September 29, 1892] 



NATURE 



5^7 



whilst that of type x is 



dK aL 



(it dx 

 We have also the additional equations 



^%1+^ 



dK 



<^^^) 



(9) 

 (10) 



(II) i 



Equations (7) to (11) were first given by myself in a 

 paper published in the Proc. Camb. Phil. Soc. for 1887 ; 

 and it will be observed that they include the equations of 

 Lagrange and Hamilton. A form of the modified 

 Lagrangian function, which is equivalent to (7), was 

 given by Dr. Routh a few years previously ; but it is not 

 of much practical use, owing to the fact that the elimina- 

 tion of the velocities % has not been performed. 



It sometimes happens that the co-ordinates of the type 

 X do not enter into the expression for the energy of the 

 system, in which case they are caWtd ignored co-ordinates ;'^ 

 under these circumstances it follows from (9), that all 

 the momenta k are absolute currents. A top spinning 

 about its point under the action of gravity is one of the 

 most familiar examples of ignored co-ordinates, and one 

 which illustrates several important dynamical theorems. 



When there are ignored co-ordinates, the steady motion 

 of the system, and the stability of the steady motion, can 

 very easily be investigated. For if we suppose that all 

 the co-ordinates 6 have constant values, (8) reduces to 



d^ . dV 



a^ + a^ = °- 



There are as many equations of this type as there are 

 co-ordinates 6, and an examination of this system of 

 equations will show whether steady motion is possible, 

 and if so, will determine the necessary conditions which 

 the co-ordinates 6 and the constant momenta k must 

 satisfy. 



It can also be shown that the steady motion will 

 always be stable when ^ + V is a minimum (see Proc. 

 Camb. Phil. Soc. May 1892). 



We have therefore the following simple rule for deter- 

 mining the steady motion of a dynamical system when 

 there are ignored co-ordinates. Eliminate all the veloci- 

 ties corresponding to these co-ordinates from the expres- 

 sion for the kinetic energy of the system, so that the 

 latter is expressed in terms of the velocities d and the 

 momenta k. Let ^ and V be that portion of the total 

 energy which does not depend upon the ^'s ; then the 

 conditions of steady motion are, that ^ + V should be 

 stationary, and the steady motion will be stable provided 

 this quantity is a minimum. 



The preceding theorem also enables us to deduce by a 

 very concise method all the results connected with the 

 steady motion of a liquid ellipsoid, which is rotating 

 about a principal axis under the influence of its own 

 attraction. It also enables us to examine the stability of 

 these different cases of steady motion, for disturbances 

 which produce an ellipsoidal displacement. 



A. B. Basset. 



THE PASSAGE OF GRANITE ROCK INTO 



FERTILE SOIL. 



LJAVING for the last three or four years paid par- 



•• ■■• ticular attention to the natural formation of soil, 



I venture to believe that a concise account, or rather 



I must confess that I do not like the phrase speed co-ordinates, intro- 

 duced by Prof. J. J. Thomson, for it conveys absolutely no meaning to my 

 mmd. I have no sympathy with the attempts, which have occasionally 

 been made, to introduce short words of Teutonic origin into scientific 

 nomenclature, as the words in question appear to me to be singularly 

 deficient in point. 



summing up, of the results of my researches, and of the 

 mass of my observations — in one typical direction — may 

 be of interest to the readers of Nature. As indicated 

 in the heading, the making of soil from granite is the 

 only section of a very large subject which will be briefly 

 considered in this paper. 



The agents concerned with the turning of granite (or 

 any other rock) into a fertile soil maybe shortly classified 

 as mechanical, chemical, and vital. The first-named 

 produce the largest results in bulk, and the principal 

 mechanical agent with which we have to deal is frost. 

 The second and third classes of forces are extremely 

 important, as it is by their actions that the raw material 

 of plant-food is prepared, though unfortunately poisons 

 also are brought into being through their activity. These 

 last-named classes, however, likewise materially aid the 

 action of frost (or, in tropical countries, of varying tem- 

 peratures) in the mechanical separation of rocky matter. 

 To render my descriptions as little confusing as possible 

 I will endeavour, without regard to classification, order, 

 or divisions, to trace the history of a granite soil as I 

 have observed it in many localities in Scotland, from the 

 practically unbroken rock into the condition in which it 

 has been made by nature fit to bear the most luxuriant 

 crops. But first of all I must remind my readers of two 

 or three geological facts about granite. It is a holo- 

 crystalline {i.e. wholly crystalline) igneous rock, com- 

 posed essentially of orthoclase, quartz, and mica. In its 

 most typical condition the last-named mineral is always 

 of the biotite or magnesia-mica species. Besides these 

 essentials we always find (in Scotch granites at least) 

 plagioclase, other species of mica than the essential, 

 apatite as an endomorph, i.e. locked up in the mass of 

 other minerals, and magnetite, and almost invariably, if 

 not always, a little pyrites, and more or less hornblende, 

 &c. 



A rough mineralogical analysis of Kemnay granite 

 taken from the lowest working of the well-known quarry 

 in Aberdeenshire gave the following percentages : — 



NO. I 196, VOL. 46] 



The first change which comes over granite is the per- 

 oxidation of some of the iron always present in its mass. 

 This sets in, and increases to the greatest extent, of course, 

 where air and water can most readily enter. The surface 

 of the rock becomes browned with the hydrated ferric 

 oxide formed, and brown skins, of a deeper colour than 

 the surface generally, coat the walls of the original rock 

 joints. But in the mass of the rock, away from these 

 primary fissures, there are areas which are more permeable 

 than others from the surface, and through these, streaks 

 of ferric oxide — anhydrous first, afterwards hydrated — 

 are produced. Those lines of rust are the beginnings of 

 a new set of joints, which have not yet been properly 

 recognized in geological literature, and which I will here 

 call weather joints to distinguish them from the primary 

 joints of consolidation and rock movements. The first 

 oxidation streaks of the coming weather fissures are 

 invisible to the eye, but can be determined under the 

 microscope. They gradually increase in width above as 

 they extend their Hnes beneath, and they afford passages 

 through which water can more readily percolate than in 

 the surrounding fresher areas, and as a consequence 

 planes along which frost can more powerfully act. By 



