8o 



NATURE 



[May 30, 1872 



and q the angle between the plane A R and a fixed 

 plane. Let /j, m, be similar co-ordinates of A' referred to 

 ]i, and let fi, «', be the values which fi and co assume at 

 the point A. Let /•' {jx a') be a function of n' and w', 

 which, when ja' and w' become ix and w, may be written 

 F {(I, a)). If S ,S' be an element of the surface of the sphere, 

 whose radius wc shall take as unity, then S 6' may be ex- 

 pressed by - S/ . S17, or by -Sfi . 801, according as occa- 

 sion reciuircs. 



It is obvious from spherical trigonometry that 



/ = /-'m'+ n'i-m" V' /' cos ('d -co') 

 and that therefore in the expansion 



, , .,' - .Ji =i+P,x-\-P.x"--\- . . . 

 (\+x--2xp) < > I • 



The quantities P^, P.. &c , satisfy Laplace's equations in \i 

 and CO, and also in \i! and co'. 



Differentiating this equality, multiplying by 2 .r, and 

 adding, we get 



Integrating each side of this equation over the whole sur- 

 face of the" sphere, and ecpiating the results, we have 



Jo j-.(i-f .r=-2.r/)- ^ ^ 



r "/!'('+ 3 /'. •^■+ 5 P. -r' -f . . ) c/^t «'co 



The first of these two integrals is readily found to be equal 

 to 4n-, being thus independent of .r. 



.-. I"J j'^_[ (I -f 3 ^. ^-+ S ^=-v' +..)<(> c/co = 4 ,r 



Now as .r approaches unity, every term in the series whose 

 limit is represented by the integral 



/:/: 



dpdq 



{l+X--2X/>) 



becomes more nearly ec|ual to zero, except the terms in 

 the immediate vicinity of the value / = i, which increase 

 in value, ;>. in the neighbourhood of the point A. Hence, 

 as we diminish x, the ratio of 



f r/! ] ^ (/^ ") {• + 3 />! + 5 i''= -V^ + • . } d^ "'" to 



Piix'o') /'7f!| {i + 3P,x+sP.x-+. .},7>c/co 



that is to 4 TT . F (fi' o)') becomes continually more nearly a 

 ratio of equality, since F(n' co') is the value towards which 

 F {fi co) continually approximates as we draw nearer to 

 the point A. Hence we have in the limit 



4TT.F(fi.'w')= /'"f^' ^(j"«) (i+A + 5^; + - -'jdixdoi 



and since /•„ satisfies Laplace's equation of the //ih 

 order in n' co' 



also satisfies it, because fi co' are constants so far as this 

 integration is concerned. Hence P" (m' <■>') is expanded in 

 a series of functions satisfying Laplace's equations. 



James Stuart 



LETTERS TO THE EDITOR 



[ The Editor does not hold himself responsible for opinions expressed 

 by his correspondents. No notice is taken of anonymous 

 communications. ] 



The Volcanoes of Central France 

 The eruptions of a.d. 458-460, whose showers of pumice or 

 ashes reached and alarmed the city of Vienne, then the capit a 



of the chief State in Gaul, and led to the institution of the 

 Rogations (now called Litany) and the " Rogation Days," can- 

 not have proceeded from the province of Auvcrgne, as Mr. 

 Green supposes (Naturk, May 16). That province, containing 

 about half the French volcanoes, is the most distant of the three 

 volcanic ones from Vienne, and moreover is held to have been 

 quiescent in that age (as well as ever since) ; because Uve eminent 

 writer, Sidonius Apollinaris, who had settled there, and wrote 

 a poem on its scenery, betrays no knowledge of its volcanic 

 plienomena. .So, at least. Sir Charles Lyell has repeatedly in- 

 sisted. It is true th.it, writing before the date of the Vienne 

 calamities, his silence proves nothing ; but as fully liulf the 

 French craters are not in the Auvergne, but between that pro- 

 vince and \'ienne, namely, in either the Velay or Vivarais, within 

 aI)out fifty miles of that city, .and ranged along almost a quadrant 

 (the S.W. quadrant) of its horizon, there can be little doubt that 

 some of them were the scene of the "portentous fires," and 

 sources of the " Sodomiiic showers" that alarmed the IJurgun- 

 dian capital, and led St. Mamertus to institute these fasts. Of 

 Mamertns himself there remain no writings, and the memory of 

 any historic eruptions in France appears to have died out from 

 that very century to the present ; though none in all history were 

 belter attested, none within many centuries of Pliny's even so 

 well. For it is strange that no later clironiclers mention anything 

 hut the earthquakes and some fires of buildings ; the sole autho- 

 rities for the eruptions being their contemporaries, the above 

 Sidonius and the bishop who succeeded Mamertus in his see, 

 and these two being the sole men in Gaul of that generation 

 whereof any document remains. The former writes to Mamertus 

 himself a very fulsome, adulatory, but necessarily a materially 

 true memoir ot the facts ; and the latter allusively recounts them 

 in a sermon to the very flock among whom the observances had 

 begun. It seems impossible to conceive belter witnesses to any 

 event whatever, and they are literally all the contemporary 

 writers extant. 



The late Sir Francis Palgrave appears the first modern to have 

 disinterred this page of forgotten history, in the Qiiartcrly 

 /v'lT'.Vri' of October 1844. See also a most extraordinary paper 

 on it in the Gcnikman's Magiizinc of May 1S65, commented on 

 by me in the Pcudcr of the next month (p. 6S3). As the original 

 passages, however, of Sidonius and Bishop .'\vitus have not been 

 reproduced, I enclose literal translations of them, if you think 

 they would interest your readers. The style of their time must 

 be allowed to be detestable, but not quite without parallels. 



E. L. Garbett 



" .Sidonius to I,ord Pa'riarch Mamertus, health ! It is reported 

 the Goths have advanced then' camps on to Roman soil ! "Tuthis 

 kind of eruption we wretched Auvergnats are always the gate. 

 For we afford to the enemies' malice peculiar satisfaction; 

 because, as they have not yet marked their bounds from Ocean 

 to Rhone by the course of the Loire, they (under Christ's mercy) 

 find their sole hindrance from our opposition only. Indeed, the 

 tracts and regions of the surrounding country the eager assault 

 of their threatening power would long ago have devoured. But 

 in this, our so bold and dangerous a resolution, we trust not in 

 our hearts either to the crumbling face of ramparts, the rotten 

 barrier care (stiidiiuii), or the failing defence of sentinels for assis'- 

 ance, but are only soothed by the comfort of the Rogations 

 introduced, of which you were the auilior ; which, being to be 

 founded and instituted, the Auvergne people has begun to practise, 

 if not with equal result, certainly with not inferior zeal, and on 

 this account does not turn its back to the surrounding terrors. 

 For it does not escape our research (laicf iios/roin scisci/n/ioiicm) 

 how, in the first times of these supplications being instituted, by 

 the terrors of what manner of prodigies the city divinely com- 

 mitted to you was being emptied. For at one time the walls of 

 the public fortifications were shaken down by the continual earth- 

 c|uakes ; at another the fires, often attended with flame, were 

 smothering (tiimidahaiil) the frail roofs with a load of showered 

 ashes (siipciicito fn'illanim nioiilc). Now the vast lairs (.>////.■«,/,; 

 ciddlii:) in the forum harboured the boldness (O portentous lame- 

 ness !) (paz'Ciida maiisiic/ndo!) of deer. When you, amid that 

 flight of the nobles and the common people, the state of the city 

 being desperate, quickly had recourse to new imitations of the 

 ancient Ninevites, lest your despair, too, should mock the divine 

 admonition. And truly at that time you could the least distrust 

 God w ithout sin, after your experience of his mercies. For once, 

 when a certain city had begun to blaze, your faith had glowed 

 more than the fire," etc. (He proceeds to relate the extinction 

 of a former conflagration by the prayers of Mamertus ; but this 



