172 



NATURE 



[June 25, 1891 



of the fact, and its close connection with the applications 

 of mathematics to physical problems, some mention 

 might have been made of the importance of the critical 

 points of a function in determining its value. A full 

 discussion of such things is doubtless impossible in an 

 elementary treatise ; but the reader should at least be 

 warned that what is given regarding the expansion of 

 functions in power-series is a mere fragment of what is 

 known on the subject. The tendency of Prof. Green- 

 hill's chapter on the expansion of functions certainly will 

 be to suggest to the mind of a beginner wrong general 

 notions on the subject. 



In § 126 we have two proofs given that 



^'''f{x,y)ldx^y = &^/{x,y)ldydx, 



both of them insufficient ; for the one rests on the assump- 

 tion that f{x + h, y -\- k) can always be expanded in 

 an integral ^-A'-power-series, the other on the assump- 

 tion that 



L Lx(/^/&) = L L x{h,^^, 



/i =: O /c =: o k =■ O h ^ O 



both of which propositions are liable to exception. 



In the discussion of single and double integrals, no hint 

 is allowed to reach the reader of the necessity of con- 

 vergency as a condition of their having any meaning at 

 all, of the precautions that must be observed in differ- 

 entiating them, or in altering the order of integration, 

 and so on. Still, the reader is given a proof of Green's 

 theorem. What use this is likely to be to one ignorant 

 of the fundamental character of the convergency and dis- 

 continuities of multiple integrals, upon which many of 

 the most important applications of the theorem in ques- 

 tion depend, it is not easy to see. Too much of the 

 work before us bears, in fact, the character of a hurriedly 

 written precis or syllabus of lectures ; witness, for ex- 

 ample, the oracular character of §§ 146, 151, 152, &c. 

 Our author makes enormous demands on the intelligence 

 of a beginner if he expects him to follow and understand 

 exposition so elliptical. 



One more example of the thing we complain of. In 

 § 183 we are introduced to Fourier's series. No proof is 

 given (none was to be expected in an elementary treatise) 

 of the conditions under which the expansion is possible, 

 but it ought to have been stated that there are such con- 

 ditions. Moreover, the method given for the determina- 

 tion of the coefficients is a mere memoria technica for 

 recollecting them. It has no demonstrative force, be- 

 cause, as the author must be very well aware, it is not 

 unconditionally allowable to replace the integral of an 

 infinite series (even if it be convergent) by the sum of the 

 integrals of its separate terms. In order that this may 

 be admissible, the series must be uniformly convergent. 



Seeing that the world is very evil, and not to be mended 

 in a day, we must put up with such things in the ordinary 

 writer of English text-books, who caters for the victims 

 of our manifold examinations ; but in a pillar of mathe- 

 matical society like Prof Greenhill they are "most tolerable 

 and not to be endured." A work with his name on its back, 

 and the impress of his vigorous personality on its pages, 

 will not remain long in a second edition. If he would 

 be at once the friend of the practical man, and a well- 

 deserver of the mathematical republic, let him, when the 

 third edition is called for, reduce his elementary work to 

 NO. T T30, VOL. 44] 



the compass of the first edition or less, and replace all 

 half demonstrations by honest statements of fact ; and 

 let him, meantime, write a larger work, to which he can 

 refer the elementary reader who takes for his motto, 

 " Thorough." G. C. 



THE GEOLOGY OF THE COUNTRY ROUND 



LI VERPOOL.. 



Geology of the Counhy around Liverpool. By G. H. 



Morton, F.G.S. Second Edition. (London : Philip 



and Son, 1891.) 

 T N this work Mr. Morton has entirely re-written the 

 -L " Geology of the Liverpool District," first published in 

 1863, by the light of the various discoveries made since 

 that time, and especially of the Geological Survey maps 

 and memoirs. He has succeeded in making a compact 

 and well-printed hand-book, which will be of great service 

 to the students of the local geology. The area described 

 extends to about 20 miles from Liverpool on every side, 

 excepting the sea on the west. The strata which he 

 describes range from the Upper Silurians of the Vale of 

 Clwyd through the Carboniferous, Permian, and Triassic 

 rocks, down to the recent alluvia. To a geologist the 

 chapter relating to the Carboniferous rocks of North 

 Flintshire and the Vale of Clwyd will be of great interest, 

 as it shows the thinning off of the strata as' they approach 

 the ancient Carboniferous land of North Wales.* The 

 Carboniferous Limestone, over 3000 feet thick in North 

 Lancashire, is reduced to 1700 feet in North Flint and 

 the Vale of Clwyd ; while the Yoredales and Millstone 

 Grits, over 9000 feet thick between Clitheroe and Burnley, 

 are represented by the Cefn-y-Fedw Sandstone, 370 feet. 

 The Lower and Middle Coal-measures, too, of South- 

 west Lancashire, 3180 feet thick, have dwindled down to 

 no more than 1000 feet as they approached the Welsh 

 Silurian Hills. It is therefore obvious that the Snow- 

 donian area was dry land while the Carboniferous sea 

 occupied the areas of Lancashire, Derbyshire, and 

 Cheshire, and that it also overlooked the forest-covered 

 morasses, now represented by the coal-seams of the same 

 region in the Upper Carboniferous age. In the table 'of 

 the rocks (p. 6) Mr. Morton gives 300 feet as the thick- 

 ness of the Millstone Grit in South- West Lancashire. It 

 is probably much more than this, and not much less 

 than 2000 feet. Mr. Morton also, we may remark, under- 

 states the thickness of the Keuper Marls, which he puts 

 down at 400 feet (p. 75). In the Lancashire and Cheshire 

 plain it is 700 -f feet, and is estimated by Prof Hull at 

 3.000 feet. 



Mr. Morton, in dealing with the deep boring at Bootle, 

 made in 1878, under the advice of the writer of this 

 review, is mistaken in supposing that it was aimed at the 

 water in the Permian Sandstone. It was intended to 

 strike the water in the Lower Bunter Sandstones, and to 

 draw upon the enormous area of water-bearing strata in 

 the Lancashire and Cheshire plain, which have their 

 outlet seawards between Prescot and the estuary of the 

 Dee. It is very likely that the Permians are not repre- 

 sented under Liverpool. We expected to strike the Coal- 

 measures at 1000 feet. The boring was successful, both 

 from the geological and the engineering point of view. It 

 proved that the Lower Bunter Sandstones below the top 



