kennelly aurthur tables of complex hyperbolic and cicular functions

TABLES OF COMPLEX HYPERBOLICAND CIRCULAR FUNCTIONS ÆTHERFORCE... HYPERBOLIC SINES, sinh pIS r1.4 i-S 45 ÆTHERFORCE... TABLE HYPERBOLIC SINES, sinh p/5 r/% CONTINUED45 ÆTHERFORCE... HYPER

OXFORDUNIVERSITY PRESS

1921

ÆTHERFORCE

Second edition, February, 1021

ÆTHERFORCE

have nothithertobeenpublished,exceptover a veryrestricted range They have

importantapplications in electricalengineering For instance,it ispossiblewith

Although the principalapplicationof these functions at the present time isin

to a numberof workers, both inmathematical andpracticalfields; and

particu-larlytoMessrs C L Bouton, W.Duddell, E V Huntington,F.B Jewett, John

A. E K.

January,1914.

I

J These areactually extensionsof the tablesI to VIalready incorporated. Ithas

volumeratherthanto recasttheoriginal tables insuch a manneras to includethe

electrical engineering to which complex hyperbolic functions may be

TABLE OF CONTENTS

PAGE

/_y 5 45 to 90 2

H. HYPERBOLIC COSINES, cosh(P/5) =r fa . " " " " 8

HI HYPERBOLIC TANGENTS, tanh(P/S) = rfy_ " " " " 14

IV CORRECTING FACTOR ^^ . " " " " 20 a V CORRECTING FACTOR ^-? " " " " 26 (7 VL FUNCTIONS OF SEMI-IMAGINARIES /(p745) 32

VII HYPERBOLIC SINES, sinh(x +iq) =u + w 42

VIII HYPERBOLIC COSINES, cosh(x +iq) =u +iv 58

IX HYPERBOLIC TANGENTS, tanh(x+iq) = +iv 74

X HYPERBOLIC SINES, sinh(x +iq) = r /_y_ 90

XL HYPERBOLIC COSINES, cosh(x+iq) = r /_y_ 106

XII HYPERBOLIC TANGENTS, tanh(* + </) =r /_y_ 122

XIII FUNCTIONS OF 4 +iq. f(4+iq)=u+iv . 138

f(4+iq) =r/7 139

XIV. SEMI-EXPONENTIALS and logio (~ XV REAL HYPERBOLIC FUNCTIONS /(x +io) = u +io 144

XVI. SUBDIVISIONS OF A DEGREE 150

EXPLANATORY TEXT 151

XVH. HYPERBOLIC SINES sinh(p/6) = r/r 5o to 45 214

XVIII HYPERBOLIC COSINES cosh(p/5) = r/r "" " " 216

XIX. HYPERBOLIC TANGENTS tanh(p/) = r[y_ "" " " 218

XX. CORRECTING FACTOR s - " " " " 220

XXI. CORRECTING FACTOR " " " " 222

a XXII FUNCTIONS OF SEMI-IMAGINARIES /(P/45) . 224

XXIII HYPERBOLIC FUNCTION FORMULAS 226

ÆTHERFORCE

TABLES OF COMPLEX HYPERBOLIC

AND CIRCULAR FUNCTIONS

ÆTHERFORCE

TABLE I. HYPERBOLIC SINES, sinh (p/S) r

TABLE I. HYPERBOLIC SINES, sinh =r

TABLE I. HYPERBOLIC SINES, sinh (pIS) r

1.4 i-S

45

ÆTHERFORCE

TABLE HYPERBOLIC SINES, sinh (p/5) r/% CONTINUED

45

ÆTHERFORCE

TABLE I. HYPERBOLIC SINES, sinh (p

45

ÆTHERFORCE

ÆTHERFORCE

TABLE II. HYPERBOLIC COSINES, cosh (p/5) r

/_y

0-3 0.445

ÆTHERFORCE

TABLE II. HYPERBOLIC COSINES, cosh (p/8) r

45

ÆTHERFORCE

TABLE HYPERBOLIC COSINES, cosh (p/5) r

1.445

TABLE HYPERBOLIC COSINES.

ÆTHERFORCE

TABLE II. HYPERBOLIC COSINES, cosh (p{8) r/_V CONTINUED

45

ÆTHERFORCE

TABLE HYPERBOLIC COSINES, cosh (p /5) r/T CONTINUED

2.6 2.7

2.945

ÆTHERFORCE

TABLE III. HYPERBOLIC TANGENTS, tanh (p[$) r

TABLE HYPERBOLIC TANGENTS, tanh (p 5) r/j CONTINUED

45

ÆTHERFORCE

TABLE III. HYPERBOLIC TANGENTS, tanh(pIS) r y. CONTINUED

45

ÆTHERFORCE

TABLE III. HYPERBOLIC TANGENTS, tanh (p

/8) =r /. CONTINUED

45

ÆTHERFORCE

TABLE III. HYPERBOLIC TANGENTS, tanh(p[S) r

45

ÆTHERFORCE

TABLE III. HYPERBOLIC TANGENTS, tanh

ÆTHERFORCE

ÆTHERFORCE

ÆTHERFORCE

1.6 1.7

ÆTHERFORCE

1ABLE IV.

2.1

ÆTHERFORCE

ÆTHERFORCE

ÆTHERFORCE

FUNCTIONS OF SEMI-IMAGINARIES /(p/4) =rfa

ÆTHERFORCE

FUNCTIONS OF SEMI-IMAGINARIES /(p/ 5) = r/y CONTINUED

Sinh Cosh

4-5

ÆTHERFORCE

FUNCTIONS OF SEMI-IMAGINARIES /(p/4s) = r /y. CONTINUED

Sinhandcosh Tanhandcoth Sechandcosech

6.05

ÆTHERFORCE

FUNCTIONS OF SEMI-IMAGINARIES f(p[*) =r CONTINUED

Sinhandcosh Tanhandcoth Sechandcosech

8.30

ÆTHERFORCE

FUNCTIONS OF SEMI-IMAGINARIES. /(p/45JO =

r/jy. CONTINUED

Sinhandcosh Tanhandcoth Sechandcosech

15-05

ÆTHERFORCE

FUNCTIONS OF SEMI-IMAGINARIES.

p Sinhandcosh Tanhandcoth

=r/jy. CONTINUED

Sechandcosech 17.30

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x +iq) u +iv

TABLE VII HYPERBOLIC SINES, sinh (x +iq) u +iv. CONTINUED

x= 0.25 x =

0.3 =0.35 x=0.4 x =0.45

o

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x + iq) u+iv. CONTINUED

x=0.5 =0.55 x=0.6 x=0.65 x=

0.7o

TABLE VII HYPERBOLIC SINES, sinh (x+iq) u +iv. CONTINUED

TABLE VII HYPERBOLIC SINES, sinh(*+iq) u +iv. CONTINUED

TABLE VII HYPERBOLIC SINES, sinh (x +iq) u +iv. CONTINUED

= 1.25 x=

1.3 *= i-35 *=1.4 =

1-45o

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x +iq) = u +iv. CONTINUED

TABLE VII HYPERBOLIC SINES, sinh (x+iq) u +iv, CONTINUED

= i-7S x = 1.8 = 1.85 *=

1.9 x= 1.95o

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (*+ig) u +iv. CONTINUED

TABLE VII HYPERBOLIC SINES, sinh(x+iq) u +iv. CONTINUED

q

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x+iq) u + iv. CONTINUED

TABLE VII HYPERBOLIC SINES, sinh (x+iq) u+iv. CONTINUED

TABLE VII HYPERBOLIC SINES, sinh (* +iq) u +iv. CONTINUED

-3.05 x=3.10 =3-iS x =3.20o.o

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x+iq) u +iv. CONTINUED

x=

3-25 x= 3.30 =3-35 x= 3-4 x=3-45

o.o

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x + iq) u +iv. CONTINUED

x =3-50 = 3-55 x=3.60 =3-65 x= 3.70

0.0

ÆTHERFORCE

TABLE VII HYPERBOLIC SINES, sinh (x+ iq) u+iv. CONTINUED

=

3-75 x=3.80 * =

3.85 x =3.90 =3-95o.o

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh(x + iq) u + iv

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh (x +iq) u +iv. CONTINUED

TABLE VTII HYPERBOLIC COSINES, cosh (x + iq) u +iv. CONTINUED

TABLE VIII HYPERBOLIC COSINES, cosh (x +iq) u + iv. CONTINUED

TABLE VIII HYPERBOLIC COSINES, cosh(x +iq) u +iv. CONTINUED

TABLE VIII HYPERBOLIC COSINES, cosh(* +iq) = u +iv. CONTINUED

9

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh (x +iq) u+ iv. CONTINUED

i-55 x= 1.6 x= 1.65 x= 1.7

o

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh (x +iq) u +n>. CONTINUED

?

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh(x +iq) u +iv. CONTINUED

9

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh(* +iq) = u +iv. CONTINUED

TABLE VIII HYPERBOLIC COSINES, cosh(x +iq) u + iv. CONTINUED

q

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh(x + iq) = u +iv. CONTINUED

TABLE VIII HYPERBOLIC COSINES, cosh(* + ig) u +iv. CONTINUED

x=3-0 3-S x=3.10 =3-iS = 3.20

o

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh (* +iq) u +iv. CONTINUED

=3-25 x=3.30 =3-35 =3-40 3-45

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh (x +iq) u +iv. CONTINUED

=3-50 =3-55 3.60 x=3-65 x=3.70

o

ÆTHERFORCE

TABLE VIII HYPERBOLIC COSINES, cosh (* +iq) = u +iv. CONTINUED

*=3-75 3-80 3-85 x=

3.90 3-95o

ÆTHERFORCE

TABLE IX HYPERBOLIC TANGENTS, tanh(* +iq) u+iv

x=0.05 x =0.15 X=0.2

ÆTHERFORCE

TABLE IX HYPERBOLIC TANGENTS, tanh(x +iq) u+ iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x +iq) +w CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) u+iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x +iq) u +iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) u+iv. CONTINUED

ÆTHERFORCE

TABLE IX HYPERBOLIC TANGENTS, tanh(x +iq) u+iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) = u +iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) =u+iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) u +iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) u +iv. CONTINUED

TABLE HYPERBOLIC TANGENTS, tanh(x+iq) =u+iv. CONTINUED

x=

2.75 x= 2.80 x= 2.85 = 2.90 x= 2.95

ÆTHERFORCE

TABLE IX HYPERBOLIC TANGENTS, tanh(x+iq) =u+iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x +iq) =u+iv. CONTINUED

=3-25 *=3-30 *=3-35 x=3.40 =3-45

o

ÆTHERFORCE

TABLE IX HYPERBOLIC TANGENTS, tanh(*+iq) =u +iv. CONTINUED

TABLE IX HYPERBOLIC TANGENTS, tanh(x +iq) u+iv. CONTINUED

3-95o

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh(x+iq) r/y_

x=0.05 x =0.15

o

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (at+iq) r

=0.25 x=0.3 =0.35 *=0.4 x=0.45

0.25261

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (x +iq) r CONTINUED

TABLE X HYPERBOLIC SINES, sinh (x+iq) r /y. CONTINUED

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (* +iq) r/r CONTINUED

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (x+iq) = r/y CONTINUED

TABLE X HYPERBOLIC SINES, sinh (*+iq) r/V CONTINUED

x =1.50 X= 1.60 x= 1.65 a;= 1.70

o

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (*+iq) = rfy. CONTINUED

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh(x +iq) r /y. CONTINUED

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (x+ig) =r

2.3 *= 2.35 x= 2.4 x= 245

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh(x+iq) /y CONTINUED

TABLE X HYPERBOLIC SINES, sinh (x+iq) = r/y CONTINUED

x=

2.75 = 2.85 * = 2.9 x= 2.95

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (x +iq) =r/y. CONTINUED

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (x +iq) =r

=3.25 =3-3 =3-35 *=3-4 *=3-45

o

ÆTHERFORCE

TABLE X HYPERBOLIC SINES, sinh (x+iq) = r

TABLE X HYPERBOLIC SINES, sinh (x+iq) = rfy CONTINUED

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh(* +ig) r/y

TABLE XI HYPERBOLIC COSINES, cosh (x+ig) rAy. CONTINUED

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh (x+iq) r/y CONTINUED

ÆTHERFORCE

TABLE HYPERBOLIC COSINES, cosh(x +iq) = r

x=0.75 = 0.85 x=o.g x=

0.95o

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh(at+iq) r/y CONTINUED

TABLE XI HYPERBOLIC COSINES, cosh(x +iq) = r

TABLE XI HYPERBOLIC COSINES, cosh (x+ig) r /y. CONTINUED

x=1.5 =

1-55 x= 1.6 *=1.65 x= 1.7

o

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh(* +iq) r/y CONTINUED

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh(* +iq) /T CONTINUED

TABLE XI HYPERBOLIC COSINES, cosh(x+iq) =

ÆTHERFORCE

TABLE XL HYPERBOLIC COSINES, cosh (x+iq) r/T CONTINUED

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh (x+iq) = r

x = 2.75 x =.2.85 *= 2.9 x= 2.95

9

ÆTHERFORCE

TABLE XL HYPERBOLIC COSINES, cosh(x +iq) r/y CONTINUED

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh(* +iq) r y. CONTINUED

=3-25 *'=

3-3 = 3-35 x=3.4 =3-459

ÆTHERFORCE

TABLE XI HYPERBOLIC COSINES, cosh(x +iq) r /y. CONTINUED

ÆTHERFORCE

TABLE XL HYPERBOLIC COSINES, cosh(x+iq) =r

3-95

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh (x+iq) r

ÆTHERFORCE

TABLE HYPERBOLIC TANGENTS, tanh (x +iq) r

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh (* +*?) r/y CONTINUED

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh (x+ig) rAy CONTINUED

x=0.75 x=0.85 x=

o.g x=0.95

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh(* +iq) =

x= 1.05 *= 1.15

1

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh(x +iq) =r CONTINUED

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh(* +iq) r/y CONTINUED

ÆTHERFORCE

TABLE HYPERBOLIC TANGENTS, tanh(* +iq) r

=

1.75 x= 1.8 *= 1.85 *=

1.9 x= 1.959

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh(* + iq) f CONTINUED

TABLE XII HYPERBOLIC TANGENTS, tanh(*+iq) r/y. CONTINUED

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh (x+iq) =r

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh (x +iq)

TABLE XII HYPERBOLIC TANGENTS, tanh(x +iq) = r CONTINUED

ÆTHERFORCE

TABLE HYPERBOLIC TANGENTS, tanh(x+iq) = r % CONTINUED

=3-25 =3-35 *= 3-4 *=3-45

o

ÆTHERFORCE

TABLE XII HYPERBOLIC TANGENTS, tanh(x+iq) r/r CONTINUED

TABLE XII HYPERBOLIC TANGENTS, tanh(x +iq) = r/y CONTINUED

ÆTHERFORCE

TABLE XIII FUNCTIONS OF 4 +ig. f(4 + ?) +

ÆTHERFORCE

TABLE XIII FUNCTIONS OF 4 +iq. /(4 +iq) =r

ÆTHERFORCE

T'ABLE SEMI-EXPONENTIALS, and logw

e* e*

-ÆTHERFORCE

TABLE XIV SEMI-EXPONENTIALS - and loglo (

FABLE XIV SEMI-EXPONENTIALS.

ÆTHERFORCE

TABLE XIV SEMI-EXPONENTIALS - and

X

ÆTHERFORCE

TABLE XV

o.oo

ÆTHERFORCE

TABLE XV

e

ÆTHERFORCE

TABLE XV

e

ÆTHERFORCE

TABLE XV

1.50

ÆTHERFORCE

TABLE XV

e

ÆTHERFORCE

TABLE XV

ÆTHERFORCE

TABLE XVI SUBDIVISIONS OF A DEGREE AUXILIARY TABLE

O.OI

ÆTHERFORCE

EXPLANATORY TEXT

ÆTHERFORCE

EXPLANATORY TEXT

INTRODUCTION

ofacomplexvariable eitherintherectangular coordinateformof that variable (x -f-iy)

circular functions of a complex variable A few formulas are added as aids to the

XIVinclusive,between which, the functions sinh (x+iy),cosh (x+ iy),tanh(x -fiy),

limitsofo and fory. Itis shown, moreover, tobe aneasy matterto extendthe

practic-ablelimits Consequently, interpolation must ordinarilyberesorted to, when three or

moresignificant digitsare neededin theresults Suchinterpolationsrequire an

at mostfour, significant digits may be needed, a separate atlas of 23 large-scale charts,

by inspection

COMPLEX QUANTITIES

funda-mental importance in connection with the Tables,for the assistance ofthose who have

com-plexnumbers For a more comprehensive discussion ofcomplex quantities, the reader

must be referred to special treatises on thesubject.

Ordinary numerical quantities, or the numbers dealt with in ordinary arithmetic,

theleft, to plus infinityon the right, Obeing the zero point Thepointx\. wouldthen

[153]

ÆTHERFORCE

EXPLANATORY TEXT

or, as thevector Ox\; ie., the straightline drawn from the originO to the point x\ and

numbers of arithmetic may be represented geometrically as vectors; but such vectors

Complex quantities, or complex numbers, cannot be completely represented by

movable point withrespect to afixedpoint as origin Thus,in Fig i, the plane XOY

-Y

2 X

FIG 2 Plane Vector 2.236 e"- 106 or

FIG i Complex quantity i+ a designated by 2.236 763 26'.

A complex number may be specified either in rectangular coordinates, or in polar

expres-sion (i +22), where the symbol i signifies measurement along 'the subordinate axis

It is shown in mathematicaltreatises that * = V i. The vector OP\ of Fig. i may

"real" axis, and YOY the "imaginary," axis; so that the ^-component of a complex number becomesthe "real component," andthe^-component the"imaginary

component." The symboli stillstands forthe imaginarycomponent. In mathematics

in this book wenecessarily considercomplex quantitiesfroma broader viewpointthan

component, perpendicularly rotated with respect to the fundamental Xaxis

[154]

ÆTHERFORCE

EXPLANATORY TEXT Complex quantities may also be expressed in polar coordinates, as in Fig 2, where

the fundamental reference axis OX is drawn in the positive direction in the reference

circular radians, in degrees-minutes-seconds, quadrants, or any other recognized unit

the modulusto thesamescale oflinearmeasureasin Fig i, and 63.26'istheargument.

If one and the same complex quantity be expressed both in rectangular and polar

psin5, ;y/a;=tan

2, yz =

i, pz = Vs =

2.236

and82 = 206.34'.

ADDITION OF COMPLEX QUANTITIES

-X

-YFIG 5 Addition of two complex quantities

free end The last named vector is the required sum. Thus, in Fig 5, the complex

ÆTHERFORCE

EXPLANATORY TEXT

to OPi = 2.236/63.26' of Fig. 2, to produce OP = 1.414/135 = p3/8zof Fig. 6.

+ xz + . -f *)(xi +iyj + (xz +iyz) + - + (xn+ fyB ) =

bothitsreal and imaginary components Reversing the sign ofa polar complex

quan-tity means changing itsargument by 180.

Y

o

FIG 7 Complex Subtraction

(i+12)-(-2-*i) 3+*3 =OP

FIG 8 Complex Subtraction, Polar Coordinates

= 4.243/45.

complex addition, isvery easily made onthe drawing board by purely geometric

proc-esses, whether the quantities are rectangular or polar. If, however, the process is to

[156]

ÆTHERFORCE

EXPLANATORY TEXT

MULTIPLICATION OF COMPLEX QUANTITIES

rules of algebra, remembering that i2 = i. Thus

(*i +fyi) (*2+ iyz) = (xixz

-yiy2 ) +i(xiy2+ xz yi) (3)

RECIPROCAL OF A COMPLEX QUANTITY

EXPLANATORY TEXT

V5/63.2 6' VS QUOTIENT OF COMPLEX QUANTITIES

their moduliand the difference of theirarguments Thatis

[158]

ÆTHERFORCE

EXPLANATORY TEXT POWERS AND ROOTS OF COMPLEX QUANTITIES

It willbe evident from the foregoing that

y~p/8/n (9)

CIRCULAR AND HYPERBOLIC FUNCTIONS GEOMETRICALLY COMPARED

REAL CIRCULAR AND HYPERBOLIC FUNCTIONS The geometryof the real circular functions sin x, cos x and tan x relates, as iswell

known, to themotion ofaradiusvectoroveracircle. The geometryofthereal

hyper-bolicfunctions sinhx,cosh x and tanh x relates tothemotion ofaradiusvectorovera

and a circular angle /3, the tangent Ef being always perpendicular to the radius

(1) By the ratio of the circular arc length 5 described during the motion, by the

dfi circularradians,by moving its terminal over an infinitesimally small circular arc ds

(10)

y

the total circularsector and circular angle generated will ofcoursebe:

/ds = ($2 si) = s circularradians (n)

theinitialpositionOA toanypositionsuchasOE,itwillsweepout acircularsectorOEA.

[159]

ÆTHERFORCE