134 



NATURE 



{Dec. 6, il 



might well fancy that the basalts marked by two birds lay 

 upon, and were newer than, the granophyre marked with 

 four. Let us all take warning thereby. 



But it is time to leave this perilous ground, and come to 

 matters on which there can scarcely be difference of 

 opinion. If it were desired to direct a student to a paper 

 from which he could gather a clear and comprehensive 

 view of the manifold forms under which volcanic products 

 present themselves, not treated in the abstract but 

 brought home to him by concrete examples, none could 

 be found better fitted for the purpose than the memoir 

 before us. And if a beginner would learn a lesson of the 

 way in which a geologist goes to work when he wishes to 

 unravel and interpret a complex group of geological 

 documents, he will here find both precept and example. 

 A point or two may be specially noticed. The enormous 

 area which is seamed across by dykes, presumably of the 

 same date, enables us to realize the importance of under- 

 ground volcanic action, which is necessarily hidden from 

 view in the case of volcanoes now in activity. I first 

 learned this lesson while traversing a similar district, 

 fully three times as large as that treated of by Dr. Geikie, 

 in South Africa In connection with the striking paral- 

 lelism of a large number of the dyke^-, reference is fittingly 

 made to the classical paper of Mr. Hopkins, which he used 

 so pathetically to complain had proved of interest neither 

 to geologists nor mathematicians. But the mention of 

 this paper again makes me lapse into criticism. When I 

 first, many years ago, made acquaintance with Mr. 

 Hopkins's investigations, two of his conclusions struck me 

 as on the face of them so improbable physically, that, 

 though I felt the presumption of the notion, I could not 

 help suspecting some hitch in his analysis. One such 

 oversight, so obvious that I can now hardly believe it to 

 have been made by so first-rate a mathematician, I then 

 detected. The other I have no doubt will reveal itself 

 to careful inquiry. But from a hasty reperusal of the 

 paper 1 do not think that either of these slips, supposing 

 both to exist, affects the conclusions appealed to by Ur. 

 Geikie; and the agreement, as far as they are concerned, 

 between theory and observation is as complete as can be. 

 The skill with which Dr. Geikie uses his pencil to bring 

 out the geological features of a landscape is well known : 

 that his right hand has not lost its cunning will be evident 

 from the two illustrations here reproduced (F"ig3. 2 and 3). 



Reference has been repeatedly made to the proofs of 

 enormous denudation since Tertiary times which the vol- 

 canic rocks we are dealing with furnish in lavish abund- 

 ance ; it has not been so o.ten noticed that denudation has 

 during the same interval made its effects felt on harder 

 and more intractable rocks. But dykes furnish proof of 

 this in a way which I believe has not been made the sub- 

 ject of comment. " The evidence of this denudation," 

 says our author, " is singularly striking in such districts 

 as that of Loch Lomond, where the difference of level 

 between the outcrops of the dykes on the crest of the 

 ridges and the bottom of the valley exceeds 3000 feet. It 

 is quite obvious that, had the deep hollow of Loch 

 Lomond lain, as it now does, in the pathway of these 

 dykes, the molten rock, instead of ascending to the 

 summits of the hills, would have burst out on the floor of 

 the valley. We are therefore forced to admit that a deep 

 glen and lake basin have in great measure been hollowed 

 out since the time of the dyke." A point this in favour of 

 the "gutter-theory." A. H. Green. 



THE THEORY OF PLANETARY MOTION.^ 

 T N the work the title of which is printed below, Dr. Otto 

 ^ Dziobek seeks to develop the theory of the motion of 

 bodies subject to attraction according to Newton's law. 

 The author, in his preface, draws attention to the objec- 



I "Die mathematischen Theotien der Plaueten-bewegungen." By Dr. 

 Otto Dziobek. (Leipzig : Johann Ambrotius Barth, 1888.) 



tionable practice of the majority of writers of the present 

 day, of treating the subject so briefly that many students 

 scarcely get beyond Kepler's laws in their knowledge of 

 the theory of the solar system. He has therefore pre- 

 pared a work which is intended not only as an introductioii 

 to the study of this branch of astronomy but especially 

 for those desiring an acquaintance with the higher pro- 

 ductions of the masters in this science. 



The book is divided into three sections. The first 

 begins with the assumption of Newton's law, and then 

 treats of the motion of two bodies about their centre of 

 gravity, giving the usual deductions relating to the motion 

 of the centre of gravity, to the projections upon the 

 three co-ordinate planes of the areas swept out by the 

 radius-vector in a given time, and to the form of the orbit 

 described. In determining Gauss's constant of attraction, 

 /', the author says that the unit of length is the major 

 axis of the earth's orbit (he doubtless means semi- axis, 

 though the statement is repeated on the same page, and a 

 like oversight occurs on pp. ii and i6) ; and then with 

 I : 354710 as the earth's mass and 365'2563835 mean solar 

 days as the length of the Sidereal- year, k is fotmd. = 

 o 017209895. This is the value found by Gauss, and given 

 in his " Theoria Motus." This constant has been incor- 

 porated in many tables, and any change in its value would 

 be attended with considerable inconvenience. But since 

 the time of Gauss more accurate values of the earth's mass 

 and of the length of the sidereal year have been found, and 

 consequently a more accurate value of k may be deduced. 

 To avoid this inconvenience, the above value of k is re- 

 tained, and with the new values of the earth's mass and the 

 length of the sidereal year the unit of length is deter- 

 mined. This unit of length is slightly greater than the 

 earth's mean distance from the sun, but differs from it 

 by less than a unit of the eighth decimal. 



A collection of formulas giving the relations between the 

 radius-vector, the mean, eccentric, and true anomalies, as 

 in Gauss's "Theoria Motus," is added, together with the 

 usual expansions in series of these quantities. The ex- 

 pressions for the expansion of the eccentric anomaly and 

 of the radius-vector by means of BesseFs functions are 

 also added. 



We next come to the general treatment of the problem 

 of the motion of any number of bodies projected in any 

 manner in space, and subjected only to their mutual 

 attractions. Here, considering n bodies, we have the usual 

 deductions relating to the invariable plane of the system, 

 and to the sum ot the products of the mass of each body 

 into the area described by its radius-vector. The author 

 then proceeds to simplify the case by discussing the motion 

 when n = 3, and thus the case of the celebrated problem of 

 the three bodies. Of this the usual outline is given, to- 

 gether with certain special cases of the problem, the lines 

 of the investigations of Lagrange and of Jacobi being 

 chiefly followed. A brief historical outhne of the problem, 

 and of the chief investigations thereon from the time of 

 Lagrange up to almost the present day, closes the first 

 section of the work. 



The second section of the book treats of the general 

 properties of the integrals introduced in the consideration 

 of the problem of Jt bodies. The investigations of Poisson 

 and Lagrange are discussed, and the development by these 

 writers of formulas for the elements of the elliptic orbit of 

 a planet is given. And here, on p. 98, we again note the 

 oversight before referred to, viz. that of putting a = the 

 major axis of the orbit instead of the semi-major axis. 

 Of course such a proceeding if it were carried on through- 

 out would have no effect upon the developments which are 

 obtained, except on their symmetry, but the author, after 

 mentioning that the quantity a represents the major axis, 

 immediately proceeds to use the quantity with its usual 

 signification, viz. the semi-major axis. The oversight 

 occurs again on p. 112, and again in discussing the canon- 

 ical constants for the eUiptic motion of a planet, and again. 



