 


495 6405 


16-920 


448 4741 


16-622 


604 1672 


16-722 


546 6731 


16822 


494 6502 


16-922 


447 57S0 


16-624 


602 9C01 


16-724 


545 5808 


16-824 


493 6619 


16-924 


446 6838 


16-626 


601 7553 


16-726 . 


544 49°7 


16-826 


492 6756 


16-926 


445 7913 


16-628 


600 5530 


16-728 


543 4028 


16-828 


491 6912 


16-928 


444 9006 


16-630 


599 3531 


16-730 


542 3171 


16830 


490 7088 


16-930 


444 °ii7 


16-632 


598 1556 


16-732 


541 2336 


16-832 


489 7284 


16-932 


443 1246 


16-634 


596 9605 


16-734 


540 1522 


16-834 


488 7499 


16-934 


442 2392 


16-636 


595 7678 


16-736 


539 073° 


16-836 


4S7 7734 


16-936 


441 3556 


16-638 


594 5775 


16-738 


537 9959 


16-838 


486 79S8 


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 


16-842 


484 8555 


16-942 


438 7154 


16-644 


591 0207 


16-744 


534 7776 


16-844 


483 8868 


16-944 


437 8389 


16-646 


589 8398 


16-746 


533 7091 


16-846 


4S2 9200 


16-946 


436 9641 


16-648 


588 6613 


16-748 


532 6428 


16-848 


481 9551 


16948 


436 0910 



232 



]\rR F. W. NEWMAN'S TABLE 
[Second Fart. Fourteen decimal places^ 



[i6-95o-i7-42o] 



X 


C-x 


X 


(,-X 


X 


e-x 


X 


(,-X 


16-950 


435 2197 


17-050 


393 8030 


17-150 


356 3277 


ir2S° 


322 4187 


16952 


434 3501 


17-052 


393 0162 


17-152 


355 C158 


17-252 


321 7745 


16-954 


433 4823 


17-054 


392 2310 


17-154 


354 9053 


17-254 


321 131S 


16-956 


432 6162 


17-056 


391 4473 


17-156 


354 1962 


17-256 


320 4899 


I6-95S 


431 7518 


17-058 


390 6652 


17-158 


353 4885 


17-258 


319 8496 


16-960 


430 8892 


17-060 


3S9 S846 


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 


38S 3282 


17-164 


351 3739 


17-264 


317 9363 


16-966 


438 3116 


17-066 


387 5523 


17-166 


35° 6719 


17-266 


317 3010 


16-968 


427 4558 


17 068 


386 7780 


17-168 


3-19 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 


17074 


384 4643 


17-174 


347 8777 


17-274 


314 7727 


16-976 


424 0498 


17-076 


383 6961 


17-176 


347 1826 


17-276 


314 1438 


I6-97S 


423 2026 


17-078 


382 9295 


17-178 


346 48S9 


17-278 


313 5162 


16-980 


422 3570 


17-080 


382 1644 


17-180 


345 7966 


17-280 


312 8898 


16-982 


421 5132 


17-082 


3S1 4008 


17-182 


345 1057 


17-282 


312 2646 


16-984 


420 6710 


17-084 


380 6388 


17-184 


344 4162 


17-284 


311 6407 


16-986 


419 8305 


17-086 


379 8783 


17-186 


343 7281 


17-286 


311 oi8o 


16-988 


418 9917 


17-088 


379 "93 


17-188 


343 0413 


17-288 


310 3966 


16990 


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 


17-096 


376 0984 


17-196 


340 3079 


17-296 


307 9234 


16-998 


414 8226 


17-098 


375 3470 


17-198 


339 6280 


17-298 


307 3081 


17 000 


413 9938 


17-100 


374 5970 


17-200 


338 9494 


17-300 


306 6941 


17-002 


413 1666 


17-102 


373 8486 


17-202 


338 2722 


17-305 


305 1645 


17004 


412 3411 


17-104 


373 1016 


17-204 


337 5964 


17-310 


303 6425 


17 006 


411 5173 


17-106 


372 3562 


17-206 


336 9218 


17-315 


302 1280 


17-008 


410 6951 


17-108 


371 6122 


17-208 


Zi<^ 2487 


17-320 


300 6212 


17-010 


409 8745 


17-110 


370 8697 


17-210 


335 5768 


17-325 


299 1218 


17-012 


409 0556 


17-112 


370 12S7 


17-212 


334 9064 


17-330 


297 6299 


17-014 


408 2383 


17-114 


369 3892 


17-214 


334 2372 


17-335 


296 1455 


17016 


407 4226 


17-116 


36S 6512 


17-216 


333 5694 


17-340 


294 6685 


17-018 


406 6086 


17-118 


367 9146 


17-218 


332 9029 


17-345 


293 1988 


17-020 


405 7961 


17-120 


3<57 1795 


17-220 


332 2378 


17-350 


291 7365 


17022 


404 9854 


17-122 


366 4459 


I7'222 


331 5740 


17-355 


290 2814 


17-024 


404 1762 


17-124 


365 7137 


17-224 


330 9115 


17-360 


288 8336 


17-026 


403 3687 


17126 


364 9830 


17-226 


330 2504 


17-365 


287 3931 


17 028 


402 5628 


17128 


364 2538 


17-228 


329 5905 


17-370 


285 9597 


17030 


401 7584 


17-130 


363 5260 


17-230 


328 9320 


17-375 


284 5335 


17-032 


400 9557 


17-132 


362 7997 


17-232 


328 2748 


17-380 


283 1 143 


'7-034 


400 1546 


17-134 


362 0748 


17-234 


327 6189 


17-385 


281 7023 


17-036 


399 3551 


17-136 


361 3514 


17-236 


326 9643 


17-390 


28a 2973 


17038 


398 5572 


17-138 


360 6294 


17-238 


326 3H0 


17-395 


278 8993 


17040 


397 7608 


17-140 


359 9089 


17-240 


325 6590 


17-400 


277 5083 


17-042 


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 7470 


17-046 


395 3814 


17-146 


357 7559 


17-246 


323 7'09 


17-415 


273 3768 


17-048 


394 5914 


17-148 


357 0411 


17-248 


323 0642 


17-420 


272 0133 



[i7"42S— i8-42o] 



OF THE DESCENDING EXPONENTIAL. 
^Second Fart. Fourteen decimal places?^ 



233 



X 


Q-X 


X 


Q-X 


X 


e-^ 


1 X 


Q-X 


17-425 


270 6566 


17-675 


210 7876 


17-925 


164 1615 


18-175 


127 8491 


17-430 


269 3067 


17-680 


209 7363 


17-930 


163 3428 


i8-i8o 


127 2115 


17-435 


267 9635 


17-685 


2o8 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 9109 


18-195 


125 317s 


17-450 


263 9741 


17-700 


205 5832 


17-950 


160 1084 


18-200 


124 6925 


17-455 


262 6575 


17-705 


204 5579 


17-955 


159 3098 


18-205 


124 0706 


17-460 


261 3475 


17-710 


203 5376 


17-960 


158 5153 


i8-2io 


123 4518 


17-465 


260 0440 


17-715 


202 5225 


17-965 


157 7247 


18-215 


122 8361 


17-470 


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 


121 6139 


17-483 


256 1725 


17-73° 


199 5073 


17-9S0 


155 3765 


18-230 


121 0073 


17-485 


254 8948 


17-735 


198 5123 


17-985 


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 


119 2057 


17-500 


251 0999 


17-750 


195 5568 


18-000 


152 2998 


18-250 


118 6112 


17-505 


249 8475 


17-755 


194 5815 


18-005 


151 5402 


18-255 


118 0196 


17-510 


248 6014 


17-760 


193 6110 


18-010 


150 7844 


18-260 


117 431° 


17-5^5 


247 3615 


17-765 


192 6454 


18015 


150 0323 


18-265 


116 8453 


17-520 


246 1278 


17-770 


igi 6845 


18-020 


149 2841 


18-270 


116 2625 


17-525 


244 9002 


17-775 


190 7285 


18-025 


148 5395 


18-275 


115 6827 


17-530 


243 6788 


17-780 


189 7772 


18-030 


147 7987 


18-280 


115 1057 


17-535 


242 4634 


17-785 


188 8307 


18-035 


147 0615 


18-285 


114 5316 


17-540 


241 2541 


17-790 


187 8889 


18-040 


146 3280 


18-290 


113 9604 


17-545 


240 0509 


17-795 


186 9518 


18-045 


145 5982 


18-295 


113 3920 


17-550 


238 8536 


17-800 


1S6 0194 


18-050 


144 8720 


18-300 


112 8264 


17-555 


237 6623 


17-805 


185 0916 


18-055 


144 1495 


i8-3°5 


112 2637 


17-560 


236 4770 


17-810 


184 1685 


i8-o6o 


143 4305 


18-310 


III 7038 


17-565 


235 2975 


17-815 


183 2499 


18-065 


142 7152 


18-315 


III 1467 


17-570 


234 1240 


17-820 


182 3360 


18-070 


142 0034 


18-320 


110 5923 


17-575 


232 9563 


17-825 


181 4266 


18-075 


141 2951 


18-325 


110 0407 


17-580 


231 7944 


17-83° 


180 5217 


i8-o8o 


140 5904 


18-330 


109 4919 


17-585 


230 6383 


17-835 


179 6213 


18-085 


139 8892 


18-335 


108 9458 


17-590 


229 4880 


17-840 


178 7255 


18-090 


139 1915 


18-340 


108 4025 


17-595 


228 3435 


17-845 


177 8341 { 


18-095 


138 4973 


18-345 


107 8618 


17-600 


227 2046 


17-850 


176 9471 


i8-ioo 


137 8065 


18-350 


107 3238 


17-605 


226 0714 


17-855 


176 0646 


18-105 


137 1192 


18-355 


106 7S86 


17-610 


224 9439 


17-860 


175 1865 


18110 


136 4353 


18-360 


106 2559 


17-615 


223 8220 


17-865 


174 3127 


18-115 


135 7549 , 


18-365 


105 7260 


17-620 


222 7057 


17-870 


173 4433 


18-120 


135 0778 


18-370 


105 1987 


17-625 


221 5949 


17-875 


172 5783 


18-125 


134 4041 1 


18-375 


104 6740 


17-630 


220 4897 


17-880 


171 7175 i 


-18-130 


133 7338 1 


18-380 


104 1519 


17-635 


219 3900 


17-885 


170 S611 I 


18-135 


1T,1 0667 


18-385 


103 6325 


17-640 


218 2958 


17-890 


170 0089 


18-140 


132 4031 


18-39° 


103 1156 


17-645 


217 2070 


17-895 


169 1610 


18-145 


131 7427 


18-395 


102 6013 


17-650 


216 1237 


17-900 


168 3173 


18-150 


131 0856 


18-400 


102 0896 


17-655 


215 0458 


17-905 


167 4778 


18-155 


130 4318 


18-405 


101 5804 


17-660 


213 9732 


17-910 


166 6425 


18160 


129 7813 


18-410 


101 0738 


17-665 


212 9060 


17-915 


165 8114 i 


18-165 


129 1340 


18-415 


100 5697 


17-670 


211 8442 


17-920 


164 9844 


18170 


128 4900 1 


18-420 


100 0681 



Vol. XIII. Pakt III. 



31 



234 



MR F. W. NEWMAN'S TABLE 



[18-425— 19-670] 



[Second Part. Fourteen decimal places^ 



X 


Q-X 


! ^ 


(,-X 


X 


(,-X 


X 


Q-X 


1 

X ■ 

[ 


C-x 


18-425 


99 5690 


\ 18-675 


77 5444 


18-925 


60 3917 


'9-'75 


47 0331 


19-425 


36 6294 


18430 


99 0724 


18-680 


77 1576 


18-930 


60 0905 


19-180 


46 7985 


19-430 


36 4467 


1 18-435 


98 5783 


18-685 


76 7728 


'8935 


59 7908 


19-185 


46 5651 


'9-435 


36 2649 


18-440 


98 0866 


18-690 


76 3S99 


18-940 


59 4925 


19-190 


46 3328 


19-440 


36 0840 


iS-445 


97 5974 


18-695 


76 0089 


'8-945 


59 '958 


i9-'95 


46 1018 


'9-445 


35 9041 


18-450 


97 1106 


18-700 


75 6298 


18-950 


58 9006 


19-200 


45 8718 


19-450 


35 7250 


18-455 


96 6263 


18-705 


75 2526 


'8-955 


58 6068 


19-205 


45 6430 


19-455 


35 5468 


18-460 


96 1444 


18-710 


74 8773 


18-960 


58 3'45 


19-210 


45 4154 


19-460 


35 3695 


18-465 


95 6648 


18-715 


74 5039 


18-965 


58 0237 


19-215 


45 1889 


19-465 


35 '93' 


18-470 


95 1877 


18-720 


74 1323 


18-970 


57 7343 


19-220 


44 963s 


19-470 


35 0176 


18-475 


94 7130 


18-725 


73 7625 


18-975 


57 4463 


19-225 


44 7392 


'9-475 


34 8429 


18-480 


94 2406 


18-730 


73 3946 


18-980 


57 '598 


19-230 


44 5161 


19-480 


34 6692 


18-485 


93 77^5 


'8-735 


73 0286 


18-985 


56 8747 


'9-235 


44 294' 


'9-4S5 


34 4963 


18-490 


93 3029 


18-740 


72 6643 


18-990 


56 5910 


19-240 


44 0732 


19-490 


34 3242 


18-495 


92 8375 


18745 


72 3°'9 


18-995 


56 3088 


19-245 


43 8534 


'9-495 


34 '530 


18-500 


92 3745 


iS-750 


7' 9413 


19-000 


56 0280 


19-250 


43 6346 


19-500 


33 9827 


18-505 


91 9138 


'8-755 


71 5825 


19-005 


55 7485 


19-255 


43 4170 


'9-505 


33 8132 


18-510 


91 4554 


18-760 


7' 2255 


19-010 


55 4705 


19-260 


43 2004 


19-510 


33 6445 


18-515 


90 9992 


18-765 


70 8703 


19-015 


55 1938 


19-265 


42 9850 


19-S'S 


33 4767 


18-520 


90 5453 


18-770 


70 5168 


19-020 


54 9'85 


19-270 


42 7706 


19-520 


33 3098 


18-525 


90 0938 


'8-775 


76 1651 


19-025 


54 6446 


19-275 


42 5573 


'9-525 


33 '436 


'8-53^ 


89 6444 


18-780 


69 8152 


19-030 


54 3721 


19-280 


42 3450 


19-530 


32 9783 


•8-535 


89 1973 


18-785 


69 4670 


'9-035 


54 '009 


19-285 


42 1338 


'9-535 


32 8139 


18-540 


88 7524 


18-790 


69 1205 


19-040 


53 8311 


19-290 


4' 9237 


'9-540 


32 6502 


18-545 


88 3098 


18-795 


68 7757 


19-045 


53 5626 


'9-295 


41 7146 


'9-545 


32 4873 


18-550 


87 8693 


18-800 


68 4327 


19-050 


53 2954 


19-300 


41 5065 


'9-550 


32 3253 


18-555 


87 43" 


18-805 


68 0914 1 


'9-055 


53 0296 


19-305 


41 2995 


'9-555 


32 1641 


18-560 


86 9950 


18-810 


67 7518 


19-060 


52 765' 


19-310 


4' 0935 


19-560 


32 0037 


18-565 


86 5611 


18-815 


67 4139 


19-065 


52 5020 


'9-315 


40 8886 


'9-565 


3' 8441 


18-570 


86 1294 


18-820 


67 0776 


19-070 


52 2401 


19-320 


40 6846 


'9-570 


3' 6852 


18-575 


85 6998 


18-825 


66 7431 


i9'o75 


5' 9796 


'9325 


40 4817 


19-575 


31 5272 


18-580 


85 2724 


18-830 


66 4102 


19-080 


5' 7203 


19-330 


40 2798 


19-580 


31 3700 


18-585 


84 8471 


18-835 


66 0790 


19-085 


51 4624 


'9335 


40 0789 


'9-585 


3' 2135 


18-590 


84 4239 


18-840 


65 7494 


19-090 


5' 2057 


'9-340 


39 8790 


'9-590 


31 0578 


'8-595 


84 0029 


18-845 


65 4215 


'9-095 


50 9503 


'9345 


39 6801 


19-595 


30 9029 


i8-6do 


83 5839 


18-850 


65 0952 


19-100 


50 6962 


19-350 


39 4S22 


X9-6oo 


30 7488 


18-605 


83 1670 


i8-3=;5 


64 7706 


19-105 


50 4433 


19-355 


39 2853 


19-605 


30 5954 


18-610 


82 7522 


18-860 


64 4475 


19-110 


50 '917 


19-360 


39 0894 


19-610 


30 4428 


18-615 


82 3395 


18865 


64 1261 


19-115 


49 9414 


'9-365 


38 8944 


19-615 


30 2910 


18-620 


8i 9288 


18-870 


63 8062 


19-120 


49 6923 


'9-370 


38 7004 


19-620 


30 '399 


18-625 


81 5202 


18-875 


63 4880 


19-125 


49 4445 


19-375 


38 5074 


19-625 


29 9896 


18-630 


81 1136 


iS-880 


63 1713 


19-133 


49 1979 


19-380 


38 3154 


19-630 


29 8400 


'8635 


80 7091 


18-885 


62 8563 


'9-'35 


48 9525 


19-385 


38 1243 


19-635 


29 6912 


18-640 


80 3065 


18-890 


62 5428 


19-140 


48 7084 


19-390 


37 9341 


19-640 


29 5431 


18-645 


79 9060 


18-895 


62 2309 


'9-145 


48 4654 


19-395 


37 7449 


19-645 


29 3958 


18-650 


79 5075 


18-900 


61 9205 


19-150 


48 2237 


19-400 


37 5567 


19-650 


29 2492 


'8-655 


79 1109 


18-905 


61 6116 


19-155 


47 9832 


19-405 


37 3693 


'9-655 


29 1033 


18-660 


78 7164 


. 18-910 


61 3044 


19-160 


47 7439 


19-410 


37 1830 


19-660 


28 9581 


18-665 


78 3238 


18-915 


60 9986 


19-165 


47 5057 


19-415 


36 9975 


19-665 


28 8.37 


18670 


77 933' ; 


18-920 


60 6944 1 


19-170 


47 26S8 


19-420 


36 8130 


19-670 


28 6700 1 



[i9'675— 2o'92°] 



OF THE DESCENDING EXPONENTIAL. 
[Secoid Pari. Fouiieai decimal places.^ 



235 



X 


(,-X 


X 


Q-X 


X 


e 


-X 


X 


Q-X 


X 


,-. 


i9'67S 


28 5270 


19-925 


22 2168 


20-175 


17 


3025 


20-425 


13 4752 


20-675 


1 

10 4945 


l9'68o 


28 3847 


19-930 


22 1060 


20-i8o 


17 


2162 


20-430 


13 40S0 


2o-68o 


10 4421 


19-685 


28 2431 


19-935 


21 9958 


20-185 


17 


1303 


20-435 


13 3411 


20-685 


10 3901 


19-690 


28 1023 


19-940 


21 8861 


20-190 


17 


0449 


20-440 


13 2746 


20-690 


10 3382 


19-695 


27 9621 


19-945 


21 7769 


20-195 


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 


19705 


27 6839 


19-955 


21 5602 


20-205 


16 


7911 


20-455 


13 0769 


20-705 


10 1843 


19-710 


27 5458 


19-960 


21 4527 


20-210 


16 


7074 


20-460 


13 0II7 


20-710 


10 1335 


19715 


27 4084 


19-965 


21 3457 


20-215 


16 


6240 


20-465 


12 9468 


20-715 


10 0830 


19-720 


27 2717 


19-970 


21 2392 


20-220 


16 


5411 


20-470 


12 8822 


20-720 


10 0327 


19725 


27 1357 


19-975 


21 1333 


20-225 


16 


4586 


20-475 


12 8180 


20-725 


9 9827 


19-730 


27 0004 


19-980 


21 0279 


20-230 


16 


3765 


20-480 


12 7541 


20-730 


9 9329 


19735 


26 8657 


19-985 


20 9230 


20-235 


16 


2949 


20-4S5 


12 6905 


20-735 


9 8S33 


19-740 


26 7317 


19-990 


20 8187 


20-240 


16 


2136 


20-490 


12 6272 


20-740 


9 8340 


19745 


26 5984 


19-995 


20 7148 


20-245 


16 


1327 


20-495 


12 5642 


20-745 


9 7850 


1975° 


26 4657 


20-000 


20 6115 


20-250 


16 


0523 


20-500 


12 5015 


20-750 


9 7362 


19755 


26 3337 


20-005 


20 5087 


20-255 


15 


9722 


20-505 


12 4392 


20-755 


9 6876 


19-760 


26 2024 


20-0I0 


20 4064 


20-260 


15 


8925 


20-510 


12 3771 


20-760 


9 6393 


19765 


26 0717 


20-015 


•20 3047 


20-265 


15 


8133 


20-515 


12 3154 


20-765 


9 5912 


19-770 


25 9417 


20-020 


20 2034 


20-270 


15 


7344 


20-520 


12 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 


20-280 


15 


5779 


20-530 


12 1320 


20-780 


9 4484 


19785 


25 5555 


20-035 


19 9026 


20-285 


15 


5002 


20-535 


12 0715 


20-785 


9 4013 


19-790 


25 42S0 


20-040 


19 8033 


20-290 


15 


4229 


20-540 


12 OII3 


20-790 


9 3544 


19795 


25 3312 


20-045 


19 7046 


20-295 


15 


3459 


20-545 


II 9514 


20-795 


9 3078 


19-800 


25 1750 


20-050 


19 6063 


20-300 


15 


2694 


20-550 


II 8918 


20-8oO 


9 2614 


19-805 


25 0494 


20-055 


19 5085 


20-305 


15 


1932 


20-555 


II 8325 


20-805 


9 2152 


19-810 


24 9245 


20-o6o 


19 4112 


20-310 


15 


1175 


20-560 


II 7735 


20-810 


9 1692 


19-815 


24 8002 


20-065 


19 3144 


20-315 


15 


0421 


20-565 


II 7148 


20-815 


9 1235 


19-820 


24 6765 


20-070 


19 2181 


20-320 


14 


9670 


20-570 


II 6563 


20-820 


9 0780 


19-825 


24 5534 


20-075 


19 1222 


20-325 


14 


8924 


20-575 


II 5982 


20-825 


9 0327 


19-830 


24 4309 


20-080 


19 0268 


20-330 


14 


81 81 


20-580 


II 5404 


20-830 


8 9876 


19-835 


24 3=391 


20-085 


18 9319 


20-335 


14 


7442 


20-585 


II 4828 


20-835 


8 9428 


19-840 


24 1879 


20-090 


18 8375 


20-340 


14 


6707 


20-590 


II 4255 


20-840 


8 8982 


19-845 


24 0672 


20-095 


18 7436 


20-345 


14 5975 


20-595 


II 3685 


20-845 


8 8538 


19-850 


23 9472 


20-IOO 


18 6501 


20-350 


14 


5247 


20-600 


II 3118 


20-850 


8 8097 


19-855 


23 8277 


20-105 


18 5571 


20-355 


14 


4523 


20-605 


II 2554 


20-855 


8 7657 


19-860 


23 7089 


20-110 


18 4645 


20-360 


14 


3802 


2o-6io 


II 1993 


20'86o 


8 7220 


19-865 


23 59°7 


20-115 


18 3724 


20-365 


14 


3085 


20-615 


II 1434 


20-865 


8 6785 


19-870 


23 4730 


20-I20 


18 2808 


20-370 


14 


2371 


20-620 


II 0879 


20-870 


8 6352 


19-875 


23 3559 


20-125 


18 1896 


20-375 


14 


1661 


20-625 


II 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 


20-635 


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 


20-145 


17 8294 


20-395 


13 


8856 


20-645 


10 8141 


20-895 


8 4220 


19-900 


22 7793 


20-150 


17 7405 


20-400 


13 


8163 


20-650 


10 7602 


20-goo 


8 3800 


19-905 


22 6656 


20-155 


17 6520 


20-405 


13 


7474 


20-655 


10 7065 


20-905 


8 3382 


19-910 


22 5526 


20-160 


17 5640 


20-410 


13 


6788 


20-660 


10 6531 


20-910 


8 2966 


19-915 


22 4401 


20-165 


17 4764 


20-415 


13 


6106 


20-665 


10 6000 


20-915 


8 2553 


19-920 


22 3282 


20-170 


17 3892 


20-420 


13 


5427 


20-670 


10 5471 1 


20-920 


8 2141 



31- 



!36 



UR F. W. NEWMAN'S TABLE 
[Se-fottd Part. Fourteen decimal places?^ 



[20-925— 22-170] 



X 


Q-X 


1 

X 


Q-m 


X 


Q-X 


X 


Q-X 


X 


Q-X 


20-925 


8 1731 


21-175 


6 3652 


21-425 


4 9572 


21-675 


3 8607 


21-925 


3 0067 


20-930 


8 1323 


21-180 


6 3335 


21-430 


4 9325 


21-680 


3 8414 


21-930 


2 9917 


20935 


8 0918 


21-185 


6 3019 


21-435 


4 9079 


21-685 


3 8223 


21-935 


2 9768 


20940 


8 0514 


21-190 


6 2705 


' 21-440 


4 8834 


21-690 


3 8032 


21-940 


2 9620 


20-945 


8 0113 


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 


21-950 


2 9325 


20-955 


7 9316 


21-205 


6 1771 


21-455 


4 8107 


21-705 


3 7466 


2 1-955 


2 9179 


30-960 


7 8920 


21-210 


6 1463 


21-460 


4 7867 


21-710 


3 7279 


21-960 


2 9033 


20965 


7 8526 


21-215 


6 II t;6 


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 


21-225 


6 0548 


21-475 


4 7155 


21-725 


3 6724 


21-975 


2 8601 


20-9S0 


7 7357 


21-230 


6 0246 


21-4S0 


4 6920 


21-730 


3 6541 


21-980 


2 8458 


20-985 


7 6971 


21-235 


5 9945 


21-485 


4 66S6 


21-735 


3 6359 


21-985 


2 8316 


2099 D 


7 6588 


21-240 


5 9646 


21-490 


4 6453 


21-740 


3 6177 


21-990 


2 8175 


20995 


7 6206 


21-245 


5 9349 


21-495 


4 6221 


21-745 


3 5997 


21-995 


2 8034 


21-000 


7 5826 


21-250 


5 9053 


21-500 


4 5090 


21-750 


3 5817 


22-000 


2 7895 


21-005 


7 5447 


21-255 


5 8758 


21-505 


4 5761 


21-755 


3 5639 


22-005 


2 7756 


21-010 


7 5°7i 


21-260 


5 8465 


21-510 


4 5533 


21-760 


3 5461 


22-OIO 


2 7617 


21-015 


7 4697 


21-265 


5 8174 


21-515 


4 5306 


21-765 


3 5284 


22-015 


2 7479 


2I-O2 


7 4324 


21-270 


5 7884 


21-520 


4 5080 


21-770 


3 5108 


22-020 


2 7342 


21-025 


7 3953 


21-275 


5 7595 


21-525' 


4 4855 


21-775 


3 4933 


22-025 


2 7206 


21-030 


7 3585 


21-283 


5 7308 


21-530 


4 4631 


21-780 


3 4759 


22-030 


2 7070 


21-035 


7 3218 


21-285 


5 7022 


21-535 


4 4409 


21-785 


3 45S5 1 


22-035 


2 6935 


21-040 


7 2852 


21-290 


5 6737 


21-540 


4 4187 


21-790 


3 4413 


22-040 


2 68oi 


21045 


7 2489 


21-295 


5 6454 


21-545 


4 3967 


21-795 


3 4241 


22-045 


2 6667 


21-050 


7 2127 


21-300 


5 6173 


21-550 


4 3747 


21-800 


3 4071 


22-050 


2 6534 


21-055 


7 1768 


21-305 


5 5893 1 


21-555 


4 3529 


21-805 


3 3901 : 


22-055 


2 6402 


21-060 


7 1410 


21-310 


5 5614 


21-560 


4 3312 


2I-8lO 


3 3732 1 


2 2 -060 


2 6270 


21-065 


7 1054 


21-315 


5 5337 


21-565 


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 


21-075 


7 0347 


21-325 


5 4786 


21-575 


4 2667 


21-825 


3 3229 


22-075 


2 5879 


2i-o8o 


6 9996 


21-330 


5 4513 


21-580 


4 2455 


21-830 


3 3064 


22-080 


2 5750 


21-085 


6 9647 


21335 


5 4241 


21-585 


4 2243 


21-835 


3 2899 


22-085 


2 5622 


21-090 


6 9299 


21-340 


5 397° 


21-590 


4 2032 


21-840 


3 2735 


22-090 


2 5494 


21095 


6 8954 


21-345 


5 3701 


21-595 


4 1823 


21-845 


3 2571 ! 


22-095 


2 5367 


21-100 


6 8610 


21-350 


5 3433 


21-600 


4 1614 


21-850 


3 2409 ' 


22-IOO 


2 5240 1 


21-105 


6 8268 


21-355 


5 3167 


21-605 


4 1406 


21-855 


3 2247 ' 


22-105 


2 5114 


21-110 


6 7927 


21-360 


5 2902 


21-610 


4 1200 


21-860 


3 2086 


22-110 


2 4989 


21115 


6 7588 


21-365 


5 2638 


21-615 


4 0994 


2I-S65 


3 1926 


22-115 


2 4864 


21120 


6 7251 


21-370 


5 2376 


21620 


4 0790 


21-870 


3 1767 


22-120 


2 4740 


21125 


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 


21-630 


4 0384 


21-880 


3 1451 i 


22-130 


2 4494 


2II35 


6 6250 


21-385 


5 1595 


21635 


4 0183 


21-885 


3 1294 j 


22-135 


2 4372 


21140 


6 5920 


21-390 


5 1338 


21-640 


3 9982 


21-890 


3 1138 


22-140 


2 4250 


2II45 


6 5591 


21-395 


5 1082 


21-645 


3 9783 


21895 


3 0983 


22-145 


2 4129 


21-150 


6 5264 


21-400 


5 0827 


21-650 


3 9584 


21-900 


3 0828 


22-150 


2 4009 


21-155 


6 4938 


21-405 


5 0574 


21-655 


3 9387 


21-905 


3 0674 


22-155 


2 3889 


21160 


6 4614 


21-410 


5 0322 


21-660 


3 9190 


21-910 


3 0521 1 


22-l6o 


2 3770 


21165 


6 4292 


21-415 


5 0071 


21-665 


3 8995 


21-915 


3 0369 1 


22-165 


2 3652 


21170 


6 3971 


21-420 


4 9S21 


21-670 


3 8801 


21-920 


3 0218 ' 


22-170 


2 3534 



[22-175— 23"42o] 



OF THE DESCENDING EXPONENTIAL. 
\Seco7id Part. Fourken decimal places."] 



237 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


22-175 


2 3416 


22-425 


I 8237 


22-675 


I 4203 


22-925 


I 1061 


23'i75 


8614 


22-180 


2 3299 


22-430 


I 8146 


22-680 


I 4132 


22-930 


1 1006 


23-180 


8571 


22-185 


2 3183 


22-435 


I 8055 


22-685 


I 4061 


2 2-935 


I 0951 


23-185 


8529 


22-190 


2 3068 


22-440 


I 7965 


22-693 


I 3991 


22-940 


I 0896 


23-190 


8486 


22-195 


2 2953 


22-445 


1 7876 


22-695 


I 3921 


22-945 


I 0842 


23-195 


8444 


22-203 


2 2838 


22-450 


r 7786 


22-700 


I 3852 


22-950 


I 0788 


23-200 


8402 


22-205 


2 2724 i 


22-455 


I 7698 


22-705 


I 3783 


22-955 


I 0734 


23-205 


8360 


22-2IO 


2 2611 ' 


22-460 


I 7639 


22-710 


I 3714 


22-960 


1 0681 


23-210 


8318 


22-215 


2 2498 


22-465 


r 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 


23-220 


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 


I 7261 


22-730 


I 3442 


22-980 


I 0469 


23-230 


8153 


22-235 


2 2053 


22-485 


I 7175 


22-735 


1 3376 


22-985 


I 0417 


23-235 


81I3 


22-240 


2 1943 


22-493 


I 7089 


22-740 


I 33^9 


22-990 


I 0365 


23-240 


8072 


22-245 


2 1833 [ 


22-495 


I 7004 


22-745 


I 3243 


22-995 


I 0313 


23-245 


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 


I 6835 


22-755 


1 3111 


23-005 


I 0211 


23-255 


7952 


22-260 


2 1508 


22-513 


I 6751 


22-760 


I 304 s 


23-010 


I 0160 


23-260 


7912 


22-265 


2 I40I 


22-515 


I 6667 


22-765 


I 2983 


23-015 


I 0109 


23-265 


7S73 


22-270 


2 1294 i 


22-520 


I 6584 


22-770 


I 2915 


23-020 


I 0059 


23-270 


7834 


22-275 


2 1188 


22-525 


I 6501 


22-775 


I 2851 


23-025 


I 0009 


23-275 


7795 


22-280 


2 1082 


22-533 


I 6419 


22-780 


I 27S7 


23-030 


9959 


23-280 


7756 


22-285 


2 0977 


22-535 


I 6337 


22-785 


I 2723 


23-°3S 


9909 


23-285 


7717 


22-293 


2 0873 


22-540 


I 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 


9810 


23-295 


7640 


22-300 


2 0665 


22-550 


1 6094 


22-800 


I 2534 


23-050 


9761 


23-300 


7602 


22-305 


2 0562 


22-555 


I 60:4 


22-805 


I 2471 


23-055 


9713 


23-305 


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 


23-065 


9616 


23-315 


7489 


22-320 


2 0256 


22-570 


I 5775 


22-820 


I 2286 


23-070 


9568 


23-320 


7452 


22-325 


2 0155 


22-575 


I 5697 


22-825 


I 2224 


23-075 


9520 


23-325 


7415 


22-330 


2 OD54 


22-580 


I 5618 


22-830 


I 2163 


23-080 


9473 


2^33,0 


7378 


22-335 


I 9954 


22-585 


I 5540 


22-835 


I 2103 


23-085 


9426 


23-335 


7340 


22-340 


I 985s 


22-590 


I 5463 


22-840 


I 2042 


23-090 


9379 


23-340 


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 53^9 


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 


1 9461 


22-610 


I 5157 


22-860 


I 1834 


23-110 


9193 


23-360 


7159 


22-365 


I 9364 


22-615 


I 5081 


22-865 


I 1745 


23'iiS 


9147 


23-365 


7123 


22-370 


1 9268 


22-620 


I 5006 


22-870 


I 1686 


23-120 


9102 


23-370 


7088 


22-375 


1 9172 


22-625 


I 4931 


22-875 


I 1628 


23-125 


9356 


33-375 


7053 


22-380 


1 9076 


22-630 


I 4856 


22-8S0 


I 1570 


23-130 


9311 


23-380 


7018 


22-385 


1 8981 


22-635 


I 4782 


22-885 


1 1512 


23-135 


8966 


23-385 


6983 


22-390 


I 8886 


22-640 


I 4709 


22-890 


I 1455 


23-140 


8921 


23-390 


6948 


22-395 


I 8792 


22-645 


I 463s 


22-895 


I 1398 


23'i45 


8877 


23-395 


6913 


22-400 


1 8698 


22-650 


I 4562 


22-900 


I 1341 


23-150 


8832 


23-400 


6879 


22-405 


I 8605 


22-655 


I 4490 


22-905 


I 1284 


23-155 


8788 


.23-405 


6844 


22-4x0 


1 8512 


22-66o 


I 4417 


22-910 


I 1228 


23-160 


8745 


23-410 


6810 


22-415 


I 8420 


22-665 


I 4345 


22-915 


I 1172 


23-165 


8701 


23-415 


6776 


22-420 

1 


I 8328 


22-670 


I 4274 


22-920 


1 1116 


23-170 


8658 


23-420 


6742 



238 



MR F. W. NEWMAN'S TABLE 
[Second Part. Fourteen decimal places.'] 



[23-4^5— 24-<>7o] 



X 


Q-X 


X 


(,-X 


X 


Q-X 


X 


Q-X 


1 

X 


Q-X 


23 "42 5 


6709 


23-675 


5225 


23-925 


4069 


24-175 


3169 


24-425 


2468 


23'43o 


6675 


23-680 


5 "99 


23-930 


4049 


24-180 


3153 


24-430 


2455 


23-435 


6642 


23-685 


5173 


23-935 


4029 


24-185 


3137 


24-435 


2443 


23-440 


6609 


23-690 


5147 


23940 


4009 


24-190 


3122 


24-440 


2431 


23-445 


6576 


23-695 


5122 


23945 


3989 


24-195 


3106 


24-445 


2419 


23-450 


6543 


23-700 


5096 


23-950 


3969 


24-200 


3091 


24-450 


2407 


23-455 


6510 


23-705 


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 


23-465 


6446 


23-715 


5020 


23-965 


3910 


24-215 


3045 


24-465 


2371 


23-470 


6414 


23-720 


4995 


23970 


3890 


24-220 


3030 


24-470 


2359 


23-475 


63S2 


23-725 


4970 


23-975 


3871 


24-225 


3015 


24-475 


2348 


23-480 


6350 


23-730 


4945 


23-980 


3852 


24-230 


3000 


24-483 


2336 


23-485 


6318 


23-735 


4920 


23-985 


3833 


24-235 


2985 


24-485 


2324 


23-490 


6286 


23-740 


4895 


23-990 


3S14 


24-240 


2970 


24-490 


2313 


23-495 


6255 


23-745 


4871 


23995 


3794 


24-245 


2955 


24-495 


2301 


23-500 


6224 


23-750 


4847 


24-000 


3775 


24-250 


2940 


24-500 


2290 


23-505 


6193 


23-755 


4823 


24-005 


3756 


24-255 


2925 


24-505 


2278 


23-510 


6162 


23-760 


4799 


24-010 


3737 


24-260 


2910 


24-510 


2267 


23-515 


613I 


23-765 


4775 


24-015 


3718 


24-265 


2895 


24-515 


2256 


23-520 


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 


24-530 


2222 


23-535 


6010 


23-785 


4680 


24-035 


3645 


24-285 


2S39 


24-535 


22II 


23540 


5980 


23-790 


4657 


24-040 


3627 


24-290 


2825 


24-540 


2200 


23-545 


5950 


23-795 


4634 


24-045 


3609 


24-295 


2810 


24-545 


2189 


23-550 


^921 


23-800 


461 1 


24-050 


3591 


24-300 


2797 


24-550 


2178 


23-555 


5891 


23-805 


4588 


24-055 


3573 


24-305 


2783 


24-555 


2167 


23-560 


5861 


23-810 


4565 


24-060 


3555 


24-310 


2769 


24-560 


2156 


23-565 


5832 


23-815 


4542 


24-065 


3536 


24-315 


2755 


24-565 


2146 


23-570 


5803 


23-820 


4519 


24-070 


3519 


24-320 


2741 


24-570 


2135 


23-575 


5774 


23-825 


4497 


24-075 


3502 


24-325 


2728 


24-575 


2124 


23-580 


5745 


23-830 


4474 


24-080 


3485 


24-330 


2714 


24-580 


2II3 


23-585 


5716 


23835 


4452 


24-085 


3467 


24-335 


2700 


24-585 


2103 


23-590 


5687 


23840 


4430 


24-090 


3450 


24-340 


2687 


24-593 


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 


24-630 


2072 


23-605 


5604 


23-855 


4364 


24-105 


3398 


24-355 


2647 


24-605 


2061 


23-610 


5576 


23-860 


4342 


24-1 10 


3381 


24-360 


2634 


24-610 


2051 


23-615 


5548 


23-865 


4320 


24-115 


33^M 


24-365 


2621 


24-615 


2041 


23-620 


5521 


23870 


4299 


24-120 


3347 


24-370 


2608 


24-620 


2031 


23-625 


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 


423s 


24-135 


3297 


24-385 


2569 


24-635 


2000 


23-640 


54" 


23-890 


4214 


24-140 


3281 


24-390 


2556 


24-640 


1990 


23645 


5384 


23895 


4193 


24-145 


3265 


24-395 


2543 


24-645 


I981 


23650 


5357 


23-900 


4172 


24-150 


3249 


24-400 


2530 


24-650 


I97I 


23655 


5330 


23-905 


4151 


24-155 


3233 


24-405 


2518 


24-655 


I961 


23 660 


5303 


23-910 


4130 


24-160 


3217 


24-410 


2505 


24-660 


I95I 


23665 


5277 


23-915 


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 



[24'G75— 25'92o] 



OF THE DESCENDING EXPONENTIAL. 
[^Second Part Fourteen decimal places.'] 



239 



X 


(,-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


g-K 


24-675 


1922 


24-925 


1497 


25-175 


I165 


25-425 


908 


25-675 


707 


24-680 


1913 


24-930 


1489 


25-180 


n6o 


25-430 


903 


25-680 


704 


24-685 


1903 


24-935 


1482 


25-185 


1154 


25-435 


899 


25-685 


700 


24-690 


1894 


24-940 


1475 


25-190 


1 148 


25-440 


894 


25-690 


697 


24'695 


1884 


24-945 


1467 


25195 


1142 


25-445 


890 


25695 


693 


24-700 


187s 


24-950 


1460 


25-200 


"37 


25-450 


885 


25700 


690 


24705 


1865 


24-955 


1453 


25-205 


1131 


25-455 


8SI 


25-705 


686 


24-710 


1856 


24-960 


1445 


25-210 


1126 


25-460 


877 


25710 


683 


24715 


1847 


24-965 


1438 


25-215 


1120 


25-465 


872 


25-715 


679 


24-720 


1838 


24-970 


143 I 


25-220 


"IS 


25-470 


868 


25-720 


676 


24725 


1828 


24-975 


1424 


25-225 


1109 


25-475 


864 


25-725 


673 


24730 


1819 


24-980 


I4I7 


25-230 


1 103 


25-480 


859 


25-730 


669 


24735 


1810 


24-985 


I4IO 


25-235 


1098 


25-485 


855 


25735 


666 


24-740 


i8or 


24-990 


1403 


25-240 


1092 


25-490 


851 


25-740 


663 


24745 


1792 


24-995 


1396 


25-245 


1087 


25-495 


847 


25-745 


659 


24750 


1783 


25-000 


1389 


25-250 


1082 


25-500 


842 


25-750 


656 


24755 


1774 


25-005 


1382 


25-255 


1076 


25-505 


838 


25755 


653 


24760 


1766 


25-010 


1375 


25-260 


1071 


25-510 


834 


25-760 


649 


24765 


1757 


25-015 


1368 


25-265 


1065 


25-515 


830 


25-765 


646 


24770 


1748 


25-020 


I36I 


25-270 


1060 


25-520 


826 


25-770 


643 


24775 


1739 


25-025 


1354 


25-275 


1055 


25-525 


822 


25775 


640 


24-780 


1730 


25-030 


1348 


25-280 


1050 


25-530 


817 


25-780 


636 


24785 


1722 


25-035 


I34I 


25-285 


1044 


25-535 


813 


25-785 


633 


24-790 


1713 


25-040 


1334 


25-290 


1039 


25-540 


809 


25-790 


630 


24795 


1705 


25-045 


1328 


25-295 


1034 


25-545 


805 


25795 


627 


24-800 


i6g6 


25-050 


I32I 


25-300 


1029 


25-550 


801 


25-800 


624 


24-805 


1688 


25-055 


I3I4 


25-305 


1024 


25-555 


797 


25-805 


621 


24-810 


1679 


25-060 


1308 


25-310 


1019 


25-560 


793 


25-810 


618 


24-815 


1671 


25-065 


I3OI 


25-315 


1013 


25-565 


789 


25-815 


615 


24-820 


1663 


25-070 


1295 


25-320 


1008 


25-570 


785 


25-820 


612 


24-825 


1654 


25-075 


1288 


25-325 


1003 


25-575 


781 


25-825 


609 


24-830 


1646 


25-080 


1282 


25-330 


998 


25-580 


777 


25-830 


606 


24-835 


1638 


25-085 


1276 


25-335 


993 


25-585 


774 


25-835 


603 


24-840 


1630 


25-090 


1269 


25-340 


98S 


25-590 


770 


25-840 


600 


24-845 


1622 


25-095 


1263 


25-345 


984 


25-595 


766 


25^^45 


597 


24*850 


1614 


25-100 


1257 


25-350 


979 


25-600 


762 


25-850 


594 


24-855 


1606 


25-105 


1250 


25-355 


974 


25-605 


758 


25-855 


591 


24-860 


1598 


25-110 


1244 


25-360 


969 


25-610 


755 


25-860 


588 


24-865 


1590 


25-115 


1238 


25-365 


964 


25-615 


751 


25-865 


585 


24-870 


1582 


25-120 


1232 


25-370 


959 


25-620 


747 


25-870 


582 


24-875 


1574 


25-125 


1226 


25-375 


955 


25-625 


743 


25-875 


579 


24-880 


1566 


25-130 


I2I9 


25-380 


950 


25-630 


740 


25-880 


576 


24-885 


1558 


25-135 


I2I3 


25-385 


945 


25-635 


736 


25-885 


573 


24-890 


1550 


25-140 


1207 


25-390 


940 


25-640 


732 


25-890 


570 


24-895 


1543 


25-145 


I20I 


25-395 


936 


25-645 


729 


25-895 


567 


24-900 


1535 


25-150 


II95 


25-400 


931 


25-650 


725 


25-900 


565 


24-905 


1527 


25-155 


I189 


25-405 


926 


25-655 


721 


25-905 


562 


24-910 


1520 


25-160 


1183 


25-410 


9?2 


25-660 


718 1 


25-910 


559 


24-915 


1512 


25-165 


I177 


25-415 


917 


25-665 


714 


25-9-5 


556 


24-920 


1504 


25-170 


II7I 


25-420 


912 


25-670 


711 

1 


25-920 


553 



240 



MR F. W. NEWMAN'S TABLE 
[Stxond Part. Fourteen decimal placcs?[ 



[25-925— 27-170] 



X 


C-* 


. X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


Q-X 


25'92S 


551 


26-175 


429 


26-425 


334 


26-675 


260 


26-925 


202 


25-93° 


54S 


26-180 


427 


26430 


332 


26-680 


259 


26-930 


201 


25'93S 


545 


26-185 


425 


26-435 


331 


26-685 


257 


26-935 


200 


25 '940 


543 


26-190 


422 


26-440 


329 


26-690 


256 


26-940 


199 


25-945 


540 


26-195 


420 


26445 


32S 


26-695 


255 


26-945 


198 


25-950 


537 


26-200 


418 


26-450 


326 


26-700 


254 


26-950 


197 


25-955 


534 


26-205 


416 


26-455 


324 


26-705 


252 


26-955 


196 


25-960 


532 


26-210 


414 


26-460 


323 


26-710 


251 


26-960 


195 


25-965 


5^9 


26-215 


412 


26-465 


321 


26-715 


250 


26-965 


194 


25-970 


526 


26-220 


410 


26-470 


319 


26-720 


249 


26-970 


193 


25-975 


523 


26-225 


408 


26-475 


318 


26-725 


247 


26-975 


193 


25-980 


521 


26-230 


406 


26-480 


316 


26-730 


246 


26-980 


192 


25-9S5 


518 


26-235 


404 


26-485 


315 


26-735 


245 


26-985 


191 


25-990 


516 


26240 


402 


26-490 


313 


26-740 


244 


26-990 


190 


25995 


513 


26-245 


400 


26-495 


3" 


26-745 


242 


26-995 


189 


26-000 


5" 


26-250 


39S 


26-500 


310 


26-750 


241 


27-000 


188 


26-005 


508 


26-255 


396 


26-505 


308 


26-755 


240 


27-005 


187 


26-010 


506 


26-260 


394 


26-510 


307 


26-760 


239 


27-010 


1 86 


26-015 


503 


26-265 


392 


26-515 


305 


26-765 


238 


27-015 


185 


26-020 


501 


26-270 


390 


26-520 


304 


26-770 


236 


27-020 


184 


26-025 


498 


26-275 


388 


26-525 


302 


26-775 


235 


27-025 


183 


26-030 


496 


26-280 


386 


26-530 


301 


26-780 


234 


27-030 


1S2 


26-035 


493 


26-285 


384 


26-535 


299 


26-785 


^2,2, 


27-035 


181 


' 26-040 


491 1 


26-290 


382 


26-540 


298 


26-790 


232 


27-040 


180 


26-045 


488 ! 


26-295 


380 


26-545 


296 


26-795 


231 


27'o45 


179 


26-050 


486 


26-300 


378 


26-550 


29s 


26-800 


230 


27-050 


178 


26-055 


484 


26-305 


377 


26-555 


293 


26-805 


228 


27-055 


178 


26-06D 


481 


26-310 


375 


26-560 


292 


26-8x0 


227 


27-060 


177 


26-065 


479 


26-315 


373 


26-565 


291 


26-815 


226 


27-065 


176 


26-070 


476 


26-320 


371 


26-570 


289 


26-820 


225 


27-070 


175 


26-075 


474 


26-325 


369 


26-575 


287 


26-825 


224 


27-075 


174 


26-080 


472 


26-330 


367 


26-580 


286 


26-830 


223 


27-080 


173 


26-085 


469 


26-335 


366 


26-585 


285 


26-835 


222 


27-085 


172 


26*090 


467 


26-340 


364 


26-590 


283 


26-840 


221 


27-090 


172 


26095 


465 


26-345 


362 


26-595 


282 


26-845 


219 


27-095 


171 


26-100 


462 


26-350 


360 


26-600 


280 


26-850 


218 


27-100 


170 


26-105 


460 


26-355 


358 


26-605 


279 


26-855 


217 


27-105 


169 


26-110 


458 


26-360 


356 


26-610 


278 


26-860 


216 


27-110 


i68 


26-115 


455 


26-365 


355 


26-615 


276 


26-865 


215 


27-1x5 


168 


26-120 


453 


26-370 


353 


26-620 


275 


26870 


214 


27-120 


167 


26-125 


451 


26-375 


351 


26-625 


273 


26-875 


213 


27-125 


166 


26-130 


449 


26-380 


349 


26-630 


272 


2 6 880 


212 


27-130 


165 


i 26-135 


446 


26-385 


348 


26-635 


271 


26-885 


211 


27-135 


164 


1 26-140 


444 


26-390 


346 


26-640 


269 


26-890 


210 


27-140 


163 


26-145 


442 


26-395 


344 


26-645 


268 


26-895 


209 


27-145 


162 


26-150 


440 


26-400 


342 


26650 


267 


26-900 


208 


27-150 


162 


26-155 


437 


26-405 


341 


26-655 


265 


26-905 


207 


27-155 


161 


26-160 


435 


26-410 


339 


26-660 


264 


26-910 


206 


27160 


i6o 


26-165 


433 


26-415 


337 


26-665 


263 


26-915 


204 


27-165 


159 


26-170 


431 


26-420 


336 


26670 


261 


26-920 


203 


27-170 


158 



[27-175—27-635] 



OF THE DESCENDING EXPONENTIAL. 
[Second Part. Fourteen decimal places^ 



241 



X 


Q-X 


X 


Q-X 


X 


Q-X 


X 


g-X 


X 


Q-X 


27-175 


158 


27-275 


143 


27-375 


129 


27-475 


117 


27-575 


106 


27-180 


157 


27-280 


142 


27-380 


128 


27-480 


116 


27-580 


105 


27-185 


156 


27-285 


141 


27-385 


128 


27-485 


116 


27-585 


105 


27-190 


155 


27-290 


140 


27-390 


127 


27-490 


"5 


27-590 


104 


27-195 


155 


27-295 


140 


27-395 


127 


27-495 


114 


27-595 


104 


27-200 


154 


27-300 


139 


27-400 


126 


27-500 


114 


27-600 


103 


27-205 


153 


27-305 


138 


27-405 


125 


27-505 


"3 


27-605 


103 


27-210 


152 


27-310 


138 


27-410 


125 


27-510 


"3 


27-610 


102 


27-215 


152 


27-315 


137 


27-415 


124 


27-515 


112 


27-615 


102 


27-220 


151 


27-320 


136 


27-420 


124 


27-520 


112 


27-620 


lOI 


27-225 


15° 


27-325 


136 


27-425 


123 


27-525 


III 


27-625 


101 


27-230 


149 


27330 


135 


27-430 


122 


27-530 


III 


27630 


100 


27"235 


148 


27-335 


134 


27-435 


122 


27'S35 


no 


27-635 


99 


27-240 


148 


2 7 '340 


134 


27-440 


121 


27-540 


no 






27-245 


147 


27'345 


133 


27-445 


121 


27-545 


109 






27-250 


146 


27-350 


132 1 


27-450 


120 


27-550 


108 






27-255 


145 


27-355 


132 1 


27-455 


119 


27-555 


108 






27-260 


145 


27-360 


131 i 


27-460 


118 


27-560 


107 






27-265 


144 


27-365 


130 1 


27-465 


118 


27-565 


107 






27-270 


143 


27-370 


130 


27-470 


117 


27-570 


106 







Vol. XTII. Part TIT. . 



32 



V. Tables of the Exponential Function. By J. W. L. Glaisher, 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 
if, e'", log,„ e"" and logj„ e'". The ranges of the four tables are as' follows : 

Table I. From a; =0001 to a- = 0100 at intervals of 0001. 

Table II. From a; = 001 to a; = 2-00 at intervals of O'Ol. 

Table III. From a^ = 0-l to a- =100 at intervals of O'l. 

Table IV. From x = l to x = .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 logj„ (e~^) are equal respectively to x logj^, e and *• logj„ (e~') it is 
evident that the logarithmic results in the tables are merely multiples of logj^e and 
logjd (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 imit, 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 
a; = 0"001 to a; = 15'350 to twelve places of decimals at intervals of 0001, from a; = 15-350 
to ir = l7'300 at intervals of 0002, and thence to a; = 27-63.5 at intervals of O'OOo to 
fourteen places. The introduction contains (pp. 148, 149) a table of e'" from a; = 0'l to 
X = 370 at intervals of O'l to eighteen places. The onlj' other tables of exponential functions 
that I know of are the following: 

(i) On p. 188 of the first volume of Schulze's Sammlung hgarithmischer trigono- 
metrischer...Tafeln (Berlin, 1778) there is a table giving the values of e", for x=\,2, 3,... 24 
to 28 or 29 figures, and for « = 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 a; = 001 
to X = 1000 at intervals of 0"01 was given in Vega's TahuloB logarithmico-trigonometrictB 
(1797), and has been retained in the later editions of this work. Kohler'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,, siuh a:, log,jCosh.T and log,„taDha; from x = 2 to a; =5 at intervals of 001 
to nine decimal places, and from x = 5 to a; = 12 at intervals of O'Ol 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-hyperholischen 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 
pubhcation in a separate form; but the scheme was not carried out and the tables 
were left in an incomplete state. In 1876 Mr Ne^vman 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 Jlr 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 figmre : the ficrure was in these cases changed so as to asrree with 
Mr Newman's more extended value. 

During the time that his large table was being printed Mr Newman sent me a 
table of e' from x = to a; = l at intervals of O'OOl 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 e"* 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 e* were recalculated. 

Table IV. The values of e" 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. VIII. pp. 195—212, 301—316; Vol. iz. pp. 81— 90, 193—208, 297—304. 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



245 



Gudermann's table (iii) mentioned above begins at x = 2, and it is for this reason 
that a; = 2 was taken as the limit of Table 11. 

With respect to the use of the tables, it may be remarked that they may be 
conveniently combined in interpolations : thus, for example, 

g.K7 ^ gf2 X gow = 6G-6863310 x 1-03769302, 
and log e*'^ = log (e") + log (e"""^) = 1-8240368240 + 0-0160688958. 

For the sake of completeness I reproduce here Schulze's table referred lo in (i). 



X 


e' 


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 01 


6 


403 


428 793 492 735 122 608 387 


180 5 


7 


1006 


633 


158 428 458 599 263 720 


238 1 


8 


2980 


957 987 041 728 274 743 592 


099 


9 


8103 


083 


927 575 384 007 709 996 


688 


10 


22026 


465 


794 806 716 516 957 900 


641 


11 


59874 


141 


715 197 818 455 826 485 


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 


187 200 844 910 003 4 




20 


485165195 


409 790 277 9G9 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 1G2 068 




^ 


__ 




72004899337 . 385 872 524 161 351 466 126 


fi" 





10686474581524 . 462 146 99C 


1 468 650 741 2 



6^°= 11420073898156842836629.5718 . 314 472 

This table was partially verified in the following manner. Since 

a"-'' - 1 



1 + a + a 



-{- a" =- 



a-1 ' 



•24(5 Mh GLAISIIER, TABLES OF THE EXPONENTIAL FUNCTION. 



; — ^ , and that UDity added to the sum of the first twenty-four powers of e is equal 



it follows that unity added to the sum of the first twelve powers of e is equal to 

e- 

e*" - 1 

to — . Retaining 23 places of decimals, we find by addition from Schulzc's table 

l+e + eV.. + e" = 2o7473.706 979 533 059 990 318 032 45, 
and by division, taking Schulze's value of e", 

^^^ = 257473 . 706 979 533 059 990 318 032 37, 
e- 1 

which verifies the values of the first thirteen powers of e to 22 places of decimals. 

Similarly by addition we find 

1 -he + e'...+e=* = 419051741 94 . 247 197 714 849 662 8, 
and by division, taking Schulze's value of e'*, 

^-^ = 41905174194. 247 197 714 849 663 0. 
e — 1 

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 
values of ^ , — , -j-., ■■■ irrr-. , which I worked out on account of their use in calculating 
the values of series having factorials in their denominators. 

The figures enclosed in brackets denote the numbers of ciphers occurring between the 
decimal point and the first significant figure. From j^^ to — the number of significant 
figures given is twenty-eight. 

|i = 0-5, ^ = ^'^^1 ^^' 

i=01G, 1 0000 198 412 6, 

:ir = 00416, ^ = 0000 024 801 5873, 

4 ! o! 

^ = 00083, ^, = 0000 002 755 731 922 398 589 065 2, 

o ! y ! 

j^, = 0-000 000 275 573 192 239 858 906 52, 
ri-, = 0000 000 02o 052 108 385 441 718 77, 

^, = 0000 000 002 087 075 698 786 809 897 921 009 032 120 143 231 254 342 365 
453 476 564 5, 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



247 



n 


1 


13 


0- (9) 


160 


590 438 


368 216 


145 993 923 771 7 


14 


O-(IO) 


114 707 455 


977 297 247 138 516 979 8 


15 


0-(12) 


764 716 373 


181 981 


647 590 113 


198 6 


16 


0(13) 


477 947 733 238 738 529 743 820 749 1 


17 


0-(14) 


281 


145 725 


434 552 076 319 894 558 3 


18 


0-(15) 


156 


192 


069 


685 862 


264 622 163 


643 5 


19 


0-(17) 


822 


063 


524 


662 432 


971 695 598 


123 7 


20 


0-(18) 


411 


031 


762 


331 216 


485 847 799 


061 9 


21 


0(19) 


195 729 410 633 912 612 308 475 743 7 


22 


0-(21) 


889 


679 


139 


245 057 


328 674 889 


744 3 


23 


0-(22) 


386 817 017 063 068 403 771 691 


193 2 


24 


0(23) 


161 


173 757 


109 611 


834 904 871 


330 5 


25 


0-(25) 


644 


695 


028 438 447 


339 619 485 


321 9 


26 


0-(2G) 


247 


959 


626 


322 479 


746 007 494 


354 6 


27 


0-(28) 


918 


368 


986 


379 554 


614 842 571 


683 7 


28 


0-(29) 


327 988 923 706 983 791 015 204 172 7 


29 


0(30) 


113 


099 


628 864 477 


169 315 587 645 8 


30 


0-(32) 


376 


998 762 


881 590 


564 385 292 


152 6 


31 


0-(33) 


121 


612 


504 


155 351 


794 962 997 468 6 


32 


0-(35) 


380 039 075 


485 474 359 259 367 089 3 


33 


0(36) 


115 


163 


356 


207 719 


502 805 868 


814 9 


34 


0-(38) 


338 


715 


753 


552 116 


184 723 143 573 3 


35 


0-(40) 


967 759 


295 


863 189 


099 208 981 


638 1 


36 


0-(41) 


268 


822 


026 


628 663 


638 669 161 


566 1 


37 


0-(43) 


726 


546 


017 


915 307 


131 538 274 


503 1 


38 


0-(44) 


191 


196 


320 


504 028 


192 510 072 


237 7 


39 


0-(46) 


490 


246 


975 


0^51 354 


339 769 415 


994 


40 


0-(47) 


122 


561 


743 


912 838 


584 942 353 


998 5 


41 


0-(49) 


298 


931 


082 


714 240 


451 078 912 


191 5 


42 


0-(51) 


711 


740 


673 


129 143 


931 140 267 


122 5 


43 


0-(52) 


165 


521 


086 774 219 


518 869 829 


563 4 


44 


0-(54) 


376 


184 


288 


123 226 


179 249 612 


644 


45 


0-(56) 


835 


965 


084 718 280 


398 332 472 


542 3 


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 



•248 Mk 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 .^^ _ , , by 
dividing it by n. 

The results should be iu all cases correct to the last figure, as several figures 
were rejected. 

By addition we find 

H.i_+1..+ I_^= 1-543 080 634 815 2-i3 778 477 905 620 757 OGl G82 6, 

l+i. + J_... + ^ = 1-175 201 193 643 801 456 882 381 850 595 600 815 2, 
<> ! 5 1 33 ! 

giving 

e =2-718 281 828 459 045 235 360 287 471 352 662 497 8, 

e-'= 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, however, a good verification of the values of the reciprocals 
of the factorials even as far as , ; for, of those beyond j^l (i^ which only 28 significant 

figures are given in the table on the preceding page) the first only, — r-^, is verified to 

its full extent, the next, , , , , is verified to only 27 figures and so on, the first figure 

14 ! 

alone being verified" in the case of 007- 

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 dififerent terms of 
the series are shown on the opposite page, and it will be seen that ten figures (3287949417) 

of - — are thus verified. In order to complete the calculation it was necessary to find the values 
50 ! 

of - — r and the subsequent terms to a few places of decimals. The last term included 
51 ! 

is that involving ^ , . 

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 

, ,^ 10' . 10^ , 10* , 10' 10\ . 
l + 10+2T+3T+4T+-5T + -6!+*^°- 



Mr GLA.ISHER, TABLES OF THE EXPONENTIAL FUNCTION. 

Calculation of e'" and e"'° from the series. 
(Even terms.) (Uneven terms.) 



249 



1 

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 I7l 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 




33 871 575 355 211 618 472 




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 


L1013 


232 920 103 323 139 721 376 087 


11013- 


232 874 703 


393 377 236 524 556 



By adding and subtracting the sums of the two columns we find 
e''' = 22026-465 794 806 716 516 957 900 643, 
e-"'= 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. 

Vol. XIII. Part III. 33 



250 Mk GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 

As an additional verification I have made the following calculation of e"'" : 

.,0 4539992976 + A 
Let 6'°= j^ii , 

then - 10 = log: (4539992976 +/()- 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 
kno\vn, for 

4539992976 = 1296 x 1763 x 1987, 

and, taking the logarithms of 1296, 1763 and 1987 from Wolfram's table*, we find 

log,4539992976 = 22-236 191 301 861 907 078 9. 
Also 141og,10-10 = 22-236 191 301 916 639 576 3. 

Putting therefore a; = 4539992976, we have 

\og,{x + h)-\og,x = Q-OW 000 000 054 732 497 355 4. 

7 72 73 

Now log, {x + h) -\og^a; = -- \ ~^ + l -3- &c., 

72 13 

and therefore li=x {log, (x + h) — log, x]+\ ^ — ^ + &c. 

By multiplication we find 

a;{log,(a; + /0-loge«]=0'248 485 153 552 351 449 550. 

Taking log, {x ■\-K)— log,a; as an approximate value of - , we have 

CO 

- = 0000 000 000 054 732 497 355, 

X 

and therefore i- = 0-000 000 000 006 800 106 505, 

whence A = 0248 485 153 5-59 151 556 055; 

and, except for last-figure errors, this value should be correct as far as it extends. We 
thus find 

e*" = 0000 045 399 929 762 484 851 535 591 515 560 6, 

which difiFers from the value found above for e"'" 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 

nambera np to 2,200 and of primes, and also of a great many the addition of six logarithms that were omitted through 

composite nnmbers, np to 10,009 to 48 places of decimals. Wolfram's death, in Vega's Thetaurut logarithmorum com- 

It was first published in Schnlze's Sammlung, referred to pletus (1791). 



jMe glaisher, tables of the exponential function. 251 

To verify the accuracy of this value of e"'° I found by division the reciprocal of 

22026-465 794 806 716 516 957 900 643, 

the result being 

0000 045 399 929 762 484 851 535 591 515 565, 

which agrees to 32 places ^vith the value of e"'° just found. 

I have thought it worth while to give this calculation of e"'" 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 x-, we then obtain log/c, and it only remains to calculate K from the formula 

A = a;{log.(a; + /0-]og.a;)+ J --^-, + &c. 
by repeated approximation. 

In calculating a table of e? 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 

twenty-eight figure number given as the value of ^^ , and thence deduced 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 ^— -r given on p. 247, we have 

1 _ 328794941-663 315 806 703 163 069 6 
50 ! ~ 10" 

whence we ought to find 

log,(50!)= 73 log„10- log, 328794941-663 315 806 703 163 069 6. 

To calculate the logarithm of 328794941-663 315 806 703 163 069 6 we notice 
that 328794943 = 17 x 19 x 569 x 1789, whence, taking the logarithms from Wolfram's table, 

log. 328794943 = 19-610 944 840 857 706 621 131 134 384 5, 

33—2 



252 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 

and putting 328794943 = x, and denoting the number whose logarithm is required hy x-h 
we have 

log,(a;-/0 =log.a;-- - i ^- i ^- &c. 

■^vherc /i = l-33G 684 193 296 836 930 4. 

T 12/2 

Thus log. (50 !) = 73 log. 10 + - + i ^ + i ^s + &c. - log. x, 

and the final steps of the calculation are as follows: 

73 log. 10 = 168088 711 788 565 334 933 313 376 191 9 

4 065 403 747 091 198 207 4 
8 263 753 813 4 

22 4 

168-088 711 792 630 738 688 668 328 235 1 
log- x= 19-610 944 840 857 706 621 131 134 384 5 

148-477 766 951 773 032 067 537 193 850 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- 

50 ! 

ceding one by division, this afi"ords a verification also of the values of the other reciprocals 
of factorials. 

A table of log,„(a;!) from x = l to a; = 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 Encydopwdia 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 

1 1 1 1 1 „ 

«=l + r!+2~l+3!+^! + *''-' 

and I used the continued-fraction formula 

e-1 111 1 



2 1 + 6 + 10 + ... 4n + 2 + ... 
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 liritinh Association for 1871 (pp. 10—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-' = 0-367 879 441 171 442 321 595 523 770 161 460 867 445 811 12. 

I 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'^ = 8886110-520 507 87, e-'''= 0-000 000 11. 

This value of e''° is correct. The value of e'° given in Schulze's table {ante, p. 245) 
is 8886110-520 507 872..., which is thus directly verified to fifteen figures. 

The modulus M of the common or Briggian logarithms is equal to logj„ e, and its value 
has been given by Professor J. C. Adams to 282 places of decimals in Vol. xxvii., p. 93 
(1878) of the Proceedings of the Royal Society. This value is reprinted in the article 
"Logarithms" in the Encyclopcedia Britannica (1882). 

It may be here remarked that Gauss's posthumous memoir "De curva lemniscata" 
{Werke Vol. in., pp. 413 — 432) contains the values of e-^ e~i", and e i" to 50 or more 

25 49 

places of decimals, and the value of e^ to 34 places. The values of e"^", e * " and e~ 4 " 
are also given to sixteen, twenty-four, and twelve significant figures respectively. 

The values of !^e 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 
e«-l 1 11 1 



2n - 1 + 6m + lOw + 14/1 + &c. ' 



254 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



TABLE I. 

Values of e', e""", logio(e"=), logi„((?~-'') from ;i; = o-ooi to .c^o'ioo at intervals of o-ooi. 



X 


log,o(^'0 


e" 


e-" 


logic (' 


3-^) 


O-OOI 


0-00043 42945 


I -00 


100 050 


0-999 


000 


500 


7-99956 


5705s 


0'002 


0-00086 85S90 


i-oo 


200 200 


0-998 


001 


999 


£-99913 


141 10 


0-003 


0-00130 28834 


I -00 


300 450 


0-997 


004 


496 


I -99869 


71166 


0-004 


0-00173 71779 


I-oo 


400 801 


0-996 


007 


989 


1-99826 


28221 


0-005 


0-00217 14724 


I-oo 


501 252 


0-995 


012 


479 


1-99782 


85276 


o'oo6 


0-00260 57669 


I-oo 


601 804 


0-994 


017 


964 


1-99739 


42331 


0-007 


0-00304 00614 


I-oo 


702 456 


0-993 


024 


443 


1-99695 99386 1 


0-008 


0-00347 43559 


I-oo 


803 209 


0-992 


031 


915 


£•99652 


56441 


0-009 


0-00390 86503 


I-oo 


904 062 


0-991 


040 


379 


1-99609 


13497 


o-oio 


0-00434 29448 


I -01 


005 017 


0-990 


049 


834 


^■99565 


70552 


o-oii 


0-00477 72393 


I -01 


106 072 


0-989 


060 


279 


1-99522 


27607 


0-012 


0-00521 15338 


i-oi 


207 229 


0-988 


071 


713 


1-99478 84662 1 


0-013 


0-00564 58283 


I -01 


308 487 


0-987 


084 


135 


1-99435 


41717 


0-014 


0-00608 01227 


I-OI 


409 846 


0986 


097 


544 


1-99391 


98773 


0-015 


0-00651 44172 


I-OI 


511 306 


0-985 


III 


940 


1-99348 


55828 


0-016 


0-00694 871 17 


I-OI 


612 869 


0-984 


127 


320 


£•99305 


12883 


0-017 


0-00738 30062 


I-OI 


714 532 


0-983 


143 


685 


1-99261 


69938 


o-oi8 


0-00781 73007 


I-OI 


816 298 


0-982 


161 


032 


1-99218 


26993 


0-019 


0-00825 15952 


I-OI 


918 165 


0-981 


179 


362 


1-99174 


84048 


O-02O 


0-00868 58896 


1-02 


020 134 


0-980 


198 


673 


1-99131 


41104 


0-02I 


0-00912 01841 


1-02 


122 205 


0-979 


218 


965 


£•99087 98159 1 


0-022 


0-00955 44786 


1-02 


224 378 


0-978 


240 


235 


1-99044 


55214 


0-023 


0-00998 87731 


1-02 


326 654 


0-977 


262 


484 


I -99001 


12269 


0-024 


0-01042 30676 


1-02 


429 032 


0-976 


285 


710 


T-98957 


69324 


0-025 


0-01085 73620 


1-02 


531-512 


0-975 


309 


912 


1-98914 


26380 


0-026 


O-01129 16565 


I -02 


634 095 


0-974 


335 


090 


1-98870 


83435 


0-027 


O-01172 59510 


1-02 


736 780 


0-973 


361 


242 


1-98827 


40490 


0-028 


o-oi2i6 02455 


1-02 


839 568 


0-972 


388 367 


1-98783 


97545 


0-029 


0-01259 45400 


1-02 


942 459 


0-971 


416 


464 


1-98740 


54600 


0-030 


0-01302 88345 


1-03 


045 453 


0-970 


445 


534 


1-98697 


11655 


0-031 


0-01346 31289 


1-03 


148 550 


0-969 


475 


573 


1-98653 


68711 


0-032 


0-01389 74234 


1-03 


251 751 


0-968 


506 


582 


1-98610 


25766 


o'033 


0-01433 I7I79 


1-03 


355 054 


0-967 


538 


560 


7-98566 


82821 


0-034 


0-01476 60124 


1-03 


458 461 


0-966 


571 


505 


1-98523 


39876 


0-035 


0-01520 03069 


1-03 


561 971 


0-965 605 


416 


1-98479 


96931 


0-036 


0-01563 46013 


1-03 


665 585 


0-964 640 


293 


1-98436 


53987 


0-037 


0-01606 88958 


1-03 


769 302 


0-963 676 


135 


1-98393 


11042 


0-038 


0-01650 31903 


1-03 


873 123 


0-962 


712 


941 


1-98349 


68097 


0-039 


0-01693 74848 


1-03 


977 048 


0-961 


750 


709 


1-98306 


25152 


0-040 


0-OI737 17793 


1-04 


081 077 


0-960 


789 


439 


1-98262 


82207 


0-041 


0-01780 60738 


1-04 


185 211 


0-959 


829 


130 


1-98219 


39262 


0-042 


0-01824 03682 


1-04 


289 448 


0-958 


869 


781 


7-98175 96318 


0-043 


0-01867 46627 


1-04 


393 789 


0-957 


911 


390 


1-98132 


53373 


0-044 


0-01910 89572 


I -04 


498 235 


0-956 


953 


957 


1-98089 


10428 


0-045 


0-01954 32517 


1-04 


602 786 


0-955 


997 


482 


7-98045 


67483 


0-046 


0-01997 75462 


1-04 


707 441 


0-955 


041 


962 


1-98002 


24538 


0-047 


0-02041 18406 


1-04 


812 201 


0-954 


087 


398 


1-97958 


81594 


0-048 


0-02084 6 1 35 1 


1-04 


917 066 


0-953 


133 


787 


1-97915 


38649 


0-049 


0-02128 04296 


105 


022 03s 


0-952 


181 


130 


1-97871 


95704 


0-050 


0-02171 47241 


1-05 


127 no 


0-951 


229 


425 


1-97828 


52759 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 255 

TABLE I. {continued). 
Values of e^, e"'', logio(e*), \og^^{e~'') from x = o-ooi to « = o*ioo at intervals of O'ooi. 



X 


lo&„(e^) 


e^ 


e-^ 


log.„(^-^) 


o'05i 


0-02214 90186 


1-05 232 289 


0-950 278 


671 


1-97785 09814 


0-052 


0-02258 33131 


1-05 337 574 


0-949 328 


867 


I-9774I 66869 


0-053 


0-02301 76075 


1-05 442 964 


0-948 380 


012 


1-97698 23925 


0-054 


0-02345 19020 


1-05 548 460 


0-947 432 


107 


1-97654 80980 


0-055 


0-02388 61965 


1-05 654 061 


0-946 485 


148 


I-9761I 38035 


0-056 


0-02432 04910 


1-05 759 768 


0-945 539 


136 


i"97567 95090 


0-057 


0-02475 4785s 


1-05 865 581 


0-944 594 


069 


1-97524 52145 


0-058 


0-02518 90800 


1-05 971 500 


0-943 649 


947 


I -97481 09200 


0-059 


0-02562 33744 


1-06 077 524 


0-942 706 


769 


1-97437 66256 


0-060 


0-02605 76689 


1-06 183 655 


0-941 764 


534 


1-97394 23311 


o-o6i 


0-02649 19634 


1-06 289 891 


0-940 823 


240 


1-97350 80366 


0-062 


0-02692 62579 


1-06 396 234 


0-939 882 


887 


1-97307 37421 


0-063 


0-02736 05524 


1-06 502 684 


0-938 943 


474 


1-97263 94476 


0-064 


0-02779 48468 


1-06 609 240 


0-938 005 


000 


1-97220 51532 


0-065 


0-02822 91413 


1-06 715 902 


0-937 067 


463 


1-97177 08587 


0-066 


0-02866 34358 


1-06 822 672 


0-936 130 


864 


£-97133 65642 


0-067 


0-02909 77303 


i-o6 929 548 


°'935 195 


201 


1-97090 22697 


0-068 


0-02953 20248 


1-07 036 531 


0-934 260 


474 


1-97046 79752 


0-069 


0-02996 63193 


1-07 143 621 


0-933 326 


680 


1-97003 36807 


0-070 


0-03040 06137 


1-07 250 818 


0-932 393 


820 


£-96959 93863 


0-071 


0-03083 49082 


1-07 358 123 


0-931 461 


892 


1-96916 50918 


0-072 


0-03126 92027 


1-07 465 534 


0-93° 530 


896 


1-96873 07973 


0-073 


0-03170 34972 


1-07 573 054 


0-929 600 


830 


1-96829 65028 


0-074 


0-03213 77917 


1-07 680 681 


0-928 671 


694 


1-96786 22083 


0-075 


0-03257 20861 


1-07 788 415 


0-927 743 


486 


1-96742 79139 


0-076 


0-03300 63806 


1-07 896 257 


0-926 816 


207 


1-96699 36194 


0-077 


0-03344 06751 


i-o8 004 208 


0-925 889 854 


£-96655 93249 


0-078 


0-03387 49696 


1-08 112 266 


0-924 964 


427 


I -9661 2 50304 


0-079 


0-03430 92641 


1-08 220 432 


0-924 039 


924 


1-96569 07359 


0-080 


0-03474 35586 


1-08 328 707 


0-923 116 


346 


1-96525 64414 


o-o8i 


0-03517 78530 


1-08 437 090 


0-922 193 


691 


1-96482 21470 


0-082 


0-03561 21475 


1-08 545 581 


0-921 271 


959 


T-96438 78525 


0-083 


0-03604 64420 


1-08 654 181 


0-920 351 


147 


1-9639S 35580 


0-084 


0-03648 07365 


1-08 762 889 


0-919 431 


256 


£-96351 9263s 


0-085 


0-03691 50310 


I -08 871 707 


0-918 512 


284 


1-96308 49690 


0-086 


0-03734 93254 


I -08 980 633 


0-917 594 


231 


1-96265 06746 


0-087 


0-03778 36199 


1-09 089 668 


0-916 677 


096 


1-96221 63801 


0-088 


0-03821 79144 


1-09 198 812 


0-915 760 


877 


1-96178 20856 


0-089 


0-03865 22089 


1-09 308 066 


0-914 84s 


574 


£-96134 779" 


0-090 


0-03908 65034 


1-09 417 428 


0-913 931 


i8s 


1-96091 34966 


0-091 


0-03952 07979 


1-09 526 901 


0-913 017 


711 


1-96047 92021 


0-092 


0-03995 50923 


1-09 636 482 


0-912 105 


150 


1-96004 49077 


0-093 


0-04038 93868 


1-09 746 174 


0-911 193 


500 


1-95961 06131 


0-094 


0-04082 36813 


1-09 85s 975 


0-910 282 


762 


£•95917 63187 


0-095 


0-04125 79758 


1-09 965 886 


0-909 372 


934 


1-95874 20242 


0-096 


0-04169 22703 


i-io 075 906 


0-908 464 


016 


1-95830 77297 


0-097 


0-04212 65647 


i-io 186 037 


0-907 556 


006 


1-95787 34353 


0-098 


0-04256 08592 


i-io 296 279 


0-906 648 


904 


1-95743 91408 


0-099 


0-04299 51537 


I -10 406 630 


0-905 742 


708 


1-95700 48463 


o-ioo 


0-04342 944S2 


I-IO 517 092 


0-904 837 


418 


1-95657 05518 



256 Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 

TABLE II. 
Values of e*, e'^ log,o(e'"), logi„(p"'^) from a; = o-oi to x = 2-oo at intervals of o-oi. 



X 

I 


log.«(e^) 


e* 


e-' 


log>o(e-^) 


O-OI 


0-00434 29448 


i-oi 005 017 


0-990 049 834 


T-99565 70552 


0'02 


000868 58896 


I-02 020 134 


0-980 198 673 


1-99131 41104 


0-03 


0-01302 88345 


1-03 045 454 


0-970 445 534 


1-98697 1 1655 


0-04 


0-0I737 17793 


1-04 081 077 


0-960 789 439 


1-98262 82207 


0-05 


0-02171 47241 


1-05 127 no 


0-951 229 425 


T-97828 52759 


o'o6 


0-02605 76689 


i-o6 183 655 


0-941 764 534 


1-97394 23311 


0-07 


0-03040 06137 


1-07 250 818 


0-932 393 820 


1-96959 93863 


o-o8 


0'03474 35586 


1-08 328 707 


0-923 116 346 


1-96525 64414 


0*09 


0-03908 65034 


1-09 417 428 


0-913 931 185 


1-96091 34966 


O'lO 


0-04342 94482 


i-io 517 092 


0-904 837 418 


1-95657 05518 


o-ii 


0-04777 23930 


i-ii 627 807 


0-895 834 135 


1-95222 76070 


0'12 


0-05211 53378 


1-12 749 685 


0-886 920 437 


T-94788 46622 


013 


0-05645 82826 


I-I3 882 838 


0-878 095 431 


T-94354 17174 


014 


0-06080 12275 


I-I5 027 380 


0-869 358 235 


I-93919 87725 


0-15 


0-06514 41723 


i'i6 183 424 


0-860 707 976 


1-93485 58277 


o'i6 


006948 71 17 1 


I-I7 351 087 


0-852 143 789 


I -9305 I 28829 


017 


0-07383 00619 


i-i8 530 485 


0-843 664 817 


7-92616 99381 


o-i8 


0-07817 30067 


1-19 721 736 


0-835 270 211 


T-92182 69933 


019 


0-08251 59516 


1-20 924 960 


0-826 959 134 


I -91 748 40484 


20 


0-08685 88964 


1-22 140 276 


0-818 730 753 


1-91314 11036 


0'2I 


0-09120 18412 


1-23 367 806 


o-8io 584 246 


1-90879 81588 


0-22 


0-09554 47860 


1-24 607 673 


0802 518 798 


1-90445 52140 


0-23 


0-09988 77308 


1-25 860 001 


0794 533 603 


1-90011 22692 


0-24 


0-10423 06757 


1-27 124 915 


0-786 627 861 


1-89576 93243 


0-25 


0-10857 36205 


1-28 402 542 


0-778 800 783 


1-89142 63795 


0-26 


0-11291 65653 


1-29 693 009 


0-77X 051 586 


1-88708 34347 


0-27 


0-11725 95101 


1-30 996 445 


0-763 379 494 


1-88274 04899 


0-28 


o-i2i6o 24549 


1-32 312 981 


0-755 783 741 


T-87839 75451 


0-29 


0-12594 53998 


1-33 642 749 


0-748 263 568 


1-87405 46002 


0-30 


0-13028 83446 


1-34 985 881 


0-740 818 221 


7-86971 16554 


0-31 


0-13463 12894 


1-36 342 511 


0-733 446 956 


7-86536 87106 


0-32 


0-13897 42342 


1-37 712 777 


0-726 149 037 


7-86102 57658 


o"33 


0-I433I 71790 


1-39 096 813 


0-718 923 733 


7-85668 28210 


o"34 


0-14766 01238 


1-40 494 759 


0-711 770 323 


1-85233 98762 


o'35 


0-15200 30687 


I -41 906 755 


0-704 688 090 


1-84799 69313 


0-36 


0-15634 6013s 


1-43 332 942 


0-697 676 326 


1-84365 39865 


0-37 


o-i6o68 89583 


1-44 773 462 


0-690 734 331 


I-83931 10417 


038 


0-16503 19031 


1-46 228 459 


0-683 861 409 


1-83496 80969 


0-39 


0-16937 48479 


1-47 698 079 


0-677 056 874 


7-83062 51521 


0-40 


0-17371 77928 


1-49 182 470 


0-670 320 046 


7-82628 22072 


0-41 


0-17806 07376 


1-50 681 779 


0-663 650 250 


7-82193 92624 


0-42 


0-18240 36824 


1-52 196 156 


0-657 046 820 


1-81759 63176 


0-43 


0-18674 66272 


1-53 725 752 


0-650 509 095 


1-81325 33728 


0-44 


0-19108 95720 


1-55 270 722 


0-644 036 421 


1-80891 04280 


0-45 


0-19543 25169 


1-56 831 219 


0637 628 152 


7-80456 74831 


0-46 


0-19977 54617 


1-58 407 399 


0-631 283 646 


7-80022 45383 


0-47 


0-20411 84065 


1-59 999 419 


0-625 002 268 


1-79588 15935 


0-48 


0-20846 13513 


i*6i 607 440 


o-6i8 783 392 


1-79153 86487 


0-49 


0-21280 42961 


1-63 231 622 


0-612 626 394 


1-78719 57039 


©•50 


0-21714 72410 


1-64 872 127 


0-606 530 660 


1-78285 27590 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



257 



TABLE II. {continued). 
Values oi (f, e'^, log,„(e'^), log,(,(e"-^) from a:: = o*oi to a;=2-oo at intervals of o-oi. 



X 


log.(<'^) 


e 


c~- 


log.o(e-^) 


0-51 


0-22149 01858 


1-66 529 120 


0-600 495 579 


7-77850 98142 


0-52 


o"22583 31306 


1-68 202 765 


0-594 520 548 


T774I6 68694 


0-53 


0-23017 60754 


1-69 893 231 


0-588 604 970 


1-76982 39246 


0-54 


0-23451 90202 


1-71 600 686 


0-582 748 252 


1-76548 09798 


0-5S 


0-23886 19650 


1-73 325 3°2 


0-576 949 810 


1-76113 80350 


0-56 


0-24320 49099 


1-75 067 250 


0-571 209 064 


1-75679 50901 


0-S7 


o'24754 78547 


176 826 705 


0-565 525 439 


1-75245 21453 


0-58 


0-25189 07995 


1-78 603 843 


0-559 898 367 


I-748IO 92005 


o"59 


o'25623 37443 


1-80 398 842 


0-554 327 285 


1-74376 62557 


o'6o 


0-26057 66891 


1-82 211 880 


0-548 811 636 


1-73942 33109 


o'6i 


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 


o'2736o 55236 


1-87 761 058 


0-532 591 801 


I 72639 44764 


0-64 


0-27794 84684 


1-89 648 088 


0-527 292 424 


1-72205 15316 


0-65 


0-28229 14132 


1-91 554 083 


0-522 045, 777 


1-71770 85868 


0-66 


0-28663 43581 


1-93 479 233 


0-516 851 334 


£-71336 56419 


067 


0-29097 73029 


1-95 423 732 


0-5" 708.578 


1-70902 26971 


0-68 


0-29532 02477 


1-97 387 773 


0-506 616 992 


1-70467 97523 


o'69 


0-29966 31925 


1-99 371 553 


0-501 576 069 


1-70033 68075 


070 


0-30400 61373 


2-OI 375 271 


0-496 585 304 


1-69599 38627 


071 


0-30834 90822 


2-03 399 126 


0-491 644 197 


1-69165 09178 


072 


0-31269 20270 


2-05 443 321 


0-486 752 256 


7-68730 79730 


073 


0-31703 49718 


2-07 508 061 


0-481 908 990 


1-68296 50282 


074 


o'32i37 79166 


2-09 593 551 


0477 113 916 


7-67862 20834 


075 


0-32572 08614 


2-1 1 700 002 


0-472 366 553 


7-67427 91386 


076 


0-33006 38062 


2-13 827 622 


0-467 666 427 


1-66993 61938 


077 


0-33440 67511 


2-15 976 625 


0-463 013 068 


1-66559 32489 


078 


0-33874 96959 


2-18 147 227 


0-458 406 oil 


1-66125 03041 


079 


0-34309 26407 


2-20 339 643 


0-453 844 795 


1-65690 73593 


o-8o 


0-34743 55855 


2-22 554 093 


0-449 328 964 


£•65256 44145 


o-8i 


0-35177 85303 


2-24 790 799 


0-444 858 066 


1-64822 14697 


0-82 


0-35612 14752 


2-27 049 984 


0-440 431 654 


7-64387 85248 


o'83 


0-36046 44200 


2-29 331 874 


0436 049 286 


1-63953 55800 


0-84 


0-36480 73648 


2-31 636 698 


0-431 710 523 


1-63519 26352 


0-85 


0-36915 03096 


2-33 964 685 


0-427 414 932 


1 63084 96904 


0-86 


0-37349 32544 


2-36 316 069 


0-423 162 082 


7-62650 67456 


0-87 


0-37783 61993 


2-38 691 085 


0-418 951 549 


7-62216 38007 


0-88 


0-38217 91441 


2-41 089 971 


0-414 782 912 


7-61782 08559 


0-89 


0-38652 20889 


2-43 512 965 


0-410 655 753 


7-61347 79111 


0-90 


0-39086 50337 


2-45 960 3" 


0-406 569 660 


I -60913 49663 


o'9i 


039520 79785 


2-48 432 253 


0-402 524 224 


1-60479 20215 


0'92 


0-39955 09234 


2-50 929 039 


0-398 519 041 


1-60044 90766 


o'93 


0-40389 38682 


2-53 450 918 


0-394 553 710 


T-59610 61318 


0-94 


0-40823 68130 . 


2-55 998 142 


0-390 627 835 


7-59176 31870 


0-95 


0-41257 9757S 


2-58 570 966 


0-386 741 023 


1-58742 02422 


0-96 


0-41692 27026 


2-61 169 647 


0-382 892 886 


7-58307 72974 


o'97 


0-42126 56474 


2-63 794 446 


0-379 083 038 


1-57873 43526 


0-98 ■ 


0-42560 85923 


2-66 445 624 


0-375 3" 099 


1-57439 14077 


0-99 


0-42995 15371 


2-69 123 447 


0-371 576 691 


1-57004 84629 


I'OO 


0-43429 44819 


27L828 183 


0-367 879 441 


^-56570 55181 



Vol. XIII. Part III. 



34 



JJS 



Mr GLATSHER, TABLES OF THE PIXPONENTIAL FUNCTION. 



TABLE n. (continued). 
A'ahies off', e'^, log,„(e'), logio(t-'"''') from a' = 0"oi to x = 2'oo at intervals of ooi. 



.r 


log.o(e') 


e^ 


er' 


log,„(e-") 


I 01 


0-43863 74267 


2-74 


560 lOI 


0-364 218 980 


1-56136 25733 


I 02 


0-44298 03715 


2-77 


319 476 


0-360 594 940 


1-55701 96285 


1-03 


o'44732 33164 


280 


106 584 


0-357 006 961 


1-55267 66836 


I 04 


0-45166 62612 


2-82 


921 701 


0-353 454 682 


1-54833 37388 


1-05 


0-45600 92060 


2-85 


765 112 


0-349 937 749 


1-54399 07940 


I 06 


0-46035 21508 


2-88 


637 099 


0-346 455 810 


1-53964 78492 


107 


0-46469 50956 


2-91 


537 950 


0-343 008 517 


1-53530 49044 


I 08 


0-46903 80405 


2-94 


467 955 


0-339 595 526 


1-53096 19595 


I 09 


04733S 09853 


2-97 


427 407 


0-336 216 494 


1-52661 90147 


I'lO 


0-47772 39301 


3-00 


416 602 


0-332 871 084 


T-52227 60700 


i-ii 


0-48206 6S749 


3-03 


435 839 


0-329 558 961 


7-51793 31251 


112 


048640 98197 


3-06 


485 420 


0326 279 795 


1-51359 01803 


113 


049075 27646 


3-09 


565 650 


0-323 033 256 


1-50924 72354 


114 


0-49509 57094 


3-12 


676 837 


0-319 819 022 


1-50490 42906 


i-'S 


0-49943 86542 


3-15 


819 291 


0-316 636 769 


£■50056 13458 


116 


0-50378 15990 


3-18 


993 328 


0-313 486 181 


1-49621 84010 


117 


0-50812 45438 


3-22 


199 264 


0-310 366 941 


7-49187 54562 


118 


051246 74886 


3-25 


437 420 


0-307 278 739 


1-48753 25114 


119 


0-51681 04335 


3-28 


708 121 


0-304 221 264 


I -483 1 8 95665 


I-20 


0-52115 33783 


3-32 


on 692 


0301 194 212 


T-47884 66217 


I-2I 


0-52549 63231 


3-35 


348 465 


0-298 197 279 


£■47450 36769 


1-22 


052983 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 


124 


0-53852 51576 


3-45 


561 347 


0-289 384 218 


I -46 147 48424 


1-25 


054286 81024 


3-49 


034 296 


0-286 504 797 


1-45713 18976 


1-26 


0-54721 10472 


3-52 


542 149 


0-283 654 027 


1-45278 89528 


1-27 


0-55155 39920 


3-56 


085 256 


0-280 831 622 


1-44844 60080 


1-28 


0-55589 69368 


3-59 


663 973 


0-278 037 301 


1-44410 30632 


1-29 


0-56023 98817 


y<^3 


278 656 


0-275 270 783 


1-43976 01 183 


1-30 


0-56458 28265 


3-66 


929 667 


0-272 531 793 


1-43541 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 


^■53 


0-57761 16609 


378 


104 339 


0-264 477 261 


1-42238 83391 


•■34 


0-58195 46058 


3-8i 


904 351 


0-261 845 669 


1-41804 53942 


1-35 


0-58629 75506 


3-85 


742 553 


0-259 240 261 


1-41370 24494 


1-36 


0-59064 04954 


3-89 


619 330 


0-256 660 777 


£■40935 95046 


I '37 


0-59498 34402 


3-93 


535 070 


0-254 106 960 


1-40501 65598 


'•38 


0-59932 63850 


397 


490 163 


0-251 578 553 


T-40067 36150 


"•39 


060366 93298 


4-01 


485 005 


0-249 075 305 


i'"39633 06701 


1-40 


0-60801 22747 


4-05 


519 997 


0-246 596 964 


1-39198 77253 


1-41 


0-61235 52195 


4-09 


595 541 


0-244 143 283 


1-38764 47805 


I 42 


0-61669 81643 


4-13 


712 044 


0-241 714 017 


1-38330 18357 


'•43 


0-62104 11091 


4-17 


869 919 


0-239 308 922 


1-37895 88909 


144 


0-62538 40539 


4-22 


069 582 


0-236 927 759 


i"3746i 59461 


'•45 


0-62972 69988 


4-26 


311 452 


0-234 570 288 


i^37o27 30012 


1-46 


063406 99436 


4-30 


595 953 


0-232 236 27s 


1^36593 00564 


147 


0-63841 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 


I 49 


0-64709 87780 


4'43 


709 552 


0-225 372 656 


1-35290 12220 


1-50 


0-65144 17229 


4-48 


168 907 


0-223 '30 160 


■"-34855 82771 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



259 



TABLE II. (continued). 
Values of e'', e"^ logj„(e''), log,„(e''^) from .k = o'Oi to a:; = 2-oo at intervals of ooi. 



X 


log:o(«1 


e" 


e-^ 


log,„(f-0 


I '5 1 


0-65578 46677 


4-52 673 079 


0-220 909 978 


^•34421 53323 


I'52 


o'66oi2 76125 


4-57 222 520 


0-2lS 711 887 


1-33987 23875 


1-53 


0-66447 05573 


4-61 817 682 


0-216 535 667 


1-33552 94427 


1-54 


0-66881 35021 


4-66 459 027 


0-214 381 lOI 


1-33118 64979 


1-55 


o"673i5 64470 


4-71 147 018 


0-212 247 973 


1-32684 35531 


1-56 


067749 93918 


4-75 882 125 


0-210 136 071 


I -32250 06082 


i'57 


0-68184 23366 


4-80 664 819 


0-208 045 182 


1-31815 76634 


1-58 


o-686i8 52814 


4-85 495 581 


0-205 975 098 


1-31381 47186 


'•59 


0-69052 82262 


4-90 374 893 


0-203 925 612 


1-30947 17738 


I -60 


0-69487 11710 


4-95 303 242 


0-201 896 518 


1-30512 88290 


r6i 


0-69921 41159 


5-00 281 123 


0-199 887 614 


7-30078 58841 


1-62 


07035s 70607 


5-05 309 032 


0-197 898 699 


r-29644 29393 


163 


0-70790 00055 


5-10 387 472 


0-195 929 574 


1-29209 99945 


1-64 


0-71224 29503 


5-15 516 951 


0-193 980 042 


1-28775 70497 


1-65 


0-71658 58951 


5-20 697 983 


0-192 049 909 


1-28341 41049 


1-66 


0-72092 88400 


5-25 931 084 


0-190 138 980 


1-27907 11600 


1-67 


0-72527 17848 


5-31 216 780 


0-188 247 066 


1-27472 82152 


1-68 


0-72961 47296 


5-36 555 597 


0-186 373 976 


7-27038 52704 


1-69 


73395 76744 


541 948 071 


0-184 519 524 


1-26604 23256 


170 


0-73830 06192 


5-47 394 739 


0-182 683 524 


7-26169 93808 


171 


074264 35641 


5-52 896 148 


0-180 865 793 


1-25735 64359 


172 


0-74698 65089 


558 452 846 


0-179 066 148 


1-25301 349" 


173 


075132 94537 


5-64 065 391 


0-177 284 410 


1-24867 05463 


174 


075567 23985 


5-69 734 342 


0-175 520 401 


1-24432 76014 


175 


0-76001 53433 


5-75 460 268 


o-'73 773 944 


1-23998 46567 


176 


0-76435 82881 


5-81 243 739 


0-172 044 864 


1-23564 17119 


177 


0-76870 12330 


5-87 085 336 


0-170 332 989 


1-23129 87670 


178 


077304 41778 


5-92 985 642 


0-168 638 147 


1-22695 58222 


179 


0-77738 71226 


5-98 945 247 


0-166 960 170 


7-22261 28774 


I -So 


078173 00674 


6-04 964 746 


0-165 298 889 


7-21826 99326 


i-8i 


0-78607 30122 


6-1 1 044 743 


0-163 654 137 


7-21392 69878 


1-82 


079041 59571 


6-17 185 845 


0-162 025 751 


1-20958 40429 


1-83 


079475 89019 


6-23 388 666 


0-160 413 568 


7-20524 10981 


1-84 


0-79910 18467 


6-29 653 826 


0-158 817 426 


7-20089 81533 


1-85 


0-80344 47915 


6-35 981 952 


0-157 237 166 


1-19655 52085 


1-86 


0-80778 77363 


6-42 373 677 


0-155 672 630 


1-19221 22637 


1-87 


0-81213 06812 


6-48 829 640 


0-154 123 662 


7-18786 93188 


1-88 


0-81647 36260 


6-55 350 486 


0-152 590 106 


I -18352 63740 


1-89 


0-82081 65708 


6-61 936 868 


0-151 071 809 


1-17918 34292 


1-90 


0-82515 95156 


6-68 589 444 


0-149 568 619 


I -17484 04844 


1-91 


0-82950 24604 


6-75 308 880 


0-148 080 387 


1-17049 75396 


I '92 


0-83384 54053 


6-82 095 847 


0-146 606 962 


£•16615 45947 


1-93 


0-83818 83501 


6-88 951 024 


0-145 148 199 


1- 16 18 1 16499 


I '94 


0-84253 12949 


6-95 875 097 


0-143 703 950 


7-15746 87051 


1-95 


0-84687 42397 


7-02 868 758 


0-142 274 072 


1-15312 57603 


1-96 


0-85121 71845 


7-09 932 707 


0-140 858 421 


1-14878 28155 


1-97 


0-85556 01293 


7-17 067 649 


0-139 456 856 


£-14443 98707 


1-98 


0-85990 30742 


7-24 274 299 


0-138 069 237 


1-14009 69258 


1-99 


0-86424 60190 


7-31 553 376 


0-136 695 426 


^-13575 39810 


2-00 


0-86858 89638 


7-38 905 610 


0-135 335 283 


1-13141 10362 



34—2 



-(H> ifK GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 

TABLE in. 
Values of e', e'', \ogj^{e''), log,„(e"') from ^• = oi to x= lO'O at intervals of oi. 



X 


log,o(e^) 


e* 


e-^ 


log.o{«'0 


o-i 


o"04342 94482 


i-io 517 092 


0-904 837 418 


1-95657 05518 


0-2 


0-08685 88964 


1-22 140 276 


0-818 730 753 


1-91314 11036 


0-3 


0-13028 83446 


1-34 985 881 


0-740 818 221 


1-86971 16554 


0-4 


0-17371 77928 


1-49 182 470 


0-670 320 046 


1-82628 22072 


o-S 


0-21714 72410 


1-64 872 127 


0-606 530 660 


7-78285 27590 


06 


0-26057 66891 


1-82 211 880 


0-548 811 636 


1-73942 33109 


07 


0-30400 61373 


2-01 375 271 


0-496 585 304 


1-69599 38627 


0-8 


0-34743 55855 


2-22 554 093 


0-449 328 964 


1-65256 44145 


0-9 


0-39086 50337 


2-45 960 311 


0-406 569 660 


1-60913 49663 


i-o 


o'43429 44819 


2-71 828 183 


0-367 879 441 


1-56570 55181 


II 


°*47772 39301 


3-00 416 602 


0-332 871 084 


1-52227 60699 


1-2 


0-52115 33783 


3-32 Oil 692 


0-301 194 212 


T-47884 66217 


13 


0-56458 28265 


3-66 929 667 


0-272 531 793 


1-43541 71735 


1-4 


o-6o8oi 22747 


4-05 519 997 


0-246 596 964 


1-39198 77253 


1-5 


0-65144 17229 


4-48 168 907 


0-223 13° 160 


1-34855 82771 


1-6 


694S7 11710 


4-95 303 242 


0-201 896 518 


1-30512 88790 


17 


0-73830 06192 


5-47 394 739 


0-182 683 1^24 


1-26169 93808 


1-8 


0-78173 00674 


6-04 964 746 


0-165 298 888 


1-21826 99326 


1-9 


o-82=;i5 95156 


6-68 589 444 


0-149 568 619 


1-17484 04843 


2-0 


0-86858 89638 


7-38 905 610 


0-135 335 283 


1-13141 10362 


21 


0-91201 84120 


8-16 616 991 


0-122 456 428 


1-08798 15880 


2 '2 


0-95544 78602 


9-02 501 350 


o-iio 803 158 


£-04455 21398 


2-3 


0-99887 73084 


9-97 418 246 


o-ioo 258 844 


1-00112 26916 


2 '4 


1-04230 67566 


ii-o 231 764 


(1)907 179 533 


2-95769 32434 


2'5 


1-08573 62048 


I 2-1 824 940 


(1)820 849 986 


2-91426 37952 


2-6 


1-12916 56529 


13-4 637 380 


(1)742 735 782 


2-87083 43471 


27 


1-17259 51011 


14-8 797 317 


(1)672 055 127 


2-82740 48989 


2-8 


I -21602 45493 


16-4 446 468 


(1)608 100 626 


2-78397 54507 


2-9 


1-25945 39975 


18-1 741 454 


(1)550 232 201 


2-74054 60025 


30 


1-30288 34457 


20-0 855 369 


(1)497 870 684 


2-69711 65543 


31 


1-34631 28939 


22-1 979 513 


(1)450 492 024 


265368 71061 


3"-' 


1-38974 23421 


24-5 325 302 


(1)407 622 040 


2-61025 76579 


3 3 


I -433 '7 17903 


27-1 126 389 


(1)368 831 674 


2-56682 82096 


3 '4 


1-47660 12385 


29-9 641 001 


(1)333 732 700 


2-52339 87615 


3-5 


1-52003 06867 


33-1 154 520 


(1)301 973 834 


2-47996 93133 


3-6 


1-56346 01349 


36-5 982 344 


(1)273 237 224 


2-43653 98651 


37 


I -60688 95830 


40-4 473 044 


(1)247 235 265 


2-39311 04170 I 


3-8 


1-65031 90312 


44-7 oil 845 


(1)223 707 719 


2-34968 09688 


39 


1-69374 84794 


49-4 024 491 


(1)202 419 114 


2-30625 15206 


1 ^'^ 


1-73717 79276 


54-5 981 500 


(1)183 156 389 


2-26282 20724 


4-1 


1-78060 73758 


60-3 402 876 


(1)165 726 754 


2-21939 26242 


4-2 


1-82403 68240 


66-6 863 310 


(1)149 955 768 


2-17596 31760 


4-3 


1-86746 62722 


73-6 997 937 


(1)135 685 590 


2-13253 37278 


4-4 


1-91089 57204 


81-4 508 687. 


(1)122 773 399 


2-08910 42796 


4-5 


1-95432 51686 


90-0 171 313 


(1)1 11 089 965 


2-04567 48314 


4-6 


'-99775 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 


121- 510 418 


(2)822 974 705 


3-91538 64869 


4-9 


2-12804 29613 


134- 289 780 


(2)744 658 307 


3-87195 70387 


5'° 


2-17147 24095 


148- 413 '59 


(2)673 794 700 


3-82852 75905 



The nmnbers in parentheses denote the numbers of ciphers between the decimal point and the 
first significant figure; for example, e~' = 000673794700. 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



261 



TABLE IIL [continued). 
Values of e-', e''', log,„(e''), logi„(e"-^) from x = o-i to x— lo'o at intervals of o'l. 



X 


log„(e^) 




e-' 


log:,(e--) 


S'l 


2-21490 18577 


164- 


021 


907 


(2)609 674 657 


3-78509 81423 


5-2 


2-25833 13059 


i8i- 


272 


242 


(2)551 656 442 


3-74166 86941 


5 '3 


2-30176 07541 


200- 


336 


810 


(2)499 159 391 


3-69823 92459 


5-4 


2-34519 02023 


221- 


406 


416 


(2)451 658 094 


3-65480 97977 


5-5 


2-38861 96505 


244- 


691 


932 


(2)408 677 144 


3-61138 03495 


5-6 


2-43204 90987 


270- 


426 


407 


(2)369 786 372 


3-56795 09013 


57 


2-47547 85468 


298- 


867 


401 


(2)334 596 546 


3-52452 14532 


5-8 


2-51890 79950 


Zi^- 


299 


560 


(2)302 755 475 


3-48109 20050 


5-9 


2-56233 74432 


365- 


037 


468 


(2) 273 944 482 


3-43766 25568 


6-0 


2-60576 6S914 


403- 


428 


794 


(2)247 87s 218 


3-39423 31086 


6-1 


2-64919 63396 


445- 


857 


770 


(2) 224 286 772 


3-35080 36604 


6-2 


2-69262 57878 


492- 


749 


041 


(2) 202 943 064 


3-30737 42122 


6-3 


273605 52360 


544- 


571 


910 


(2) 183 630 478 


3-26394 47640 


6-4 


2-77948 46842 


601- 


84s 


038 


(2) 166 155 727 


3-22051 53158 


6-5 


2-82291 41324 


665- 


141 


633 


(2) 150 343 919 


3-17708 58676 


6-6 


2-86634 35806 


735- 


095 


189 


(2) 136 036 804 


3-13365 64194 


6-7 


2-90977 30288 


8i2- 


405 


82s 


(2) 123 091 190 


3-09022 69712 


6-8 


2-95320 24769 


897- 


847 


292 


(2) III 377 515 


3-04679 75231 


6-9 


2-99663 19251 


992- 


274 


716 


(2)100 778 543 


3-00336 80749 


7-^ 


3-04006 13733 


109 


6-63 


316 


(3)911 881 966 


4-95993 86267 


7-1 


3-08349 08215 


121 


1-96 


708 


(3)825 104 923 


4-91650 91785 


7-2 


3'i2692 02697 


^2,7, 


9 43 


077 


(3) 746 585 808 


4-87307 97303 


7-3 


3-17034 97179 


148 


0-29 


993 


(3)675 538 775 


4-82965 02821 


7-4 


3-21377 91661 


163 


5-98 


443 


(3) 611 252 761 


4-78622 08339 


7-5 


3-25720 86143 


180 


8-04 


242 


(3)553 084 370 


4-74279 13857 


7-6 


3-30063 80625 


199 


8-19 


590 


(3) 500 451 433 


4-69936 19375 


77 


3-34406 75107 


220 


8-34 


799 


(3)452 827 183 


4-65593 24893 


7-8 


3-38749 69588 


244 


0-60 


198 


(3)409 734 979 


4-61250 30412 


7 9 


3-43092 64070 


269 


7-28 


233 


(3)370 743 540 


4-56907 35930 


8-0 


3-47435 58552 


298 


0-95 


799 


(3)335 462 628 


4-52564 41448 


8-1 


3-51778 53034 


329 


446 


808 


(3)303 539 138 


4-48221 46966 


8-2 


3-56121 47516 


364 


0-95 


031 


(3)274 653 570 


4-43878 52484 


8-3 


3-60464 41998 


402 


3-87 


239 


(3)248 516 827 


4-39535 58002 


8-4 


3-64807 36480 


444 


7-06 


675 


(3) 224 867 324 


4-35192 63520 


8-5 


3-69150 30962 


491 


4-76 884 


(3)203 468 369 


4-30849 6903S 


8-6 


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 


3-82179 14407 


663 


4-24 


401 


(3) 150 733 075 


4-17820 85593 


8-9 


3-86522 08889 


733 


1-97 


354 


(3) 136 388 926 


4-13477 91111 


9"o 


390865 03371 


810 


3-08 


393 


(3) 123 409 804 


4-09134 96629 


91 


3-95207 97853 


895 


529 


270 


(3) III 665 808 


4-04792 02147 


92 


3-99550 92335 


989 


7-12 


906 


(3) loi 039 402 


400449 07665 


9-3 


4-03893 86817 


109 


38-0 


192 


(4)914 242 315 


5-96106 13183 


9 '4 


4-08236 81299 


120 


88-3 


807 


(4) 827 240 656 


5-91763 18701 


95 


4-12579 75781 


133 


59-7 


268 


(4) 748 518 299 


5-87420 24219 


1 9-6 


4-16922 70263 


147 


64-7 


816 


(4)677 287 365 


5-83077 29737 


' 97 


4-21265 64744 


163 


17-6 


072 


(4)612 834 951 


5-78734 35255 


9-8 


4-25608 59227 


180 


33-7 


449 


(4)554 515 994 


5-74391 40774 


9 '9 


4-29951 53708 


199 


30-3 


704 


(4) 501 746 821 


570048 46292 


lo-o 


4-34294 48190 


220 


26-4 


658 


(4)453 999 298 


5-65705 51810 



The numbers in parentheses denote the numbers of ciphers between the decimal point and the 
first signiBcant figure; for example, e"'° = 0-0000453999298. 



•JG-2 Mr GLAISHER, tables OF THE EXPONENTIAL FUNCTION. 

TABLE IV. 
Values of c', e~', log,„(('"'), log,„(e~-') from i to 500 at intervals of unity. 



X 


log.oCe') 


^ 


e 


X 


log>o(e-*) 


I 


■43429 44819 


2-71 S28 


183 




0-367 


879 441 


^•56570 55181 


2 


•86858 89638 


7-38 905 


610 




.0-135 


335 283 


1-13141 10362 


3 


1-30288 34457 


20-0 855 


369 




(i) 497 


870 684 


2-69711 65543 


4 


173717 79276 


54-5 981 


500 




(0 183 


156 389 


2-26282 20724 


5 


2-17147 2409s 


148-413 


159 




(2) 673 


794 700 


3-82852 75905 


6 


2-60576 68914 


403-428 


793 




(2) 247 


875 218 


3-39423 31086 


7 


3-04006 13733 


109 6-63 


.^16 




(3) 911 


881 966 


4-95993 86267 


8 


3-47435 58552 


298 0-9S 799 




(3) 335 


462 628 


4-52564 41448 


9 


3-90865 03371 


810 3-08 


393 




(3) 123 


409 804 


4-09134 96629 


10 


4"34294 48190 


220 26-4 


658 




(4) 453 


999 298 


5-65705 S1810 


II 


477723 93009 


598 74-1 


417 




(4) 167 


017 008 


5-22276 06991 


12 


5'2ii53 37828 


162 754- 


791 




(5) 614 


421 23s 


6-78846 62172 


13 


5-64582 82647 


442 413- 


392 




(5) 226 


032 941 


6'354i7 17353 


14 


6-08012 27466 


120 260 


4-28 




(6) 831 


528 719 


7-91987 72534 


15 


6-51441 72285 


326 901 


7-37 




(6) 305 


902 321 


7-48558 27715 


16 


6-94871 17105 


888 611 


0-52 




(6) 112 


535 175 


7-05128 82896 


17 


7-38300 61924 


241 549 


52-8 




(7) 413 


993 772 


8-61699 38076 


18 


7-81730 06743 


656 599 


69-1 




(7) 152 


299 797 


8-18269 93257 


19 


8-25159 51562 


178 482 


301- 




(8) 560 


279 644 


9-74840 48438 


20 


8-68588 96381 


485 165 


195- 




(8) 206 


115 362 


9-31411 03619 


21 


9-12018 41200 


131 881 


573 


I 


(9) 758 


256 043 


10-S7981 58800 


22 


9-55447 86019 


358 491 


285 


I 


(9) 278 946 809 


^-44552 13981 


23 


9-98877 30838 


974 480 


345 


I 


(9) 102 


618 796 


IO-OII22 69162 


24 


10-42306 75657 


264 891 


221 


2 


(10) 377 


513 454 


1^-57693 24343 


25 


10.85736 20476 


720 048 


993 


2 


(10) 138 879 439 


11-14263 79524 


26 


11-29165 65295 


195 729 


609 


3 


(11) 510 


908 903 


12-70834 34705 


27 


11-72595 10114 


532 048 


241 


.3. 


(11) 187 


952 882 


12-27404 89886 


28 


12-16024 54933 


144 625 


707 


4. 


(12) 691 


440 on 


■^3-83975 45067 


29 


12-59453 99752 


393 133 


430 


4 


(12) 254 


366 565 


13-40546 00248 


30 


13-02883 44571 


106 864 746 


.5 


(13) 935 


762 297 


14-97116 55429 


31 


13-46312 89390 


290 488 


497 


.5. 


(13) 344 


247 711 


14-53687 10610 


32 


13-89742 34209 


789 629 


602 


S 


(13) 126 


641 656 


14-10257 65791 


33 


14-33171 79028 


214 643 


580 


6' 


(14) 465 


888 615 


1566828 20972 


34 


14-76601 23847 


583 461 


743 


'6 


(14) 171 


390 843 


T5-23398 76153 


35 


15-20030 68666 


158 601 


345 


i 


(15) 630 


511 676 


16-79969 31334 


36 


15-63460 13485 


431 123 


155 


7 


(15) 231 


952 283 


£6-36539 86515 


37 


16-06889 58304 


117 191 


424 


8 


(16) 853 


304 763 


17-93110 41696 


38 


16-50319 03123 


318 559 


318 


'8' 


(16) z^z 


913 279 


77-49680 96877 


39 


16-93748 47942 


865 934 


004 


8' 


(16) 115 


482 242 


77-06251 52058 


40 


17-37177 92761 


235 385 


267 


'9 


(17) 424 


835 426 


18-62822 07239 


41 


17-80607 37580 


639 843 


493 


.9, 


(17) 156 


288 219 


7S19392 62420 


42 


18-24036 82399 


173 927 


494 


10 


(18) 574 


952 227 


^9-75963 17601 


43 


1867466 27218 


472 783 


947 


10 


(18) 211 


513 104 


19-32533 72782 


44 


19-10895 72037 


128 516 


001 


II 


(19) 778 


113 224 


20-89104 27963 


45 


19-54325 16856 


349 342 


711 


1 1 


(19) 286 


251 858 


20-45674 83144 


46 


'9-97754 61675 


949 611 


942 


11 


(19) 105 


306 174 


20-02245 38325 


47 


20-41184 06495 


258 131 


289 


12' 


(20) 387 


399 763 


21-58815 93505 


48 


20-84613 51314 


701 673 


591 


12 


(20) 142 


516 408 


21-15386 48686 


49 


21-28042 96133 


190 734 


657 " 


13' 


(21) 524 


288 566 


^-71957 03867 


50 


21-71472 40952 


518 470 


553 . 


'3. 


(21) 192 


874 985 


22-28527 59048 



The nimiht-rff in h(|uare braokots denote the miinlifis of figuros between tlie last figure given 
and the (leeiin.al point; for example, the first nine figure.s of «'" are 518470553, and there are 
13 additional tigures before the dicirnal point is reaehed. The numbers in jiarentheses denote the 
numbers of ciphers between the decimal point and the fir.st significant figure ; for example, in e~" 
there are 21 ciphers between the decimal point and the figures 192874985. 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



263 



TABLE IV.. {continued). 
Values of e^ e"", logi„(e'^), logi„(e"^) from i to 500 at intervals of unity. 



X 

51 


log,„(e^) 


^ 


e-' 


log.„(e-^) 


22-14901 85771 


140 934 


908 


.^4 


(22 


) 709 547 


416 


23-85098 14229 


.52 


22-58331 30590 


383 100 


800 


14 


(22 


1 261 027 


907 


23-41668 69410 


53 


23-01760 75409 


104 137 


594 


.15 


(23 


1 960 268 


005 


24-98239 24591 


54 


23-45190 20228 


283 075 


iio 


15 


(23 


) 353 262 


857 


24-54809 79772 


55 


23-88619 65047 


769 478 


527 


.15 


(23 


) 129 958 


143 


24-11380 34953 


56 


24-32049 09866 


209 165 


950 


i6' 


(24 


) 478 089 


288 


25-67950 90134 


57 


247547S 54685 


568 572 


000 


16 


(24 


175 879 


220 


25-24521 45315 


58 


25-18907 99504 


154 553 


894 


17' 


(25 


647 023 


493 


26-81092 00496 


59 


25'62337 44323 


420 121 


040 


17 


(25 


) 238 026 


641 


^•37662 55677 


60 


26-05766 89142 


114 200 


739 


ri8' 


(26 


) 875 651 


076 


27-94233 10858 


61 


26-49196 33961 


310 429 


794 


[18^ 


(26 


322 134 


029 


27-50803 66039 


62 


26-92625 78780 


843 83s 


667 


'18 


(26, 


ri8 506 


487 


27-07374 21220 


63 


27-36055 23599 


229 378 


316 


'19 


(27 


435 961 


000 


28-63944 76401 


64 


27-79484 68418 


623 514 


908 


.19 


(27 


160 381 


089 


28-20515 31582 


65 


28-22914 13237 


169 48S 


924 


2-0 


(28 


590 009 


054 


29-77085 86763 


66 


28-66343 5S056 


460 718 


663 


20" 


(28 


217 052 


201 


29-33656 41944 


67 


29-09773 0287s 


125 236 


317 


b\ 


(29 


798 493 


425 


30-90226 97125 


68 


29-53202 47694 


340 427 


605 


21 


(29 


293 748 


2X1 


30-46797 52306 


69 


29-96631 92513 


925 378 


172 


21 


(29) 


108 063 


928 


3003368 07487 


70 


30-40061 37332 


251 543 


867 


22" 


(30 


397 544 


974 


31-59938 62668 


71 


30-S3490 82151 


683 767 


123 


22" 


(3°; 


146 248 


623 


31-16509 17849 


72 


31-26920 26970 


185 867 


175 


[23- 


(31 


538 018 


616 


3^-73079 73030 


73 


31-70349 71789 


505 239 


363 


.23 


(31. 


197 925 


988 


32-29650 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 


33-42791 38573 


76 


33-00638 06246 


loi 4S0 


039 


25 


{ii 


9S5 415 


469 


34-99361 93754 


77 


33'44o67 51066 


275 851 


346 


25' 


(i3 


362 514 


092 


34-55932 48934 


7S 


33'S7496 95885 


749 841 


700 


25 


(Zi 


133 361 


482 


34-12503 04115 


19 


34-30926 40704 


203 828 


107 


26' 


(34. 


490 609 


473 


35-69073 59296 


80 


3474355 85523 


554 062 


238 


26' 


(34 


180 485 


139 


35-25644 14477 


81 


35'i7785 30342 


150 609 


731 


27' 


(35) 


663 967 


720 


36-82214 69658 


82 


35-61214 75161 


409 399 


696 


27 


(35 


244 260 


074 


36-38785 24839 


83 


36-04644 19980 


111 286 


376 


28 


(36) 


898 582 


594 


37-95355 80020 


84 


36-48073 64799 


302 507 


732 


28' 


(36) 


330 570 


063 


37-51926 35201 


85 


36-91503 09618 


822 301 


271 


28' 


(36) 


121 609 


930 


37-08496 90382 


86 


37-34932 54437 


223 524 


660 


29 


(37) 


447 377 


931 


38-65067 45563 


87 


37-78361 99256 


607 603 


023 


29 


(37 


164 581 


143 


38-21638 00744 


88 


38-21791 44075 


165 163 


626 


30 


(38) 


605 460 


189 


39-78208 55925 


89 


38-65220 88894 


448 961 


282 


3° 


(38 


222 736 356 1 


39-34779 11106 


90 


39-08650 33713 


122 040 


329 


.31 


(39) 


819 401 


262 


40-91349 66287 


91 


39-52079 78532 


331 740 


010 


31 


(39) 


3or 440 


879 


40-47920 21468 


92 


39-95509 23351 


901 762 


841 


.31 


(39) 


no 893 


902 


40-04490 76649 


93 


40-38938 68170 


245 124 


554 


32 


(40) 


407 955 


867 


41-61061 31830 


94 


40-82368 12989 


666 317 


622 


32 


(40) 


150 078 


576 


4T-17631 87011 


95 


41-25797 57808 


181 123 


908 


a 


(41) 


552 108 


228 


42^-74202 42192 


96 


41-69227 02627 


492 345 


829 


M. 


(41) 


203 109 


266 


42-30772 97373 


97 


42-12656 47446 


133 833 


472 


.34. 


(42) 


747 197 


234 


43-87343 52554 


98 


42-56085 92265 


363 797 


095 


.34 


(42) 


274 878 


501 


43-43914 07735 


99 


42-99515 37084 


988 903 


032 


34 


(42 


loi 122 


149 


43-00484 62916 


100 


43-42944 81903 


268 811 


714 


.35 


(43) 


372 007 


598 


44-57055 18097 



The numVjers in squai-e 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. 



HU Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 

TABLE IV. (continued). 
Values of e^, e~', \og^„{e''), \og^^{e''-^) from i to 500 at intervals of unity. 



X 


log,„(e^) 


<f 




e-^ 


44-13625 73278 


101 


4386374 267- 


730 705 


998 


35 




(43) 136 853 


947 


102 


4429803 71541 


198 626 


484 


36 




(44) 503 457 


536 


45-70196 28459 


103 


4473233 16360 


539 922 


761 


36' 




(44) 185 211 


676 


45-26766 83640 


104 


45-16662 6II79 


146 766 


223 


37 




(45) 681 355 


682 


4683337 38821 


105 


45'6oo92 05998 


398 951 


957 


37 




(45) 250 656 748 1 


46-39907 94002 


106 


4603521 50S17 


108 446 386 1 


38' 




{46) 922 114 


642 


47-96478 49183 


107 


46-46950 95636 


294 787 


839 


38' 




(46) 339 227 


019 


4753049 04364 


108 


46-90380 40456 


801 316 


426 


38. 




(46) 124 794 


646 


4709619 59544 


109 


47-33S09 85275 


217 820 


388 


39 




(47) 459 093 


847 


48-66190 14725 


no 


47-77239 30094 


592 097 


203 


39. 




(47) 168 891 


188 


48-22760 69906 


III 


48-20668 74913 


160 948 


707 


40 




{48) 621 315 


959 


49-79331 25087 


112 


48-64098 19732 


437 503 


945 


40 




(48) 228 569 368 


49-35901 80268 


"3 


49-07527 64551 


118 925 


902 


41 




(49) 840 859 


712 


50-92472 35449 


114 


49-50957 09370 


323 274 


119 


41 




(49) 309 335 


001 


5049042 90630 


"5 


49'94386 54189 


878 750 


164 


41. 




(49) 113 797 


987 


55-05613 45811 


116 


5o'378iS 99008 


238 869 


060 


42 




(50) 418 639 


400 


51-62184 00992 


J17 


50-81245 43827 


649 313 


426 


42' 




(50) 154 008 


829 


51-18754 56173 


118 


51-24674 SS646 


176 501 


689 


43, 




(51) 566 566 


818 


52-75325 11354 


"9 


51-68104 33465 


479 781 


333 


43 




(51) 208 428 


284 


52-31895 66535 


120 


52'ii533 78284 


130 418 


c88 


44 




(52) 766 764 807 


53-88466 21716 


121 


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 


i-'3 


53-41822 12741 


261 951 


732 


.45 




(53) 381 749 


719 


54-58177 87259 


124 


53-85251 57560 


712 058 633 


45 




(53) 140 437 


873 


54-14748 42440 


125 


54-28681 02379 


193 557 


604 


46 




(54) 516 642 


063 


55-71318 97621 


126 


5472110 47198 


526 144 


118 


■46- 




(54) 190 061 


994 


55-27889 52802 


127 


5515539 92017 


143 020 


800 


.47. 




(5S) 699 199 


000 


56-84460 C7983 


128 


5S'.'^8969 36836 


388 770 


841 


47 




(55) 257 220 


937 


56-41030 63164 


129 


56-02398 81655 


105 678 


871 


48^ 




(56) 946 262 


947 


57-97601 18345 


130 


56-45828 26474 


287 264 


955 


48" 




(56) 348 no 


684 


57-54171 73526 


131 


56-89257 71293 


780 867 


107 


48; 




{56) 128 062 


764 


57-10742 28707 


132 


57-32687 16112 


212 261 


687 


49 




(57) 471 116 


580 


58-673x2 83888 


133 


57-76116 60931 


576 987 


086 


.49^ 




(57) 173 314 


104 


58-23883 39069 


»34 


58-19546 05750 


156 841 


351 


.50] 




(58) 637 586 


958 


59-80453 94250 


135 


58-62975 50569 


426 338 


995 


,5°. 




(58) 234 555 


134 


59-37024 49431 


136 


59-06404 95388 


115 890 


954 


51 




(59) 862 880 


116 


60-93595 04612 


137 


59-49834 40207 


315 024 


275 


51. 




(59) 317 435 


855 


60-50165 59793 


'38 


59-93263 85026 


856 324 


762 


51 




(59) 116 778 


125 


60-06736 14974 


139 


60-36693 29846 


232 773 


204 


52 




(60) 429 602 


713 


61-63306 70154 


140 


60-80122 74665 


632 743 


171 


52 




(60) 158 042 


006 


^-19877 25335 


141 


61-23552 19484 


171 997 


426 


53 




(61) 581 404 


049 


62-76447 80516 


142 


61-66981 64303 


467 537 


479 


53 




(61) 213 886 


597 


62-33018 35697 


'I43 


62-10411 09122 


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 


»45 


62-97269 98760 


939 074 


129 


54 




(62) 106 487 


866 


63-02730 01240 


146 


63-40699 43579 


255 266 


8.4 


55 




(63) 391 746 966 


64-59300 56421 


147 


63-84128 88398 


693 887 


142 


55 




(63) 144 115 


65s 


6415871 11602 


148 


64-27558 33217 


188 618 


081 


56 




(64) 530 171 


867 


65-72441 66783 


149 


64-70987 78036 


512 717 


102 


:56 




(64) 195 039 


330 


65-29012 21964 


'50 


65-14417 22855 


139 370 


958 


.57 




(65) 717 509 


597 


66-85582 77145 



The numbers in square brackets denote the numbers of figures between the last figure given 
Hiid the decimal point ; for example, the first nine figures of e"" are 730705998, and there are 
35 additional figures tefore the decimal point is reached. The numbers in parentheses denote the 
numters of ciphers between the decimul |>oint and the first significant figure ; for example, in e '"' 
there are 43 ciphers between the decimal point and the figures 136853947. 



Mr GLAISHEE, TABLES OF THE EXPONENTIAL FUNCTION. 



2G5 



TABLE IV. {continued). 
Values of e", e"-", log,„(e''), logi„(e"-^) from i to 500 at intervals of unity. 



X 


^ogM) 


e' 


e 


-X 


log:o(e-^) 


rSi 


65'57846 67674 


378 


849 543 


57 




(6s) 263 


957 030 


66-42153 32326 


152 


66'oi276 12493 


102 


981 983 


58" 




(66) 971 


043 646 


67-98723 87507 


153 


66-44705 57312 


279 


934 052 


58. 




(66) 357 


226 994 


67-55294 42688 


154 


66'88i35 02131 


760 


939 648 


58 




(66) 131 


416 467 


67-11864 97869 


15s 


67'3i564 46950 


206 


844 842 


59. 




(67) 483 


45:4 164 


^■68435 53050 '■ 


156 


6774993 91769 


562 


262 575 


59, 




(67) 177 


852 848 


68-25006 0823! 


157 


68-18423 365S8 


I,'52 


838 814 


60 




(68) 654 


284 062 


69-81576 63412 


158 


68-61852 81407 


415 


458 971 


60' 




(68) 240 


697 655 


69-38147 18593 


159 


69-05282 26226 


112 


933 457 


61" 




(69) 885 


477 1S8 


70-94717 73774 


160 


69-48711 71045 


306 


984 964 


'61' 




(69) 325 


74S 853 


70-51288 28955 


161 


69-92141 15864 


834 


471 649 


ei 




(69) 119 


836 306 


70-07858 84136 


162 


70-35570 60683 


226 


832 912 


62' 




(70) 440 


853 133 


7T-64429 39317 


163 


70-79000 05502 


616 


595 783 


62' 




(70) 162 


iSo 804 


71-20999 94498 


164 


71-22429 50321 


167 


608 III 


63' 




(71) 596 


629 837 


72-77570 49679 


165 


7i'658s8 95140 


455 


606 083 


.63' 




(71) 219 


487 851 


72-34141 04860 


166 


72-092S8 39959 


123 


846 574 


64 




(72) 807 


450 679 


73-90711 60041 


167 


72-52717 84778 


336 


649 891 


64 




(7i) 297 


044 505 


73-47282 15222 


i68 


72-96147 29597 


915 


109 280 


•64 




(72) 109 


276 566 


73-03852 70403 


169 


73'39576 744i6 


248 


752 493 


.65: 




(73) 402 


006 022 


74-60423 25584 


170 


73-83006 19236 


676 


179 381 


.65 




(73) 147 


889 751 


74-16993 80764 


171 


74-26435 64055 


183 


804 612 


66 




(74) 544 


055 988 


7"573564 35945 


172 


74-69865 08874 


499 


632 738 


'66' 




(74) 200 


147 013 


75'3oi34 91126 


173 


75'i3294 53693 


135 


814 259 


■67- 




(75) 736 


299 712 


76-86705 46307 


174 


75'S6723 9851^ 


369 


i8i 433 


67 




(75) 270 


869 527 


76-43276 0148S 


I7S 


76-00153 43331 


100 


353 918 


68 




(76) 996 


473 301 


77-99846 56669 


176 


76-43582 8S150 


272 


790 232 


■68' 




(76) 366 582 041 


77-56417 11850 


177 


76-87012 32969 


741 


520 730 


68' 




(76) 134 


857 996 


77-12987 67031 


178 


77"3044i 777S8 


201 


566 233 


69 




(77) 496 


114 844 


7S-69558 22212 


179 


77-73871 22607 


547 


913 828 


.69' 




(77) 182 


510 451 


78-26128 77393 


l83 


78-17300 67426 


148 938 420 


70 




(78) 671 


418 429 


79-82699 32574 


181 


78-60730 12245 


404 


856 601 


70 




(78) 247 


001 036 


79-39269 87755 


182 


79-04159 57064 


no 


051 434 


71 




(79) 9=8 


666 032 


80-95840 42936 


183 


79-47589 01883 


299 


150 814 


.71 




(79) 334 


279 552 


80-52410 98117 


1 84 


79-91018 46702 


813 


176 221 


.71 




(79) 122 


974 575 


80-08981 53298 


185 


80-34447 91521 


. 221 


044 214 


72 




(80) 452 


398 179 


8I-65552 08479 


186 


80-77877 36340 


' 600 


860 471 


72 




(80) 166 


427 989 


81-22122 63660 


187 


81-21306 81159 


163 


330 Sto 


.73 




(81) 612 


254 357 


8^-78693 18841 


18S 


81-64736 25978 


443 


979 173 


73 




(8i) 225 


235 791 


82-35263 74022 


189 


82-0S165 70797 


120 


686 052 


74 




(8a) 828 


596 168 


85'9i834 29203 


190 


82-51595 15616 


32S 


058 702 


74 




(82) 304 


823 495 


83-48404 84384 


191 


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 


,76" 




(84) 558 


303 706 


S5-74687 05108 


19s 


84-68742 39711 


486 


882 283 


76 




(84) 205 


388 455 


85-31^57 60289 


ig6 


85-12171 84530 


132 


348 326 


77 




(85) 755 


581 902 


86-87828 15470 


197 


85'556oi 29349 


359 


760 050 


77 




(85) 277 963 048 


86-44398 70651 


198 


85"99o3o 74168 


977 


929 206 


77 




(85) 102 


256 891 


86-00969 25832 


199 


86-42460 18987 


265 


828 719 


78 




(86) 376 


182 078 


87-57539 81013 


200 


86-85889 63807 


722 


597 377 


78 




(86) i^^ 


389 653 


87-14110 36194 



The numbers in square brackets denote tbe numbers of figures between the last figure given 
and the decimal ]ioint ; for example, the first nine figures of e'*' are 378849543, and there are 
57 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 65 ciphers between the decimal point and the figures 263957030. 

Vol. XIII. Part III. 35 



•JG(3 Mr GLAISIIEK, TABLES OF THE EXPONENTIAL FUNCTION. 

TABLE IV. (continued). 
Values of e\ c"-"', log,o(e'), log„((^"-') from i to 500 at intervals of unity. 



X 
201 


log.o(«') 


e' 


- e--^ 


log:„(e 


-) 


87-29319 08626' 


196 


422 


332 


79; 


(87) 


509 107 


081 


S870680 


91374 


202 


8772748 53445 


533 


931 


25s 


79 


(87) 


187 290 


028 


ss-2725 I 


46555 


203 


88-16177 98264 


145 


137 


563 


80 


(88) 


689 OOI 


510 


89-83822 


01736 


204 


8S-59607 43083 


394 


524 


Soo 


8o' 


(88) 


253 469 


490 


S9-40392 


56917 


205 


89-03036 87902 


107 


242 


960 


81' 


(S9) 


932 462 


145 


90-96963 


12098 


206 


89-46466 32721 


291 


516 


588 


81' 


(89) 


343 033 


653 


90-53533 


67279 


207 


89-89895 77540 


792 


424 


244 


'8i' 


(89) 


126 195 


029 


90-10104 


22460 


208 


90-33325 22359 


215 


403 


242 


"82' 


(90) 


464 245 


566 


^-66674 77641 1 


209 


90-76754 67178 


585 


526 


719 


'82' 


(90) 


170 786 


399 


9I-232-1S 


32822 


210 


91-20184 11997 


159 


162 


664 


:83: 


(91) 


628 288 


051 


92-79815 


88003 


211 


91-63613 56816 


432 


648 


977 


.83 


(91) 


231 134 


257 


92-36386 


43 1 84 


212 


92-07043 01635 


117 


606 


185 


84 


(92) 


850 295 


414 


93-92956 98365 1 


213 


92-50472 46454 


319 


686 


757 


■84 


(92) 


312 806 


202 


93-49527 


53546 


214 


92-93901 91273 


868 


998 


701 


84 


(92) 


"5 074 


971 


93-06098 


08727 


215 


93"3733i 36092 


236 


218 


338 


85: 


(93) 


423 337 


159 


94-62668 


63908 


216 


93-80760 80911 


642 


108 


015 


85 


(93) 


155 737 


037 


94-19239 


19089 


217 


94-24190 25730 


174 


543 


°55 


86 


(94) 


572. 924 


543 


95 75809 


74270 


218 


94-67619 70549 


474 


457 


215 


86' 


(94) 


210 767 


161 


95-32380 


29451 


219 


95-11049 15368 


128 


970 


843 


87' 


(95) 


775 369 


053 


96-88950 


84632 


220 


95'54478 60187 


350 


579 


098 


87' 


(95) 


285 242 


334 


9I-45521 


39813 


221 


95-97908 05006 


952 


972 


790 


87 


(95) 


104 934 


791 


96-02091 


94994 


222 


96-41337 49825 


259 


044 


862 


88 


(96) 


386 033 


520 


97-58662 


5017s 


223 


96-84766 94644 


704 


156 


941 


■88' 


(96) 


142 or3 


796 


97-15233 


05356 


224 


97-28196 39463 


191 


409 


702 


89: 


(97) 


522 439 


558 


98-71803 


60537 


225 


97-71625 84282 


520 


305 


514 


89 


(97) 


192 194 


773 


98-28374 


15718 


226 


98'iSo5S 29101 


141 


433 


702 


.9°. 


(98) 


707 045 


056 


9984944 


70899- 


227 


98-58484 73920 


384 456 663 


.90. 


(98) 


260 107 


340 


99-41515 


26080 


228 


99-01914 18739 


104 


506 


156 


91 


(99) 


956 8S1 


429 


100-98085 


81261 


229 


99'45343 63558 


284 


077 


185 


.91. 


(99) 


352 017 


006 


105-54656 36442 1 


230 


99-88773 08377 


772 


201 


850 


.91. 


(99) 


129 499 


819 


100-11226 


91623 


231 


100-32202 53197 


209 


906 


226 


.92. 


(100) 


476 403 


211 


ioT-67797 


46S03 


232 


100-75631 98016 


570 


5S4 


279 


.92. 


(100) 


175 258 947 


101-24368 


01984 


233 


101-19061 42835 


155 


100 


888 


93 


(lOl) 


644 741 


635 


102-80938 


57165 


234 


101-62490 87654 


421 


607 


925 


.93. 


(lOl) 


237 187 


193 


102-37509 


12346 


235 


102-05920 32473 


114 


604 


916 


.94. 


(102) 


872 562 


919 


103-94079 


67527 


236 


102-49349 77292 


311 


528 


461 


.94 


(102) 


320 997 


959 


103-50650 


22708 


237 


102-92779 22111 


846 


822 


154 


.94 


(102) 


118 088 


550 


103-07220 


77889 


238 


103-36208 66930 


230 


190 


127 


.95. 


(103) 


434 423 


497 


104-63791 


33070 


239 


103-79638 11749 


625 


721 


640 


.95. 


(103) 


159 815 


473 


104-20361 


88251 


240 


, 104-23067 56568 


170 


088 


776 


.96 


(104) 


587 928 


270 


105-76932 


43432 


241 


104-66497 01387 


462 


349 


230 


.96 


(104) 


216 286 


723 


105-33502 


98613 


242 


105-09926 46206 


125 


679 


551 


97 


(105) 


795 674 


389 


106-90073 


53794 


243 


i°5-53355 91025 


341 


632 


440 


.97. 


(105) 


292 712 


250 


706-46644 


08975 


244 


105-96785 35844 


928 


653 


253 


.97. 


(i°5) 


107 682 


819 


106-03214 


64156 


245 


106-40214 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 


10770503 i5'2o 


507 


027 


496 


.99. 


(107) 


197 227 


962 


168-29496 84880 1 


249 


108-13932 59939 


137 


824 


363 [ 


[OO 


(108) 


725 561 


126 


109-86067 


40061 


250 


108-57362 04758 


374 


645 461 [ 


00' 


(108) 


266 919 


022 


109-42637 


95242 



The numbers in square bracket."! denote the numbers of figures between the last figure given 
and the decimal point ; and the numbers in jiarentheses denote tlic numbers of ciphers between 
the decimal point and the first significant figure. 



Mr GLAISHEE, TABLES OF THE EXPONENTIAL FUNCTION. 

TABLE IV. {continued). 
Values of e-^, e"-^, log,„(e''), log,o(e"-') from i to 500 at intervals of unity, 



267 



X 



251 
252 
253 
254 
25s 
256 
257 
258 
259 
260 
261 
262 
263 
264 
265 
266 
267 
268 
269 
270 
271 
272 
273 
274 

275 
276 

277 
278 

279 
283 
281 
282 
283 
284 
285 
286 
287 
288 
289 
290 
291 
292 

293 
294 

295 
296 

297 
298 
299 

300 



log>o(eO 



09'oo79i 
09-44220 
09-87650 
0-31079 
0-74509 
1-17938 
1-61368 
2-04797 
2-48227 
2-91656 
3'35o8s 

378515 
4-21944 

4-65374 
5-08833 

5'52233 
5-95662 
6-39092 
6-S2531 

7-25951 

7-69380 

8-12809 

8-56239 

8-99668 

9-4309S 

9-86527 

20-29957 

20-73386 

21-16816 

21-60245 

22-03674 

22-47104 

22-9=533 

23-33963 

23-77392 

24-20822 

24-64251 

25-07681 

25-51110 

25-94539 
26-37969 
26-81398 
27-24828 
27-68257 
28-11687 
28-55116 
28-98546 
29-41975 
29-85405 
30-28834 



49577 
94396 

39215 
84034 
28853 
73672 
18491 
63310 
08129 
52948 
97767 
42587 
87406 
32225 

77^44 
21863 
666S2 
11501 

56323 
01139 

45958 
93777 
35596 
80415 

25234 
70053 
14872 

59691 
04510 

49329 
94148 
38967 
83786 
28605 

73424 
18243 
63062 

07881 
52700 

97519 
42338 
87157 
3197V 
76796 
21615 
66434 
11253 
56072 
00891 
45710 



loi 839 195 
276 827 633 
752 495 525 
204 549 491 
556 023 165 
151 142 767 
410 848 636 
III 68d 238 

303 578 362 
825 211 544 

224 315 755 
609 753 439 
165 748 169 

450 550 237 
122 472 252 
332 9'i4 098 
904 954 342 
245 992 094 
668 675 840 
181 764 939 
494 088 330 

134 307 133 
365 084 638 
992 402 938 
269 763 087 
733 292 09S 
199 329 459 
541 833 645 
147 285 655 
400 363 920 
108 830 197 



295 



147 



804 152 430 
218 591 294 

594 192 742 
161 518 333 
439 052 350 
119 346 803 
324 418 245 
88 I 860 219 

239 714 461 
651 611 463 
177 126 360 
481 479 366 
130 879 661 
355 767 804 
967 077 157 
262 878 826 

714 57S 737 
194 242 640 



lOI 


(109) 


lOI 


(109) 


lOI 


(109) 


102 


(no) 


102 


(no) 


133 


(in) 


103 


(III) 


104 


(112) 


104 


(112) 


104 


(112) 


105 


("3) 


.i°5. 
106 


("3) 
(114) 


106 


(114) 


107 


(115) 


107 


("5) 


II07J 
108 


(IIS) 
(116) 


108 


(116) 


109 


(117) 


109 


(117) 


no 


(118) 


no 


(118) 


no 


(118) 


III 


(119) 


I II 


(".9) 


112 


(120) 


1 12 


(120) 


113 


(121) 


113 


(121) 


114 


(122) 


114 


(122) 


114 


(122) 


.lis. 


(123) 


II!: 
116 


(123) 
(124) 


116 


(124) 


117 


(125) 


117 


(125) 


117 
"iiS 


(125) 
(126) 


■118 


(126) 


119 


(127) 


."9. 
120 


(127) 
(128) 


120 


(128) 


120 


(128) 


121 


(129) 


121 


(129) 


122 


(130) 



981 

361 
132 
488 
179 
661 
243 

895 

329 

121 

445 
164 
603 

221 
816 
300 

no 

406 
149 

550 
eo2 

744 
273 
100 

370 
136 
501 
184 
678 
249 
918 

338 

124 

457 
168 
619 
227 

837 
308 

"3 

417 
153 
564 
207 
^64 
281 

103 
380 

139 

514 



940 205 

23s 614 
891 156 
879 241 
848 622 
626 io6 
398 642 

413 564 
404 242 
181 048 
8oo 163 
000 715 
324 914 
95° 832 
511 481 

377 787 
502 813 

517 129 
549 294 

161 108 
392 961 
562 094 
909 0S7 
765 522 
695 639 
371 304 
681 993 

558 491 
952 746 

772 757 
862 622 
030 668 
354 533 
474 762 
295 560 
ii4 764 
763 272 
894 253 
244 070 
396 656 

162 985 
465 6«6 
568 707 
693 220 
o6o 659 
082 2o8 
404 366 
4^3 403 
944 591 
820 022 



logioC^^""^) 



0-99208 

1-55779 
0-12349 
1-68920 

1-25490 
2-82061 
2-38631 
3-95202 
3-51772 
3-08343 
4-64914 
4-21484 
5-78055 

5-34625 
6-9II96 
6-47766 

5'o4337 

7-60907 

7-17478 

8-74048 

S-30619 

"9-87190 

9-43760 

9-00331 

20-56901 

20-13472 

21-70042 

21-26613 

22-83183 

^■39754 

^3-96325 

53-52895 

23-09466 

24 66036 

24-22607 

25-79177 

25-35748 

26-92318 

26-48889 

26-05460 

27-62030 

27-1S601 

28-75171 
28-31742 
29-88312 
29-44883 

29-01453 
30-58024 

30-14594 
31-71165 



50423 
05604 
60785 
15966 
71147 
26328 
81509 
36690 
91871 
47052 
02233 

57413 
12594 

67775 
22956 

78137 
33318 
88499 
43680 
98861 
54042 
09223 
64404 

195S5 
74766 
29947 
85128 
40309 

95490 
50671 

05852 
61033 
16214 

71395 
26576 

81757 
36938 
92119 

47300 
02481 
57662 
12843 
68023 
23204 

78385 
33566 
88747 
43928 
99109 

54290 



Tlie numbers in square brackets denote the numbers of figures bet-sveen the last figure given 
and the decimal point; and the numbers in parentheses denote the numbers of ciphers between 
the decimal poiut and the first significant figure. 

35—2 



■2G8 



Mr GLAISIIER, TABLES OF THE EXPONENTIAL FUNCTION. 



TABLE IV. (continued). 
Values of e^, c''', log,„(c'-'), log,„((^~'') IVom i to 500 at intervals of unity. 



X 


log,„(e') 


e' 


e-^ 


log,„(c' 


-') 


301 


13072263 90529 


528 


006 237 


122] 


(130) 189 391 


702 


131-27736 


09471 


30-' 


131-15693 35348 


143 


526 976 


'123 


(131) 696 733 


135 


132-84306 


64652 


2,°i 


131-59122 80167 


390 


146 771 


.123] 


(131) 256 313 


796 


132-40877 


19833 


3^4 


132-02552 24986 


106 


052 888 


124 


(132) 942 925 


762 


133-97447 


75014 


305 


I32-459SI 69805 


28S 


281 638 


124 


(132) 346 S83 


002 


133-54018 


30195 


306 


I32-894II 14624 


783 


630 737 


124 


(132) 127 611 


125 


133-10588 85376 1 


337 


133-32S40 59443 


213 


012 919 


.125 


(133) 469 455 


094 


134-67159 


40557 


308 


13376270 04262 


579 


029 148 


.125. 


(133) 172 702 


878 


134-23729 


95738 


309 


134-19699 49081 


157 


396 441 


126 


(134) 635 338 


381 


135-80300 


50919 


310 


134-63128 93900 


427 


847 886 


126 


(134) 233 727 


929 


135-36871 


06100 


311 


i35'o6558 3S719 


116 


301 113 


127' 


i^ZS) 859 836 


997 


136-93441 


61281 


■ 312 


135-49987 83538 


316 


139 203 


127 


(135) 316 316 


354 


136-50012 


16462 


313 


135-93417 28357 


859 


355 450 


127 


(13s) 116 366 


284 


136-06582 


71643 


3J4 


136-36846 73176 


233 


507 031 


128' 


(136) 42S 087 


634 


^7-63153 


26824 


315 


136-S0276 17995 


634 


982 563 


128- 


(136) 157 48^ 


640 


137-19723 


82005 


316 


137-23705 62814 


172 


606 156 


}-9. 


(137) 579 353 


612 


138-76294 


37186 


317 


137-67135 07633 


469 


192 178 


129 


(137) 213 132 


283 


138-32864 


92367 


318 


138-10564 52452 


127 


539 657 


130 


(138) 784 069 851 


139-89435 47548 1 


319 


138-53993 97271 


346 


688 732 


130 


(138) 2S8 443 


179 


139-46006 


02729 


323 


138-97423 42090 


942 


397 682 


.130 


(138) 106 112 


315 


139-02576 


57910 


321 


139-40852 S6909 


256 


170 249 


.131. 


(139) 390 365 


393 


140-59147 


13091 


322 


139-84282 31728 


696 


342 934 


131 


(139) 143 607 


403 


I40-I57I7 


68272 


2,^ 


140-27711 7654S 


189 


285 634 


.132. 


(140) 528 302 


1 10 


141-722S8 


23452 


324 


140-71141 21367 


514 


531 700 


132 


(140) 194 351 


485 


. 141-28858 78633 1 


3-'5 


141-14570 66186 


139 


864 217 


133 


(141) 714 979 


157 


142-85429 


33814 


3-6 


141-58000 11005 


3S0 


190 360 


,133 


(141) 263 026 


133 


142-41999 


88995 


327 


142-01429 55824 


103 


346 455 


134] 


(142) 967 619 


067 


143-98570 


44176 


328 


142-44859 00643 


280 


924 790 


.134 


(142) 355 967 


162 


143-55140 


99357 


329 


142-88288 45462 


763 632 751 


134 


(142) 130 953 


001 


143-11711 


54538 


330 


143-31717 90281 


207 


576 903 


.135 


(143) 481 749 


167 


144-68282 


09719 


331 


143-75147 35100 


564 


252 524 


,135 


(143) 177 225 


614 


744-24852 


64900 


332 


144-18576 79919 


153 


379 738 


,136 


(144) 651 976 


599 


145-81423 


20081 


333 


144-62006 24738 


416 


939 355 


136 


(144) 239 84S 787 


145-37993 


75262 


334 


145-05435 69557 


"3 


m 149 


137 


(145) 882 354 


377 


146-94564 


30443 


335 


145-48865 14376 


308 


071 439 


.137] 


(HS) 324 600 


035 


146-51134 


85624 


3i(> 


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 


r^^ 


146-79153 48833 


618 


778 027 


138 


(146) 161 608 


841 


147-20846 


51167 


339 


147-22582 93652 


168 


201 307 


.139. 


(147) 594 525 


703 


14S-77417 


06348 


340 


147-66012 38471 


457 


218 555 


,139 


(147) 218 713 


783 


14^-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 


148-96300 72928 


918 


348 018 


140 


(148) 108 891 


181 


149-03699 


27072 


344 


149-39730 17747 


249 


632 S73 


141 


(149) 400 588 


267 


150-60269 


82253 


345 


149-83159 62566 


678 


572 502 


141- 


(149) 147 368 


188 


150-16840 


37434 


346 


150-26589 07385 


184 


455 130 


142 


(150) 542 137 


266 


151 -73410 


92615 


347 


150-70018 52204 


SOI 


401 028 


142 


(150) 199 441 


155 


151-29981 


47796 


348 


151-13447 97023 


136 


294 931 


•43 


(15O 733 703 


005 


152-86552 


02977 


349 


151-56877 41842 


370 


488 033 


143 


(151) 269 914 


251 


152-43122 


5^15^5 


350 


152-00306 86661 


100 


709 089 


144J 


(152) 992 959 


040 


153-99693 


13339 



The numbers in Sfiiiare brackets denote the numbers of figures between the last figure given 
and the decimal point ; and the nunil.ers in parentlieses denote the numbers of ciphers between 
the decimal point and the first significant figure. 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 269 



TABLE IV. (continued). 
Values of c^ e-^ log,„(e^), log„(e-^) from i to 500 at intervals of unity. 



X 


log.(e^) 


e" 


e" 


log.o(e-^) 


351 


i52"43736 31480 


273 755 686 


144 


(152) 365 289 217 


153-56263 68520 


352 


152-87165 76299 


744 145 106 


.144 


(152) 134 382 393 


153-12834 23701 


353 


i53'3o595 21118 


302 279 612 


.145. 


(153) 494 365 196 


154-69404 78882 


354 


15374024 6593S 


549 852 994 


.145. 


(153) 181 866 792 


154-25975 34062 


355 


154-17454 10757 


149 465 540 


146 


(154) 669 050 538 


155-82545 89243 


356 


154-60883 55576 


406 289 462 


■146' 


(154) 246 129 938 


155-39116 44424 


357 


i5S'043i3 0039s 


no 440 926 


,147 


(155) 905 461 441 


156-95686 99605 


358 


i55'47742 45214 


300 209 562 


147. 


(155) 333 100 649 


156-52257 54786 


359 


I55'9ii7i 90033 


816 054 198 


147 


(155) 122 540 8S1 


156-08828 09967 


360 


156-34601 34852 


221 826 530 


148' 


(156) 450 802 707 


157-65398 65148 


361 


156-78030 79671 


602 987 025 


■148 


(156) 165 841 048 


157-21969 20329 


362 


157-21460 24490 


163 908 S67 


149 


(157) 610 095 120 


158-78539 75510 


363 


157-64889 69309 


445 550 495 


149 


(157) 224 441 452 


158-35110 30691 


364 


158-08319 14128 


121 113 182 


150 


(158) 825 673 958 


159-91680 85872 


36s 


158-51748 58947 


329 219 761 


150. 


(158) 303 748 474 


159-48251 41053 


366 


158-95178 03766 


894 912 093 


150 


(158) HI 742 819 


15904821 96234 


367 


159-38607 48585 


243 263 328 


.151. 


(159) 411 078 858 


160-61392 51415 


368 


159-82036 93404 


661 255 566 


.151 


(159) 151 227 461 


160-17963 06596 


369 


160-25466 38223 


179 747 899 


.152. 


(160) 556 334 737 


161-74533 61777 


370 


160-68895 83042 


488 605 447 


.152. 


(160) 204 664 112 


161-31104 16958 


371 


161-12325 27861 


132 816 731 


,153. 


(161) 752 917 192 


162-87674 72139 


372 


161-55754 72680 


361 033 306 


^53. 


(161) 276 982 756 


162-44245 27320 


373 


161-99184 17499 


981 390 275 


['53. 


(161) 101 896 262 


162-00815 82501 


374 


162-42613 62318 


266 769 535 


,154. 


(162) 374 855 397 


163-57386 37682 


375 


162-86043 07137 


725 154 779 


154 


(162) 137 901 594 


1^-13956 92863 


376 


163-29472 51956 


197 117 506 


.155. 


(163) 507 311 614 


164-70527 48044 


377 


163-72901 96775 


535 820 935 


^55. 


(163) 186 629 513 


164-27098 03225 


378 


164-16331 41594 


145 651 231 


156 


(164) 686 571 609 


165-83668 58406 


379 


164-59760 86413 


395 921 094 


:i56. 


(164) 253 575 580 


165-40239 13587 


380 


165-03190 31232 


107 622 512 


157 


(165) 929 173 632 


166-96809 68768 


381 


165-46619 76051 


292 548 318 


.157. 


(165) 341 823 876 


166-53380 23949 


382 


165-90049 20870 


795 228 776 


157 


(165) 125 749 977 


166-09950 79130 


383 


166-33478 65689 


216 165 593 


'158 


(166) 462 608 311 


167-66521 34311 


384 


166-76908 10508 


587 599 004 


>s8: 


(166) 170 184 087 


167-23091 89492 


385 


167-20337 55328 


159 725 970 


159 


(167) 626 072 268 


I6S-79662 44672 


386 


167-63767 00147 


434 180 200 


159. 


(167) 230 319 116 


168-36232 99853 


387 . 


16807196 44966 


118 022 415 


160 


(168) 847 296 678 


169-92803 55034 


388 


168-50625 89785 


320 818 186 


i6o 


(168) 311 703 028 


169-49374 IO2I5 


389 


168-94055 34604 


872 074 245 


160" 


(168) 114 669 136 


169-05944 65396 


39° 


169-37484 79423 


237 054 357 


161" 


(169) 421 844 176 


17062515 20577 


391 


16980914 24242 


644 380 552 


i6i' 


(169) 155 187 800 


170-19085 75758 


392 


170-24343 69061 


175 160 794 


162' 


(170) 570 904 oil 


i7Jt-75656 30939 


393 


170-67773 13880 


476 136 404 


162" 


(170) 210 023 848 


171-32226 86120 


394 


171-11202 58699 


129 427 294 


163- 


(171) 772 634 560 


172-88797 41301 


395 


171-54632 03518 


351 819 860 


163' 


(171) 284 236 370 


172-45367 96482 


396 


171-98061 48337 


956 345 533 


163- 


(171) 104 564 717 


172-01938 51663 


397 


172-41490 93156 


259 961 668 


164 


(172) 384 672 096 


173-58509 06844 


398 


172-S4920 37975 


706 649 079 


164 


(172) 141 512 956 


173-15079 62025 


399 


173-28349 82794 


192 087 13s 


l6:^' 


(173) 520 597 071 


174-71650 17206 


400 


173-71779 27613 


522 146 969 


^6l] 


(173) 191 516 960 


174-28220 72387 



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. 



•270 :hr glaishee, tables of the exponential function. 

TABLE IV. (continued). 
Values of e^, e~', logi^(e'), log,(,(c'"') from i to 500 at intervals of unity. 



X 


log,„(e^) 


e' 


e~- 


c 


logic {e 


-') 


401 


174-15208 72432 


141 


934 


262 


i66' 


(174) 704 


551 5" 


i75'8479i 


27568 


402 


17408638 17251 


385 


817 


325 


i66' 


(174) 259 


190 020 


i75'4i36i 


82749 


403 


i75"02o67 62070 104 


876 


022 


167 


(175) 953 


506 796 


176-97932 


37930 


. 404 


I75"4S497 06889 285 


082 


586 


167 


(175) 350 


775 547 


176-54502 


93111 


405 


175-88926 51708 774 


934 


812 


167- 


(175) 129 


043 112 


176-11073 


48292 


406 


176-32355 96527 21° 


649 


122 


1 68' 


(176) 474 


723 081 


177-67644 


03473 


407 


176-75785 41346 572 


603 


680 


168; 


(176) 


174 


640 862 


177-24214 


58654- 


40S 


177-19214 86165 


15s 


649 


818 


169 


(177) 


642 


467 826 


178-80785 


13835 


409 


177-62644 309S4 


423 


100 


071 


169 


(177) 


236 


350 70s 


178-37355 


69016 


410 


178-06073 75S03 


115 


010 


524 


170' 


(178) 


869 485 652 


179-93926 


24197 


411 


178-49503 20622 


312 


631 


016 


i7o_ 


(178) 


319 


865 896 


179-50496 79378 


412 


178-93932 65441 


849 


819 


210 


170 


(17S) 


117 


672 087 


179-07067 


34559 


413 


179-36362 10260 


231 


004 


812 


171. 


(179) 


432 


891 416 


180-63637 


89740 


414 


17979791 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 


00102 


416 


180-66650 44718 


463 985 567 


172 


(180) 


215 


523 945 


i«i-33349 


552S2 


417 


181-10079 89537 


126 


124 


354 


173 


(i8i) 


792 


868 285 


182-89920 


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 


i^3'5963i 


76006 


421 


182-83797 68813 


688 


615 


639 


174 


(182) 


145 


218 892 


183-16202 


31187 


422 


183-27227 13632 


187 


185 


138 


.175 


(1S3) 


534 


230 448 


184-72772 


S6368 


423 


183-70656 58451 


508 


821 


958 


.175. 


(1S3) 


196 


532 399 


184-29343 


41549 


424 


184-14086 03270 


138 


312 


148 


176 


(184 


723 


OC2 290 


185-85913 


96730 


425 


184-57515 480S9 


375 


971 


399 


:i76- 


(184 


26s 


977 679 


185-42484 


51911 


426 


185-00944 92908 


102 


199 


622 


JT!. 


(1S5) 


978 


477 197 


186-99055 


07092 


427 


185-44374 37727 


277 


807 


376 


177 


(•85) 


359 


961 645 


180-55625 


62273 


428 


185-87803 82546 


755 


158 


743 


17V 


(185 


132 


422 489 


186-12196 


17454 


429 


186-31233 27365 


205 


273 


429 


■178' 


(166 


487 


155 lii 


I87-68766 


72635 


430 


186-74662 72184 


557 


991 


031 


176 


(,86^ 


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 


188-48380 51460 


304 


652 


780 


'180' 


(188 


328 


242 532 


TS9-51619 


48540 


435 


188-91809 96279 


828 


132 


"7 


■i8o' 


(188 


120 


753 679 


189-08190 


03721 


436 


189-35239 41098 


225 


109 


648 


iSi' 


(189 


444 


227 960 


190-64760 


58902 


437 


189-78668 85917 


611 


911 


467 


"iSi' 


(.89 


) 163 


422 334 


■J90-21331 


140S3 


438 


190-22098 30736 


166 


334 


782 


■182' 


(190 


) 601 


197 168 


191-77901 


69264 


439 


190-65527 75555 


452 


144 


816 


'162' 


(19b 


221 


168 078 


191-34472 


24445 


440 


191-08957 20374 


122 


905 


704 


>83' 


(191 


) 813 


631 891 


192-91042 


79626 


441 


191-52386 65193 


334 


092 


341 


183 


(191 


) 299 


318 445 


192-47613 


34807 


442 


191-95816 10012 


908 


157 


139 


183 


(191 


110 


113 102 


192-04183 89988 


443 


192-39245 54831 


246 


862 


705 


184' 


(192 


) 405 


083 466 


193-60754 


45169 


444 


192-82674 99650 


671 


042 


405 


184' 


(192 


) M9 


021 879 


193-17325 


00350 


445 


193-26104 44469 


182 


408 


237 


>85: 


(193 


) 548 


220 856 


194-73895 


55S3I 


446 


i93'69533 89289 


495 


836 


997 


i8s. 


t (193 


) 201 


6-/9 182 


194-30466 


10711 


447 


194-12963 34108 


134 


782 


470 


186 


(194 


) 741 


936 248 


195-87036 65892 


448 


i94'56392 78927 


366 


376 


739 


186' 


(194 


) 272 


943 092 


195-43607 


21073 


449 


194-99822 23746 


995 


915 


232 


1 86' 


(194 


1 100 


411 052 


195-00177 


76254 


450 


195-43251 68565 


270 


717 


828 


•187' 


(195 


) 369 


388 307 


196-56748 


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 firet significant figure. 



Mr GLAISHER, TABLES OF THE EXPONENTIAL FUNCTION. 



271 



TABLE IV. (continued). 
Values of e"", e"^ logi„(e''), log,o(e"'^) from i to 500 at intervals of unity. 



X 



logi„(e^) 



451 
452 
453 
454 
455 
456 
457 
458 

459 
460 

461 
462 
463 
464 
465 
466 
467 
468 
469 
470 

471 

472 

473 
474 

475 
476 

477 
478 

479 
480 
481 
482 

483 
484 

485 
486 
487 
488 
489 

49° 
491 
492 
493 

494 

495 
496 

497 
498 

499 
500 



195-86681 
I96-30IIO 
19673540 
I97'i6969 
197-60398 
198-03828 
198-47257 
198-90687 
199-34116 
19977546 
200-20975 
200-64405 
201-07834 
201-51263 
201-94693 
202-38122 
202-81552 
203-24981 
203-68411 
204-11840 
204-55270 
204-98699 
205-42128 

2o5'85558 
206-28987 
206-72417 
207-15846 
207-59276 
208-02705 
208-46135 
208-89564 
209-32994 
209-76423 
210-19852 
210-63282 
21 1-06711 
211-50141 
211-93570 
212-37C00 
212-80429 
213-23859 
213-67288 
214-10717 

2i4'S4i47 
214-97576 
215-41006 

2i5'84435 
216-27865 
216-71294 
217-14724 



13384 
5S-03 
03022 
47841 
92660 

37479 
82298 
27117 
71936 

16755 
61574 
06393 
51212 
96031 
40850 
85669 
30488 

75307 
20126 

64945 
09764 
54583 
99402 
44221 
89040 

33859 
78679 

23498 
68317 
13136 

57955 
02774 

47593 
92412 

37231 
82050 
26869 
71688 
16507 
61326 
06145 
50964 

95783 
40602 
85^21 
30240 

75059 
19878 

64697 
09516 



logio(e"1 



735 8S7 352 


[187] 


(195) 


200 034 922 


18a 


(196) 


543 751 292 


'188' 


(196) 


147 806 926 


'189' 


{197) 


401 780 880 


'189 


(197) 


109 215 367 


190 


(198) 


296 878 146 


190' 


{198) 


806 998 471 


190 


(198) 


219 364 928 


191' 


(199) 


596 295 697 


-I91. 


(199) 


162 089 976 


192 


(200) 


440 606 236 


;i92^ 


(200) 


119 769 192 


.193 


(201) 


325 566 419 


.193 


(201) 


884 981 282 


193 


(201) 


240 562 854 


194 


(202) 


653 917 634 


194 


(202) 


177 753 242 


.195 


(203) 


483 183 408 


.'95. 


(203) 


131 342 868 


196 


(204) 


357 026 931 


196" 


(204) 


970 499 818 


196 


(204) 


263 809 202 


;i97' 


(205) 


717 107 760 


197 


(205) 


194 930 099 


198' 


(206) 


529 874 947 


"198 


(206) 


144 034 944 


;i99; 


(207) 


391 527 571 


199 


(207) 


106 428 228 


200 


(208) 


289 301 919 


200 


(208) 


786 404 148 


200 


(208) 


213 766 811 


201 


(209) 


581 078 436 


201' 


(209) 


157 953 496 


"202 


(210) 


429 362 117 


203 


(210) 


116 712 724 


[203; 


(211) 


317 258 077 


203 


(211) 


862 396 865 


.203' 


(211) 


234 423 773 


204 


(212) 


637 229 881 


204 


(212) 


173 217 041 


.205 


(213) 


470 852 734 


.205 


(213) 


127 991 043 


206 


(214) 


347 915 727 


206 


(214) 


945 732 997 


206 


(214) 


257 076 882 


>o7' 


(215) 


698 807 417 


207 


(215) 


189 955 550 


208' 


(216) 


■516 352 721 


'208' 


(216) 


140 359 222 


209 


(217) 



135 

499 
183 
676 
248 

915 
336 
123 

455 
167 
616 
226 
834 
307 
1 12 

415 
152 
562 
206 
761 
280 
103 
379 
139 
513 
188 

694 

255 

939 

345 
127 

467 
172 

633 
232 
856 
315 

115 
426 

156 

577 

2 12 

781 
287 

105 
388 

143 

526 

193 

712 



890 364 
912 711 
907 609 
558 284 
S91 883 
622 070 

838 535 
915 972 
861 386 
702 032 
941 298 
960 020 
939 253 
156 986 
996 740 
691 777 
924 4S9 
577 643 
960 749 
366 047 
090 916 
039 690 
061 834 

449 056 
004 407 

723 775 
275 967 
409 855 
600 347 
659 650 
161 079 

799 467 
093 807 

097 734 
903 640 
804 611 
200 801 

955 895 
577 897 

929 239 
310 406 
380 629 
304 673 
425 926 
738 089 
988 692 
100 942 

438 947 
666 066 

457 641 



196-13318 86616 



197-69889 
7^-26459 
198-83030 
198-39601 
199-96171 
799-52742 
199 -09312 
200-65883 
200-22453 
201-79024 

2oi'35594 
20292165 
202-48736 
202-05306 
253-61877 
203-18447 
204-75018 
204-31588 
205-88159 
205-44729 
205-01300 
206-57871 
206-14441 
207-71012 
207-27582 
208-84153 
208-40723 
209-97294 
209-53864 
209-10435 
210-67005 
2T0-23576 
211-80147 
21T-36717 
212-93288 
212-49858 
212-06429 
213-62999 

2i3'i957° 
214-76140 
2T4-32711 
215-89282 

2^-45852 
215 -02423 
216-58993 
216-15564 

21772134 
2^7-28705 
218-85275 



41797 
96978 

52159 
07340 
62521 
17702 
72883 
28064 

83245 
38426 
93607 
48788 
03969 
59150 
14331 
69512 
24693 
79874 

35055 
90236 

45417 
00598 

55779 
10960 

66141 

21321 

76502 
31683 
86864 
42045 
97226 

52407 
07588 
62769 
17950 
73131 
28312 

83493 
38674 
93855 
49036 
04217 
59398 

14579 
69760 
24941 
80122 

35303 
90484 



The numbers in square brackets denote the numbers of figures between the hist 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. 2i4- 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 pcrhaj^s convey the impression that no erroi's 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 tiio discrepancy amounts to at least 3 in the last 
tigure) were found in Vega's Tabiihv lojarithmico-trir/oiiometricw : 

X = O-iG, logj^e' is given as 0-199175.) instead of 0-lf)977o.5 

„ 1-27, e' „ 3-oG0S(J0 „ 3\iC0S5r> 

„ 1-46, „ „ 4-3059.50 „ 4-305960 

„ 1-71, » ,; 5-0289G4 „ 5-52S9G1 

„ 2-30, „ „ 9-974185 „" !)-974182 

„ O-30, „ „ 200-3371 „ 200-33G8 

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 M. J. M. Hill, M.A. 

[Read JanxMry 29, 1883.] 
The objects of tlii.s paper are 

(I) To develope a series of functions *, here called Normal Functions, of [i — 1) 
variables analogous to Tesseral Harmonics, and 

(II) To show how to expand any function of [i— 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 formnlae of transformation 

a-j = r sin ^j sin ^2 sin d^_^ sin Q^^ 

«„ = r sin ^, sin 0„ sin Q, „ cos B, , 



.r ^^ = r sin O^nmB^ sin ^__,„ cos d._ 

x._^ — r sin 6^ sin fl„ cos Q^ 
'"*",_, = '' sin ^1 cos 6„ 
X = r cos Q, 



the equation -, — 5 + -,— s + + -r^. = is transformed into the equation 

ax, dx' die' 



fd^ i--l du\ 1 

Ur' "*" r dr) '^ ? 



d'u ,. „, , - du 

d07 + ^'-">"'^^^d0. 



1 (d^u ,. „, ^ „ du 

+ . ■ 2a iTST + (« - 8 cot 0., -jjj 

r sm 0, ld0J ■ d0„ 



+ 



r'%m^e,^m^0„ sin^ 



+ (,_,. _1) cot ^„^ 



+ 



+ • 



r sm 



0„ 



I ^u , n du 



sin' 



d0, 



+ 



sm 



+ 






* 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 

Vol. XIII. Part III. 36 



•-'74 Mk hill, on functions OF MORE THAN TWO VARIABLES 

•2. A solution u = /-^ (sin ^,)'''0. (sin ^,„)''"'0,„ (sin 6>,./'-= 0,.^ (^''^ 1\., 0,.) i=^ 



sin" 



tVnnul. where 0„, is a function of 6^^^ only, satisfying the equation 



(f© f/0 

^'"' ^". ^W" + ^-P'« + ' - '« - 1) sin ^„. cos 0„. ^+ (/)„.., -^)„.) (p,„.j +p„ + t - m - 1) sin=^„. 0,„ = 0. 

The indices p, p^, 1\, p^.^ are all integers sueh that no one is greater than any 

one preceding it in the series. 

The function (sin ^, )'"0, (sin 0„f°' ©.„ (sin d,_f'~%l, ^^ p.„_ 0..] will be re- 



ferred to in this paper as a Normal Function of the equation -r „ + "i-^ + + t ^ = 0- 

' «.;■," dx„ (Lv. 

II 
.'}. Putting cos 0^ = fi, i — m — l= n, and denoting the expansion of (1 — 2fih + li') i 

in powers of h by 

Qo+QJ^+ +Qp.-X"'' + ■ 

it is shown that I j- 1 (3;,,„_, is a possible value of 0^. 

4. If n be an even integer it is shown that 

1 fd\V{{i'"'-+l 



'■^I'm-i („ _ 2) (w- 4) 4.2 \dfj,J n 

Pm-l + 5 

If ?! be an odd integer, and P^ the r"" Legendre's coefficient, 

^''■"-' " (n - 2) (n - 4) S.lKdJJ ' ^0>,„-.+"y)' 

.3. If 0„ = 0,; when /\,., =;/„.,, but = ©„" when iV^ =7/',,,., ; and if ©,„', 



d0,., 



©„", .^"' are all Hnite when ^„=0 or tt ; then 



I (sin^„.r^'"'+"0„'0„"(Z^„, = O. 
.'0 



0. It is .shown that 



/ (sin O^'""-(0j'd^^= :" ±|^.i±|a_ . ^I?-'- . ["(sin 0^ {Q„„..yd0„^. 



ANALOGOUS TO TESSERAL HAEMONICS. 275 

7. It is shown that 



J u 






where F is the sj'mbol of the second Euleriao integral. 

8. If «,=(sia^/'©. (sia^^r-0^ (sin ^^./^'^ 6.., Q°^V.-. ^..) ; 

«; = (sin e/' ©; (sin e,f- ©-,„ (sin e,y'-^@\^_ (^°^^pv. &.-) ; 

and dS = (sin ^J--= (sin 0^)''' (sin ^, J rZ^^ dd^ dd^d0^_^ , 

ju^n;dS=0 (A), 



then 



unless ?<, = »/ identically, the linaits of integration for d._^ being and Stt, but and tt for 
the other variables. 

9. It is shown that 
j".'^'S = trr^ • ,,._,,.. I.;-.. • ' !„-. multiplied by 



P^ |j^,-4-p,-s |j>. -;^2 |2^-jo. 



1 1 1 ^a 



TT^ 



P.-3 + ^ /.+^ ^^^•2-r(ji-i).^.,^ (2^ 

unless 2^-2 vanish, when this value must be doubled. 

10. It is shown that * 

'-'(sin ^f ®. (siu^J-'"" e,„ (sin^^J"-©.^ , C^'p , \ 

is a rational integral homogeneous function of the variables x^x,, cc. of degree w. 

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 
^,*'.. ^i of the same degi-ee which satisfy the equation 

dxy dxy ^ dx^~^- 

Part II. 1. Preliminary propositions. 

i 

(a) If B^ be the coefficient of h" in the expansion of (1 - 2A cos + /r)"-^^\ then 
R^ is greatest when cos 6=1, i being a positive integer not less than 3. 

36—2 



•27G Mi; iiu.L. o:j^ functions of moke than two variables 

(/3) If /( <1, aud i a positive integer not less tlian 2, 

^'^(l-2hcosd + hy •'» 
•2. lu general any arbitrary function of the variables 0^', 6,^ ... 6\_^, 

F^e:e.:...e\_,) = ^^fi^j'^J ^-/j£^m^.- ^^.) iide,dd,...dd,_.M-M-. -(B), 

where It = fs'(2/) + ? - 2) i?^,] (sin 6',)'-=(,sin e.y-' («in 9, J' (sin 61,.,), 

p = 

„ -'■ +1 
and where B^ is tlie coefficient of h'' in the expansion of (1 — 2/i cos 6 + Ir) ^ , and 

cos 6 = -J—* -, —'- . 

rr 

The relation between the variables x^x,^...x\ and r'6^ ...d\_^ is the same as that 
between x^oc^...x. and r9^...6^_^. 6'-_^ lies between and 'lir; 6^ ...6\_.. between and tt. 

3. In general any arbitrary function of the variables 0^6,^ ... 6\^ 
F {9^6.;... 6'. J ^ ' ^ 

= S (sin ^,')".©; ... (sin 9JP"'®,,; ... (sin 6'';..,)''-= 0',_, {C cos ;),_., 9'._, + D smp,_J\_,], 

where C = 

r...rr {sin ^/'--e.... (sin ^„.)''-™©„.... (sin 9,,,r>-^^®,_,cosp,J^,F{e,9,...d,_,)dd,d9,...d9,., 
Jo JoJ 



}C. 



r... f T "(sin 0,)=''.«-=e,= . . . (sin 9X-*'-"'' ®J ■ ■ ■ («'» ^. ■f"-'"'®\-^ (cos ?)._, 9^_f dd,d9^ . . . d9^_^ 
Jo .'o-'o 

and B = 

j\..n\in0y^*'--€i^...ism9J''''''---'@„...(sm9^J''>^^'@,^^^^^^ 

Jo JoJo 

r...r ['"(sin 0f^-*'-^@^' ... (sin ^^J^"-*'-"'-' 0„,^ . . (sin 6'._ J^«-«-^' ©V, (sin p,_., 9^yd9, d9,... d9,_^ 
Jo Jo Jo 

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 ^j with which that term 
is connected nor any index greater than any one which precedes it. The limits between 
which 6^9,^... 9\_^ lie are the same as in the last article. 



ANALOGOUS TO TESSERAL HARMONICS. 277 



PART I. 

1. The formulae of transformation for x^x^...x^ given in the abstract are equivalent 
to the following : — 



0. = tan 



+ ...+^.J)- 



9„, = tan 






}■ 



"— ■f^}. 



, = tan-' P 



p du _ da dr da dd^ da dd._^ 

dx^ dr ' dx^ dd^ dx^ ' ' ' dO__^ dx^ ' 

. d'^a _ dr fd'hi dr d'a dO^ d'u dd\ du d^r 

dx^ dx^\dr' dx^ drdO^ dx^ '" drdO._^ dx^J dr' dx^ 

dd, I d-u dr^ d'a dO^ d'a dO.A da d^d^ 

"^ dx^ [drdf, dx^ "^ d£; dx^ +' ' ■"*" de^dJ^^ d^/ "*" 5^ ' ^ 



+ . 



d0._^/ dhi dr^ dhi dd, dru^ dd^\ _da^ d%_^ 

"*" dx^ [drd0,_^ dx^ "^ d0^_^d0, dx^'^'"'^ de,_^^ dxj^ d0~, Ih^ ' 



m, r du d'a d'a 

1 hereiore ,— „ + -5-^, + . . . + -^-^ 

dx, dx' dxr 



' dr' \\dxj +■ • ■ + [d^J j + 5^ VfT^'' +• • ■+ dx^ 



^ d'u /d0, dr ^ ^\, (JV /d0,_, dr (W^ dj-^ 

drd0^\dx^ dx^ '" dx. dx.) '" drd0._^\dx^ dx^ '" dx. dx^/ 

d0md0„\dx^ dx^ '" dx. dx.) 

^d0^ \dxy-^ dx,') +•••+ d0,_,[dx;' +■■•+ dx^ J ■ 



•J 7 8 



Mu HILL, ON FUNCTIONS OF MORE THAN TWO VARIABLES 



Aiul iVoin tlio lonmilae of trausformation it may be sliowu that 

rdrV . rdr 



©-■H 



= 1, 



dy _ i-i 

da'''^"-^dx'~ r 



d°r 

dx. da; '" da-, dd\ ' 



dds' , , fde,y ^ 1 



/d8 
I ' 
\dx 

dx^ dx^ '" dx, dx, ' 



1 I 






Therefore 



d^u d'u d^u 

dx^ dx.^ '" dx^ 



dSi i—\ du\ 1 {d 



dr) ' r'ldt). 



+ -A7f,-^. + {^-^)oote^ 



dit\ 

dej 



+ 



+ 



1 [d'^u ,. .,, . „ dn\ 



da 



r''sin'0,...sm''0.„ , c?0. - ^ ' •" dO , 



111-1 V-*""! 



i-3 ^ (-2 



d'a 



d'u . '^^ 1 r 

7-^siii-'^.....sm^^.. , Vrf^'~ ■*" ^° '-^ rf^J r^^in'-' ^, . . . sin'-' 6, „ \dd. 



(-2 N""!-! 



J = o 



.(I). 



i. Put in the equation (I) zi = r^.u^, where ?(, is a function of 6^9,^... 6^^^ only. 
After dividing out by r''^; it becomes 






do:- 



sin'^....sin-'^„.,V/^„. 






'rf^. 



■ sin'^,. . .sin^^,.3 Vrf^,./ + '^°*^'-^ dej + sin'^^,. . .sin''^,.,, U^,?/ . 



.(II), 



Now put in the equation (II) m, = (sin^,)''' . 0, . i/^, Avhere •©, is a function of 6^ only, 
and Uj of 6.fi^...6^_^ only. After dividing out l)y (sin^,)'''"-0,, it becomes 



((sin0.)P.^,'+^2;,, + i-2)(sin^,)''.->cos^//^' + (7)-;).)(;;+i>. + i-2)(sin^,)".®. | 

«j ^^^i r--^^ — ^ ^r^(l\+^-^ 

= 1 (sint^.r-'H, ) 



ANALOGOUS TO TESSERAL HARMONICS. 27!i 



sin 



(I'll , , ^ du„ ' 



1 / d-ii^ du, \ 1 (dru^ \ _ 



If 6, be chosen so that the coefficient of u.^ is ^\(Pi + « — 3), then ©^ must satisfy 
the equation 

siii'^.7^+(2A + *'-2)sin^,cos^,^ + Q;-^;JQj+^,, + t-2)sin'^,.0, = O (1), 

and Mj satisfies 



J 



, 1 (d'u., diK\ , 1 (dhi.^\ ^ 

^ sin^0,... sin=^,.3 W,_; "*" "^"'"'-^ f/^,.,^ + sin-=03...sin^d;,., VdT.V " " 

(HI). 

The equation (III) is strictly similar to (II). 

If therefore in it u^ be put = {sin6^''-^M„.v^, where ©„ is a function of 6^ only 
and u^ is a function of 6^...6-_^ only, then proceeding as before but employing the quantity 
p.^(p^-\-i — ^) instead of p, (^^, + 1 — 3), ©^ and u^ satisfy the equations 

fP© f/0 

sin' S, -T0 + (2i5, + i - 3) sin 6^ cos ^, = + (i^, - p.) (p, + f, + i - 3) sin^ ^, . ©, = 0. . . (2), 



diL 1 1 
cW 



^ J 3/ 3 Tfl—1 V 7/1 

...sin^^. , W / "^ ''°'' ^-^ fT^J + sin^^, sin^^. , (fTi9.7V " *^ J 



!--(iV). 



sin^ 



Proceeding in this way the solution of the transformed ecjuation will be 
u = r" (sin ef ©, (sin 6 J'"' ©„. (sin 0^_J"" ©,_, . ?r., ; 

where the quantities ©g...©;,, satisfy the equations (.3) ... (j — 2), similar to the equations 
(1) and (2) 

r?''R d<F> 

sm'e,'^ + {2p,+ i-4)sm0,cose,^^^+(p,-p,)(p._+p, + i-i)s,m%.B, = O (:}). 



r/-© d& 

sin^0„^r + (2p,„+{-m-l)sin0„,cos^,,,^°' + (p„,_.-;;J(p,„_, + ;7„,+z-m-l)sin^^„..@„=O (w). 



sin^ 0,_, 1^%' + (2p,., + 1) sin 0^_, cos^,., y=? + (^;,_3 - p,.,] {p,_, +p,_, + 1) sin^^,., . ©,., = ...(*- 2). 



•J so Mr hill, on FUNCTIONS OF MORE THAN TWO VARIABLES 

The qviantities similar to u^, »„, ti.^, viz. ?(.,...?', ",-o ",., will satisfy equations similar 

to the equations II., III., IV. the last two of wliicli will be 

((T'u,, , ^ (hi.\ 1 d-u,„ ,, 

V.-. (;',-3 + 1) ".= + {^e:} + ^°^ ^- dO + sin^ ^,:, ^ = «' 

Hence u = / (sin 0/' 0, (sin O^" 0„ (sin ^,../'-^ ©,_, {^^p,_, ^,.,) . 



/cos 



Any function of the form (sin ef 0, (sin 6 J"' 0„, (sin 6, J '" ©,., f .^^ /?,., 0,.^ 

will lio referred to in this paper as a Normal Function of the equation 

d^n d'u d'-ii 

or more briefly as a Normal Function. 

3. Certain properties of the functions will now be proved. The equations (1), 

(2) («'— 2) which they satisfy are all of the same form, and may be typified by the 

equation (m). 

Putting cos 6,„ = fj, and i — m — I = n for brevity, this becomes 

It is now possible to obtain further information of the nature of 0,„. 

It will be shown that if 
1 



-„= '?o+ OJ' + QJ''+ + ^y.»,-. ''"'"- + 



/d\^ 
then (;,— I Qpm-\ satisfies the equation in 0„,. 

i.e. it is required to show that 

But this is (|-p^,l-;.=)^|^-(n + l)/x''^--'+p,^,(^,,„.. + ,OQ.^^ 

It will be sufficient therefore to sliow that the term in brackets vanishes. Multi- 
plying it by /i^"', it may be arranged thus: — 



ANALOGOUS TO TESSERAL HARMONICS. 281- 

This will vanish if 

('-"•) I-- ("+ ')4+ ("a("a))+''''s!(TTi;i;i9-°' 

because the expression which must be demonstrated to vanish is the term containing 

A " on the left-hand side of the last equation. It is necessary therefore to show 
that 



■ = 0, 



(1 - ;u') n (n + 2) h' (1 - 2fjiJi + h'f'-" - n (« + 1) hfi (1 - 2^11 + h-)'^'~ 
+ {n (1 - 2pih + h'p~' [h/j, - 2h') + n {n + 2) h' (/x - h)' (1 - 2fi7i + ]rp' 
+ i>Vt{/j.-h){l-2fih + h')~"'~' 

i.e. (1 - 2,xh+iep~"- {n {n + 2) (A' - /*>') + n{n + 2) (/(,y - 2A> + h')] ] 
+ (1 - 2^/i + /(,') - {- n {n + 1) hfj. + n {hfi - 2h') + n'h (fi - k}] J 

i.e. (1 - 2fih + h^)'"'""' 71 (n + 2) A' (1 - 2fih + A') + (1 - 2M + Iv'p'' (- 2nh' - nVt') = 0, 
which is identically true. 

4. The form of Qj,,„_^ must now be more closely examined. 
Firstly, let n be an even integer. 

Since log (1 - 2p,h + h'} = -2(h cos &„, + j cos 20„ + ^ cos 3^„, + . . . 



after differentiating both sides ^ times with regard to fi, and comparing the coefficients 
with the coefficients in the equation 

it appears that 



1 ,Jeos](,.„., + g)g„ 

Vi'-i („_2)(„-4) 4.2Vc' 



■(«-2)(«-4) 4.2Wm/ „ ,« 

Secondly, let ?i be an odd integer. 

Since ^- i = P„ + Pyt + +P,A'+ 

(1 - 2M + hy 

11 — \ . 
where P, is the r* Legendre's coefficient, after differentiating both sides --^ times 

Vol. XIII. Part III. 37 



282 Mr hill, ON FUNCTIONS OF MORE THAN TWO VARIABLES 

with regard to /i, aiul dividiug by k ' , this becomes 

{1-3.5 («-2)}. 1 -„=(J^)Mp...,+ +P r"-' + . 

Comparing with the series 

„-Q.+ Q.h + + Q.„.-.^"""+ 



n-l 

it appears that (^,„.. = ^„,_2)(«_4) 5.3.1 (1) ' ^(,„._„-) " 

111 both cases Qp„,_, is a rational integral function of /j, of the degree p,„_, con- 
taining only even or only odd powers of fj, ; and therefore ©,„ is also a rational integral 
function of /j, of the degree ■p,^^i—2K>- 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 iJ„,_i =y„,_,, 0„, = 0,,,' satisfies the equation [m) in article 
2; but if i\,_, = /j",„-i. 0m = ®i»" satisfies it; it is required to determine the value of 






Since the equations 
Bin'^„. -^ + (2i^„. + n) sin 6,^ cos 6^ ^^^ + {p\_,-p„) {p\„_, +p,^ + n) sin=^,„ . O,,.' = 0, 

sin'^„. '^ + {2p„, + n) sin ^,„ cos ^„. ^®"^' + (p"„^_,-p„) (i/'„,_, +;>„. + n) sin^^„. . O,,." = 0, 

hold good; it appears that, multiplying the first of them by (sin ^„_)2""'+»"2©J', and the 
second by (sin ^J^^"""*"""-©,,,' and subtracting, the result may be arranged as follows: 

J-[(sinO''^*"{0/'^f-0J-^Y|]+(iC.-/;,-O(/^ 

Integrating with regard to ^,„ between the limits and tt, 

[(sin^J^^™+"|0/^^'-©,:-®^']jV(/,„.,-i/;,.,)(;/,„..+;/'„^,+n)J^^^^^ 

, d€> ' „ cK") " . 

Now ©„', -v-j— , ©„,", , '"- are finite when 6^^ = or tt since ©,„'©„," are rational 

"I'm "fc'm 

integral functions of cos 9„ by the last article ; therefore the term in brackets is zero 
at both limits: ;/„_,- i>",„., is not zero, since ;/,^, and ;/'^, are different: 2''m-i+/'"^i + 
is not zero, since ;/,^, p"„,_, are positive integers or zero, and n is a positive integer 
the least value of whicli is L 



H 



Therefore the equation shows that f (sin ^J2i'"'+" ©„'©,„" (?^,„ = 0. 

Jo 



ANALOGOUS TO TESSEEAL HAEMONICS. 283 

6. The value of the integral / (sin dJ^-V'+n &JcW^,^ can now be determined. 

J 

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 n even, the second to n odd. 

It has been shown that 

siii'^. '^l-l + i^Pn, + n) sin ^„. cos ^„. ^' + (p_, -pj (p„., +p,„ + «>sin^^,„ . ©„ = 0. 

Multiplying by (sin ^„_)2;,„,+n--2 0^^ a^ml integrating between the hmits and tt, the result 
may be an-anged thus 

{P^, -P.n) (P„,-^ +P„. + «) f"(sin ej'P'r'+^^&^dd^ 

Jo 

= -\l {(sin 0'^'"+» 0,,.^" + {^Pr. + n) (sin ^J2p»+.-i ^os ^,„0„, ^»1 dO^ 

= - (sin O-^'"+"0„.y^"J^ + J^ (sin ej^P-^+n (^j dd,„; 

••■ (P,n-. - VJ (p.-. + v.. + ") ^(sinO=P"'+"®^„cZ^„ = f \sin ^J2i'.»+"+2 ('_J^» V ,10 

Jo Jo \~smd^ddj 

•■■ 0'.-, -i'.) (r.,-, + )'., + >o/%iii CJ"'-*" {( .si,*^ jo )"' %.-.}'■'». 

-/.'(™ «-'"-"*' {(^„ca«.)"" «'-F '*'- 

Put * (,J =/;(si„ <,.)*.« |(_j^ J-g,..,}' ,,.. 

Then the last equation may be written 

(P,n-i - PJ (i\.-. +;'« + ")* (pj = * (P,. + 1 )■ 
This is an equation which is true for all positive integral values of p„ less than j) ,_ . 

Hence changing ;>,„ successively into p„-l, p,„- 2, 2, 1 a series of equations is 

obtained, whence it can be shown that 

$(„)=l^±^^^.bti?^ J&^ 
^^"^ \^P....^ \P.-,-P.'^'' 



(sinO^-^"0JcZ^„=^=^J^^ . ^^ . /''(sin^J"(Q^.,yc^^„,. 

37—2 



284 Mr hill, ON FUNCTIONS OF MORE THAN TWO VARIABLES 

7. Let V be an even integer. 

Then since Q;,„,., satisfies the equation satisfied by 0,„ when 7),„ = 0, it aiipcars that 

■■■ p.-^ (p..r + «)//sin ej" [{-^J COS \{p^_,+ 1) eJ^'J dd,„ 

This is an equation which is true whatever positive number p^,^ may be and what- 
ever even integer n may be. Changing n into n — 2, and p,^,, into ^>,„., + 1, the equation 
becomes 



(;>«-. + 1) ip.-.+ « - i)//sin ej"-- [(£)''' cos |(^_.+ 1) e 

= f "(sin ^ J" 

Jo 



d0„ 



cos -1. 1 /?„,_, + 5 



dd.. 



Proceeding in this way there arise finally (by putting in the last equation n = i, 
and for jj„_i. iVi + 2~-' ^^^^ " = "■ '^^^^ ^°'' P-^-i' Pm-i + 2~^) *^^® *^° equations 
(p.. + 1 - l) (2^..+ 1 + l)/;sin^ ^„ [|^ cos |(^„..+ g ^„}]^ J^„ 

(P.-. + 1) (i^.. + l)/; [ cos {{p... + l) 0]]d0^ =/;sin' ^,„ [|^ cos {(,,,,.. + 1) ^^ 

/ , n\ \ p,.-i - 



rf^.. 



--' j^ cos ^{iJ,„_i + ^ 



(/^„ 



2' 



.C<"°''->-'g'-''"^-- |(-2)(.-4) «.2r • -^ • ^^^ ■ f^ 



.'. if n be an even integer • 

f » 'n 4- V + V — 1 

^0 \P^-,-Pn, 



n- {{n-2)(n-4) 4.2}^ ' 2' 

Z^m-i ' 2 



ANALOGOUS TO TESSERAL HARMONIC'S. 

But if n he an odd integer 
d V" 



285 






dd 



1 



jTTir/.'*"" *"'■[© ""^ 



| w + p„-i-l \p,n-i-Pm {(n-2)(w-4)... 

I w+jP^-i + P^-l la,^ 1 . . ,. , , 

— ' q — • 1 • f"7 i^TT n — „ -, la multiplied by 

In — l n — 1 

9 ri'»i-i H 2 



(p.,.-.+v) 



f^^„. 



w-1 



71-1 „ //i - 1 



+ F,n-l - — ,— 2 ( — ^ + p,„_, ) + 1 



by a kno^\^l property of Legendre's coefficients. 

.•. if 71 be an odd integer 

' n + p„-i + p,„-l 1 



(\smdJ-P"'+"&Jd9„ = - 
Jo 



\P,n-l-p,. 



Pn,-1 + 



11 {(n-2)(n-i) .5.3.1}''' 



Both forms are included in 



f %in e,„yp" 

Jo 



■+n@jde, 



m fn 



\Pm-l-P«, 

where T is the symbol of the second Eulerian integral 



n 2"-'ir(i")r 

I'm-l ^ 2 



8. SupiDose now that there are two solutions of the equation in u^ of Art. 2, \-iz. 



Mj, ^t^ where 



and 



«,= (sin ejP^e, (sin d,_^P'-^B,_, (shi^^-^-' 

u\= (sm9,) P.'©; (sin^,., )^'-©',.., (^^Jh-A.) 



and if dS be a differential element having the same relation to the variables 
x^ X., a\ as an element of area of a sphere has to the three variables a^j a'^ ■''. so 



that 



(r'-'drdS = Idx^ dx.^ dx. = {jdrd6^ 



de.. 



dd. 



where J is the Jacobian of j\x,^ x. with regard to rd^6.^. 



that 



dS = ^, Jde^dd, d0._^ = (sin ey-'dd^ (sin eX'de, (sin 0..,)^ d0^, (sin 6I,.J dd,,J0,_^ . 



Then 



|«,!«\fZS=0 



un 



less u, u, are identical ; the limits of inte- 



gration for 0,_, being and 2-n; and for the rest of the variables 
and TT. 



^ (A). 



236 Mr niLL, 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. 



/:i:°>.-.*-.}(::"W'"'- 



and this vanishes if p,_„}y\,^ be different positive integers; and if Pi.^p'i.^ be the same, 
tlie integral will still vanish unless the terms in brackets be both sines or both cosines. 

It is therefore only necessary to consider the integral when }\.^p'i_, are the same. 
Supposing this to be the case, but JhsP'i-s different; then by Art. o, 

f " (sin ^,J2p'-+i R.,eV,ci^,_= 0. 
Jo 

It is therefore only necessary to consider the integral in the cases in which p,_^Pi_^ 
are respectively equal to i^'j.jiVs- ^^ ^'^'^ ^^ ^^' ^^'^ i*.-4 different from p',._, ; then 
again by Art. 5, 

f" (sin ^J2p<-3+2 0.._3 0^3^^,-3 = 0- 
Jo 

And so on, it appears that the integral in (A) will vanish unless the quantities i',-2i^,-3'--i'i 
be respectively the same as J^Vzi^'i-s-'-Pi* '^'^^® corresponds to the conjugate property of 
Tesseral Harmonics. 

9. It is now required to determine the value of \u'' clE, i.e. 
Supposing Pj_j not to vanish, this is 



> /A-3 + ;',-» 1 -rr \ / \pi+P, + i 



P.^'^ ^-\r"-'-^' 



multiplied by /^-+^. + *'-^^ ' 



1/^. p^^ 2-»jr^^"= 



1^, + 7Vj _ \P^±Pi-±tl \h±h'^\~^_ . l^ + ^'±.^^~'^ . _ 1 



P,-»-P^, \P,-4-pi-:, \Pl-P, P-P, Pi-,+ h 



IT IT TT TT 



,y^ ■ „, '-^ ■ ir(4)l- -sira,,. -^-ir,,,!- 2-.{r(L;')}' ' ^'-\vl^^)\ 



ANALOGOUS TO TESSERAL HARMONICS. 287 



But 77 "^ 



{r{h)f 2{V{i)r 2Mr(i)r 2-^r(^'^)r 2-{rf^)^' 



1 1 



1 1 



r(^) H^)r(^j}{r(^)r(^)}...{r(f)r(i)}{r(i)r(i)jr(i) 



■ 2i+«s+...+i^-4Wi-) A-2\ (i_4 i-o i-6 1 ,-|fi-5 i-6 1 



^ C"2~) f"¥~'^-^ - 2^4f~^-V^ ■■■\'^''^\ - ^^>/''f '^/'^i ^^ 



Jl+2t3+...+(i-4) ^l; 



i-4 i- 3 ... 2 ,1 



2.««^...+(-4)^(.-3) ^ /,-_2s ^^ |i__5 . . . ^ |1 . {Jnry-^ 2'-3 r f'l" 

, . 1 ^ ^^^ ^ — — . ' multiplied by 

Ka-j'i-^ \ Pi-t-Pi-s \Px-P. \p-Pi 



1 i-3 i-2 2'^'r(it-l) .|i-41i-o 12 11 ' 

i^i-s + g P^+^r -?' + ~2~ — ^ — 

where F is the symbol of the second Eulerian Integral. If p,.^ = 0, this value must be 
doubled. 

10. To show that 

rP (sin 6'JP' 0, (sin e^)P^ ©., (sin O^Jp.-^ ©._^ ^^."^ j^-^ ^.-i) 

is a rational integral homogeneous function of x^x.^ x. of degree p. 

It is homogeneous and of degree p because ?•' is of degree ^j in these variables, and 
all the quantities sin 9, (s) are of degree zero. 

Moreover it was shown in Art. 4 that 0„ contains (cos ^„,)J'"'-i~*'" and lower powers 
of cos ^„ whose indices are positive integers differing from p^^.^—Pm t)y some even number 
which may be denoted by 2q,„, so that the general term in ©,,^ is a numerical multiple 
of (cos ^JJ'™-'-i'"-'-'^". 

In like manner the general term in cos J*,..^^,.! is a numerical multiple of 
(cosd.^yPi-^'-'ii-i, and that in sin^j^,^.,^ is a numerical multiple of 

(sin 6',,,) {cos e,_,y''-'--^'!i-i-\ 

It is sufiScient therefore to show that the quantities included in the form 
rP (sin d^)P^ (cos 0^)p-p^-2i>- (sin e^)P' (cos e^^-P^-^i, (sin d^^)P'-^ (cos 0^^)pi-*-pt-^-2i'-^ 

Itiplled by (sin e^„)"'-- (cos d._^)Pi-^-pi-^-^'ii-' (cos ^,..,)^'-°-'^""'"^f'^°''^,-, 



mu 
are rational integral functions oi x^x.^ x^. 



sin 



•288 5[r hill, on functions OF MORE THAN TWO VARIABLES 

Now j-P (cos ^Ji'-i'i-^'A = (r cos ^J^'-i'.-^'i. . j-i'i+i-/, = (,- cos ey'-i'^--i'- . i-^''' . r''' ; 
.-. r'' (sin ^,)P> (cos 0^y-i>i-^i^ = {,■ cos dy-i'^-^'n . I'-C' . (r sin ^j)*"' ; 
.-. J-'' (sin dJP' (cos ^j)P-i'i-23i (cos ^Jj'i-J'«-23. 

= (r cos ey-P'--'ii 7--'"- (?• sin ^, cos ^JJ'i-i'^-av. (,■ sin ^ Jft+2!?» 

= (r cos e^)'>-"^-^< r-'h {r sin 0, cos ^Ji'i-J'^-a-i, (^ sin ^J^,, (,. g;,) ^^);;, . 

.-. 7-''(sinS,)'''(cos5,)''-"'-25>(sin0J''«(cos^,)'"'-^'»-29» 

= (r cos ^,)''-i'i-2'ii }•-''■ (r sin ^, cos ^JJ'i-^'^-S'a (,• sin ^,)2'i» (c sin 9^ sin flj'''. 

Proceeding in this mauuer it appears that 

J* (sin ^,)P' (cos ^J" -Pi -2^1 (sin d^f^ (cos 0^)pi-p^-''Js 

(sin e^^y-' (cos ^^^)j'.-*-i'i-»-2'yi-3 (sin ^.JP'-= (cos (^^Jz-.-s-i'-^-Z'/'-^ 

= (r cos ^,)"--P. - 2i. r^-'h (,- sin ^, cos d^)Pi-r.-i'i. [r sin ^J^r/, 

(?• sin 0^ sin 0., sin 0,_^ cos ^^Ji'.-^-i'.-3-29.-s 

X (r sin ^, sin ^, sin 0^)^'-^ (r sin ^, sin 0, sin ^,_, cos 0._^)Pi-='Pi-^-2'u-"- 

multiplied by (r sin 0^ sin ^,, sin ^..g) -'J'-"- (/• sin ^, sin 0., sin ^,.3 sin 0,_y'--. 

Evcrj' term except tlie last is a rational integral function of the variables. It 
remains therefore to show that the expressions included in the form 

(r sin 0^ sin ^„ sin 0, , sin 6, ,)?•-= (cos 0^\m-^-"~'u-^-i I ^°^ 0. 

are rational integral functions of x^x„ a-^. Tiiis form may be an-angcd thus: 

(/• sin 0^ sin 0.^ sin ^^, cos g'.JP'-a-a?.-!-! (,. gin ^, sin ^^ sin 0^.)^'"-' ' 

(COS \ 

;• sin ^j sin ^2 sin 0._^ . 0,_, 1 

and every term in this is a rational integral function of x^x^ w^. 

The result is that 

rP(sia ^,)P" (cos ^,)p-P'-29. (sin ^j)P"(cos 0^p^-V2-^-12 

• {sm0^P>-^(cos0^_^P^-*-p>-^-^-'ii->{s\n0^y'-^{cos0^_.y'-'-P>-'-'^v-'.{^^^ 

= (x,)P-P.-2^. (^.^ + < + + x^t^ (ar^,)p.-p.--"73 {x^ + x^+ + x._l)i^ 

11. To determine tlie number of different functions included in the form 
tp (sin ^,)P. 6. (sin ^Jp« 0, (sin 0,_^v,-^ ©,., Ch,P^-^ ^'-) ' 

which can be obtained by giving all possible integral values to the exponents jh, i>, ;'._, 

wliifh 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 HARxMONICS. 289 

The possible values of p, are 

0, 1, 2, S,...p (a). 

When p, = 0, the only possible value of ^j, is ; 

/?, = 1, the possible values of p^ are 0, 1 ; 
\p, = 2, „ „ „ „ 0, 1, 2; 

Pi'^'^} J) » » 1) ^) ■') ^) •* > 

and so on. 

Thus the values of p.^ giving rise to different functions may be arranged thus : 

0; 0, 1; 0, 1, 2; 0, 1, 2, 3; ; 0, 1, 2, 3...p (/S). 

(It should be noticed that though 2^2 '^^7 have the same value twice over, it gives 
rise to different functions. Thus there is a function corresponding to p.^ = 0, ^', = ; whilst 
a distinct function corresponds to j)^ = 0, ??! = 1 ; and so on.) 

Again the values of p^ giving rise to diflferent functions may be arranged thus : 

0; 0,0,1; 0,0, 1, 0, 1,2; 0,0, 1,0, 1, 2,0, 1, 2, 3; ; 

0, 0, 1, 0, 1, 2, 0, 1, 2, 3, 0, 1, 2, S...P (7). 

The law of formation is seen to be that the numbers which stand in the n''^ place 
of the row (7) include all the numbers which stand in the first n places of the row (ff). 
The relation of the numbers in the row (/S) to the numbers in the row (a) is the same 
as the relation of the numbers in the row (7) to those in the row {/3). 

It is necessary therefore to find the number of the numbers standing in the row 
corresponding to p,_2, the number of zeros being counted separately, because if ^^.2=0, 
sin|)j_2^,._, = 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 
p, is 1, 



p, is 1+1 + 1+...+ 1 (p + 1 terms), 



p^ is 1 + 2 + 3+.. .+ {p + l) {p + 1 terms), 
and so on. 

The number of other figures in the row corresponding to 

/), is 1 + 1 + 1+...+ 1 (p terms), 
j9j is 1 + 2 + 3+...+^ (j3 terms), 

p, is l + 3 + 6+...+ ^^y^^ ip terms), 
and so on. 

Vol. XIII. Part 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+2 + 3+...; 

, „ third „ 1+3 + G+...; 

and so ou, the law of formation being the same as in the above. 

Therefore the number of zeros in the row corresponding to 2)._^ is the sum of the 
first (jJ+1) terms of the figurate numbers of order i — 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 z — 2. 

Therefore the number of zeros is the {p + 1)*"" figurate number of order i - 2, and 
the number of other figures is the p^^ figurate number of order t'-l. 

|?i + r-2 
But the J)"" figurate number of r"" order is ' — 



i-1 r-1 



Therefore the number of functions arising from the numbers in the row corresponding 
|p + t-3 |» + i — 3 
top,., is 1. =17^+ 2- ' 



|y h'-3 '\p_ 



1-1 I- 2 



It is required to show that this number is equal to the number of independent 
rational integral homogeneous functions of x^x^.-.x^ of degree p, which satisfy the equation 

d^u dhi (Tu _ 

The most general form of a rational integral homogeneous function of i variables of degree 

\p + i — l 
n contains ', , . constants. If it satisfy 

d^u d'u d'^ii _ 

d^^^d^^^-^dbo^~' 

then since ^ + -i-^o +.••+ ^-^ is a rational integral homogeneous function of degree 
ax, ax, ax. 



2 



\P + 1-^ ... 

n - 2, there must exist A— ^-, .- -,- linear relations amongst the arbitrary constants. In 

' 1;) — 2 u— 1 

I jj J- ^' 3 

virtue of these, ' cTT^-^ ^ of the constants may be expressed as linear functions of the 
\p — z U — 1 

remainder. Therefore the most general form of a rational integral homogeneous function 

... . dhi d^u d^ii . 

of i variables of degree p which satisfies the equation j^i + ^'2 +•■•+ 7~2 = ^ contains 

\p + i—l \p + t — 3 . , , • 1- 

'-'^ =-- arbitrary constants involved in a linear manner. 

\ p + i-3 \ p + i~S _ \ p + i~l \p+i-^ 

^"* \p |t-3 ^ ^ \p-l\i-2 = lp\i-l ~ \]^ \i^ • 



ANALOGOUS TO TESSERAL HARMONICS. 291 

Therefore the number of functious of degree p of the form considered in this article, 
each of which is known to satisfy the equation 

dJ^'^dx^'^-^dxf~^' 

is equal to the number of the independent rational integral homogeneous functions of 
x^x^...Xi of degree p which satisfy it. But the functions considered in this article are all 
different, and are all rational integral homogeneous functions of x^x^...x^ of degree j). 

Therefore the functions included in the form 

rP (sin e,y^ 0, (sin e,y^ 0,. . .(sin ^,.,)^.-. %,_, {^^^ p,., 0..,) 

are all the independent rational integral homogeneous functions of x^x,^...x. of degree p 
which satisfy the equation 

d'u d^w d'^xi _ 

d^''^d^:'^'"'^d^'~- 



PART II. 

The expansion of an arbitrary function of the variables 6^0^... O^^^ will now be considered. 
The proof of this expansion for any rational integral function of the quantities 

cos^j, sin^j cos^2> ^i^^^i ^i^^a cos^,, , sin^, sin^2...sin^i_2 cos^,._,, sin^, sin^2...sin^,_j sin^^.j, 

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. 

(a) To show that if Bj, be the coefficient of W in the expansion of 

1 

(1 - 2h COS0 + h'f'' 
then Bf is greatest when cos^=l, i being a positive integer not less than 3. 

Suppose that ^ ^zj- = Q,+ Q,h+ QJi' +...+ QJi" +..., 

{l-2hcose + h'')'' ' 

and j = P, + Ph + P^h'+...+ P,h'+.... 

{l-2hcose + hy 

38—2 



2D2 Mr hill, ON FUNCTIONS OF MORE THAN TWO VARIABLES 

1 1 1 



ThcD since 

{l-2h cosd + h'f {l-2hcose + k'y" {l-2hcosd + K') 



-"i"„ .. . . ,„J 



.-. {R, + EJi + liji' +. ..+ BJi' +...) 

Therefore R, = P,Q„ + P^,Q, + P,.,Q,+...+ P,Q,. 

Now it is known that each of the quantities P„, P,, P,y..Pj, is greatest when cos^ = 1 ; 
if therefore this property hold also for Q„, Q,, Q^...Qj, it will hold for i?^; i.e. if this property 

hold for the coefficients in the expansion of jr^ — , then it holds for tlie 

(l-2Acos6l + ;t')'^"' 

coefficients in the expansion of — ; but it does hold for the coefficients in 

(I-2/tcos6' + 7jy'' 

the expansion of 3— ; therefore it holds for the coefficients in the expansion 

(l-2Acos6l + A'')-^'' 

of j— , and so on universally. 

(l-2/icos^ + ;0' ' 



^-1 



The proof holds good only when i is a positive integer not less than 3. 

(/3) To show that 

'(l-;i^(sin^y^d^ 



i 



il-2kcos0 + hy ^° 



where h<l, and i is a positive integer not less than 2, 

Let r=E„ + i2,;j+ +R,h' + 

(l-2Acos^ + /i7'' 

/ • nx cos 9 —k r, -r, T r^t 

:. (1-2) i =-S.+ +pRJi'^'+ 

(1 - 2h cos e + h'y 

... (i _2) 1^ ^ = {{- 2 + 2 . 0)R,+ ii-2 + 2 .l)hR,+ ... + (i - 2 + 2p) h-R, + ... 

(1 - 2h cos e + h'y 

. (,■ o) r (1-^0 (sin gy^rfg 
jo (1 - 2h cos + hy 

= (i - 2 + 2 . 0) ["(sin 0)'-^ R,d0 + {i-2 + 2.\)h ['(sin 0y-^R^d0 

Jo JO 

+ + (i-2 + 2.p) h" ["(sin 0y-^R^d0+ 

Jo 

But putting n = i—2, 2^m-i = P> Pw, = ^' wi = 1, 0^=0, in equation {m) of Art. 2, (f)^ 
becomes ii,. 



ANALOGOUS TO TESSERAL HARMONICS. 293 

Therefore sin'^f? ~l + (i - 2) sin ^ cos ^ ^' +p{p + i- 2) sm^e .B^ = 0. 

Multiplying by (sin ^)'"* and integrating between the limits and tt with regard 
to the variable 6 



(sin^y-'^'^ +p(p + i-2)j (smey-'R^de=o, 

["(sin ey-^R^de = O a ph& not zero. 

Jo 

If p be zero, then since R^ = \ 

r(smey-'R,de= ["(sin ey-'d0 -, 

Jo Jo 

= (sin6')'" 
Jo 



f 

Jo 



(l-h')(smdy-'d0_, ,„.„^v-.^^. 



/o (l-2/tcos6' + /i7' 

The restriction i 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 x^w^ x.. 

2. Suppose now that 

«;.'+< + +x^ = r\ a;7 + .r'\+ +< = r'', 

da- = r'-' (sin ey^ (sin d^' (sin e,_,y (sin 0,.,) d9,d9^ d9,_, de^_^ rZ^,., 

W = (^7, - a;\)' + («, - .r',)' + +{x^- x\y = J-" - 2rr' cos 61 + r'^ 

F{6^9^ ^i_j) any function of 6^6.^ 6._^ whatever, 

u = I ^ s^f '-^ , the limits of integration for 0._^ being and 

27r ; for 9^6.^ 9,_2 the limits being and tt. 

Suppose that ?•' is infinitely nearly equal to r but greater than it, then every 
element of the integral u is very small except those which make I) infinitely small; 
therefore for all elements which do not make D infinitely small, anything may be sub- 
stituted for F [9^9^ 9._^ 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 F{9^9^ 0^^ which is 

very nearly equal to F{9'^9\ ^',_,) cannot be replaced by anything else. It is possible 

then to replace F{9^9^ 9-_^ by F(9^6^' 9'._j) in all the elements of the integral. 

(Since it has been assumed that 9^9^ 9^_^ can be put equal to 9\9\ 0',_, 

respectively, the limits between which 9" ^9'^ 9'._^ lie must be the same as those 

between which 9^9^ 0,_, are contained.) 

Therefore j i^''-^'^^^^ ^.-) d. = F^S'^,^ ^'.j/^^<^ 

= F{9;9; 9\_,) [ '-^^^^ , r'-'S'd9,d9, d9,_,d9,_d9,., 

} (r" - 2rr' cos 9 + rj 
where S' = (sin 9^)'-' (sin ^,)'-' (sin 9._,Y (sin ^..,),. 



294 Mr hill, ON FUNCTIONS OF MORE THAN TWO VARIABLES 

Let the variables a; a; x. be transformed by an orthogonal transformation to new- 
variables y, yj Vi, anii let i/, 2/, I/i ^^ transformed into a set of variables 

,.A <^^ <^^_^ by a transformation similar to that given in the abstract of Art. 1 of 

Part I. Moreover let ^, = 6. Then 

»•'- (sin ey^ (sin 0,)-' (sin 6,., f (sin ^, J dOJd, d0,., de,., d^,., 

= ?■■-■ (sin ey^ (sin (^,)'-' (sin 4>,_,y sin ^,.Jdd4>.^ d<}>,.,d<f),.,d^,.^; 

and the limits for the new variables are the same as those for the old. 

Therefore 

i D' 'Jo (r'= -2n'' cos 6> +?•')= 

I (sin</)jy"'rf^j sin(^..jd0...J rf<^._,. 

j - .' 

Hence by Preliminary Proposition (/3) 
[ {r^-r^F{e fi^...e,.^_da ^ ^ ^^,^, ^^^^ >;^ J'^ ^.^j^^ qj., ^q multiplied by 

rir rir ,-2ir 

{sin ^Y~'d^, sin ^i.j fZ^..j c?^,., 

.'o 'jo .'0 

= F(r0; ... ^...) Cf". ■• f ' f '(sin^J-=(sin er' (sin ^,_.) d^.«'^, d0,.,dd.., 

* 7' J J ■' 

and therefore as r' approaches indefinitely near to r 

where 5" has the same value as on previous page. 

A similar result would have been obtained if r had been infinitely nearly equal 
to r but less than it. 

Both results may be expressed thus: — 

F (e:e: ...e, ;) {\ . . f " r (sin ey-' (sin 0,y-' (sin d,_,Y (sin e,.,) de.de, . . . de,., de,., de,., 

' Jo -0 .'0 

= r ...fJl^F {efi, . . . 9..,) . -^ . r'-= S'de,d9, . . . d0,., de,_, . 

where it is supposed that jy is expanded, and r put =r after the expansion lias 
been performed. 

'^=-.45^'.4(zJ^.)-('-)i^.}- 



Also 



ANALOGOUS TO TESSERAL HARMONICS. 295 

1111 



^" ir"-2rr'cos0 + i^j^'' ''""' (l -2'-,cos9 + T-X)^" 
Similarly ~ = -1 h+R^'^+.^+R^ (LJ +..1 iir'<r. 



\\-&mede) i-2 



"^^•"' ^' = (^•-4)(^•-6)...4.2 l^^i^^i ^ T2 '^ ' ^' ^^'"' 

but = 7^^ — T^T": — -^ o -, I -. — -nja ' P/ i-sx if «' ^6 odd ; 

(t — 4i)(i — 6) ...0.3.1 V-sm yd^/ (p+V) 

therefore if )■' > r. 



but if r < r, 

r^-r" _ 1 
D' ~i-2 



l^\^ii-2 + 2.0)R, + {i-2 + 2.1)'^R,+ ... + {{-2 + 2p)(ffR, + | 



The series are convergent* except when ?• = ?•'; and in this case the unespanded 
value of u having been shown to be finite and equal to 

F{e\e\ e\_,) rT.-.r ['"(sin ex' (sin 0,.,) d0, cw,,, de,_„ 

Jo J J 

it follows that the sum of either series, obtained by substituting for — j-^ the above 

values in u, approaches more and more nearly to this value as ?•' approaches to equality 
with r. 



■•• F{e\e', e\J f... [^"'(sin^.y-^Csin^J- {sm0^J'(sm0,^)d9M d6^.de^,d0^, 

Jo J oJ 

= .^2 Jl- ■ ■ Jliy (^'^^ ^-'^ ^'^^'^^^^ d0,_M-M-^ 

where R = {"^(2^) + i-2) i? 1 (sin ^,y-^(sin 0X' (sin 0^3)'(sin 0,_,). 

* Because by Preliminary Proposition (a) Up is greatest the ratio of which to the preceding term can be made less 
when COB 9 = 1, and then the general term is than unity by taking p suificiently great. 

/r\p\ i+p-S 



296 Mr hill, ON FUNCTIONS OF MORE THAN TWO VARIABLES 

This may be written 

Fie\ff, ^'..) = 2(^,^ r (i /;. . . /X^(^.^, o^j Rdo,w, . . . de,_,do,,,do,_, . . . (B) 

where R has the same value as ou previous page. 

The above reasoning holds good except when 6'._, is either or Stt, or one of the 
other variables say ^,„ is or tt. 

In the first of these exceptional cases, it may be sho^^-n that for F{0^d^' ... 0'-_^) 
on the left-hand side of the last equation there must be substituted 

h {F{e:d.: 0\., o)+F {ex e\_, 2^)}. 

In the second if ^'„. = ^ , for F{e^e„^ &,_,) must be substituted 

\\ . . f " f ' V {e;0.: ^'„,., I ^„,„ e\ (sin ^,„,)'— (sin e,j de,„^^ rf^,_, ^^,„ 

\\.. f (■"'(sin ^,„„y-'^ (sin e,_,) de„^, de,_, de,,. 



3. It remains to show that the general tenn 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 ?<, for 

f_^+-i!,+ + '^\\ I -.= 0; 

■ (^ JL+ +j^\ \ = 

\dx'^ dx'^ dx!'] . ,i „ , ^ ,'^ 

' ' ' (r — 2?-?- cos ^ + ?• ) - 

where cos0=— ^— ' , - '—^, and the relation between the symbols a:,' a*/ 

and r'0^ ^Vi is the same as that between a;, a-^ and r^, ^,.,. 

■•■(^^-^- -B^i/'-^^^ .^,0--)-, 

This being true for all values of r 

••• L(^" + — dr'j + 7- iU'' + (* - 2) cot ^. ^^) + 

1 d} 1 

^sin'^; sin'^^, d0'\ ' ^ ' '~ ' 



assummg r < r. 



ANALOGOUS TO TESSERAL HARMONICS. 297 



d' 



sin' (9; sin"^^, d6''V, 



r=o 



where V= f'... f f''^(^A ^.-i) • ^p («>" ^i)'"' («in ^, o) dff, cW, .jW, ,; 

J i) J oJ ' 

.". the general term in the expansion of F{9^0.^ 6'■_^) satisfies the equation in y^. 

Secondly, it is a linear function of the Normal Functions considered in this paper. 

i?p is a rational integral function of cos 6 of degree j), containing onl\' even or only 
odd powers of cos ; 

.". Rj,r'' is a rational integral function of os^'w^' x- of degree p satisfying the 

d'u d'u d'u 

equation ^+^,+ + ^. = ; 

.'. by Art, 11 of Part I., it is a linear function of functions of the form 

r" (sin 0;y^ ©; (sin e'^"'-"-®'^, (^2p,_ ff,,^ . 

Let one of these functions be denoted by r'^V, then i?^=S^l. U'; where ,4 is a 
function of - , — , -' only, and therefore of 6^6^ ^,_, only. 

Substituting this in the general term of the expansion, it becomes 

2Z7' r... rr 2j. + z-2 _ ^ ^^ ^ de,_je,,M-. 

Jo J OJ I — ^ 

= S U'A', where A' is a constant. 

But U' is a normal function of the form considered in this paper; .'. F(0^'0^' 6',_^) 

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 of i?^ in 
Normal Functions. 

i?y is by the foregoing argument the sum of such terms as 

(sin 0:r ©.' (sin e\_,r-'- 0',, (^^ P,., ^..,) 

multiplied by some quantity which is independent of the variables 0^'6^' 6\_^. But 

since i?, is a symmetrical function of the accented and unaccented variables, this quantity 

Vol. XIII. Part III. 39 



298 Mr hill, ON FUNCTIONS OF MORE THAN TWO VARIABLES 

(COS \ 

,■ i5,_2^,_,) fvs a factor. Thus B^ is the sum 

of such functions as 

B (sin ^;)P. e; (sin d\^,r-' e',, ('°^'/;._, ^,_,) (sin ^.)p. 0, (sin ^.. J^'- 0,., g";;,., ^,..) 

wliere £ is a constant, 

.-. F \e;e; 0',.,) \\..\' f '%in ey-°- (sin ^,.)'-' (sin d,.y (sin 0, j fZ^^cz^,, d6,^ cie, .. de, , 



=A| - ^m. ^.-.)2 



(2i; + i-2)iJ, 



S'de dO, 



Now 6^6.^ 0',_, are independent of 6fi^ ^,_, ; 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 

(sin ^,')P.+-2 ©; (sin 6\.,Y'-'-+^ 0',.. i^^^lK, ^' .,) , 

and integrate with regard to 6',_, from to Stt, and with regard to the other variables 
from to TT ; 

••■ \l- ■ ■ \l\y^^^^: ^'.-.) (sin ^,y-+'-^ e; (sin ^',^^'-=+1 ©V. (J^'i^.., 6\.) dO; d& ,_, 

multipKed by \^ ...\'\ "(sin ^,)'"' (sin ^, .,) fZ^, de,_^ 

J {) J oJ 

i — -iJo JaJo \''in / 

multiplied by f. • . I" f "(sin 6l,')2''.+'-2 ©/^ (sin e\ „)2p'-=+i0'V, P^ Pi-. ^', .)' d6,' d6\ , . 

- J oJ Vsin ■ / 

Ail 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 + 1 — 2) 
times the first integral of tlie first member ; 

• g f "■ • • f ' f "(sin er ^-i>' ^,-J de, de,_, 

T}_^~^ 'no .'o 



2p + i-2- I".. r|\i^^^).„..-.e,^ (sin^,.,p--i©V,r"^^._,0^d^..... 



de, 



Observing that {\..\' \ '"(sin 6^-' (sin e,^de. dd, , = 2 ^,~. ; it follows by 



Part I. Art. 9, that 



ANALOGOUS TO TESSERAL HARMONICS. 299 

B = 2 |i-4 \i-h^ |3 |2 (2;v3 + 1) (2jj,., + 2) (2p, + i - 8) multiplied by 



P<-3 - Pi-^ \Pi-, - P,-, \P-P^ 



except when 2^1-2 = 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^^j..,^,., or sm'p,_„6,., be employed to 
calculate the integral in the denominator. 



T, f ... r f%mdj-"- ... [sine^Jde^ ... 

Jo Jo '0 



_. „ „ cW. 



^^+*'~^ ["...[" r (sin ^,)'-^i'.+--^@/ ... (sin^,_,)2/^-+i0,./Q°^'2^..,e,.,)'c/^. ... de,_, 
where ?;= (sin 0,>e; ... (sin e\_„)P^~'-@\_,{sm0^)P'@, ... (sin ^,_„>-=®..2Cos^,_,(^i-,- ^'.-,). 

The summation extends to all possible positive integral values of the indices p^ ...pi..^ 

not greater than p, and such that in any one term no index is gi-eater than one which 

/cos \^ . 

precedes it. If ^..^ be zero, the factor ( . jJ,.™^;., ) is to be replaced by 1. 

Substituting this value of R^,, it appears that 

where 

r ... r f'\sm0;)P^+i-m ... (sin^. .V'^-=+i0, „ (cos jj,.,6l,„.) i^(6l^^., ... 0.^,)de, ... de,_, 
p _ .h J Jo '_ 



and 



i) = 



r... r r{smd;j-'p'+'-m^' ... {sme,_.ypi-^+'^e,j {cos2},.a.,y dd^ ... de,_, 

Jo J J 
r... r fVn ^,V"+'"'0, ••■ ism0,_.;)p<-'-+'&,_,{sm2},_,e,.,)F{e,e„ ... e,_,)d6^ ... dO,^, 

' » ■' n .' n " 

r... ["/'"(sin^J-'-. + '-s©^" ... (sin0,_,)2/^-=+i©,_/(sin^,_,6',_,)'(^6l, ... de,^, 
J J J J 



(C). 



The summation in this case extends to all possible positive integral values of the 
indices ^)j ... jj,_;, such that in any one term no index is greater than the index 2> with 
which that term is connected, nor any index greater than any one which precedes it. 



CAMBRIDGE : PRINTED BY C. J. CLAT, M.A. & SON, AT THE UNIVERSITT PRESS. 



^ 



INDEX TO THE TRANSACTIONS 



OF THE 



C;nnkitr0e ^Ijiksnpljual Sntlcljr. 



VOLS. I— XII. 



I. NAMES OF AUTHORS. 



Adams, Prof. J. C, Note on Sir G. B. Airy's memoir 

on the resolution of a certain Trinomial : Nov. 

23, 1868 ; XI. 444—445. 
Airy, Sir G. B., On the use of silvered glass for the 

Mirrors of Reflecting Telescopes: Nov. 25, 

1822; II. 105—118. 

On the Figure assumed by a Fluid Homogeneous 

Mass, &c., &c.: March 15, 1824; II. 203—216. 

On the Achromatism of the Eye-pieces of Tele- 

scopes, and of Microscopes : May 17, 1824 ; 
II. 227—252. 

— '■ — On the defect in the Eye, and a mode of cor- 
recting it: Feb. 21, 1825; ii. 267—271. 

On the form of the teeth of Wheels : May 2, 1825; 

II. 277—286. 

On Laplace's investigation of the Attraction of 

Spheroids difl'ering little from a Sphere : May 8, 
1826 ; II. 379—390. 

On the Spherical Aberration in the Eye-pieces of 

Telescopes : May 14, 21, 1827 ; in. 1—63. 

On Pendulums and Balances, and the Theory of 

Escapements: Nov. 26, 1826; in. 105—128. 

On the Longitude of Cambridge Observatory : 

Nov. 24, 1828 ; III. 155—170. 

On a means of correcting the length of a Pen- 

dulum by a Ball suspended by Wire : Nov, 16, 
1829 ; III. 355—360. 

On the conditions under which Perpetual Motion 

is possible : Dec. 14, 1829 ; III. 369—372. 

Vol. XII. 



AiRT, Sir G. B., On the Nature of the Light in the two 
Bays produced by the Double Refraction of 
Quartz : Feb. 21, 1831 ; iv. 79—123. 

Addition to this memoir : Apr. 18, 1831 ; I v. 199 

—208. 

On a remarkable modification of Newton's Rings : 

Nov. 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. 409^24. 
— ^- 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. 379^02. 

On Triple Algebra : Oct. 28, 1844; viii. 241—254. 

Supplement to this memoir : May 8, 1848 ; vili. 

595—599. 

On a new construction of the Going-Fusee : 

March 2, 1840; vil. 217—225. 

On an Eye aflected by a mal-formation : May 25, 

1846 ; VIII. 361—362. 

73 



u 



INDEX TO TRANSACTIONS I— XII. 



Airy, Sir G. B., Further observations on the same: 
Feb. 12, 1872 ; xil. 392—393. 

On the substitution of Oniinary Geometry for 

the genenil Doctrine of Proportions : Dec. 7, 
1S57 ; s. 166—172. 

Suggestion of a Proof that every Equation has 

a Root : Dec. 6, 1858 ; s. 283—289. 

Supplement to this memoir: Dec. 12, 1S59 ; x. 

327—330. 

On the factorial Resolution of the Trinomial 

.r" - 2 cos n« + - : Nov. 9, 1 868 ; XI. 426—443. 

Akix, C. K., Ou the origin of Electricity: Dec. 7, 1863; 

XI. 6—20. 
Aldersox, Jas., On a Spermaceti Whale, stranded in 

Yorkshire : May 16, 1825 ; ir. 253—266.* 

Ou an Artificial formation of Plimibago : Feb. 21, 

1825; II. 441—443. 
Ansted, D. T., On some Fossil Midtilocular Shells 
found in Cornwall : Feb. 26, 1838 ; vi. 415— 
422. 

Ou a portion of the Tertiary Formations of 

Switzerland: May 20, 1839; vil. 141—152. 

On some Phenomena of the Weathering of Rocks : 

March 2, 1868; si. 387—395. 
6.\BBAGE, C, On the notation employed in the Calculus 
of Functions: May 1, 1820; i. 63—76. 

On the General Term of a New Class of Infinite 

Series: May 3, 1824; ii. 218—225. 

Ou the influence of Signs iu Mathematical Reason- 

ing : Dec. 16, 1821 ; ii. 325—377. 
Baxter, H. F., Ou Organic Polarity : March 8, 1858 ; 

X. 248—260. 
Bevaji, B., Experiments on Percussion : Nov. 10, 1825 ; 

II. 444. 
Bond, Prof., Statistical Report on Addenbrooke's 

Hospital for 1836 : March 13, 1837 ; vi. 301 

—377. 

The same for 1837 : Apr. 30, 1838 ; vi. 505—575. 

Boole, Prof. George, Of Proiwsitions numerically defi- 
nite : March 16, 1808; xi. 396—411. 

Brewster, Dr., On the Brazilian Topaz; its colour, 

structure, and optical properties : May 6, 1822 ; 

II. 1—9. 
Brodie, p. B., On Land and Freshwater Shells, and 

Bones of Animals found near Cambridge : 

Apr. 30, 1838 ; viil. 138—140. 
Cayley, Prof., On the Theory of Determinants : Feb. 20, 

1843; VIII. 75—88. 

On the Theory of Involution : Feb. 22, 1864 ; xi. 

21—38. 

On a case of the Involution of Cubic Curves : 

Feb. 22, 18C4; xi. 39—80. 

On the classification of Cubic Curves: Apr. 18, 

1864 ; XI. 81— 12a 



Catley, Prof., On Cubic Cones and Curves : Apr. 18, 
1804; XI. 129—144. 

Ou cei-tain Skew Surfaces, otherwise Scrolls : 

Nov. 11, 1867; XI. 277—289. 

On the Six Coordinates of a Line : Nov. 11, 1867 ; 

XI. 290—323. 

On a cei-tain Sextic Torse: Nov. 8, 1869; xi. 

507—523. 

On the Ceutro-surface of an Ellipsoid : March 7, 

1870; XII. 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 ; xii. 306—383. 

On the geometrical Representation of Cauohy's 

theorems of Root-limitation: Feb. 16, 1874; 

XII. 395—413. 

Cecil, W., On Hytb-ogen Gas, as a moving power in 
Machinery, &o.: Nov. 27, 1820; i. 217—239. 

Ou an apparatus for gi-iudiug Mirrors and Object 

Lenses: Deo. 11,1822; ii. 85—99. 
Challis, Prof., On the extension of Bode's Law to the 
distance of Satellites from their Primaries : 
Dec. 8, 1828; in. 171—183. 

On the small Vibratory Motions of Elastic Fluids : 

March 30, 1829 ; in. 269—320. 

On the general Equations for the Motion of Fluids, 

&c., &c. : Feb. 22, 1830; in. 383-^16. 

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. 

Ou the Differential Equations applicable to the 

Motion of Fluids: Apr. U, 1842; vii. 371—396. 

On a new Equation in Hydrodynamics : March 6, 

1843 ; VIII. 31—43. 

On the Theory of Luminous Rays on the hy- 

pothesis of Undulations: May 11, 1846; viii. 
363—370. 

Ou the Theory of the Polarization of Light, on the 

same hypothesis: May 25, 1846; viii. 371 — 
378. 

On the Transmissi(jn of Light, and on Double 

Rufriiction ; on the same hypothesis ; May 17, 
1847 ; VIII. 524—5.32. 

On the Mathematical Theory of Luminous Vibra- 

tions : March 6, 1848 ; vill. 584—594. 

On the Aurora Borealis of Nov. 17, 1848 : Nov. 27, 

1848; VIII. 621-632. 

Ou the Deteriniuation of the Longitude of Cam- 

bridge Observatory by Galvanic Signals : May 
15, 1854; IX. 487—514. 



I. NAMES OF AUTHORS. 



lU 



Christie, S. H., On the Laws hj which Masses of Iron 

affect Magnetic Needles : May 15, 1820 ; I. 

147—173. 
Clark, Prof. W., On a case of Human Monstrosity: 

May 16, 1831 ; iv. 219—255. 
Clarke, Prof. E. D., Inaugural address at the first 

general meeting of the Society : Dec. 13, 1816; 

I. (1-7). 

On the Puqile Precipitate of Cassius: May 15, 

1820; I. 53—61. 

On a deposit of Natron in the tower of a Church : 

Nov. 27, 1820; i. 193—201. 

On the Crystallization of Water, &c. : March 5, 

1821; I. 209—215. 
Clifton, R. B., Note on Prof De Morgan's Memoir on 

the history of Signs + and - : Feb. 13, 1865; 

XL 213—218. 
CoDDlNGTON, Eev. H., On the improvement of the 

Microscope: March 22, 18.30; lii. 421 — 428. 
Cox, HoMERSHAM, On Impact on Elastic Beams : Dec. 

10, 1849; IX. 73—78. 

On the Deflection of Imperfectly Elastic Beams, 

and on the Hyperbolic Law of Elasticity: 
March 11, 1850; IX. pt. ii. 177—190. 
CUMMING, Prof, On the connexion of Galvanism and 
Magnetism : Apr. 2, 1821 ; l. 269—279. 

On Magnetism as a Measure of Electricity : May 

21, 1821 ; L 281—286. 

On a large Human Calculus in the Library of 

Trinity College : Nov. 26, 1821 ; i. 347— 
349. 

On the development of Electro-Magnetism by 

Heat: Apr. 28, 1823; n. 47—76. 

See Alderson, J. 

De Morgan, Prof, On.the general Equation of Curves 
of the Second Degree: Nov. 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 ; vi. 185—193. 

On a Question in the Theory of Probabilities : 

Feb. 26, 1837; vl 423 — 130. 

On the Foimdation of Algelira : Dec. 9, 1839 ; 

viL 173—187. 



Do. 




Do. 


Vll. 


287- 


-300. 


Do. 




Do. 


VIII 


139- 


-142. 


Do. 




Do. 



Do. 



Do. 



Nov. 29, 1841; 
Nov. 27, 1843; 



On Triple Algebra : Oct. 28, 
1844; VIII. 241—254. 

On Divergent Series, &c., &c. : March 4, 1844; 

viil 182—203. 

On the Structure of the Syllogism, and its ap- 

plication, &c. : Nov. 9, 1846; viii. 379 — 408. 



De Morgan, Prof, On Integrating Partial Differential 
Equations : June 5, 1848 ; viii. 606—61.3. 

On the Symbols of Logic, the Theory of the Syllo- 

gism, &c., &c. : Feb. 25, 1850; ix. 79—127. 

On some points of the Integral Calculus : Feb. 24, 

1851 ; IX. pt. ii. 107—1.38. 

On some points in the Theory of Differential 

Eqiiations : March 27, 1854; ix. 515 — 554. 

On the singular points of Curves, and on Newton's 

method of Co-ordinated Exponents : May 21, 
1855 ; IX. 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. iil, and on Logic in 

general: Feb. 8, 1858; x. 173—230. 

On the Syllogism, No. IV., and on the Logic 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 : Nov. 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; XL 145—189. 

On a Theorem relating to Neutral Series : May 16, 

1864; XL 190—202. 

On the Early History of the Signs + and - : Nov. 

28, 1864; XI. 203— 218. 

On the Root of any Function: and on Neutral 

Series, No. ii. : May 7, 1866 ; xl 239—266. 

Note on the same: Oct. 26, 1868; XL 447—460. 

Denison, E. B., On Clock Escapements : Nov. 27, 1848 ; 

vin. 633—638. 

On TuiTet-clock Remontoirs : Feb. 26, 1849; viii. 

639—641. 

On some recent Improvements in Clock-Esoape- 

meuts: Feb. 7, 1853; ix. 417 — 430. 
Donaldson, Dr. J. W., On the Structm'e of the Athe- 
nian Trireme: Nov. 6, 1856; x. 84 — 9.3. 

On the Statue of Solon mentioned by yEschines 

and Demo.sthenes : Feb. 22, 1858; x. 231 — 239. 

On Plato's Cosmical System : as exhibited in Tho 

Pu'jiuMic: Book x. : Feb. 28, 1859; x. 305—316. 

On the Origin and Use of the word Argument : 

Nov. 28, 1859; x. 317—326. 
Earnshaw, Rev. S., On Fluid Motion, as expressed by 
the Equation of Continuity : March 21, 1836 : 
VI. 203—233. 

73—2 



IV 



INDEX TO TRANSACTIONS I— XII. 



Earssha-w, Kev. S., On the Diffraction of an Objoct- 
Gli«s with a triangxilar Aperture: Dec. 12, 
lS36;vi. 431 — 442. &e Airy. 

On the Nature of the ilolecular Forces of Lumi- 

uiferoius Ether: Miwch 18, 1839; vil. 97— 112. 

On the Values of the Sine and Cosine of an Infinite 

Angle : Dec. 9, 1844 ; viil. 255—268. 
On two great Solitary Waves of the First Order : 

Dec. 8, 1845; viii. 326—341. 
Ellis, E. L., On the Foimdation of the Theory of 

Frobabilities : Feb. 14, 1842; vili. 1—6. 

On the method of Least Siiuares : March 4, 1844 ; 

vm. 204—219. 

Remarks on the Theory of Slatter : Jlay 22, 1848 ; 

VIII. 600—605. 

On the Fundamental Principle of the Theory of 

Probabilities: Nov. 13^ 1854; ix. 605—607. 

Fabish, Prof., On Isometrical Perspective : Feb. 21, 
March 6, 1820; 1. 1—19. 

Fexsell, C. a. M., On the First Ages of a Written 
Greek Literature : Nov. 23, 1868 ; xi. 461— 
480. 

Fisher, Rev. Osmond, On the Purbeck Strata of Dorset- 
shire: Nov. 13, 1854; IX. 555—581. 

On the elevation of Mountains by lateral pres- 

sure, &c., &c.: Apr. 27, 1868; xi. 489—506. 

On the InequaUties of the Earth's Siurface viewed 

in connection with the secular cooling: Dec. 1, 
1873; xn. 414 — 133. 

On the same, as produced by lateral pressure, on 

the hypothesis of a hquid substratum : Feb. 22, 
1875; xn. 434 — 454. 

Glaisher, J. W. L., Tables of the first 250 Bernouilli's 
Numbers, and of their logarithms : May 29, 
1871 ; XII. 384—387. 

Supplement to the same memoir : March 11, 1872 ; 

xiL 388—391. 

GoDFBAY, HrcH, On a Chart and Diagram for facilitat- 
ing Great Circle Saihng: May 10, 1858; x. 
271—282. 

GoODE, Hesrt, On a peculiar Defect of Vision : Nov. 9, 
1846; May 17, 1847; viil 493 — 496. 

Goodwin, Rev. H., On the Connesion between Me- 
chanics and Geometry: Feb. 10, 1845; vin. 
269—277. 

On the Pure Science of Magnitude and Direction : 

May 12, 1845; vin. 278—286. 

On the Geometrical Representation of the Roots of 

Algebraic Equations : Apr. 27, 1846 ; vill. 342 
—360. 
Gbees, George, On the Laws of Equilibrium of Fluids 
analogou-s to the Electric Fluid : Nov. 12, 1832 ; 
V. 1—63. 

On the Exterior and Interior Attractions of Ellip- 

soids, &c., &c. : May 6, 1833; v. 395 — 429. 



Green, George, On the Reflexion and Refraction of 
Sound: Dec. 11, 1837; vi. 403—113. 

On the Motion of Waves in a variable Canal of 

small depth and width: May 15, 1837; VL 
457—462. 

Note on the motion of Waves in Canals : Feb. 18, 

1839; vii. 87—95. 

Memoir on the Laws of Reflection and Refraction 

at the common Surface of two non-crystallized 
Media: Dec. 11, 1837; vn. 1—24. 

Supplement to the memoir : May 6, 1839 ; vil. 

113—120. 

On the Propagation of Light in Crystallized 

Media : May 20, 1839 ; vn. 121—140. 
Gregory, Dr. Olinthus, On some Experiments to 

determine the Velocity of Soimd : Dec. 8, 1823; 

n. 119—137. 
Hailstone, Rev. J., On an extraordinary depression 

of the Barometer in Dec. 1821, &c., &c. : 

Feb. 25, 1822; 1. 453—458. 
H.iViLAND, Dr., On the solution of the Stomach by 

Gastric Juices: Dec. 11, 1820; i. 287—290. 
Hat^vard, R. B., On a direct method of estimating 

Velocities, &c., &c., with reference to Axes 

moveable in Space: Feb. 25, 1856; x. 1—20. 
Hesslow, Prof., On the Geology of Anglesea : Nov. 26, 

1821 ; I. 359—452. 

On a hybrid Digitalis: Nov. 14, 1831; IV. 257— 

278. 

On the Monstrosity of the Common Mignionette : 

May 21, 1832; v. 95—100. 
Herschel, Sir J. F. W., Deviation in Crystals from 
Newton's scale of Tints, with Polarized Light : 
May 1, 1820; L 21—41. 

Planes of Polarization, as aflected by Plates of 

Rook Crystal: Apr. 17, 1820; l. 43—52. 

Functional Equations, Reduction of, to Equations 

of Finite Differences : March 6, 1820; i. 77—87. 

Apophyllite, On the Refraction of coloured Rays 

in : May 7, 1821 ; I. 241—247. 

On a Machine for resolving by Inspection Trans- 

cendental Equations: May 7, 1832; iv. 425 — 

440. 
HiERN, W. P., A Monograph of Ebenaceos : March 11, 

1872 ; XII. 27—300. 
HOLDITCH, Rev. H., On RolUng Curves : Dec. 10, 1838 ; 

VII. 61—86. 

On Small Finit« Oscillations : May 15, 1843; viil. 

89—104. 
HorKiNS, 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 Motifin of Glaciers: May 1, 1843; vm. 

50—74. 



I. NAMES OF AUTHORS. 



HoPKXSS, W., On the Motion of Glaciers (second me- 
moir) : Dec. 11, 1843; VIII. 159—169. 

On the Transport of Erratic Blocks : Apr. 29, 

1844; VIII. 220—240. ' 

On the Internal Pressure of Rook Masses, &c., &c. : 

May 3, 1847 ; viil. 456-^70. 

On the External Temperature of the Earth, and 

the other Planets of the Solar System : May 21, 

1855; IX. 628—672. 
Humphry, Prof. G. M., On the growth of the Jaws : 

Nov. 9, 1863; xi. 1—5. 
Jarrett, T., On Algebraic Notation : Nov. 12, 1827 ; 

III. 65—103. 
Jebb, Prof E. C, On the place of Music in Education 

as conceived by Aristotle in his " Politics :" 

May 17, 1875 ; xn. 523—530. 
jESrxs, Rev. L., On the Ornithology of Cambridge- 
shire : Nov. 28, 1825 ; II. 287—324. 

On Pennant's Natterjack ; with a list of the Rep- 

tiles of Cambridgeshire: Feb. 22, 1830; in. 
37a— 381. 

Monograph on the British species of Cyclas and 

Pisidium: Nov. 28, 1831 ; rv. 289—312. 
Kelland, Prof, On the Dispersion of Light, on the 
theory of Finite Intervals : Feb. 22, 1836 ; vi. 
153—184. 

On the Motion of a System of Particles, with 

reference to Sound and Heat : May 16, 1836; 
Ti. 235—288. 

On the transmission of Light in Cry.stallized 

Media: Feb. 13, 1837; vi. 32.3— 352. 

Supplement to the same : May 1, 1837 ; vi. 353 — 

360. 

On Molecular Equilibrium: March 26, 1838; vil. 

25—59. 

On the Quantity of Light intercepted by a gi'ating 

placed before a Lens ; and on the effect of the 

Interference: March 30, 1840; vil. 153—171. 
Kemp, George, On the Nature of the Biliary Secretion : 

March 6, 1843; VIII. 44-^9. 
King, J., A new demonstration of the Parallelogram of 

Forces: Apr. 14, 1823; II. 45 — 46. 
Lee, Prof, On the Astronomical Tables of Mohammed 

Abibeker Al Farsi : the MSS. of which are in 

the Public Library: Nov. 13, 1820; I. 249— 

265. 
Leslie, Prof, On the Sounds excited in Hydrogen Gas : 

Apr. 2, 1821 ; I. 267—268. 
Lowe, R. T., Primitive Faimae et Florre Maderte et 

Portus Sancti: Nov. 15, 1830; iv. 1 — 70. 

Piscium Maderensium Species {see M. Young) : 

Nov. 10, 1834 ; VI. 195—201. 

NovitijB Florae Maderensis : or Gleanings from 

Madeiran Botany : May 28, 1838 ; vi. 523— 
551. 



Lubbock, Sir J., On the Calculation of Annuities, and 
on some points in the Theory of Chances: 
May 26, 1828; in. 141—154. 

Comparison of various Tables of Annuities : 

March 30, 1829 ; in. 321—341. 

LuxN, F., Phosphate of Copper from the Rhine, Analy- 
sis of: March 5, 1821 ; I. 20.3—207. 

Mandell, W., On the improved methods of procflring 
Potassium: Nov. 26, 1821; i. 343—345. 

Maxwell, Prof J. Clerk, On the Transformation of 
Surfaces by Bending: March 13, 1854; ix. 
445—470. 

On Faratlay's lines of Force : Dec. 10, 1855, 

Feb. 11, 1856; x. 27—83. 

On Boltzmanu's Theorem on the average distri- 

bution of energy in a system of material points : 
May 6, 1878; xn. 547—570. 
Miller, Prof., On the Crystals of Boracic Acid ; 
Nov. 30, 1829; in. 365—367. 

On Crystals found in Slags: March 22, 1830; in. 

417—420. 

On the position of the Ases of Optical Elasticity 

in certain Crystals: Dec. 8, 1834; v. 431 — 
438. March 21, 1836; vil 209—215. 

On spurious Rainbows: March 22, 1841; vii. 

277—286. 
Moore, A. A., On a difficulty in Analysis noticed by 

Sir AVm. Hamilton: May 1, 1837; vi. 317—322. 
MoRTOX, Pierce, On the Focus of a Conic Section : 

March 2, 1829 ; in. 185-190. 
Moseley, Rev. H., On the Equilibrium of the Arch : 

Dec. 9, 1833; v. 293—313. 

On the Theory of the EquiUbrium of Bodies in 

X'ontact: May 15, 1837; vi. 463 — 491. 

MtJNRO, Rev. H. A. J., On a Metrical Latin Inscrip- 
tion at Cirta in Algeria: Feb. 13, 1860; x. 
374—408. 

MuBPHT, R., On the General Properties of Definite 
Integrals: May 24, 1830 ; in. 429 — 443. 

On the Resolution of Algebraical Equations: 

March 7, 1831 ; iv. 125—15.3. 

On the inverse method of Definite Integrals, with 

Physical Applications : March 5, 1832 ; iv. 
35.3 — 408. 

Second memoir on the same: Nov. 11, 1833; v. 

113—148. 

Third memoir on the same : March 2, 1835 ; v. 

31.5—39.3. 

On Elimination between an Indefinite number of 

unknown Quantities : Nov. 26, 1832; v. 65 — 75. 

On the resolution of Equations in Finite Dif- 

ferences: Nov. 15, 1835; VL 91—106. 
O'Brien, Rev. M., On the Propagation of Luminous 
Waves in the Interior of Transparent Bodies : 
Apr. 25, 1842 ; vii. 397—437. 



VI 



INDEX TO TRANSACTIONS I— XII. 



O'Briex, Rev. M., On the Reflection and Refraction of 
Light at the surface of an Uncrystallized Botly : 
Nov. 28, 1842 ; viii. T— 26. 

On the possibility of accounting for the Absorption 

of Liglit, &c., &c. : Feb. 14, 1843 ; viii. 27—30. 

On a Xew Notation to be used in Geometry, &c., 

&c.: Nov. 23, 1846; viii. 415—428. 

On a System of Symbolical Geometry and Me- 

chanics: March 15, 1847; viii. 497 — 507. 

On the Equation for the Vibratory Motion of an 

Elastic Medium : March 15, 1847 ; viu. 508 — 
523. 
Okes, J., On the remains of a Fossil Beaver found in 
Cambridgeshire: March 6, 1820; i. 175 — 177. 

On a dilatation of the Ureters, &c. : Nov. 12, 1821 ; 

I. 351—358. 
OwEX, Richard, F.R.S., Description of an extinct 

Lacertiau Reptile: Apr. 11, 1842; vii. 355— 

3C9. 
Paget, Prof. G. E., On some remarkable Abnormities 

in the Voluntary Muscles ; March 8, 1858 ; x. 

240—247. 
Palet, 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; XI. 360—386. 
Phear, J. B., On the Geology of some parts of Suffolk, 

particularly of the Valley of the Gipping: 

Feb. 27, 1854; IX. 431 — 144. 
PlERSOS, R., On the Theory of the Long Inequality of 

Uranus and Neptune: 1852; ix. Appendix, 

pp. Ixvii. 
Potter, R., Mathematical considerations on the Prob- 
lem of the Rainbow : Dec. 14, 1835 ; vi. 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 Aurorse Boreales, &c., 

&c. : Dec. 8, 1845 ; vrii. 320—325. 
Power, Rev. J., On the principle of Virtual Velocities : 
March 21, 1825 ; ii. 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- 
hsioas: May 29, 1841 ; vii. 301—317. 

On the Truth of a Theorem in Hydrodynamics : 

May 9, 1842; vii. 455—464. 
RlGAUD, Prof., On the relative Quantities of Land and 

Water on the Globe: Feb. 13, 1837; vi. 289 

—300. 
RoHBs, J. H., On the Oscillation of a Suspension Chain : 

Dec. 8, 1831 ; ix. 379—398. 



Ronns, J. IL, On the Motion of Beams, and thin Elastic 
Rods: Apr. 2.3, 1860; x. 359—373. 

On the Strains to which Ordnance are subject, 

and on the Vibrations of Solid Bodies : Apr. 18, 
1864; XI. 324—359. 
RoTHMAN, R. W., On Variations of Magnetic Intensity, 
as computed and observed: Nov. 10, 1825; 
II. 445. 

On an Ancient Observation of a Winter Solstice : 

Nov. 30, 1829 ; in. 361—363. 

An accoimt of Observations of Halley's Comet : 

Dec. 11, 1837; vi. 493—506. 
Salter, J. W., On Crotalocriniis rugosus, Miller : a 
Crinoid in the Woodwardian Museum : Feb. 8, 
1869; XI. 481—484. 

Diagram of the relations of the LTnivalve to the 

Bivalve : and of this to the Brachiopod : 
Feb. 8, 1869 ; xi. 485—488. 
Sedgwick, Prof., On the Primitive Ridge of Devonshire 
and Cornwall : March 20, 1820; i. 89—146. 

On the Phy.sical Structure of the Lizard district 

in Cornwall : Apr. 2, May 7, 1821 ; i. 291—330. 

On some Trap Dykes in Yorkshire and Dm'ham : 

May 20, 1822; ll. 21 — 44. 

On the Association of Trap Rocks with Mountain 

Limestone in Tees-Dale : May 12, 1823; 
March 1, 15, 1824; ll. 139—195. 

Note on a memoir by Dr Brodie on Land and 

Freshwater Shells, &c. : Apr. 30, 1838 ; viil. 

139—140. 
Smith, Archibald, On the Equation to Fresnel's Wave- 

Siu-face: May 18, 1835; vl 85—89. 
SpiLSBURT, F. G., On the Magnetism evolved by a single 

Galvanic combination, &c., &c. : Nov. 25, 1822 ; 

II. 77—83. 
Stephens, J. F., Description of Chiasogiiathus Grantii, 

a Lucanideous Insect : May IG, 1831 ; IV. 209 

—216. 
Stokes, Prof G. G., On the Steady Motion of Incom- 
pressible Fluids : Apr. 25, 1842 ; vii. 439 — 453. 

Memoir on some cases of Fluid Slotion : May 29, 

1843; VIIL 105—137. 

Supplement to this memoir : Nov. 3, 1846 ; viii. 

409 — 414. 

On the Internal Friction of Fluids in Jlotion : 

and the Equilibrium and Motion of Elastic 
Solids : Apr. 14, 1845 ; vnL 287—319. 

On the Theory of Oscillatory Waves : March 1, 

1847; viu. 441^55. 

On the Critical Value of the Sums of Periodic 

Series: Dec. 6, 1847; vin. 533 — 583. 

On the central Spot of Newton's Rings: Dec. 11, 

1848; VIIL 642—658. 

On the Variation of Gravity at the Earth's Sur- 

face: Apr. 23, 1849; vin. 672— 695. 



I. NAMES OF AUTHORS. 



vu 



Stokes, Prof. G. G., On an Equation relating to the 
breaking of Railway Bridges: May 21, 1849; 

VIII. 707—735. 

On the Dynamical Theory of Diffraction: Nov. 26, 

1849; IX. 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: Deo. 9, 1850; 
IX. pt. ii. 8—106. 

On the Colours of Thick Plates : May 19, 1851 ,- 

IX. pt. ii. 147 — 176. 

On the Composition and Resolution of Streams of 

Polarized Light from different soiu-ces : Feb. 16, 
March 15, 1852; IX. 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. 412^25. 

Thompson, Prof. W. H., On the genuineness of the 
Sophista of Plato, and on some of its philoso- 
phic bearings: Nov. 2.3, 1857; x. 146—165. 

ToDHUNTEB, I., On the Method of Least Squares : 
May 29, 1865; XI. 219—238. 

On the Arc of the Meridian measured in Lapland : 

May 1, 1871 ; xii. 1—26. 

On the equation determining the fo?Tn of the 

strata in Legendre's and Laplace's Theory of 
the Figure of the Earth: Oct. 16, 1871; xii. 
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; vii. 189 — 
196. 

J., On the Force of Testimony in Legal Evidence : 

Nov. 27, 1843; vill. 14.3—158. 

AVallace, William, Geometrical Theorems and For- 
mula, as applicable to Geodesy: Nov. 30, 1835 ; 
VI. 107—140. 

Warburton, H., On the Partition of Numbers ; Com- 
binations and Permutations: March 1, 1847; 
VIII. 471—492. 

On self-repeating series: May 15, 1854; ix. 471 

—486. 
Warren, Rev. J. W., Exercises in Curvilinear and 

Normal Co-ordinates: May 22, 1876; May 7, 

1877; xn. 455— 522; 531—545. 
Wedgwood, H., On the Knowledge of Body and Space : 

March 11, 1850; ix. 157—165. 
Whewell, Dr, On the Apsides of Orbits of great 

excentricity : Apr. 17, 1820; i. 179—191. 



Whewell, Dr., On the double Crystals of Fluor Spar: 
Nov. 26, 1821 ; l. 331—342. 

On the Rotatory motion of Bodies: May 6, 1822; 

II. 11—20. 

On the Angle made by two Planes, or two straight 

lines, referred to three oblique Co-ordinates: 
Nov. 24, 1823; n. 197—202. 

Note on Mr Cecil's memoir on Grinding Mirrors, 

&c.: Dec. 11, 1822: ii. 100—103. 

On Crystalline Combinations: Nov. 13, 1826; li. 

391—425. 

On a Notation to de.signate the Planes of Crys- 

tals: Feb. 11, 1826; n. 427—439. 

A Mathematical Exposition of some doctrines of 

Pohtical Economy: March 2, 14, 1829; ill. 
191—230. 

Second memoir on the same subject : ix. Apr. 15, 

1850; 123—149. 

Third memoir on the same subject : Nov. 11, 1850 ; 

IX. pt. ii. 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; vl 301—315. 

Demonstration that all Matter is heavy : Feb. 22, 

1841 ; viL 197—207. 

Discassion whether Cause or Effect are simul- 

taneous or successive: March 14, 1842; vii. 
319—331. 

On the Fundamental Antithesis of Philosophy : 

Feb. 5, 1844; vin. 170—181. 

Second memoir on the same subject : Nov. 13, 

1848; vin. 614—620. 

On the Intrinsic Equation to a Curve, &c., &c. : 

Feb. 12, 1849 ; vin. 659—671. 

Second memoir on the same subject: Apr. 15, 

1850; IX. 150—156. 

On Hegel's Criticism of Newton's Principia: 

May 21, 1849; vill. 696—706. 

On Aristotle's account of Induction: Feb. 11, 

1850; IX. 63—72. 

On the Transformation of Hypotheses in the 

History of Science: May 19, 1851 ; ix. pt. ii. 
139—146. 

On Plato's Survey of the Sciences : Apr. 23, 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; is. 598—604. 

Of the Platonic Theory of Ideas: Nov. 10, 1856; 

X. 94—104, 



VIU 



INDEX TO TRA.NSACTIONS I— XII. 



Willis, Prof., On the pressure jn-oJuccd by a stream 
of Air on a flat plate, &c. &c. : Apr. 21, 1828; 

III. 129—140. 

On Vowel Sounds; and on Reed Organ- Pipes: 

Nov. 24, 1S2S; March 16, 1829; iii. 231—268. 

On the Mechanism of the Larynx: May IS, 1829; 

IV. 323—352. 



Young, J. R., On the Principle of Continuity, in refer- 
ence to Analysis : Dec. 7, 1846 ; viil 429 — 
440. 

Young, M., Piscium Maderensium Species, Iconibus 
illustratoD: Nov. 10, 1834; vl 195. 



II. INDEX OF SUBJECTS. 



Aberration in the Eye-pieces of Telescopes : May 14, 21, 

1827 ; III. 1—58. 
Achromatic Eye-pieces, and Achromatism : May 17, 

1824; II. 227—252 ; ill. 59—63. 

Object-Glass, New Correction for: Apr. 30, 1838; 

VI. 553—564. 
Addenbrooke's Hospital, Report on, for 1836 : March 13, 
1837 ; VL 361—377. 

Ditto, Ditto, for 1837: Apr. 30, 

1838; VI. 565—575. 

Al Farsi, Astronomical Tables of: Nov. 13, 1820; J. 

249—265. 
Algebra, Foundations of. Part I.: Dec. 9, 1839; vn. 

173—187. 

Ditto, Ditto, II.: Nov. 29, 1841; vn. 

287—300. 

Ditto, Ditto, III.: Nov. 27, 1843; VUL 

1.39—142. 

Ditto, Ditto, IV.: Oct. 28, 1844; viii. 

241—2.54. 

Algebraic Equations, Geometrical representation of their 
Roots: Apr. 27, 1846; vin. 342—360. 

Notation: Nov. 12, 1827; liL 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, Obsen-ations with a new: May 1, 1837; 
VI. 301—315. 

Angle, Memoir on, as referred to three oblique Co- 
ordinates: Nov. 24, 1823; ll. 197—202. 

Anglesea, Geological description of: Nov. 26, 1821 
L 359—452. 

Animals, Occurrence of Extinct, near Cambridge 
Apr. 30, 1838; viil. 1.38—140. 

Annuities, Calculation of, with Tables : May 26, 1828 
in. 141—154. 

Compari.son of various Tables of ; March 30, 1829 

IIL 321—341. 
AntitbesLs, Fundamental, of Philosophy : Fob. 5, 1844 
viu. 170—181. 



Antithesis, Second memoir: Nov. 13, 1848; viii. 614 

—620. 
Apophyllite, Refraction in Rays from: May 7, 1821; 

L 241—247. 
Apsides of Orbits of Great Eccentricity : April 17, 1820 ; 

L 179—191. 
Arbitrary Constants, Discontinuity of, &c. : May 11, 

1857 ; X. 105—128. 

Ditto, Ditto, Supplement to this memoir : May 

25, 1868; XL 412 — 425. 
Arch, Equilibrium of the : Dec. 9, 1833; v. 293—313. 
Argument, Use and meaning of the word : Nov. 28, 

1859; X. 317—326. 
Aristotle's account of Induction: Feb. 11, 1850; ix. 

63—72. 
Atmospheric temperature, Decrement of : Feb. 13, 1837 ; 

VL 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 ; vill. 320 — 

325. 

of Nov. 17, 1848: Nov. 27, 1848; vin. 621—632. 

Axes, Moveable, On Velocities, &c., &c., relative to : 

Feb. 25, 1856; x. 1—20. 

Barometer, Extraordinary Depression of: Feb. 25, 1822 ; 

L 453^458. 
Beams and Elastic Rods, Theory of: Apr. 23, 1860; 

X. 359—373. 
Beaver, Fossil Remains of: March 6, 1820; l 175—177. 
Bernouilli's Numbers, Tables of, &c., &c. : May 29, 

1871 ; XII. 384—391. 
Biliary Secretion, Nature of : March 6, 1843 ; viii. 44 

—49. 
Bivalve, see Univalve. 
Bode's Law, Extension of, to Satellites : Dec. 8, 1828; 

in. 171—183. 
Boltzmann, see Material points. 
Boracic Acid, on the Crystals of : Nov. 30, 1829 ; in. 

365—367. 



II. INDEX OF SUBJECTS. 



IX 



Brachiopod, see Univalve. 

Brazilian Topaz, Colour, Structure and Optical Pro- 
perties of: May 6, 1822; il. 1—9. 

Calculus of Functions, Notation employed in : May 1, 
1820; I. 63—76. 

Human, specimen of : Nov. 26, 1821 ; i. 347—349. 

Cambridgeshire, On the Ornithology of : Nov. 28, 1825; 

II. 287—324. 

List of Reptiles found in: Feb. 22, 1830; in. 

373—381. 
Cassius, Constituents of Purple Precipitate of : May 15, 

1820; I. 53—61. 
Cause and Effect, simultaneous or successive : March 14, 

1842; Til. 319— 331. 
Caustic, Intensity of Light near : May 2, 1836, March 26, 

1838; VI. 379—402. 

- Ditto, Supplement to this memoir : May 8, 
1848; VIII. 595—599. 

Chances, Some points in the Theory of : May 26, 1828 ; 

III. 141—154. 

Chiasognathus Grantii, Description of: May 16, 1831; 

IV. 209—217. 

Clock Escapements: Nov. 27, 1848; viii. 633—638. 

Improvements in: Feb. 7, 1853; ix. 417 — 430. 

Turret Remontoirs : Feb. 26, 1849; viil. 639—641. 

Colours of Thick Plates: May 19, 1851; is. [147— 

176.] 
Combinations and Permutations: March 1, 1847; viii. 

471—492. 
Comet, Observations of Halley's: Dec. 11, 1837; vi. 

493—506. 
ComiJosition of Forces, General Principles of : March 14, 

1859; X. 290—304. 
Conic Section, Focus of: March 2, 1829; lii. 185—190. 
Consonances, Beats of Imperfect: Nov. 9, 1857; x. 

129—145. 
Continuity, Principle of, with reference to Analysis : 

Dec. 7, 1846 ; viii. 429—440. 
Co-ordinates, Six of a Line: Nov. 11, 1867; xi. 290 — 

323. 

Curvilinear and Normal: May 22, 1876; xil. 455 

—522. 

Ditto, Ditto, May 7, 1877; sii. 531 

—545. 

Copper, Analysis of Phosphate of : March 5, 1821 ; i. 

203—207. 
Cornwall, Fcssil Shells: Feb. 16, 1838; vi. 415—422. 

Lizard District of : Apr. 2, May 7, 1821 ; i. 291— 

330. 

Primitive Ridge of: March 20, 1820; i. 89—146. 

Crotalocrinus rugosus : Feb. 8, 1869; xi. 481 — 484. 
Crystalline Combinations, on their Classification, Nov. 

13, 1826 ; II. 391—425. 
Crystallization of Water : March 5, 1821; i. 209—215. 

Vol. XII. 



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 2 1 , 

1836 ; vil. 209—215. 

Planes of, on a Notation to designate: Feb. 11, 

1826; II. 427—439. 

as affecting Planes of Polarization : Apr. 17, 1820; 

I. 43—52. 

found in Slags: May 22, 1830; III. 417—420. 

Variation in Tints developed by: May 1, 1820; 

I. 21—41. 
Cubic Cones and Curves: Apr. 18, 1864; xi. 129—144. 
Cubic Curves, Involution of: Feb. 22, 1864; xi. 39 

—80. 

Classification of: Apr. 18, 1864; xi. 81—128. 

Siu'face, with 27 lines, by Dr Wiener: May 15, 

1871 ; XII. 366—383. 
Curve, Intrinsic Equation to: Feb. 12, 1849; viii. 659 
—671. 

Second memoir : Apr. 15, 1850; ix. 150 — 156. 

Rolling: Dec. 10, 1838; vil. 61—86. 

of the Second Degree, General Equation to : 

Nov. 15, 1830; IV. 71— 78. 

Singular points of: May 21, 1855; IX. 608—627. 

Cyclas and Pisichum, on the British Species of : Nov. 28, 

1831 ; IV. 289—312. 
Cyhndrical Tubes, Aerial Vibrations in : May 20, 1833; 
v. 231—270. 

Decrement of Atmospheric Temperature : Feb. 1 3, 1 837 ; 

VI. 443—455. 
Definite Integi'als, Inverse method of, with applications : 

March 5, 1832; iv. 353—408. 

Ditto, Ditto, Ditto, Nov. 11, 

1833; V. 113—148. 

Ditto, Ditto, Ditto, March 2, 

1835 ; V. 315—393. 

Numerical calculation of: March 11, 1850; ix. 

166—187. 

Properties of: May 24, 1830; ill. 429—443. 

Determinants, Theory of: Feb. 20, 1843; viii. 75—88. 
Devonshire, Primitive Ridge of: Mar. 20, 1820; I. 

89—146. 
Difierential Equations, Theory of : March 27, 1854; ix. 
515—554. 

Supplement to this paper: Apr. 28, 1856; x. 

21—26. 
Diflfraction, Dynamical Theory of; Nov. 26, 1849; ix, 
1—62. 

74 



INDEX TO TRANSACTIONS I— XTT. 



Diffraction, Xe«-ton's Espci-iiueuts on : May 7, 1833 ; v. 
101—111. 

of ail Object-glass with Circular aperture : Xov. 24, 

1834; V. 283—291. 

Ditto, Ditto, Triangdar Ditto: Dec. 12, 

1836; VI. 431—442. 

Digitalis, Hybrid: Nov. 14, 1831 ; iv. 257—278. 
Discoutinuous Constants, Use of: May 16, 1836; vi. 

185—193. 
Dispersion of Light, Hypothesis for : Feb. 22, 1836 ; 

VI. 153—184. 
Divergent Developments, see Arbitrary Constants. 

Series: March 4, 1844; viil. 182—203. 

Double-Sixer, Construction of: May 15, 1871; xir. 

366—383. 
Durham, see Yorkshire. 

E;irth and Planets, External Temperature of; ISIarch 21, 

1855 ; IS. 628—672. 
Earth, Theory of the Figiu-e of; Oct. 16, 1871 ; xil. 301 

—318. 

Inequalities in the Surface of: Dec. 1, 1873; 

XII. 414 — 133. 
Ebenacea;, Monograph of: March 11, 1872; sil. 27 — 

300. 
Elastic Beams, Deflection of, &c. : March 11, 1850; IX. 

[177—190.] 

Impact of: Dec. 10, 1849; ix. 73—78. 

Fluids, Vibratory Jlotions of: March 30, 1829; 

III. 269—320. 

Medium, Effect of Vibrations on a Sphere : 

April 26, 1841 ; vii. 333—353. 

Medium, Vibratory Motion of: March 15, 1847; 

VIII. 508—523. 

Rods, see Beams. 

Solids, Motion of; Apr. 14, 1845; viii. 287— 

319. 
Electric Fluid, Equilibrium of Fluids analogous to : 

Nov. 12, 1832; v. 1—63. 
Electricity, Origin of: Dec. 7, 1863; xi. 6—20. 
Electro- MagnetLsm, Development of by Heat: Apr. 28, 

1823 ; II. 47—76. 
Elevation of Mountains by Lateral Pressure : Apr. 27, 

1868 ; XL 489—506. 
Elimination between Unknown Quantities : Nov. 26, 

1832; V. 65—75. 
Ellipsoid, Centro-Surface of: March 7, 1870; xii. 319 

—365. 
Ellip8oid.s, Exterior and Interior attractions of: May 6, 

1833; V. 395—429. 
Endosmose and Exosmose, Explanation of: March 17, 

1834 ; V. 205—229. 
EquaLty, Sign of: May 16, 1864; xi. 145—189. 
Equation, Algebraic, Proof of a root in everj- : Dec. 7, 

1857 ; X. 261—270. 



Equation, Algebraic, Another proof: Dec. 6, 1858; x. 
283—289. 

Ditto, Supplement to this memoir ; Dec. 

12,1859; X. 327—330. 

relating to the breaking of bridges : May 21, 1849 ; 

viii. 707—735. 

to a Curve, The Intrinsic; Feb. 12, 1849; viii. 

659—671. 

Ditto, Second memoir: IX. 150 — 160. 

General, to Surfaces of the second degree : Nov. 12, 

1832; V. 77—94. 

Integration of Partial differential : June 5, 1848 ; 

VIII. 606—613. 

Machine for resolving by Inspection ; May 7, 1832 ; 

IV. 425—440. 
Equihbrium of the Arch: Dec. 9, 1833; v. 293—313. 

of Bodies in Contact : May 15, 1837 ; vi. 463—491. 

Molecular: M;iroh 26, 1838; vil. 25—59. 

Erratic Blocks, Transport of: Apr. 29, 1844; vili. 

220—240. 
Errors of Observation, Theory of; Nov. 11, 1861; x. 

409—427. 
Escapements, Theory of : Nov. 26, 1826; iii. 105—128. 
Exponents, Newton's method of: May 21, 1855; ix. 

COS— 627. 
Extinct Lacertian Reptile, Traces of: April 11, 1842; 

VII. 355—369. 
Eye, Change in the State of: May 25, 184C; vin. 361 

—362. 

Defect in, and how cured; Feb. 21, 1825; il. 

267—271. 

Further observations on: Feb. 12, 1872; xii. 

392, 3. 

Fauna and Flora of Madeira; Nov. 15, 1830; iv. 1—70. 
Figure assumed by a Fluid Homogeneous Mass : 

March 15, 1824; ll. 203—216. 
Finite Differences, Resolution of Equations in : Nov. 15, 

1835; VI. 91—106. 
Flora of Madeira, Notes and Gleanings : May 28, 1838 ; 

VI. 523—551. 
Fluid Motion, on : March 21, 1836 ; vi. 203—233. 

Ditto, On some cases of: May 29, 1843; viil. 

105—1.37. 

Ditto, Supplement to this memoir: Nov. 3, 

1846 ; VIII. 409—414. 
Fluids, Equilibrium of Certain: Nov. 12, 1832; v. 
1—63. 

General Equations of the motion of, &c., &c. ; 

Feb. 22, 1830; in. 383—416. 

Motion of Incompressible ; April 25, 1842 ; vii. 

439—453. 

Theory of the motion of; v. 173—203. 

in motion. Internal Friction of; Apr. 14, 1845 ; 

VIII. 287-319. 



II. INDEX OF SUBJECTS. 



XI 



Fluids, Effect on Pendulums of Internal Friction of: 

Dec. 9, 1850; is. [8—106.] 
Fluor Spar, Double Oj-stals of: Nov. 2G, 1821 ; I. 331 

—342. 
Focus of a Conic Section : JIarch 2, 1829 ; iii. 185—190. 
Force, Faxaday'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 : 

JIarch 6, 1820; I. 175—177. 

Shells, New Genus of: Feb. 26, 1838; vi. 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; I. 77 — 87. 

Galvanism, as connected with Magnetism : Apr. 2, 1821 ; 

I. 269—279. 
Gas, Hydrogen, as a moving Power in Jlacliinery : 

Nov. 27, 1820; 1.217—239. 

on Sounds excited in ; Apr. 2, 1821 ; I. 267, 268. 

Gastric Fluids, Solvent effect of, on the Stomach after 

Death: Dec. 11, 1820; I. 287—290. 
Geodesy, Geometrical Formidje applicable to : Nov. 30, 

1835 ; VI. 107—140. 
Geology, Researches iu Physical : May 4, 1835 ; vi. 

1—84. 
Geometry and Mechanics, Symbolical : March 15, 1847 ; 

VIII. 497—507. 

Substitution of, for the doctrine of Proportions : 

Dec. 7, 18.57; x. 166—172. 
Gipping, Geology of the Valley of: Feb. 27, 1854; ix. 

431—444. 
Glaciers, Motion of: May 1, 1843; viii. 50 — 74. 

Ditto, Dec. 11, 1843; vm. 1.59— 169. 

Globe, Relative Quantities of Land and Water on : 

Feb. 13, 1837; VI. 289—300. 
Going-Fusee, New Construction of: March 2, 1840; 

VII. 217—225. 

Gravity, Variation of, at the Earth's Surface : Apr. 23, 

1849; vm. 672—695. 
Great Circle Sailing : May 10, 1858 ; x. 271—282. 
Greek Literature, First Ages of written : Nov. 23, 1868 ; 

XL 461—430. 

Halley's Comet, Observations of: Dec. 11, 1837; vi. 

493—506. 
Heat, see ilotion of Particles. 
Hegel's criticism of Newton: May 21, 1849; viil. 696 

—706. 
Homeric Tumuli: March 12, 1866; xi. 267—276, 
Human ilonstrosity. Case of, with Comnieutary : 

May 16, 1831 ; rv. 219—255. 



Hybrid Digitalis, Examination of: Nov. 14, 1831 ; iv. 

257—278. 
Hydrodynamical Theorem, Investigation of: May 9, 

1842 ; VII. 455—464. 
Hydrodynamics, New Fundamental Equation in : 

March 6, 1843 ; \^^. 31—43. 
Hyperbolic Law of Elasticity: March 11, 18.50; ix. 

[177—190.] 
Hypotheses, Transformation of: May 19, 1851 ; ix. 

[139—146.] 

Ideas, Platonic Theory of: Nov. 10, 1856; x. 94 — 104. 
Iliad and Odyssey, Late date, &c., &c., of: Nov. 26, 

1866 ; XI. 360—386. 
Induction, Ari.stotle's account of: Feb. 11, 1850; IX. 

63—72. 
Infinite Angle, Sine and Cosine of: Dec. 9, 1844; vm. 

255—268. 

Series, Use of Discontinuous Constants in, &c., 

&c.: May 16, 1836; vi. 185—193. 

Ditto, General Term for a new Class of: May 3, 

1824; II. 217—22.5. 
Infinity, On: May 16, 1864; xi. 145—189. 
Inscription, Metrical Latin, from Algeria : Feb. 13, 

1860; X. 374—408. 
Integral Calcidus, On some points of: Feb. 24, 1851 ; 

IX. [107—1.38]. 
Integrals, General Properties of Definite : May 24, 1830 ; 

III. 429—443. 

Inverse method of, with Applications : March 5, 

1832; IV. 353—408. 
Internal friction of fluids: Apr. 14, 1845; vm. 287 — 

319. 
Involution, Theory of; Feb. 22, 1864; xi. 21—38. 
Involution of Cubic Curves: Feb. 22, 1864; xi. 39—80. 

Jaws, Growth of: Nov. 9, 1863; xi. 1 — 5. 

Knowledge of Body and Space: March 11, 1850; ix. 
157—16.5. 

Laminated Pressure of Rock Masses: May 3, 1847; 

■^^II. 456 — 470. 
Land, see Globe. 
Laplace, on his Theory of the Attraction of Spheroids : 

May 8, 1826 ; n. 379—390. 
Lapland, Arc of the Meridian measured in : May 1, 

1871 ; XII. 1—26. 
Larynx, On the Mechanism of: May 18, 1829; iv. 

323-352. 
Latitude of Cambridge Observatory: Apr. 14, 1834; 

, v. 271—281. 
Least Squares, Method of: March 4, 1844; VIII. 204— 

219. 
— - Ditto, Ditto : May 29, 1865 ; xi. 219—238. 

74—2 



XII 



INDEX TO TRANSACTIONS I— XII. 



Light, Nature of, from the Double Refraction of 
Quartz: Feb. -21, 1831; iv. 79—123. 

Nature of, from the Double Refraction of Quartz : 

Apr. 18, 1831 ; IV. 199—208. 

on the Disiiersion of: Feb. 22, 183G; \l. 153—184. 

Transmission of, in ccrtsvin Me<Iia : Feb. 13, 1837 ; 

VI. 323—352. 

Ditto, Supplement: May 1, 1837; vi. 

353— 3G0. 

Intensity of, near a Caustic : May 2, 1836, 

March 26, 1838 ; vi. 379—402. 

Ditto, Supplement to this memoir: May 8, 

1848 ; viii. 595—599. 

Reflection and Refi'action of, &c. : Dec. 11, 1837; 

VII. 1—24. 

Ditto, Supplement to tliis memoir : May 6, 

1839; ■^^I. 113—120. 

Projiagation of, in Crystallized Media : May 20, 

1839; VII. 121—140. 

Quantity of, &c., absorbed by a Grating placed 

before a Lens: March 30, 1840; vii. 153—171. 

Reflection and Refraction of: Nov. 28, 1842; 

VIII. 7—26. 

Absorption of, &c. : Feb. 14, 1843 ; viii. 27—30. 

Transmission tlu-ough Transparent media : Jlay 

17, 1847 ; VIII. 524—532. 

Polarized: Dec. 8, 1851 ; ix. 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; xil. 434—454. 
Lizard district of Cornwall, Physical Structure of: 

Apr. 2, May 7, 1821; I. 291—330. 
Logic, in general: Feb. 8, 1858; x. 173—230. 

of Relations: Apr. 23, 1860; x. 331—358. 

Symbols of, &c., &c.: Feb. 25, 1850; ix. 79—127. 

Longitude of Cambridge Observatory : Nov. 24, 1828 

in. 155 — 170. 

Ditto, Ditto, May 15, 1854 

IS. 487—514. 

Luminiferoua Ether, Constitution of: March 18, 1839 

VII. 97—112. 
Luminous Rays, Theory of: March 11, 1846; viii. 

363—378. 

Vibratioas, Theory of: March 6, 1848; viii. 584 

—594. 

Waves, Propagation of: April 2.5, 1842; vii. 397 

—437. 

Machine for resolving Equations: May 7, 1832; iv. 
425 — 140. 

Machinery, Influence of, on the Wealth of a Com- 
munity : May 14, 1838 ; vi. 507—522. 

Madeira, Fauna and Flora of: Nov. 15, 1830; iv. 1—70. 

Fishes of: Nov. 10, 1834; vi. 195—201. 



Madeira, Flora of, Notes and Gleanings : May 28, 1838 ; 

VI. 523—551. 
Magnetic Intensity, observed Variations of: 1825; li. 

445. 

Needles, as aSccted by Masses of Iron: May 16, 

1820; I. 147—173. 
Magnetism, Connection of, with Galvanism : Apr. 2, 
1821 ; I. 269—279. 

as a Measure of Electricity: May 21, 1821; i. 

281—286. 

evolved by a single Galvanic Combination, Ex- 

tract from Memoir on: Nov. 25, 1822; ll. 

77—83. 
Magnitude and Direction, Pure Science of: May 12, 

1845; VIII. 278—286. 
Material points. Energy in a system of : May 6, 1878 ; 

XII. 547 — 570. 
Mathematical Reasoning, Influence of Signs on : Dec. 16, 

1821 ; II. 325—377. 
Matter, Demonstration that it is heavy : Feb. 22, 1841 ; 

VII. 197—207. 

Remarks on the Theory of: May 22, 1848; viii. 

600—605. 
Mechanics and Geometry, Connection between : Feb. 10, 

1845 ; VIII. 269—277. 
Microscope, Improvement of: March 22, 1830; in. 

421 — J28. 
Mirrors and Object-Lenses, Apparatus for Grinding : 

Dec. 11, 1822; ii. 85—103. 

Use of Silvered Glass for: Nov. 25, 1822; n. 

105—118. 
Molecular Equilibrium : March 26, 1838 ; vii. 25—59. 
Mon-strosity, Human, Case of: May 16, 1831; iv. 219 

—255. 

of the Common Mignionettc : May 21, 1832; v. 

95—100. 
Motion of Fluids, on the : Nov. 24, 1828 ; in. 383—416. 

Ditto, Ditto, March 3, 1834; v. 173—203. 

of Fluids, Difiereutial Equations to: April 11, 

1842; vn. 371—396. 

Incompressible: April 25, 1842; vn. 439 — 453. 

Truth of the Laws of: Feb. 17, 1834; v. 149— 

172. 

of Particles, as affecting Sound and Heat : May 16, 

1836 ; VI. 235—288. 

of Waves in a small Canal: May 15, 1837; vi. 

457—462. 

of Waves in Canals: Feb. 18, 1839; vn. 87—95. 

Motive Power, Hydrogen Gas as a : Nov. 27, 1820 ; 

I. 217-239. 
Mountains, Elevation of, by Lateral Pressure : Apr. 27, 
1868 ; XI. 489—506. 

Second memoir: Feb. 22, 1875; xn. 434—454. 

Music in Education, place of, according to Aristotle : 

May 17, 1875; xn. 523—530. 



II. INDEX OF SUBJECTS. 



xni 



Natron, remarkable deposit of: Nov. 27, 1820; I. 193 

—201. 
Natterjack, Habits and Character of: Feb. 22, 1830; 

III. 373—381. 
Neptune, .see Uranus. 

Neutral Series, Theory relating to; May 16, 1864; xi. 
190—202. 

Ditto, Note on this paper: Oct. 26, 1868; xi. 

447 — 160. 
Newton's method of Co-ordinated Exponents : May 21, 
1855 ; IX. 608—627. 

Experiments on Diffraction: May 7, 1833; v. 

101—111. 

Principia, Criticism of: May 21, 1349; viii. 696 

—706. 
. — — Rings, Remarkable change in: Nov. 14, 1S31 ; 

IV. 279—288. 

On some Phenomena of: March 19, 1832; 

IV. 409—424. 

Central spot of: Dec. U, 1848; viii. 642 



—658. 

See Hegel. 

Non-Residence of Landlords, Influence of: March 16, 

1840; VII. 189—196. 
Notation employed in the Calculus of Functions : May 1, 

1820; I. 63—76. 

Algebnuc: Nov. 12, 1827; iii. 65—103. 

to designate the Planes of Crystals: Feb. 11, 

1826; II. 427—439. 

a New, iu Geometry, &c., &c. : Nov. 23, 1846; 

vni. 415—428. 
Numbers, Partition of: March 1, 1847; viii. 471 — 492. 

Object-Glass with circular aperture, Difiraction in : 
Nov. 24, 1834; v. 283—291. 

with triangular aperture, Difiraction in : Dec. 12, 

1836; VI. 431—442. 

Achromatic, Correction for: Apr. 30, 1838; vi. 

553—564. 
Observatory at Cambridge, Longitude of: Nov. 24, 
1828; m. 155—170. 

Ditto, Latitude of: Apr. 14, 1834; 

V. 271—281. 

Ditto, Longitude of, by Galvanic 

Signals: May 15, 1854; ix. 487—514. 

Odyssey, see Iliad. 

Onymatic System, on various points of: May 4, 1863; 

X. 428—487. 
Optical Elasticity, Axes of in certain Crystals : Dec. 8, 

1834 ; v. 431—438. 

Ditto, Ditto, (second memoir) : March 21, 

1836 ; VII. 209—215. 
Orbits of great Exceutricity, Position of their Apsides : 

Apr. 17, 1820; i. 179—191. 
Ordnance, Strains upon: Apr. 18, 1864; xi. 324 — 359. 



Ornithology of Cambridgeshire: Nov. 28, 1825; ii. 

287—324. 
Oscillations, on small Finite: May 15, 1843; viii. 89 

—104. 

of a suspension Chain: Dec. 8, 1851; ix. 379 — 

398. 
Oscillatory Waves, Theory of: March 1, 1847; viil. 
441—455. 

Parallelogram of Forces: New Demonstration of: Apr. 

14, 1823; II. 45—46. 
Partial difierential Equations, Method of integrating : 

June 5, 1848; viil. 606—613. 
Pendulum, Correction of, by a Ball suspended by a 

wire: Nov. 16, 1829; ill. 355—360. 
Pendulums, Disturbances of: Nov. 26, 1S26; in. 105^ 

128. 

Effect of Internal Friction on: Deo. 9, 1850; ix. 

[8—106.] 
Percussion, Experiments on : 1825 ; ll. 444. 
Periodic Serie.s, Critical Values of: Dec. 6, 1847; viii. 

533—583. 
Perpetual Motion, How possible : Dec. 14, 1829 ; in. 

369—372. 
Perspective, Isometrical: Feb. 21, Mar. 6, 1820; i. 

1—19. 
Philosophy, Fundamental Antithesis of: Feb. 5, 1844; 

VIII. 170—181. 

Second memoir: Nov. 13, 1848; viil. 614—620. 

Phosphate of Copper from the Rhine : March 5, 1821 ; 

I. 203—207. 

Physical Geology, Researches in: May 4, 1835; vi. 

1—84. 
Piscidium, see Cyclas. 
Piscium Maderensium Species, &c., &c. : Nov. 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 ; ix. 590—597. 

Ditto, of the Intellectual Powers: Nov. 12, 1855; 

IX. 598—604. 

Genuineness of the Sophista of: Nov. 23, 1857; 

X. 146—165. 

Cosmical system : Feb. 28, 1859 ; x. 305—316. 

Platonic Theory of Ideas: Nov. 10, 1856; x. 94—104. 
Plumbago, on the artificial formation of: Feb. 21, 1825; 

II. 441 — 143. 

Polarity, Organic : March 8, 1858 ; x. 248—260. 
Polarization, Use of a new Analyzer in : March 5, 1832 ; 

IV. 313—322. 
Polarized Light, Certain effects in Crystals exposed to : 

May 1, 1820; I. 21^1. 
Ditto, as affected by Rotation : Apr. 1 7, 

1820; I. 43—52. 



XIV 



INDEX OF TRANSACTIONS I— XII. 



Polarized Light, Composition aiul Resolution of: 

Feb. 16, M.vch 15, 1852 ; is. 390—416. 
Political Economy, Mathematical discussion of: March 

2, 14, 1S29 ; III. 191—230. 
Ditto, as expouiulod by Ric.ardo. First 

memoir: Apr. 18, May 2, 1S31 ; iv. 155 — 

198. 
Ditto, Mathematical Theory of, Second 

memoir: Apr. 15, 1850; is. 128—149. 
Ditto, Ditto, Third memoir: Nov. 11, 

1850; IS. [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 ; III. 129—140. 
Primitive Ridge of Devonshire and Cornwall : March 20, 

1820; I. 89—146. 
Probabilities, Question in the Theory of: Feb. 26, 1837 ; 

n. 423 — J30. 

Foundation of Ditto: Feb. 14, 1842; viii. 1—6. 

Fundamental principle of the Theory of: Nov. 13, 

1854; IS. 605—607. 
Proportions, see Geometry. 
Propositions numerically definite: March IC, 1868; 

XI. 396—411. 
Purbeck Strata of Dorsetshire: Nov. 13, 1854; ix. 

555 — 581. 

Quartz, Nature of the Light produced by : Feb. 21, 
1831 ; IV. 79—123. 

Ditto, Ditto, Ditto, Apr. 18, 

1831 ; IV. 199—208. 

Railway Accidents, Causes of Fatal, &c. : Nov. 29, 1841 ; 

VII. 301—317. 
Railway Bridges, Equation relating to their breaking: 

May 21, 1849; viu. 707—735, 
Rainbow, Problem of, Mathematically considered: 

Dec. 14, 1835; vi. 141—152. 
Rainbows, Spurious : March 22, 1841 ; vii. 277—286. 
R«ed Organ Pipes: Nov. 24, 1828, March 16, 1829; 

III. 231—262. 
Reflection and Refraction of Light: Dec. 11, 1837; 

VII. 1—24. 

Supplement to this memoir : May 6, 1839 ; 

vn. 113—140. 

Ditto, Ditto, &c.: Nov. 28, 1842; 

vra. 7—26. 

Refraction, Theory of Double: May 17, 1847; viii. 

524—532. 
Kicardo, tee Political Economy. 
Rock Masses, Internal pressure of: May 3, 1847; viii. 

4.56—470. 
Rocks, Weathering of: March 2, 1868; sj. 387—395. 
Boot of any Function : May 7, 18C6 ; XL 239—266, 



Root-limitation, C'auchy's Theorems of: Feb. 16, 1874; 

SII. 395—414. 
Rotatory Motion of Bodies: May 6, 1822; ll. 11—20. 

Secular Cooling of the Eai-th : Dec. 1, 1873; sil. 414 

— 133. 
Series, on Divergent: March 4, 1844; viu. 182—203. 

Critical Value of Periodic: Dec. 6, 1847; vill. 

533—583. 

Numerical calculation of Infinite : March 11, 1850 ; 

IX. 106—187. 

Self-repeating: May 15, 1854; ix. 471—486. 

Theorem on Neutral : May 16, 1864 ; xi. 190—202. 

Ditto, Part II.: May 7, 1866; xi. 239—266. 

Note on Ditto: Oct. 26, 1868; xi. 447—460. 

Sextic Torse, On a certain: Nov. 8, 1869; xi. 507 — 

523. 
Shells, Occurrence of, in Gravel: Apr. 30, 1838; viu. 

138-140. 
Signs, Influence of, in Mathematical Reasoning ; Dec. 16, 

1821 ; II. 325—377. 

+ and -, Early History of: Nov. 28, 1S04; xi. 

203—212. 

Note on this Memoir : Feb. 13, 1865 ; si. 213—218. 

Skew Sui-faces, or Scrolls : Nov. 11, 1867 ; xi. 277—289. 
Slags, Crystals found in : March 22, 1830 ; in. 417 — 420. 
Solid Bodies, Vibrations of: Apr. 18, 1864 ; si. 324—359. 
Solitary Waves, Mathematical Theory of; Deo. 8, 1845 ; 

VIII. 326—341. 
Solon, Statue of: Feb. 22, 1858; x. 231—239. 
Sound, Experiments on the Velocity of: Dec. 8, 1823; 

II. 119—137. 

sec Motion of Particles. 

Reflection and Refi-action of: Deo. 11, 1837; 

VI. 403—413. 
Spar-Fluor, Double Crystals of: Nov. 26, 1821; I. 

331— .342. 
Spermaceti Whale, account of: May 16, 1825; ii. 

253—266. 
Sphere, Motions ot, acted on by Vibrations of an Elastic 

Me(hum: April 26, 1841; vil. 333—353. 
Spherical Aberration in Eye-pieces of Telescopes : May 

14, 21, 1827; iii. 1—63. 

Spheroids difleriug little from a Sphere, on Laplace's 
Theory of; May 8, 1826; II. 379—390. 

Squares, Method of Least: March 4, 1844; viii. 201 
—219. 

Ditto, Ditto : Maor29, 1865; si. 219— 238. 

Suffolk, see Gipping. 

Surfaces of the second degree, Equation to : Nov. 1 2, 
1832 ; V. 77—94. 

Transformation of, by Bending : March 13, 1854 ; 

15. 445—470. 

Suspension Chain, Oscillations of: Dec. 8, 1851 ; is. 
379—398. 



II. INDEX OF SUBJECTS. 



XV 



Switzerland, Tertiary Formations of: May 20, 1839 

VII. 141—152. 

Syllogism, Theory of the structure of: Nov. 9, 1846 

VIII. 379—408. 



rt. II.: Feb. 25, 1850 
Pt. III.: Feb. 8, 1858 
Pt. IV.: Apr. 23, 1860 
Pt. v.: May 4, 1863 



Ditto, Ditto, 

IX. 79—127. 

Ditto, Ditto, 

X. 173—230. 

Ditto, Ditto, 

X. 331—358. 

Ditto, Ditto, 

X. 428^87. 

Symbolical Geometry and Mechanics: March 15, 184 
VIII. 497—507. 

Tertiary Formations of Switzerland : May 20, 1839 ; 

VII. 141—152. 
Testimony, Measure of Force of: Nov. 27, 1843; vill. 

14.^—158. 
Theory of Probabilitie.s, Question in : Feb. 26, 1837 ; 

VI. 423—430. 
Topaz, see Brazilian Topaz. 
Transcendental Equations, Machine for resolving : May 7, 

1832; IV. 425— 440. 
Trap Dykes in Yorkshire and Durham : May 20, 1822 ; 

II. 21—44. 

Rocks, as associated with Mountain Limestone : 

May 12, 1823: March 1, 15, 1824; ll. 139— 
195. 
Trinomial, Resolution of a certain: Nov. 9, 1868; XI. 
426—443. 

Note on this memoir: Nov. 23, 1868; xi. 444, 

445. 
Trireme, Structure of the Athenian: Nov. 6, 1856; 

X. 84—93. 
Tumuli, Homeric: March 12, 1866; xi. 267—276. 

Undulations, Theory of, applied to Luminous Waves : 

May 25, 1846 ; vrii. 371— 37a 
Univalve, Relations of, to the Bivalve, and to the 

Brachiopod : Feb. 8, 1869 ; xi. 485—488. 
Uranus and Neptune, Long Inequahty of: 1852; ix. 

Appendix. 
Ureters, Dilatation of: Nov. 12, 1821 ; i. 351—358. 



Velocities, &c., referred to Moveable Axes : Feb. 25, 

1856; X. 1—20. 
Velocity of Sound, Experiments on: Dec. 8, 1823; ll. 

119—137. 
Vibrations in Cylindrical Tubes: May 20, 1833; v. 

231—270. 

Theory of Luminous : March 6, 1848; viii. 584 — 

594. 

of Solid Bodies: Apr. 18, 1864; xi. 324—359. 

Viljratory Motion of Elastic Medium : March 15, 1847 ; 

Vlll. 508—523. 
Virtual Velocities, Demonstration of their principle : 

March 21, 1825 ; II. 273—276. 
Vision, Peculiar defect in: Nov. 9, 1846, May 17, 1847; 

VIII. 493—496. 
Voluntary Muscles, Abnormities in : March 8, 1858 ; r. 

240—247. 
Vowel Sounds, On the : Nov. 24, 1828, March 16, 1829 ; 

III. 231—268. 

Water, Crystallization of: March 5, 1821 ; I. 209—215. 

see Globe. 

Wave Surface, Equation to Fresnel's: May 18, 1S35; 

VI. 85—89. 
Waves, Motion of, in a small Variable Canal: May 15, 

1837; VI. 457 — 462. 

in Canals, Motion of: Feb. 18, 18.39; vii. 87—95. 

Theory of the two great Solitary : Dec. 8, 1845 ; 

VIII. 326—341. 
Wealth of a Community, Influence of Machinery on : 
May 14, 1838 ; VI. 507—522. 

Ditto, Effect of Non-Residence of 

Landlords on: March 16, 1840; vil. 189—196. 

Weathering of Rocks: March 2, 1868; XI. 387—395. 
Wheels, On the Forms of the Teeth of: May 2, 1825; 

II. 277—286. 
Wiener, see Cubic Surface. 

Winter Solstice, Ancient Observation of: Nov. 30, 1829 ; 

III. 361—363. 

Written Greek Literature, First Age of: Nov. 23, 1868 ; 
XI. 4G1— 480. 

Yorkshire and Durham, Trap Dykes in : May 20, 1822 ; 
II. 21 — 44. 



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