EXCHANGE 
 
ELECTROMAGNETIC 
 
 OSCILLATIONS FROM A 
 
 BENT ANTENNA 
 
 ROBERT CAMERON COLWELL 
 
 / 
 
ELECTROMAGNETIC OSCILLATIONS 
 FEOM A BENT ANTENNA 
 
 A DISSERTATION 
 
 PRESENTED TO THE 
 
 FACULTY OF PRINCETON UNIVERSITY 
 
 IN CANDIDACY FOR THE DEGREE 
 
 OF DOCTOR OF PHILOSOPHY 
 
 BY 
 ROBERT CAMERON COLWELL 
 
 PRESS OF 
 
 THE NEW ERA PRINTING COMPANY 
 LANCASTER. HA. 
 
 1920 
 
Accepted by the Department of 
 Physics, February, 1918 
 
 '. '.;', ! f ,' r ' t ; -', 
 
ELECTROMAGNETIC OSCILLATIONS FROM A BENT 
 
 ANTENNA 
 
 The purpose of this investigation is to find the mathematical 
 equations for the electromagnetic oscillations from a bent an- 
 tenna, which is known to send out directed waves. The method 
 used is that of Pocklington 1 which has been developed by M. 
 Abraham 2 and particularly by G. W. Peirce 3 who has recently 
 published a remarkable research on the radiation resistance of a 
 flat top antenna. This article is based upon the work of Peirce 
 and Abraham, but the equations are worked out for the funda- 
 mental and not for the forced vibration. The application of the 
 formula is new. 
 
 The following assumptions are made: 
 
 1. That any antenna may be considered to be made up of a 
 large number of Hertzian doublets placed end to end. 
 
 2. That the earth is a perfectly conducting plane. 
 
 3. That the waves propagated high into the air eventually 
 return to the earth. The reason for this assumption will be 
 shown in the section dealing with the horizontal part of the 
 antenna. 
 
 Let a flat top antenna have a vertical part h and a horizontal 
 part d, Fig. 1. It will be necessary to discuss the effect of the 
 
 EARTH'S SURFACE 
 
 FIG. 1 
 
 radiation from this antenna in three parts. 
 I. The radiation due to the vertical part h and its image h. 
 
 1 Pocklington H. C., Camb. Phil. Soc. Proc., 1898, p. 325. 
 
 2 Abraham, Theorie der Electrizitat, Vol. II. 
 
 3 G. W. Peirce, "Radiation Characteristics of an Antenna," Proc. Am. 
 Acad. Arts and Sciences, Vol. 52, No. 4, October, 1916. 
 
 3 
 
ELECTROMAGNETIC OSCILLATIONS 
 
 II. The 'radiation from the horizontal part d and its image d. 
 III. The mutual action between the vertical and horizontal parts. 
 
 I. RADIATION FROM THE VERTICAL PART 
 Let the point where the vertical part enters the ground be the 
 origin of co-ordinates, and choose the axes as shown in Fig. 2 
 
 FIG. 2 
 
 where the x axis is parallel to the direction in which the free 
 end of the horizontal antenna points and the vertical antenna 
 coincides with the direction OZ. The azimuth and co-latitude 
 angles have their usual designations. Let P be any point in 
 space whose polar co-ordinates are r, <f>, 6, where r is very great 
 compared to the height h of the vertical part of the antenna. 
 P lies at a distance r from the point P r which is the position of 
 one of the Hertzian doublets postulated in the first assumption. 
 The effect of the doublet at P f on the point P is given by the 
 theory of Hertz 1 
 
 CD 
 
 where f(t) = the moment of the doublet edz, r length PP f and 
 V = velocity of light. E is expressed in electrostatic and H in 
 electromagnetic units; r Q and 6 appear in place of r and 6', as is 
 legitimate because of the great magnitude of r in comparison 
 with OP. 
 
 If the total length of the antenna is / = h + d there must be 
 a node of current at + / and /. For the fundamental vibra- 
 
 1 Hertz, Electric waves, Chap. IX, Trans. D. E. Jones; Bateman, Elec- 
 trical and Optical Wave Motion, p. 8. 
 
FROM A BENT ANTENNA 5 
 
 tion, the current i at any time expressed in terms of I the maxi- 
 mum current must be 
 
 i = I sin pt cos -r (2) 
 
 A 
 
 where X = the wave length of the system, 
 
 2x7. 
 p = r is the angular velocity 
 
 A 
 
 No 
 
 w 
 
 and the current i is the rate of change of the charge e on any 
 doublet, that is 
 
 de di d 2 e 
 
 l = dt' Tt = df 
 Therefore 
 
 d>e di 2x71 2x7* 2x2 
 
 ^ = ^ = -r~ cos ^r cos ir (3) 
 
 Substituting (3) in (1) we get 
 
 p/sinfl / r zcos0\ 2xz , 
 
 dE a = tyi cos p (t -- y - 1 cos -r- dz (4) 
 
 in which -r is given the shorter form p, and r z cos 6 is 
 A 
 
 written for r as these are approximately equal. 
 
 The doublet 1 at OP f has an image at z and this doublet 
 will have an effect at P given by the equation 
 
 pi sin 6 ( r + z cos 6 \ 2x2 
 
 dEi = p cos p [t -- y - Icos -y- dz (5) 
 
 The effect due to the two doublets on the point P is found by 
 adding (4) and (5), and the action of all the doublets and their 
 
 1 Theory of Images: Jeans, Electricity and Magnetism, Chapter VIII; 
 Maxwell, Electricity and Magnetism, Chapter XI; Webster, Electricity and 
 Magnetism, p. 303. 
 
6 ELECTROMAGNETIC OSCILLATIONS 
 
 images, by integrating from to h. Then 
 
 ind C h 2irz[ \ ( r \ , pz cos 6 
 - -y~ 
 
 & cose 
 
 The terms in the square brackets are of the form [Cos (x + y) 
 cos (x y)] may be simplified by use of a well-known trigo- 
 nometrical formula: then 
 
 2plsm8 / r \ C h pzcosd 2vz , 
 
 ~^^y cos-r-ds (7) 
 
 The integral in (7) is a standard form and is found in any table 
 of integrals. Integrating and putting in the limits we get for 
 the integral of (7) 
 
 fph cos0 2Trh\ ( , 2Trh\ 
 
 in I y r 1 sm I ph cos H r- I 
 
 This reduces to 
 
 sin -^- (cos 1) sin -^ (cos 
 
 
 ~ (cos 0-1) v (cos 0+1) 
 
 A A 
 
 Adding these fractions we get for the integrated term 
 
 X J 
 
 ^rx i cos 
 
 2?r sm 2 0[ 
 
 Substituting in (7) 
 27 
 
 (2<ITh _\ . 27T/1 /27T/J \ 27T/1 
 
 I "T~ cos ^ ) sm ^v -- sln I "T~ cos ) cos "T~ cos 
 \X / X \X / X 
 
 cos ~~ cos 
 
 \ . 
 
 ) s 
 / 
 
 Vhmt". X v Iy I V X 
 
 - cos sin ( ^^: i cos z__- L (8) 
 
 Equation (8) is the basis for the following brief discussion 
 of a vertical antenna. Poynting's Vector Theorem states that 
 
FROM A BENT ANTENNA 7 
 
 the power radiated along the radius at any point of a sphere is 
 V/4:Tr(E X H) per unit area. In the case of the vertical E = H. 
 The energy radiated is directly proportional to E . The wave 
 length is equal to 4&. 27/Fr is constant. The average value of 
 cos 2 2ir/\(Vt TQ) is |. Therefore the expression for the power 
 radiated at an angle 6 takes the simple form 
 
 (COS 
 
 P. = K 
 
 (fcosfl) 
 
 ' (9) 
 
 sin 6 
 
 The angle at which the greatest amount of energy is radiated 
 is given by 
 
 fir \ 
 
 7p cos I cos 6 \ 
 
 n T7 \ / 
 
 dS " sin B 
 
 sin ( ^ cos ) ^ sin 2 6 cos ( ^ cos 8 } cos 
 
 ^ /2 \_? ) . 
 
 L sm 6 ] 
 
 The solution B = ir/2 makes d 2 P ldd 2 negative, and therefore 
 represents the direction of maximum radiation of energy. 
 
 Therefore a vertical antenna sends out its maximum energy 
 along the horizontal. Table 1 is developed from (9), K being 
 given an arbitrary value. Diagram A is a graph of Table I, 
 showing the approximate energy distribution. The distribution 
 of energy is such that nearly all the radiation lies within the 30 
 angle with the horizon. These equations (8) and (9) throw some 
 light on the action of tall antennas. The radius of action is 
 increased as the antenna height is increased for two reasons. 
 First, because the lengthening of the antenna automatically in- 
 creased the wave length and the earth is a better conductor for 
 long waves than for short ones. Secondly, because the waves 
 from a high antenna are not brought to earth so quickly (see as- 
 sumption three) as those from a short one. Just as soon, how- 
 ever, as the wave length is reached for which the earth is a 
 perfect conductor, no further improvement in conduction is 
 effected by increasing the wave. Similarly there is a limit to 
 
8 
 
 ELECTROMAGNETIC OSCILLATIONS 
 
 the second action, so that there is no reason for increasing the 
 height of the sending antenna indefinitely. 
 
 Angle 
 
 
 15 
 30 
 45 
 60 
 75 
 90 
 
 TABLE I 
 
 V 
 0.00 
 
 .078 
 
 .144 
 
 .422 
 
 .672 
 
 .902 
 1.00 
 
 *$ 
 
 0.00 
 .22 
 .42 
 .65 
 .82 
 .95 
 
 1.00 
 
 II. THE HORIZONTAL PART 
 
 In order to apply Hertz's theory to the horizontal antenna, 
 it is necessary to measure the co-latitude from the axis X. The 
 co-latitude is designated by \[/ and the azimuth by % The 
 origin remains at the point where the vertical antenna enters 
 the earth, Fig. 3. It should be noticed here that the action of 
 
 FIG. 3 
 
 the horizontal part is dependent upon the length of the vertical 
 for the latter determines the phase of the current at any point 
 in the former. Neither in theory nor in practice can the action 
 of the horizontal be dissociated from the vertical. The positions 
 of the doublets and doublet images are denoted in Fig. 4. The 
 small letters x, y, z (or x, o, h) refer to the doublets, while the 
 position of the point in space P is given by large X, Y, Z. 
 
FKOM A BENT ANTENNA 
 
 Hertz's theory may now be applied to the effect at P of the 
 doublets at Q and Q'. Then 
 
 Now 
 
 n - 
 
 = V(X - z) 2 + p + (Z + 
 
 At great distances 
 
 FIG. 4 
 
 From (11), (12), (13) 
 
 - 2Zh 
 
 2r 
 
 - 2Xx 
 
 As before in (3) 
 
 2r 
 
 2irV _ 27T. rr 
 - / cos -^- \ Vt 
 
 Also, 
 
 w(* r2 ^ (^ M 
 
 fit T7I= pi COS p I t 57 I 
 \ V ) \ V J 
 
 2ir 
 
 
 cos 
 
 27T 
 cos - 
 
 (10) 
 
 (ID 
 (12) 
 
 (13) 
 
 (14) 
 (15) 
 
 . 
 
 (16) 
 
10 ELECTROMAGNETIC OSCILLATIONS 
 
 2ro j &-j- (i+*)<fe (17) 
 
 Substituting (16) and (17) in (10), simplifying by the trigono- 
 metrical formula for the sum of two cosines; neglecting a; 2 + h 2 
 in comparison with 2Xx, we obtain the expression 
 
 Jrt , T7 47r7 sin \// . 2irZh 
 dEj, = dHy = T^T sin ~T 
 
 TQ V A A7*o 
 
 Vt - r + x cos ^} cosy (h + x)dx 1 (18) 
 
 The integration of (18) is very similar to that of (7) and gives 
 finally 
 
 21 . 2wZh [ . 2ir s f 2nd 
 
 E* = TZ . 7 sm-r sm-r- (Vt r ) i cos-^- 
 
 r<>V sm ^ Xro |_ A X 
 
 cos ( -T cos \l/ } H- cos -r- (Vt r ) 
 
 f 2^ /2^ \11 
 
 X j cos ^ sm -T- sm I -r cos ^ 1 (19) 
 
 Upon equations (8) and (19) are based all the conclusions regard- 
 ing a bent antenna. The total flux of force is obtained by com- 
 pounding EQ of (8) with the E^ of (19) in such a way as to get 
 the complete electric intensity E, and the corresponding magnetic 
 intensity H. Although the doublets which give rise to E e are 
 perpendicular to those producing E$, it by no means follows that 
 E# and E e are at right angles; in fact, they make an angle (a) 
 with one another which varies for every point on the particular 
 sphere under consideration. Professor Peirce, of Harvard Uni- 
 versity, has shown that this difference in direction gives rise to 
 a third term in the power radiation { the other two come of course 
 from (8) and (19) } of the form 
 
 sin \{/ cos 8 cos <j> 
 
 2 cos aE & E&, where cos a. = 1 r-^-r x 
 
 1 sm 2 8 cos 2 4> 
 
 The average value of this term, when multiplied by F/47T, he 
 
FROM A BENT ANTENNA 
 
 calls the mutual power. In the problem under discussion 
 2K cos cos 4> . 2irhZ J 
 
 11 
 
 4?r 
 
 X 
 
 sm 
 
 sin 0(1 sin 2 cos <f>) "" L \r Q [ 
 sin ( 
 
 cos sin 
 
 cos 
 
 2irh 2wh cos 
 ^: cos T --- cos 
 A A 
 
 2irh . ir2h cos (9 . /rtm 
 -r- sm - r- f (20) 
 A A 
 
 where for convenience K is put equal to 7 2 /7rFr 2 . This equation 
 is true only for the average value of (V/2ir) cos aE e E^ so that 
 the term 
 
 cos 2 -T- (Vt r ) = \ and sin (Vt r ) cos r- (Vt r ) = 
 
 A A A 
 
 SECOND PART NUMERICAL CALCULATION 
 
 The power radiated per unit area in certain zones at distances 
 2,000, 6,000 and 9,000 meters above the earth's surface, and at 
 a distance 10,000 meters from the origin will now be calculated 
 by means of equations (8), (19) and (20). In these equations 
 let h = 10 meters, d = 100 meters, X = 4(& + d) = 440. Then 
 the power radiated from the vertical part may be expressed by 
 (8) in the form : 
 
 V E? 
 
 47T 
 
 = K jcosf -r- COS0J sin-r- cos sin ( ~- cos 0) cos ~^- 
 
 10,000 M. 
 
 jwoo jr. 
 
 "^6000 M., 
 
 ASOOOM. 
 
 FIG. 5 
 
 The values in Table II are calculated from this equation: 
 
12 
 
 ELECTROMAGNETIC OSCILLATIONS 
 
 X = 440M 
 h = 10M 
 r = 10000M 
 
 VERTICAL TABLE II 
 
 Height 
 
 9000M 
 6000M 
 
 5000 
 2000 
 
 
 .0049K 
 .0160 
 
 .0164 
 .0196 
 .0196 
 
 The calculations of E 9 2 (V/4:ir) for the horizontal part are by no 
 means as simple as this. The value of the angle in formula (19) 
 changes for every point on the zone at the height 9000, 6000 
 and so on. This change must be calculated. In Fig. 6 ABC 
 
 FIG. 6 
 
 is a spherical triangle with sides 8, \l/, irj '2 and angle < given. 
 Then from a well-known formula in the trigonometry of triangles: 
 
 Cos \l/ = sin 6 cos <j> 
 
 The power radiated from the horizontal antenna through the 
 zone Z = 9000, 6000, etc., will vary for different angles of <j>, 
 that is, the bent part of the antenna has a directive effect on the 
 oscillations. 
 
 In equation (19) 
 
 5in , f 2rZh } 
 
 VW I Xrn I f 2nd 2ird 
 
 j cos 2 -T- r 1 + cos 2 \l/ sin 2 -r 
 
 47T 
 
 = K 
 
 sin 2 \l/ 
 
 2ird 
 2 cos -r cos 
 
 \ 
 
 1 2 
 
 cos 1 2 cos \f/ sin 
 
 sm 
 
 cos 
 
 XX 
 
 From this equation the values of Table III are calculated for 
 the different zones and different directions. The operations are 
 long and tedious but not difficult. The horizontal part of the 
 
FROM A BENT ANTENNA 
 
 13 
 
 antenna has a slight directive effect perpendicular to the direc- 
 tion in which it points. This is contrary to what one would 
 expect: because all experiments show a directive effect in the 
 direction away from the free end of the antenna. However, it 
 has been shown by Pocklington that a circular wire radiates 
 more power perpendicular to its area than in any other direction 
 and the form of equation (19) shows that Pocklington's method 
 of doublets applies to this problem and that the solution is 
 correct to the approximations made. 
 
 HORIZONTAL PART TABLE III 
 
 Height 
 
 < 
 
 7? 2 V 
 
 ^5 
 
 
 
 9000 
 
 
 
 .0097 
 
 180 
 
 
 30 
 
 .0094 
 
 150 
 
 
 45 
 
 .0092 
 
 135 
 
 
 60 
 
 .0097 
 
 120 
 
 
 90 
 
 .0107 
 
 90 
 
 6000 
 
 
 
 .0022 
 
 180 
 
 
 30 
 
 .0028 
 
 150 
 
 
 45 
 
 .0031 
 
 135 
 
 
 60 
 
 .0040 
 
 120 
 
 
 90 
 
 .0047 
 
 90 
 
 2000 
 
 
 
 .0004 
 
 180 
 
 
 30 
 
 .0001 
 
 150 
 
 
 60 
 
 .0003 
 
 120 
 
 
 90 
 
 .0000 
 
 90 
 
 It will now be shown that the power arising from the mutual 
 effect (eq. 20) tends to modify the directive effect at high points 
 in the atmosphere in such a way as to give a fore and aft directive 
 effect. 
 
 The values of Table IV are calculated from equation (20). 
 
 MUTUAL EFFECT TABLE IV 
 
 Height. 
 
 
 
 5^ 
 
 47, 
 
 4> 
 
 9000 
 
 
 
 .0027K 
 
 180 
 
 
 30 
 
 .0022 
 
 150 
 
 
 45 
 
 .0014 
 
 135 
 
 
 60 
 
 .0005 
 
 120 
 
 
 90 
 
 
 
 90 
 
 6000 
 
 
 
 .0030 
 
 180 
 
 
 30 
 
 .0023 
 
 150 
 
 
 45 
 
 .0015 
 
 135 
 
 
 60 
 
 .0009 
 
 120 
 
 
 90 
 
 
 
 90 
 
 2000 
 
 
 Negligible 
 
 
14 
 
 ELECTEOMAGNETIC OSCILLATIONS 
 
 Adding up the vertical, the horizontal and the mutual effects 
 contained in Tables II, III and IV, we obtain the complete 
 power radiated through unit area of the zones. The results are 
 given in Table V, in which K has been set equal to an arbitrary 
 value. Plate B is plotted from Table V. In Plate C, the dis- 
 tribution at 9000 M is compared to a curve obtained experi- 
 mentally by Fleming. 
 
 TOTAL EFFECT TABLE V 
 
 Height 
 
 
 
 p 
 
 4> 
 
 9000 
 
 
 
 .0173 
 
 180 
 
 
 30 
 
 .0165 
 
 150 
 
 
 45 
 
 .0155 
 
 135 
 
 
 60 
 
 .0151 
 
 320 
 
 
 90 
 
 .0156 
 
 90 
 
 6000 
 
 
 
 .0212 
 
 180 
 
 
 30 
 
 .0211 
 
 150 
 
 
 45 
 
 .0206 
 
 135 
 
 
 60 
 
 .0209 
 
 120 . 
 
 
 90 
 
 .0207 
 
 90 
 
 2000 
 
 
 
 .0206 
 
 180 
 
 
 30 
 
 .0197 
 
 150 
 
 
 60 
 
 .0199 
 
 120 
 
 
 90 
 
 .0196 
 
 90 
 
 The symmetry of the curves developed from the theory can 
 be reconciled with the asymmetrical curve found in the experi- 
 ments by supposing that the electrical waves brought back to 
 the earth from high in the atmosphere are more intense toward 
 the bend in the antenna than at the free end. Zenneck (Zenneck, 
 Phys. Zeitsch., Vol. 9, p. 553, 1908) has shown that this difference 
 may be due to imperfect conductivity in the earth's surface. 
 
 SUMMARY 
 
 First : The equations developed by Pocklington, Abraham 
 and Peirce have been applied to an antenna with vertical and 
 horizontal parts in such a way as to find the energy given out 
 for the fundamental vibration: 
 
 X = 4(fc + d) 
 
 Second: The intensities so obtained are plotted and are shown 
 to have a fore and aft directive inclination. The intensity fore 
 and aft is symmetrical and not asymmetrical as required by the 
 experiments. 
 
FROM A BENT ANTENNA 15 
 
 Third: Close to the antenna the intensities are symmetrical 
 in azimuth agreeing with experiment. 
 
 Fourth: The conclusion is that the fore and aft asymmetry 
 of a bent antenna is caused by the difference in conductivity 
 between the atmosphere and the earth for the electromagnetic 
 oscillation. 
 
 Fifth: The forms of the resulting equations show that Peirce's 
 assumptions regarding the current satisfy Pocklington's criterion 
 for the use of Maxwell's Equations. 
 
 ADDENDUM 
 
 The Integration of Equations (18), p. 10. 
 The integrable part of (18) is 
 
 I sin { Vt r Q -f- x Cos \//} cos -r- (h + x)dx 
 
 Expand and multiply, thus: 
 
 n. 27T. T . 27rxCost , 27r.__ . 27rxCost~] 
 
 sin-r- (Vt r ) cos -- r -- + cos (Vt r ) sin -- 
 A A A A 
 
 2irh sin 2 
 
 [2irh 2irx 2ir 
 
 cos-^cos^ -sm 
 
 r2ir ,. 2wh 2irx Cos \[/ 2irx . 
 
 sm (Vt r Q ) cos^r cos - r - COS-T ax (21) 
 A A A A 
 
 2ir 2wh C d . 2irx Cos \l/ 2irx , 
 
 + cos -r- (Vt fo) cos -r- sm - r - cos -T" dx (22) 
 
 A A ,/Q A A 
 
 . 2ir . 27rh C d 2wx Cos ^ . 2wx 
 
 sm^r- (Vt TQ) sin^r cos -- 7 - sm-^r- dx (23) 
 
 A A J A A 
 
 27r /T _ . 2irh r . 2jrxCos\l/ . 2wx 
 cos -T- (p t r ) sm -r- sm r - sm dx (24) 
 
 A A t/o A. A 
 
 The integration of (21) is the same as that of (7) on page 4 and 
 gives 
 
 X 2wdcos\l/ . 2wd . 2wdcos\l/^ 27rd~\ 
 
 2ir sin 2 \// \_ X X X X J 
 
 The integration of the integral part (22) comes under Form 360 
 
16 
 
 ELECTROMAGNETIC OSCILLATIONS 
 
 in Peirce's Table of Integral and simplifies into 
 
 2wd 
 
 cos 
 
 . 
 + sin 
 
 . 2irh 
 sin - cos 
 
 (26) 
 
 (23) comes under form (360) but in a different way from (22). 
 The complete integration follows: 
 To find 
 
 2irx 2irx Cos 
 
 r 
 
 sin cos 
 
 (Form 360) 
 
 sin mx cos nx ax = 
 
 cos (m n)x cos (m + ri)x 
 
 Let 
 
 2(m - n) 
 2ir 2ir cos 
 
 2(m 
 
 Then integration gives 
 
 47T 
 
 2w sin' 
 
 _X 
 
 47T 
 
 27T(Z COS \I/ 
 (1 + COS \l/) \ COS COS -- r - 
 A , A 
 
 2nd . 2irdcos\l/ 
 + sm ^ sin - T 
 
 A A 
 
 2nd 2ird cos ^ 
 (1 cos \p) \ cos -r- cos - - - 
 A A 
 
 2wd . 2ird cos \1/ 
 
 sm -r sin 
 A 
 
 2ir sin 2 
 
FROM A BENT ANTENNA 17 
 
 2ird 2ird cos , 
 
 (27) 
 
 2ir sin 2 \p . 2ird . 2nd cos ^ 
 
 + cos \f/ sin -r sin r 1 
 
 A A 
 
 The integrable part of (24) 
 
 r2wx Cos \l/ . 2irx , 
 sin r sin dx 
 A A 
 
 is also a standard form and becomes finally 
 
 2ird cos \l/ . 2ird . 2ird cos \L 2nd ] 
 >s^cos r sm^r sm COS-T ) (28) 
 
 A A A A J 
 
 Substitute 25, 26, 27 and (28) in (21), (22), (23) and (24). Then 
 insert the result in (18) and obtain: 
 
 4wl sin \// . 27rZh X 
 
 f-^Ji ~~~ Tr-\ Sin 7% . a T 
 
 , 27T 2irh I 2nd cos ^ . 2ird 
 sm -T- (Vt r Q ) cos ^ j cos sin ^~ 
 
 , 2nd Cost Ivd] 
 cos \l/ sin cos -r f 
 
 A A J 
 
 27r /T . 2irh I 2nd cost 
 
 COS y (Vt r ) COS -r j COS ^ COS 
 
 . 2,ird cos \[/ . 2wh 
 + sin -- r- -- sin -- cos \[/ f 
 
 , 2<jr .__ , 2irh f 2<7rd 2ird cos 
 
 + sin (Vt r ) sin -r j cos -r- cos -- r 
 
 2ird . 2ird cos 
 + cos t sin ~" sin -- r 
 
 1 
 
 2ir ,- . 2irh f , cos 2nd cos \l/ . 2wd 
 
 + cos -:- (Vt TO) sin ^ j cos \l/ ~Y~ ~ sm "T~ 
 
 {~] 
 | 
 
 , 2ird cos t 2<jrd 
 sm -- ^ -- COS ~X 
 
18 ELECTROMAGNETIC OSCILLATIONS 
 
 Now 
 
 . X 2irh TT 2wd 
 = 4'''T~ = 2'T~ 
 so that 
 
 2irh 2nd 
 
 cos -T = sin -T 
 
 A A 
 
 2ird 
 
 sin -r = cos -r 
 
 A A 
 
 When these expressions are substituted in (29), the terms in 
 sin (2irl\)(Vt TO) and those in cos (2ir/\)(Vt r ) may be 
 added together and give the simple form: 
 
 21 , 27rZh[ . 27r f 2wd cos ^ 
 
 .- sin -T sin L TO) cos 
 
 T7 . . 
 r V sin ^ Xr L 
 
 1 27r. T . . 
 
 cos -r f + cos -r- ( Vt r ) j cos \f/ sin 
 
 A J A 
 
 which is equation (19), page (9). 
 
/ 
 
 S g 8 
 
 i 
 
 S 
 
 o 3 
 s I 
 
 s 
 
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UNIVERSITY OF CALIFORNIA LIBRARY