214 



NA TURE 



[December 29, 1892 



forms assumed by crystals under different conditions. Petrol )- 

 gists h.ive lo.ij; been familiar with the tendency of crystals de- 

 veloping in a viscid medium to excessive growth in one or more 

 directions. Felspar is a familiar instance, the lath-like forms 

 which It frequently assumes being due to elongation along one 

 axis (jr), ihe length of pri>ms measured along this axis ofien 

 exceedini4 by ten times that along the axis y or z. The cause 

 of this need not now be discussed ; it will tie sutficient to add 

 that the phenomenon is not special to felspar, but is of quite 

 general occurrence. With this tendency is connected the origin 

 of curvilinear forms. We may consider the molecules formmg 

 the growing face of a long prism ; the spheres of influence of 

 these lie half within and half without the substance of the 

 cry-<tal. Considering this influence as attractive (directly or in- 

 direct 1>), we may say that the attraction of the molecules lead- 

 ing to further deposition is one-half their total attracti m. If 

 now Irom the face we pass to the edge between two faces at 

 right angles, only one-quarter of the sphere will be immersed, 

 and the a'traciion may be spoken of as three-quarters of the 

 whole ; while if Irom the edge we pass to a corner, only one- 

 eighth is immersed, and the attraction becomes seven-cighihs. 

 From this it follows that growth should be more rapid 

 at the edges than over the surface of the face, and still 

 more rapid at the corners. In accordance with this we find 

 young growing prisms in a viscid medium increasing so rapidly 

 at the edges as to leave a space in and a'>out the axis filled with 

 the medium in a non-cr)Stadine state. I deed, a viscid medium 

 is not necessary ; hollow prisms are of common occurrence 

 whenever crystallization takes place with rapidity. Further, in 

 quite embryonic crystallites, V gelsang figures elongated prism- 

 like lorms, in which the four corners are pnduced parallel to 

 the lung axis into processes resembling spines. There is an 

 additional reason pointed out to me by Prof. Fitzgerald why 

 gro\*th should l>e more rapid at the edges and corners than over 

 the general surface, and that is that these parts are m .re exposed 

 to molecular bombardment. 



If crystals are more readily built up along edges and corners, 

 we should ex()ect them to be more readily unt'Uilt in these 

 region^, anl this is in accoidance wiih observation; the zonal 

 felspars ol igneous rocks, in the formation of which intervals of 

 solution have alternated w th periods uf grov\th, usually present, 

 in ihe outlines of each resulting envelope, rounded corners. 



The influence of corners is well seen in some glassy rocks 

 where small prisms of felspar (andesite) may be observed, with 

 five or six slenderer but longer prisms springing from a corner 

 in radiate divergence. 



From this it is but a step to curvilinear growth. Let a prism 

 tend to rapid rectilinear growth, and any check immediately in 

 front will lead to a lor*aid grow h from a corner in a slightly 

 different direction ; even the competition of molecules (or this 

 centre of atiraciion may t>y ovcrcro^vding bring aiiout this 

 result, and thus both branching and curvilinear forms may 

 arise. This is beautifully exemplified in the spherulites of 

 many igneous rocks, where we find in the centie oi a ndiately 

 crystallized sphere a long prism of fcl>pir serving as a nucleus, 

 and tiom the en<{s of this slen ler, almost linear, |.risms diverge 

 towards a spherical surface which by repeated branching and 

 associated curving they everywhere r. ach, leaving ab.jut the 

 sides of the nucleus a spherical >pace almost devoid of crystal 

 slruc ure. The whole arrangement in median longitudinal 

 sectiun presents a remarkable resemlil ince to the lines of force 

 as shown by iron-filings atiout a bar magnet. 



Kvidenily in rapid c ysiallizition wiih a tendency to linear 

 growili, divergence may be repeated at such frequent intervals 

 as to produce (orms which to the unaided eye appear to be 

 Continuous curves. 



SOCIETIES AND ACADEMIES. 

 London. 

 Mathematical Society, December 8.— Mr. A, B. Kempe, 

 F k. S., Piesident, in the chair. — fne fnllo>ving communica- 

 tions were made : — On a thenrem in diffcrentiaiioji, ai.d its 

 application to spherical harmonics, by Ur. Hobson. — On 

 Cuichy's condensation lest lor the coiivergency of series, by 

 Prof M. J. M. Hill. Cauchy's c(jnden-auun test for the con- 

 vcrgency of scries is as loUows : — It f(n) i)e positive /or all 

 values of «, and c mstanily decrease as n increases, then ihe 

 series 2/i«) and 'S.a"J\a") are i.oth conveigent or both divergent, 



NJ. 1209, VOL. 47] 



where a is any positive integer not less than 2. There is a clear 

 reason why a cannot be un ty, for then ■S,a"f{a») = 2/(i), which 

 is always infinite. It is proved in Chry^tal's " Algebra" that 

 the theorem is also true if a have any positive fiaciional value 

 not less than 2, see part 2, chapter xxvi., § 6, cor. i. The 

 proof there given when a lies between the c msecuiive positive 

 integers/ and/ -f- i is based on Cauchy's proof for the two cases 

 a — p and a = / -f I. But this proof will not apply when 

 i<a<2, necause Cauchy's proof will not apply w hen a = I. 

 V et it does not seen possible to as«ign a reason for excluding 

 values of a between I and 2, for Cauchy's method appears to 

 de,<end on this— viz. that for increasing values of n the ex- 

 pression /(a«) occupies more and more advanced positions 

 amongst the terms of the series 2/(«) ; but this is possible if 



l<a<2, as well as when a>2. If a < i, then this is no longer 



true. The problem then considered in this paper is so to recast 

 the proof f>r fractional values of a as not 10 exclude the case 

 r<a<2. The complete theorem will then stand thus:— If/(«) 

 be positive for all values of n, and constantly decrease as n in- 

 creases, then the series 2/(«) and 'S,a''f{(i") are doth convergent 

 or both divergent if a > i. The demonstration depends on the 

 following theorems. 



I. If "Za^f^a ") be convergent, then — 



2/(«)</(0+ . • . +y(M 



+ {a - I + a-n\ 2 «"/(«") - 2 ««/(««) 



r «=" 

 L_« = I 



] 



where s is any integer so large that 



■a^>l. 



and / is the greatest integer in a^, a being greater than i 



(A). If SayCos") be divergent 

 increases bey Ji.d a certain value, 



II (A). If SayCrt") be divergent, and if «"/(«") diminish 

 as n increases beyond a certain value, then 



(I - 





"yia") - 2 ^y,a") 



] 



II (B.) If %a^f{a^) be divergent, and if aV {^^) do not 

 diminish as n increases beyond a certain value, then— 



2 /(«) > 2 /(«) + 



2 A«" 



where s is an integer taken large enough, and A is some finite 

 quantity.— Additional note on secondary Tucker circles, by 

 Mr. J. Griffiths. — Notes on determinants, by Mr, J. E. 

 Campbell. In accordance with the late Prof Smith's notation, 

 a determinant ol the /*•> class may be written 



I aijk • • • I 

 The fact that a determinant of the second class (an ordinary 

 determinant) is not altered if the vertical columns be written 

 horizontally is expressed by the identity 



I «y I - I «/V I 

 For determinants of higher class it is known that any of the 

 suffixes can be interchanged, txcept the first : and il the class 

 be even, the first suffix can also t)e interchanged witli any other, 

 but for deteiminanis of odd class thn, is not true. By con- 

 sidering a cubic determinant as an ordiiaiy determinant in alter- 

 nate numbers, the author trics to explain this essential distinc- 

 tion between (]et«.riiiitiants of odd and even classes. Il the element 

 = -a. , and ^.^^= o, the determinant is called 



skew symmetrical. It is easily s.cn that skew symmetrical de 

 lem.inants of even clas^ and odd de-iec vanish identically. 

 This is analogous to the well-kno«n theorem in ordinary deter- 

 ndnanis; but th.re is no corresp.mding analogue 10 the theorem 

 that skew symmetrical determinants of the becond_clabS and even 



