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By curvatures os a sui saeo I mean sum os curvaturos in uiualty Perpendicular norma sections at ny oint; no C. VSF 's curvatura integras, hicli is the pro luet os the curvatur in the tivo principat norma secti usu or sections os reatos and east
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sequi that the rhombio doduca hedron is tho solution of ur problem fortho age of ali ho elis qua in Volume, and very par of the Oundaryo th group ither infinitely distant frona tho place considered, o Soadjusted iis noto interfero illi the homogeneougnes of the interior distribution o dolis. 0rtainly the rhombi do lueti hedron is a sol utionos the minimum, or equilibris miroblem; nil certain it is that o ther plane-fidei polyhedron an e a solution. . ut it has seemodo mo on puroly theoretica consideration, that thoe notrahedra ingles of the rhombi dodeca hedron, giving, hen
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On the Division o Spaeo illi Minimum Partitional rea. 23
tW equat curve sides meotin a a corner of the contra curvilinearsquare in a plane perpendicular o iis plurie. It is in the plano throughan elige and tho centro of the cube. The angies of this plane curvi linea triangle are respectivel IO9 28 , ut the oint of meetin of the two curvilinea sidus and ach of the w other hal of this, or 54ῆ I4 .
former, ut in a plane perpendi utar to iis plano nil to the dii i et ion ostii blast. Thus, guin uni agnin may the filius e transformed O usto rende the litti contra curvilinear quare parallel to ne or ther of the three uir os quar aperturus os the culti ' frame . rhus e seothat the wolvo plura siluis m00t in in tho centro of the tabo is u COD-sguration os uiast ubi uis uilii, rium hi h may ho fullon Dinia in three disserunt ab S.
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se by symmetry pullud illi qua foretis in opposite directioris, undthoro re is no requiret for tho quilibrium, uni it is olear that theequilibrium is stubio ithout them.
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provide the quar o glas is furnishod illi vortieul alis for hieli stips of 00d reconvenient), as h0wn in plan by the lae bordor of tho diagram These Walis are necessar tu maintain the nequalit os ut in different diroetions hiel the in equalit of the fides of the exagon implios B inspectio of the diagram e seu that the puli is
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Tho filius seen in the PLATEAU cubo hoW ne complete mali quadrilaterat, our alvos o Dur of the large quadrilaturuis, and ighthulves of eight of the sexagoris, bolongin to si contiguous colis allinathematicalty correet in very pari suppositi tho silui and the cubes ram to b infinitoly thin . Thus o se ut the elemonis required foran Xaut constructior of the complete tetrahat lueuhedron. y multinga lay mode of hut e actuali see, We have ni to complute asymmetrica figure by symmetricatly complo ting ach hal quadrilaturaland uel half-hexagon, and putting the wolve properi together, Withthe completo mali quadrilaterat, nil nother ille it a tho fur side of the a 4-face figure thus have a correet soli modet. S. Consider no a cubi portior of pace containing a largo numberos uel colis, and of course re large, ut a comparatively mali, number
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On the Division os paeo illi minimum Partitional Aroa. 27
9. Whilo the equuligation o puli in the three principa directionsis thus produced, orti is dono by the sim noli movin Wire-work of the age, and the tota area o sim is diminished by an amount equallo in donoto th Whol ivor dono, and T li pult of thesilui per unit readth. The change of ghape of the age uin supposedio e performe infinite ly lowly go that the sim is always in equilibrium througho ut tho tota area is ut euch instant a minimum, subjectio tho conditionsi Thut tho volume of euch coli is the iven amount; a That ver par of the wir has area edged by it; and 3 That no portio os ure has any ree edge. 10. Consider no v tho figure of the celi stili of eo urso a tetra-kaidodahedron when the pulis in the thre principat directions are qua-liged a doseribe in It must e perseetly sotropic in respecto
thes three directioris Heneo the air of mali quadrilaterat mus have beeome enlarged to equalit With the wo air of turge ones, hielimus have beeo me malle in the deformationa proces describo in I. O euch exagon three edges coincide With dges o quadrilatera faces of ne celi; und uel os the three ther coincides illi edges of throe of the quadrilaterais os ne of the contiguous elis. Hunc the 36 dges of the sotropic tetrahat ducaliodron a re equul and simila Plane res: 0ach of courge symmetrical bout it mi id te poliat. very angle fmeetin of edges is essent inlly IO0 28 to mali trihedra anglos et w0en tangent planus of the luis moetiri ni QO' . Symmetry shows that the
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Wood, properly ut the three sector of O fro in tho centre to theo vertiangin edges of the adjacent quadrilateruis. Hollow ut symmetriculi the ther throe sectorf, and tho hin is done The rosuit is hown in orthogonal projection, o fur a the edges are concerned in g. 4; uni us the orthogona projections tiro equul uni simitur on three planes ut right ungies to ne nother, this diagram suffice to allo perspec
position during the oldering The circulur Curvatur of the re is notmathematicali correct . ut the error duo to it is, no do tibi hard ly peredi tibi to the ye. 15. t ut tho tria forin os tho Qui vel od ges of the i iudi i lutertii plane siluis, and of the non- plane sui faces of the e Xagona sinas may