oli^kbls n. k.ll8, d.80., i'.r.l. - royal...

440

On ^ 6 o/' t/r,6 Lln^/«ees r'n tô n NSKN ônMss ck s oT' SnrUr'nAS cr-rc on on tê ^/ettSNT'oinenk 0/^4t-nos^>^67no /ôtontrcr^ 6ôc<Frents.

OlI^kbLS n . K.LL8, D.80., I'.R.L.

(Received Na^ 10, 1915.)

H1. In oousideriuZ tbe most suitable arrauAemeuts lor recording tbe variations ok tbe atmospberio potential gradient at tile Hast London OolleZe, wbere no larZe borirroutal surkaoe is available, I bave dad oooasion to oalonlats tbe distribution ok potential in tbe ueiZbbourbood ok avails and buildings ok siinple sbapes. ^ s tlie results ean be applied in praotiee it seeins advisable to put tbem on reoord kor tbe use ok otber observers.

Observation sbows tbat during 6ne weatber tbe potential at a point in tbe atinospbere over a level portion ok tbe eartb's surkaoe in tbese latitudes inoreases as tbe point is raised, at tbe rate ok about 150 volts per metre. Ibis rate ok inorease diminisbes slowly as tbe point aseevds, owiuZ to tbe sliZbt 6X0688 ok positive over negative ions in tbe air near tbe eartb's surkaoe, and at an altitude ok a kilometre is reduced to about 25 per cent, ok its value at tbe surkaoe.*

Ibe potential dilkerenoes between points on tbe eartb's surkaoe 1000 metres apart or between a point on tbe surkaoe ok a building and one on tbe Around near it are kound to be small compared to tbose present in tbe atmospbere.

Ror tbe present purpose we sball neZleot tbe small ekleot ok tbe ions near tbe eartb's surkaoe on tbe potential gradient kor tbe brst kew metres above tbe surkaoe and sball treat tbe eartb's surkaoe as plane and tbe eartb and buildinAs as beinA conductors.^ In order kurtber to simplik^ tbe problem as muob as possible, tbe walls ok tbe buildings are taken to be lonA in comparison to tbeir beiZbts, so tbat tbe elkeots ok tbe corners on tbe distribution ok

* l b s article " ^.tmospbsric Dlsctricit^," bv 6brss, in tbs * Dnc^clopssdia Lritannica/ l l t b edition, or tbat b^ Oerdien in tbs * Dandbucb der Db^silc,' vol. 4, p. 687, or blacbe and von Lcbweidler's ^ tm ospberiscb s Dlsktririitat,' cbap. I, ma^ be consulted kor accounts ok tbe rnetbods used and tbe results obtained. I'or recent results obtained at Lew, Obres ('l?bil. Drans.,' vol. 206, p. 299 (1906), and vol. 215, p. 133 (1915)) and Dobson 0 Droc. Db^s. 8oc. Don.,' vol. 26, p. 334 (1914)) ma^ be consulted.

t Denndork (° w ien er Ler.,' vol. 109, x. 923 (1900), vol. 115, p. 425 (1906)) bas determined ivitb tbs sains simplibcations tbe cbanges in tbs vertical potential gradients near a long plateau witb rounded edges, a circular plateau, and an ellipsoidal column. 8ir d. Darmor and d. 8. D. Darmor 0 Lo^. 8oc. Droc.,' vol. 90, p. 312 (1914)) give diagrams or potential surkaces, etc., kor an ellipsoidal column and an sartb-connected spbers. ^

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441r-r -rec^ ^o-r^ Il^cr^s.

Potential near tbe middle ok tbeir lengtb ma^ be neglected. H is avails are kurtber talren to bo vortical, and in tbe case ok more tban one to bo parallel to eacb otbor. Looks aro taben to bo borirontal.

8 2. Ocrss / .—^ long tbin vortical wall projects krom a borirontal surkace abovo wbicb, at a considerable distance krom tbs wall, tbo potential gradient is constant. 1o kind tlio distribution ok potential near tlio wall.

Lake tbo s plane, wbere r — s?-i-r2/, vortical and perpendicular to tbo wall, tbo Lv axis along tbo boririontal plane, and tbo 2/ axis up. tbo wall. Ik is tbo beigbt ok tbo wall, tbo Zebwarsian transkorination ^ wbere 227 — 2 -1- 22), converts tbo Lv and ^ axes into tbo axis ok 22 in tbo 22) piano and tbo krst quadrant in tbo korrner into tbo brst two quadrants in tbe latter plane. Integrating, we bavo

---(22?/«^—1) or 22?/« — 1. (1)Ik L is a point in tbe air wbose bi-polar co-ordinates aro 6, kroni tbo

top ok tbo wall and kroni tbo imago ok tbo top in tbo piano respectively, tbe last equation gives us at L

22 — v /'(^ ) - oos (6 -i- 6"), 2) — . sin H (6 -1- 6 ). (2)Ibus tbs potential 2) at an^ point wboss bi-polar co-ordinates are given is

easily calculated.I?o calculate tbs potential at a point wboso co-ordinates are n, 2/, or to draw

tbs surkacos ok ocjual potential, it is more convenient to use equation (1), wbicb gives on equating separately tbe real and unreal parts

(2 — 2?)/«^ — (a? — 2/2-i-H2)/H2 2L2)/« —

and on eliminating 22,—2/2-i-^?)/H2.

Ibus tbe potential at tbe point M, 2/, is given bv2? --- («2/ 2 ^2) ^ - 2/2 -i- 2)2^ 4 ^ 2-j^ _ 2 ^2^ (Z)

and tbe kactor b wbicb tbs observed potential 2) at tbe point a?, 2/, sbould be multiplied to give tbs potential at tbe beigbt 2/ above an inbnite plane is

_______________________ A_______________________

lb s SHuipotential lines are readily drawn krom tbe equation

Ibe^ are sbown in bg. 1* kor tbs case — 1, « — 1.* I bs,vs 1i0 tliLnIc V. Ls.rQ68, lOtti Ds-st 8urrs^s, 0U6 ok senior students, kor

dravinA tbess ourves.



442 Dr. O. D . Dess.

1-—Lection ok tlie e^uipotential surkaces near tlie middle ok a lonA tliin v a il.

3. Ik a? 18 eonstant we Lave, sines ^ — («/^)^/(s2 - j - ^

/9tt>X , . /9vX — « — -l-2Lvv9 ^ /L v9 ^ /- ! ^ -l- 2 2LV^)'

Ik, kurtlier, 2/ — 0, tlisn

E ) - - o Z <°>wlien a; 18 inknitel^ larAe.

Ilie last two equations sliow tliat tlie ratio ok tlie vertieal potential gradient at a point on tlie Around to tlis norinal vertical gradient is e^ual to tlie oosine ok tlie anZle subtended at tlie point b^ tlie lieiAlit ok tlie wall.

Hie kollowinA liable* sliows tlie vertioal gradient at points on tlie Around wliose distanees kroin tlie koot ok tlis wall are Ziven in terms ok tlie lieiAlit ok tlie wall.

* 'I'liis la li ls and tlie eorrespondinZ ones on pp. 446 and 449 are added at tlie suZZestion ok Or. Oliree so as to be available kor discussion ok tbs eLects ok vertical potential gradients on plants and animals.



r'-r -recr /)onF sâ^s. 44L

I'able ok Ratios ok Vertival Oradient to tbe Normal Vsrtival (Gradient at Roints on tbe (Ground near tbs Middle ok a bonZ ^bin Vertioal ^Vall.

I)i8taQ66 kr0VL Loot Vertioal gradient Di8tanoe krom Loot Vertioal gradient NormalLsigLt Normal Heiglrt

0 0 0 0 1 5 0 8320-2 0 1W 2-0 0 8940-4 0-371 3-0 0 9490-6 0 SIS 4-0 0 9700-8 0 626 6-0 0 9811-0 0 707 10-0 0 996

^4. Ik z/ is vonstant ws bavs in tbe Lams wa^

/9r X ./9vX _« ________ ar-l-r^_______Voar/r, Vob'/v X. /lLv^—

^.t a? — 0 tbis Zives, ik ^ 18 less tban /r,,

- - 0 ,VM/z, I ds

L'be borixontal potential gradient at 7/ on tire wall is tberekore identioal witb tbe normal vertioal gradient it'

tbat i8 ik ^ — /r/^/2 — 0 707^..8 we proveed outwards krom tbe wall tbs borirontal potential gradient

deoreaeee, dnt at tlie beigbt 0 707^ we ma^ move outwards a distanvs 0 l . witbout tlie potential being 1 per vent. 1s88 tlian tliat wbieb would bavs been kound at tlie point ik tlie borisontal gradient liad remained oonetant and equal to tlie normal vertival gradient, ^ t a distanvs 0 25^ krom tlie wall tlie potential is 5 per vent. 1e88 tlian tlie normal gradient would give.

^.t larger distanees krom tlie wall tlie ebange ok potential sbould bs oalvulated direvtl^ krom eitlier ok tlie expre88ion8 (2) and (3) kor it in terms ok tlie vo-ordinat68 ok tlie point.

§ 5. Oase H —^ long vertival retaining wall 86parate8 kroni eavli otlier two lioriôntal plane 8urkav68 over wliivli tlie potential gradient at a vonsiderabls distanvs krom tlie wall i8 tlie earns and independent ok beigbt above tlie planes. 1o determine tlie distribution ok potential near tlie wall.

lairing tbs s plane, wbere s — rv -i- vertival and perpendivular to tbs retaining wall, tbs sr axis along tbs lower plane, and tbs axis up tbs surkavs ok tbs wall, we bave tbs 8vbwarîan transkormation



444 vr . O. L . 1.668.

vvkiok converts tlie louver doundar^ ok tlie atinospliere in tlie s x1g.no into tlis axis ok -L in tlie plans, fliers ^ -l- -iv.

writing rv/« — eosli tlie sanation deeonies s — «^(eosli ^d-1)ci^, tlie Integral ok wliieli is

L — « (sink ^d- ). (7) i

Expanding and separating real and unreal parrs, we liave^ /« — eosli eos2/, v /« — sinli s in 2/ ^ V

and « /« — sinli E o o s d - A 2//« — oosli sin 2- d- -? / 'Ik 2) — 0 we inust liave eitlier — 0 or 2) — 0 or 27-.In tlie 6rst ease sr — 0, in tlie seoond r/ — 0, in tire last 2//<r — ?r. Hie

lieiglit ok tlie wall is tlierekore ecûal to K?r, tliat is « — H,/?r, and tlieequations connecting w and s rna lie written

2L)/« — eosli A 77-2-/ — sinli ^d- (9)1 o kind tlie potential v at a given point M, it is necessary to solve kor2 , tlie ssoond set ok transcendental equations (8) or (9), and sulistitute tlie

values ok and 2) in tlie equation kor v.T'o draw tlie eûipotential surkaees we use tlie equations:—

?r — (v/«) eot 2) d- argsinli (v/« sin 2/)^ (^/^) — v / sin d-») / '

s.nd assign to 2/ all values kroni 0 to 77- and to v values exceeding 0.Hie curves obtained are sliown in 6g. 2 kor tlie ease in wliieli tlie tieiglit ok

tlis eûipotential surkaee ok potential unit)'' is ecûal to tliat ok tlie wall, ttiat is « — 1/?r, and — 1.

^KZ. 2.—Lection ok tlie 6<^nipot6nti8l 8urk8.668 near tlie middle ok 8.IONA retaining wall.



445

f On dikkerendiadinZ ec^uadions (9) wo kavs(1 /«) <2L — sink 60S 17 ^ — oosk sin 7/( 7, ^(1 /« ) — 008k sin 7 -i- sink E 008 7) ek7/(77-/^ ,)^ — oosk 608 —sink sin 7- <?7) 4- ^(77-/^ ) — sink s in 7/ ^ 4- oosk 0087/ ^ 4-

8 6. Ik Lv 18 oonstand wo kaveoosk 008 7) —sink sin 7- c?7/ 4- — 0.

1>U,» (1 /- ) -r« - / °° -k k -in k ,k M ^ ^ 00b > ^V 608Q 608 1 /

and (77- /^ ) ^ — / ^— >-oosk oos 7 4-1^7-.Voosk oos 7, 4-1 /

(11)

IIonos dke verdioal podendial Zradiond at dke poind oorrospondinZ do 7), is Zivon d^

9v «77' sink ^^ 00sK^4-608 7) (12)

«7r sink E

'9vX

^ 008k ^4-1-271- sink ^^ 008k ^—1

ik 7) — 0,

Ik 7, TI'.

In eaok 03,80 ^ ik 18 larZo.

lk o verdioal podondial Zradiond is dkerekore «?i-//r, over kodk planes ad oonsideradle disdanoes kroni dko rodaininZ wg.11. ^d snialler disdanoes id is lo88

dkan dks norinal over dks lower and Zreader dlian id ovsr dko upper plans.

ladle ok Verdioal Ors-diend ----- uear dke Middle ok a k-onZ IdedaininZHornial Verdioal (Iradiond

IVall or llad-rooked LuildinZ.

I O Sr 811178,66. Uppsr Lliirkaos.

Di8l)S.Q06 krom tool) Oradiend Ol8d8.uo6 krom lop ( radiSudL iAtilr Normal Heigtilr I 0I7HL8.1

! 0 0 0-0 0 0 -»0 127 0-100 0 066 2 160 258 0 198 0 2M 1 680 393 0 292 0 618 1 -310 537 0 380 2 23 1-110-690 0 462 22 0 1 -011 154 0 6361 79 0 7614 13 0 909

25 2 0-987>



446 Dr. O. R . D668.

His preceding âble gives in terms ok tbe normal gradient tbe vertical gradient at points wboss distances krom tbe toot and top ok tbe retaining wall on eacb ok tbe borisontal surkaoos are expressed in terms ok tbe beigbt ok tbswall. '

§ 7 Ik r/ is constant we bave

sinb sin c^4- (cosb cos 4-1) — 0.

and (-Tr/ ) — ^cosb cos 4-14- oosk^k cos^ -^ l) W

Hence tlie boriôntal potential gradient at tbe point corresponding to is given b^

_«?r sin --xOar/r, ^ costly4-eos^

— ^ ^ ; ik k — 0, tbat is wben a? — 0. (13)k 14-oos^ ^

It will lie noticed tbat tlie ratio ok tbe vertical to tlie li oriental gradient at tlie point corresponding to is sinb^/ sin

Hie boriôntal potential gradient close to tlie wall will lie identical witb tlie vertical gradient over tlie planes at great distances krom tlie wall ik

sin^ — cos ^4-1,

tliat is ik — 7r /2 .

Hie point ^ on tlie wall corresponding to — 0, — ?r/2, is given b^

-- ^(sin^4-»-)/7r — /t(l4-7r/2)/?r.

Hence tlie point on tlie wall at wbicb tlie boriôntal gradient outwards is ec ual to tlie normal vertical gradient is at a lieiglit equal to (1 -j- ?r/2) /?r ok tlie total lieiglit ok tlie wall, tliat is to 0 818 ok tlie total lieiglit.

^.s we proceed outwards krom tlie wall at tins lieiglit tlie boriôntal potential gradient decreases, lrut at a distance krom tlie wall not exceeding 01 ok tlie lieiglit ok tlie wall tlis potential is less tlian 2 per cent, smaller tlian it would lre ik tlie boriôntal gradient liad been equal to tlie normal vertical gradient kor tlie wliole distance.

H 8. H / .— Iwo long tliin vertical walls parallel to eacti otlier rise to tliesame lieiglit above a l>oriental plane. I?o kind tlie distribution ok potential in tbe space^between tbe walls.



Srrr/«06S r-r irecrT' ônA R^cri/s. 447

I?o silliplik^ tl»e calculation we sliall talcs tlie avails as beins two consecutive walls ok a regular series extendinZ on dotli sides to inLnit^. lalcins tlis 2

plans, wliers 2 — vsrtioal and perpendicular to tlis ssriss ok planes,tlis ar axis alonZ tlis liorisontal plans, and ttis ^ axis vsrtisal tlirou^li tlis point on tlis M axis lialk wa^ between tlis planes, tlis transkorination sin 22- — « sin (2/cr),* wlisrs « and cr are constants, « bsinZ less tlian unit^» converts tlis lower boundary ok ttis atinosplisrs into tlis axis ok n in tlis

plans, wlisrs 22) — 22-I- 22).LxpandinZ tlis circular functions we bavs

sin 22 cosli v — « sin (25/a ) cosli (z//a) and cos 22 sinli 2) — « cos (cv/a) sinli (^/ a)

Ik v — 0, cos (ar/a)sinb(^/a) — 0, and sin 22 — « (sin(25/a) cosb (^/a)). Hius eitlisr 2/ — 0 or (25/a ) — dl(2^d- 1)?r/2, wliers -r. is an integer.

Ik ^ — 0, sin A — « sin (25/a), and 22 and 25 increase kroni 2sro toZstlisr till ar — a?r/2 and 22 — arcsin «.

Ik 25/a — ?r/2, sin 22 — « cosli (r//cr), and 2/ increases kroin 2ero to a arZcosb 1 / «, wliile 2- increases kroni arcsin « to ?r/2.

Hie distance 8 ok ttie walls apart is tberet'ors — a?r, and ttieir bei^bt ^ is a arZcosb (1/«). Renee a — 5/?r, « — 1/cosli (?r^/L), and ttie equation con- nectinZ 22) and 2 ina^ lie written

sin ro — sin (7rr/8)/cosb (v^/L). (15)

Hie equation to tlie eûipotential lines is tlisrekors

sin (var/ L) cosli (?r^/L) cosli 2)

cos (2?^/ L) sillli (^ /L ) sinli^v

cosli (?r^/L), (16)

a quadratic in sinli 2) and cosli 2) krorn wliicli v at an^ point ar, lnaz) be calculated and tlis reducing kactor to convert readings taken at «, 2/, into readings in tlie open inav lie kound.

In drawing tlis eûipotential curves it is best to use tlie equation in tlis korin

* I k s sorasvIiLt mors A6N618.1 tl'anskorruLtion 81Q («>//3) — a sin (r/cr) nra^ 1)6 treated in tlis sains tllronZIlout.



448 Dr. O. D. Dsss.

Ik s 6ouipot6ntiai curves are siiovvn ill 6A. 3 kor tils 09.86 9 — 77- ..

I'is. 3.—Lection ok tlie e^nipotential surkaces near tlie middle ok one ok a series ok tlna parallel avails. Ik e tliiclr dotted vertical line is inidlva^ between t^vo planes.

8 9. Ik ar is oon8tg.nl; ws iiavs, kroin sanation (15)/9rtX . / 9vX _ - ?r_________008 7? (a; -I- r^)/9 _________9^ V9z//L 9 ( oosti (-n-Zr./ 9) —sin 7r (^-t- rr/)/ 9^

cv — 0 tiiis rsduoss to

— ---0 nnd ^ ________ oosii <7rz,/9),9 //r ' >9z//L 9 v'^eo8ti^(?r-^/9)-i-8inti^(7r2//9)^'

I'or 1arZ6 values ok 7/ tiiis Zives tii6 norinai vertioai potential Aradient^.t Z/ — 0 it Z1V68

/9vX _77-_________ 008 (77-3?/9)_________vdr//r 9 0 0 8 ^ ( 77-^/9) —sin^(77-a:/9)^'

tiiat 189vX _ 77-______________ 1_________Zr//- 9 sinii (77-/r./9) / 608 (7W/ 9) -p 1 '

. .t tlis point on tlie Zround liaik->va Iretween tire planes77- 19 608ll (77-^/9)

77-/9.

(17)

(18)

H 1U8 tii6 ratio ok ttie vertioai Aradient on tlie Around lialk-iva^ detween tds planes to tde norinai vertical Arad; ent is 1/oosli(77-H,/9).

Id s kollowinA lad le A1V68 tde values ok tills ratio kor dikkerent values ok tde ratio ok deiZdt ok pianos to distance apart.



^i/^/croes rVr. -r ecr l^cr^s. 44S

kable ok____^ r k isal Grad ien t— on tbs Oround lVlidwa between IvvoNormal vertical (Gradient

Vertical planes tor DiLerent patios ok Neigbt ok planes do Distance apart.

Lsixlit V srkioai Aradisntr.

Neigiit; Vsrtrieaigradient;.

>Distsuos sparti

Vsrkioaigradient;.! Diskanoe apart; i Disdanos apart;

0 0 1 -0 0-382 0 562 1 273 0 -03660 064 0-980 0 672 0 322 1 592 0 01350 127 0 925 0 763 0 180 1 910 0 00500-191 0 844 0 964 0 099

Itis vsrtioal gradient at otber points on tbs ground between tbs planes is given b^ tbs sanation (17) ado vs, and is tabulated below kor several valnss ok tbs ratio ok beigbt ok tbs planes to distanos apart.

-r»dls °t o„ krounck Kekve^ Iv.o Vsrti^Il>orinal potential (Gradient

planes.

Diskanos ok point; LtoiZiit; ok pianosDistanvtt ok pianos apart; Distnnoo apart:

kroin piano. I roin rniddio. 0 064. 0 -191. 0 382. 0 763.

0-0 0 6 o-o o-o o-o o-o0 1 0 4 0-844 0 445 0 201 0 0660 2 0 3 0 946 0 679 0 366 0 107y-3 0 2 0 970 0 783 0 473 0-1460 4 0 1 0 978 0 831 0 633 0 1720 5 0 0 0 980 0 844 0 552 0 180

Ilisse kigures are sukkicient to sbow bow great is tlis ekkect ok tbs walls on tde vertioal gradient on tbs Zronnd between tbein. pbs^ ina^ be taken as representing witb a kair degree ok accuracy tbe vertieal gradient on tbs ground in a street witb buildings on eaob side ok it.

H 10. Ik A is constant we bavs/3nX . /9vX _?r_________ oos 7r(Lv-pr )/L _________

V9»:/r, L eosb^(?7- /L)—sin (ar-^r^)/8^)

^.t rv — 8/2 tbis beooines/9nX , ^ _______ —r sinb (?r^/8)_______V9cv/v 8 (cosb^(?r^./8)—cosb (?rr//8))



450

Ik ^ is less tdan /,., tdis Fives

/3--X „ / ^ - - - ^ _________ »i»l>(7^/r>^8r/x ' x3-r/» 8 ^ / ôost? (7I-L/8)—L0S (?>-N/L)^ (IS )

Ik tde dorirrontal potential gradient ad z/ on tde wall is to be sûal to tire normal vertioal gradient ?r/8, we must dave

8ind^(?rz/'/8) — cosd^(7r^/8)—cosd (-Trr/ /8),

tdat is siud (?rz/'/8) — (1 /^ /2 ) sind (-n- /8).

^Vden tde deigdt becomes small compared to tde distance apart 8 ok tde planes, tdis Fives tlie distance 2/ up tlie plane as 0 71 times tlie lieiAlit, as was kound d^ direct ealoulation.

Hie kollowinF lad le Fives tde relation detween tlie ratio ok tde deiZlit ok tds planes to tdeir distance apart and tde fraction ok tlie deiZdt at wdied tde doririontal potential gradient outwards is eûal to tde normal vertical potential gradient over a plane surkaee.

HsiglikVistanoo apark'

Iâeklon ok kioiZIiir. Distanoo apart

Iraotion ok

0-0 0-707 0-445 0 7900-064 0 708 0 509 0 8070 127 0 717 0-572 0 8220 191 0 727 0 636 0 8360 264 0 741 — —

0-318 0 757 0 795 0 8640 382 0 773 0 954 0 886

^s tde planes approaed eaed odder tde potential surfaces near tdeir summits deeome more nearly doriôntal, so tdat tde potential Fradient is more nearly vertical tdan doriôntal.

11. It kollows from tde previous word tdat wders plane doririontal surfaces ok consideradle extent are not available kor tde determination ok tde normal vertical potential Fradient in tde atmospdere, odservations in tds nsigdbour- dood ok duildinFS. ean de utilised, tds value ok tde redueinF kaotor deinF calculated krom tde korms ok tds duildinFS in tds simple cases dealt witd. In man^ cases odservations ok tde dorisontal potential Fradient outwards krom tde walls ok duildinFS wdied are not too close toFetder ma^ de made, and, ik tde position ok tds point ok observation is properly cdosen, tds doriôntal Fradient observed will be identical witd tds normal vertical gradient over a dorirontal surkacs. I'or a long wall ok a building witd a Lat rook or witd a parapet, tds dorisontal gradient outwards sdould be measured



Ken'es o/' r-res r-r ^^a^r'rre 451

at a poind near tire middle ok dbe lengdb ok dbe wall and ad a distance up id wkioli ia generally about dbres-ciuarders ok dbe bsigbd. Ike borirondal gradient kor a distance outwards nod exceeding 1/10 dlis bsigbd ok dbe wall will nod dikker b^ inore dban 2 per vend, krom dbe norinal vertical gradient over a large bori^ondal area.

On ^ 6 L^^rnes rn o /

L^ l! . HiOL8, 8c.v., L.R.8.

(Received 15, 1915.)

Ik s problem ok dbe limits and nurnerioal relations between dbe lines ok dbe enbanced 8erie8 ok doublets in dbe alkaline eardbs lias kor long been a dikkculd do epeedroeeopisds. Rid2* in 1908 gave arrangements kor dbe 8barp series kroni lVlg do La inolusive, and proposed series kormulse kor Oa, 8r, Da, in wbieb alone be bad dbree lines krom wbieb do oaleulade dbe oonsdands. Lbs absence ok exdra lines rendered id impossible do desd bis kormulse, bud dbs values ok dbe consdands obdained kor bis kormulse were c uide oud ok line widb dbose ok dbe analogous eonsdands in odber series, and produced an insdincdive doubd as do wbedber id gave dbe correcd reladion. It is now possible do test bis limits b^ considering wbedber dbe denominator dikkerences wbicb give dbe observed separations bave an^ reladion do dbe oun or nod. Hie result ok dbis consideration is deklnidel adverse. In none ok dbs dbres is id possible do make dbe dikkerences multiples ok dbe oun widboud supposing observation errors in dbe doubled separations wbicb are Huide inadmissible; and even dben in dbe cases ok 6a and La b^ taking odd multiples ok Li, wbicb is never dbe case kor 8 doublets in an^ odber known series.

Ibere can be little doubd bud dbad Lowler^ bas ad last settled dbis question b taking dbe Rydberg numerator constant do be 41s in place ok Is, dbus combining in one set lines wbicb on dbs old supposition would be arranged in two series, depending on 8barp and principal sequences. Tbe object ok dbe present node is dbe determination ok dbe connection ok dbsss series widb certain laws wbicb bavs been arrived ad in previous communications^ do dbis 8ocied/

* * l?li^s. 2oitsclir.,' vol. 16, p. 521.d ' ktiil. trails.,' vol. 214, p. 225 (1914).I kdil. Ir-Lvs.,' vol. 210, p. 57 (1909) ; vol. 212, p. 33 (1912) ; vol. 213, p. 323

(1913)—rsksrrod 1o in tbs kollovinZ as (I), (II), and (III) respectively.



oli^kbls n. k.ll8, d.80., i'.r.l. - royal...

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