Page 80

NA TURE
[May 30, i872
and q the angle between the plane A 0 R and a fixed
plane. Let u, co, be similar co-ordinates of R referred to
P, and let A', a', be the values which 4 and co assume at
the point 1. Let F (MA' a') be a function of u' and co',
vhich, when u' and o' become us and co, may be written
F (/, co). If 8 S be an element of the surface of the sphere,
whose radius we shall take as unity, then 8 S may be ex-
pressed by - 8p . a q, or by - 8 tL . 8 Go, according as occa-
sion requires.
  It is obvious from spherical trigonometry that
         tp=P' +                   CoIs- s/l - c (- co,)
and that therefore in the expansion
       (I +X--2 Xp)2    I+1A+P
The quantities PI, P., &c, satisfy Laplace's equations in j
and c, and also in ,A'-and co'.
  Differentiating this equality, multiplying by 2 A- and
adding, wve get
( I +_1"-2_2 1 = I + 3 P1.l+ 5 P2*a+ .. + 2n+ 1) p an-
Integrating each side of this equation over the whole sur-
face of the sphere, and equating the results, we have
           f21 f + I    I_-X__ _ "o
                   f o f ~ x ~ m ~ x 2 - ~ 2 a j d)  d d q   -
        2 7rf  ( I + 3 Px  + 5P, . + *    1_ * )4
      ,, O ,  -  I
The first of these two integrals is readily found to be equal
to 4 7r, being thus independent of x;
  *. .f:Tfi2(I + 3 P, x + 5 P2, -+ * . ) dc dco = 4 r
Now as xapproaches unity, every term in the series those
limit is represented by the integral
             fto fiiI (f+  I  I  -p) dX  d
         beoms  or nary  qul o  er, xcptthtemsi
becomes more nearly equal to zero, except the
the immediate vicinity of the value p = I, which
in value, i.e. in the neighbourhood of the point -.
as we diminish i, the ratio of
terms in
increase
Hence,
   102     If      F1 co) {+±3Pi+P5 r-1'+       d    Azd to
   F(/A/ C) )  SO +' +    3 P, X-+ 5 P2 X2+  *}
that is to 4 or . F (IA' ad) becomes continually more nearly a
ratio of equality, since F (O' co') is the value towards which
F (u a) continually approximates as we draw nearer to
the point A. Hence we have in the limit
           J, 2 7I - I F (p) (I +P1 + 5 P2 +  ) dodo
and since P,, satisfies Laplace's equation of the nth
order in IA', cl'
               * * g / ~F (k @ Pn dl- d1
also satisfies it, because ftu' a' are constants so far as this
integration is concerned. Hence F (pA' caV) is expanded in
a series of functions satisfying Laplace's equations.
                                         JAMES STUART
          LETTERS TO THE EDITOR
[The Editor does not hold himself responsiblefor 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 Auvergne, as Mr.
Green supposes (NATURE, May i6).  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 the eminent
writer, Sidonius Apollinaris, who had settled there, and wrote
a poem on its scenery, betrays no knowledge of its volcanic
phenomena. So, at least, Sir Charles Lyell has repeatedly in-
sisted.  It is true that, writing before the date of the Vienne
calamities, his silence proves nothing; but as fully half the
French craters are not in the Auvergne, but between that pro-
vince and Vienne, namely, in either the Velay or Vivarais, within
about 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 " Sodomitic showers " that alarmed the Burgun-
dian capital, and led St. Mamertus to institute these fasts. Of
AMamertus 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
better attested, none within many centuries of Pliny's even so
well. For it is strange that no later chroniclers mention anything
but 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 of 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 better 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 Quar terly
Rev:L'Cw of October iS44. See also a mcst extraordinary paper
on it in the Gent/c man's Afa,-17gaine of May i865, commented on
by me in the Reader of the next month (p. 683). As the original
passages, however, of Sidonius and Bishop Avitus 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. GARBEL  T
  "Sidonius to Lord Patriarch MIamertus, health!  It is reported
the Goths have advanced their camps on to Roman soil ! To this
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 (studium), or the failing defence of sentinels for assist-
ance, but are only soothed by the comfort of the Rogations
introduced, of which you were the author; 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 (lat/ct niostram sciscitationem)
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-
(lualkes; at another the fires, often attended with flame, were
smothering (tiumulalbant) the frail roofs with a load of showered
ashes (superjectofavil/ar-um monte).  Now the vast lairs (stupenda
cubitia) in the forum harboured the boldnesi (O portentous tame-
ness !) (pavcnda mansuctudo!) 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 without 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," &c. (He proceeds to relate the extinction
of a former conflagration by the prayers of Mamertus; but this
8o