December 21, 18S3.] 



SCIENCE. 



795 



tion, as indicated by tlie arrows, would set in from 

 JD to S, one from C to D, one from S to some possibly 

 South American Cambrian locality, and one, bringing 

 a Permian or some later-day fauna, from an unknown 

 locality towards C. Were this order of migration to 

 continue here, or at other portions of the earth's sur- 

 face, in this or in a similarly consecutive manner, the 

 results obtained would be in perfect consonance with 

 the facts presented by geology. But is there any 

 reason whatever for the continuance of this order of 

 migration? Surely no facts that have as yet been 

 brought to light argue in favor of a continued migra- 

 tion in one direction. Why, then, it might justly 

 be asked, could not just as well a migration take place 

 from *' to D, and impose with it a Silurian fauna 

 upon a Devonian? What would there be to hinder 



a migration from S to C, placing the American 

 Silurian fauna upon the carboniferous of Africa ? 

 Why, as I have asked, has it just so happened that a 

 fauna characteristic of a given period lias invariably 

 succeeded one which, when the two are in superpo- 

 sition all over the world (as far as we are aware), 

 indicates precedence in creation or origination, and 

 never one that can be shown to be of a later birth ? 

 Surely these peculiar circumstances cannot be ac- 

 counted for on the doctrine of a fortuitous migration. 

 And it certainly cannot be claimed that through a 

 process of transmutation or development, depend- 

 ing upon the evolutionary forces, a fauna with a .Silu- 

 rian facies will, in the course of a possible migration 

 toward a carboniferous locality, have assumed a car- 

 boniferous or Permian character. 



The facts of geology and paleontology are, it appears 

 to me, decidedly antagonistic to any such broad coii- 

 tem|)oraneity or non-contemporaneity as has been 

 assumed by Professor ITuxlcy; and their careful con- 

 sideration will probably cause geologists to demur to 

 the statement that "all competent aulhorltlcs will 

 probably assent to the proposition that physical geol- 

 ogy does not enable us in any way to reply to this 

 question: Were the British cretaceous rocks deposited 

 at the same time as those of India, or are they a mil- 

 lion of years younger or a million of years older ? " 

 Anoelo Heilpbin. 



Academy of natural acienccs, 

 Philadelphia, Dec. 8. 



THOMSON AND TAIT'S NATURAL 

 PHILOSOPHY.^ — n. 



Before proceeding to an account of the rest 

 of tlie work, we shall add a few more words of 



' Concluded frum Xo. 36. 



explanation upon tlic harmonic solutions of the 

 diliereiitial equation ((">), expressed in polar 

 co-ordinates. On attempting to integrate this 

 equation, it is found that there is an infinite 

 number of particular solutions, as was before 

 stated must necessarily be the fact ; and each 

 of these solutions is the product of three fac- 

 tors. One fitctor is an arbitrary constant ; 

 another factor is the radius vector raised to 

 an}- integral power, positive or negative ; and 

 the remaining factor is a function of the angular 

 co-ordinates, dependent for its form upon the 

 exponent of that power of the radius vector bj- 

 which it is multiplied. It is this last factor, or 

 coefficient, whicli gives the name of • spherical 

 harmonics' to the solution : indeed, these func- 

 tions of the angular co-ordinates are them- 

 selves surface-harmonics. 



If we restrict ourselves, as is usuallj' done, 

 to real integral powers of the radius vector r, 

 positive and negative, then, from the well-known 

 principle that a general solution is obtained by 

 taking the sum of particular solutions, we should 

 have tlie most general possible solution by tak- 

 ing the sum of a series of particular solutions, 

 such as have just been described, in which the 

 powers of r have all integral values between 

 -+- 00 and — « . But since it is found, upon 

 computing the functions of the angular co-ordi- 

 nates which constitute their coefficients, that 

 the coefficients of r' and ?•"'' ' " are identical, 

 it will be more convenient to write the gen- 

 eral solution ill the form — 



a.i/„ («,(>) + (a,r + b,r--)f, (»,<p) 

 + (a.r^ + 6^r-')A, (H,^) -I- . . . 



-f- («,)•• -I- 6, r - f + ")/■,(», ^S) -I- 



(8) 



In a])plying this to any given case, either all 

 the arbitrary constants a vanish, or all the con- 

 stants h; thus giving rise to the two general 

 forms of solution })efore mentioned, in which 

 there is a series of terms, either in ascending 

 integral powers of r, or of descending integral 

 powers of r. 



A value of F consisting of several terius is 

 a compound spherical harmonic of the degree 

 (positive or negative) of its numerically high- 

 est power of r. A value of V consisting of 

 a single term is a simple harmonic. 



Returning, now, to the consideration of chap- 

 ter vii. p. 9.S, entitled • .Statics of solids and flu- 

 ids,' the subject of rigid solids is disposed of iu 

 the course of thirty pages, nearly half of which 

 is occupied with inextensible strings iu the form 

 of catenaries of various kinds. 



The authors hasU'ii on to the more intricate 

 matter of thistle solids. As is well known to 

 students of this subject, the general problem 



