TUANSACTIONS OF SECTION A. 



()27 



f>f knilV 

 itiiMi I'or 

 jri's will; 



biilaiK'o. 

 ike 1hi)s>- 

 1', cuttiiiL' 



\^ VdllSliU 



liciiliir t ' 



onliiian 



y. Now. 



V, so til ill 



lie nxis 111 



tlywiifi 1, 



e iind luii 



lll(Xlli tllf 



\)Xv^, liaiiL' 

 ■d iHisitiiin 

 IS runnulii 

 ,iitiiisuro lit 



1 n1)i>vo. il 

 lis ptissibli' 

 h lilt) i'nci'r 



,• wluMl til' 



V( it alii 111 "1 

 nil ortlinnv} 



ulijivi"!- ''"' 

 If North, 11- 

 .>s lit' pliK'i'd 

 .'uvlirs axis 

 (.li's and i'- 

 ii-ition ail'! 

 liy which 



ihviuni v.i:, 

 lU'ult with, 

 ilanci' wi;h 

 ill osfiilat. 

 sine of till- 

 I',, cdp'^ i- 

 f dotiiiiieh 

 icily "I" '''I 



I luivi-' "•'■ 



on a lixt'l 

 II art jn-t a- 

 (that is ^" 



, ; ff th.c avrn 



of ;j:yrati"'i- 



; of tlie pliico. 



oav. rotational North) insteiul ol" ' iiia;.'nt'ti(; North.' T al.so .'^liowrd a iin'thoil of 

 iiiiiiniliiiL' il jrynLStat .<() as to h'avf it iVoe lo turn roiiml a truly vfitiral ax's, ini- 

 p,.iliiil hy so littli' of IVii.Miuiial inlliirnei' as not to jirovont tho rralisatiun nf Iho 

 iilca. ihi' lui'lhod, Imwijvi'r, proinisi'd lo hr sonicwluit troiihli'-nnn', ;uiil I iiavo 

 since foiuul that tho olijt'ct of jircdiieiiiu' a ^■•yrostiitic nioiirl of tho nui-iriu'tic 

 compass niiiy, with a vitv ri'inarkablo dynamical luoditiciiliun, ht; nuicii inoro 

 siiiiply attaineil hy mon-ly siispcndinir tho Lryioslat liy ii vei'v lonjj lini' win- or even 

 by llnatinjj: it witli siillicii'nt stjibiliiy im ii jimpfrly jilaniuil iloati-r. To invistiLi-ato 

 till' tlii'ory of this arriiiii-'iMncnl, li't us first supposi- a ^'vroslat with tho axis of its 

 tivwiiei'l hori/oiitiil, to bo huiij.' by a Vfi'V lino wirt' attadu'd to its frauicwnrk at 

 11 point, iis far as can conviMiionlly bo arraniri'd for, itbovo tlu! ccr.tro of ,LMiivity of 

 tlywhiH'l and franiew;.rk, and let tho ujiprr I'lid of the wire bo uttiifhcd to a 

 torsion hi'iid, capablo ol' jjfinir turiu'd round a ii.xi'd vortical nxis as in n L'oulonib's 

 torsion balanco. First, for siiiii)li('it\'. lot us sii])pi)st' the earth to bo imt rotiitinii^. 

 Till' llywhof'l bi'iiijj: .«i't into rapid rotation, let tho iryrostat ))i' huni;- liy the wire, 

 ami alter boino; steadied as carefully as possible l)y haiiil, lot it be lelt to itself. If 

 it bi> obsorved to coninienco turning' a/.iniuthiilly in eiiher direction, check this 

 motion by tho torsion head; that is to say, turn the toisioii liead iriuitly in a 

 iliri'ctioii o])posite to tin,' oliserved iizininlhal inotion until this motion ceases, 'I'hcn 

 do nothing' to tho tiirsion head, and obsorve if a reverse a/imnthal niolion super- 

 viiics. If it does, check this motion also by opii.isinLC it by torsinn, but more gently 

 tlian liefore. Goon until Avhen the torsion head is lolt untoucliod tino ^'yrostat 

 ri;'niaiiis at. rest. The process frnno throuirli will have been undistin^-U!shiil)lo from 

 what would have had to be performed if, inst'iul of t!io iryrostiil with its rotating'' 

 tlywlieol, a rijrid body of the .same -weiirht, but witu much oroater inoment of 

 iiu'i'tia about tho vorticiil axis, had l)cun in its place. Tho formula for tho 

 inif,niienteil moment of inertia is as foUow.s. Denote by — • 



W, tho whole suspended weii_'ht of flywheel and framework. 



K, tho radius of ;:yralion roimd the vertical throuofh tho ci ntro of L'liivity of 

 tho whole muss rcLrarded lor a moment as one rigid body, 



w, iho ma.ss of the tl ywheel, 



k, tho radius of ^'vration of the ilywlioel, 



«, tho distance of tho point of attachment of tho wire above tho ccnfro of 

 f^ravity of ily wheel and framework, 



(/, the force of jrravity on unit lua-s, 



o), tho anu'ular vi.'locity of ihe Ily wheel ; tho virtual moment of inertia round 

 a vertical axis is 



Vn ix- 



1 





■ "•' ) 



(1) 



The proof in very easy, ilerc it is. Denote hv — 



(/), tho anprle between a lixed vertical piano and the vertical plane containing 



tho axis of the tlywheel at any time f, 

 0, the aiifflo (supposed to bo inlinitely small and iu the plane of </)) at which 



tho line <i is inclined to the vertical at time /, 

 H, tho moment of tho torf|ue round tho vertical axis exerted by the bearing 



wire on th<! suspended tlywheel and framework', 

 I'y the law of o'oiieratioii of moment of momentum round an axis pcrpen- 

 ili'.-ular to the axis of rotation re'[iiisite to turn tho axhs of rotation with an angular 

 velocity (I(j)'iU, we have 



'f .: ,/WaO (2) 



(Ii 



ifk- 



liccause i/WtiO is the moment of tho couple in tlu> MMlical plane throucrli tho axis 

 by wdiic'u the aii<rular motion (hj) tlf iu tho horizontal plane is produced. Apain.by 

 the same principle of frdu'ration of moment of inomeiitum taken in counoction 

 with tho eiomentary principle of acceleration of angular velocity, wo have 



if/k-ft) , + ^\Tv , 



(If d: 



,, rfV/, 



II 



(3) 



.s s 



