NA TURE 



609 



THURSDAY, OCTOBER 26, 1893. 



ANAL YTICAL MECHANICS. 

 A Treatise on Analytical Statics. With numerous 

 Examples. Vol.11. By Edward John Routh, Sc. D., 

 LL.D., M.A., F.R.S. (Cambridge: at the University 

 Press, 1892.) 



THIS volume finishes Dr. Routh'swork on the subject 

 of analytical statics, the first volume of which 

 was reviewed in Nature, June 16, 1892. It contains, in 

 three long Sections or Books, the subjects of Attraction, 

 Bending of Rods, and Astatics, left over from Vol. I. 



In Attraction a start is made with the Newtonian Law, 

 and the Gravitation Constant is introduced. 



The experimental redetermination of the numerical 

 value of the Gravitation Constant is engaging the attention 

 of Mr. Poynting (who has just been awarded the Adams 

 Prize for his Essay on this subject) and of Mr. C. V. Boys. 

 But we cannot hope to obtain, with the greatest refine- 

 ments, an accuracy of determination within limits of error 

 of less than one per cent. ; the Astronomical Unit of 

 Mass, defined in § 3, would be subject to thi same limits 

 of error, which are far beyond what is permissible in 

 careful measurements with the Balance. 



The only reason for the introduction of the Astrono- 

 mical Unit of Mass is to save the trouble of writing down 

 /•, the Gravitation Constant, in our equations ; but we 

 agree with Prof. Minchin, in his Analytical Statics, in 

 th inking that it tends to clearness if we take the trouble 

 to write k in its proper place, so as always to measure in 

 in such well-determined units as the gramme or kilo- 

 gramme. 



Nowadays the theorem? of Attraction receive their 

 most appropriate interpretation, analytical and experi- 

 mental, from the subject of Electrostatics ; the theorems 

 on the Potential of Laplace, Poisson, and Gauss, on 

 Tubes of Force, Green's Theorem, Inversion, Laplace's 

 Functions, and on the Attraction of Ellipsoids of 

 Chasles, all present themselves as fundamental in the 

 Electrostatical chapters of Maxwell's "Electricity and 

 Magnetism ;" insomuch that Maxwell ventured to present 

 a demonstration of some of the most abstruse analytical 

 results of Laplace's Functions, founded on physical 

 principles of Electrostatics, and thereby excite the ire 

 of certain mathematicians of the purest proclivities. 



For instance, the complicated theorems on Centrobaric 

 Bodies, discussed in §jj in, 116, become self-evident 

 when interpreted as the analogues of the electricity 

 induced on an uninsulated closed surface by an electrical 

 point in the interior. The external electrical effect being 

 zero, the potential of the induced electricity is equal and 

 opposite to that of the point, and therefore the surface 

 has an electrical coating which is centrobaric, the function 

 which represents the superficial density being Green's 

 Function for the surface and the point. 



If the dielectric in the interior is stratified, an elec- 

 trical concentration is distributed throughout the space, 

 and thus the analogue of the centrobaric body is 

 obtained ; but incidentally the electric analogy shows 

 that the strata of equal density in the centrobaric body 

 are each separately centrobaric, so that the centrobaric 

 NO. 1252, VOL. 48] 



body is built up of centrobaric shells. The sphere is the 

 homogeneous centrobaric body, as Newton showed in the 

 "Principia"; and an application of Sir W. Thomson's 

 powerful geometrical method of electrical inversion 

 deduced the fact that a solid sphere whose 

 density varies inversely as the fifth power of the 

 distance from an external point O' is centrobaric with 

 respect to the interior inverse point O. So also for a 

 spherical shell, either this or composed of a series of con- 

 centric strata ; and this by inversion leads to the theorem 

 that a shell bounded by two excentric spheres of which 

 the limiting points are O and O' is centrobaric if the 

 density at any point P in it is 



OP-5(f(OP/OT/) 



The discovery of Green's function for a given surface, 

 or rather the discovery of surfaces for which Green's 

 function can be assigned, is one of the most difficult and 

 baffling of modern analysis ; and it has so far only been 

 effected for some few simple cases. 



The British Association met recently at Notting- 

 ham, the birthplace of George Green in 1793. There 

 must be people still living there who remember him, 

 and could supply now, before it is too late, some 

 interesting details of the causes which led to the 

 development of his wonderful mathematical genius, at 

 a time too when little encouragement was vouchsafed 

 to such abnormal proclivities. In France a statue would 

 long ago have arisen in his honour ; but at least an in- 

 teresting paper on the subject of Green's life could be 

 communicated to Section A. 



The theorems of Chasles and Maclaurin on the attrac- 

 tion of homoeoids and focaloids are fully discussed in 

 §182 ; the homosoids receive ample illustration in 

 electrical phenomena ; but Maclaurin's theorem on the 

 attraction of confocal homogeneous solid ellipsoids is 

 rendered more convincing by supposing the smaller 

 confocal to be scooped out of the larger so as to form a 

 thick focaloid, the matter which is scooped out being 

 condensed homogeneously with the rest of the substance. 

 The effect of this operation is to leave unaltered the ex- 

 ternal potential, and the original matter may thus ulti- 

 mately be condensed into a thin focaloid, in which the 

 thickness is inversely proportional to the perpendicular 

 on the tangent plane ; and this focaloid will have the 

 same external equipotential surfaces as the solid 

 ellipsoid. 



Part ii., on the Bending of Rods, does not assume any 

 new experimental knowledge beyond that of the pro- 

 portionality of the curvature to the bending moment, an 

 assumption which we know from Prof. Karl Pearson's 

 " History of Elasticity ' to be only a first rough approxi- 

 mation to the truth. 



The analytical consequences of the hypothesis are, 

 however, very elegant and instructive, and Dr. Routh 

 has brought together an interesting collection ot 

 illustrative examples. 



He does not, however, develop the elliptic function 

 solution of the plane Elastica or associated Lintearia, 

 curves which can now be drawn with great accuracy and 

 rapidity by Mr. C. V. Boys's scale. He also restricts 

 himself to the uniform helix in the tortuous Elastica ; 

 but the student who wishes to pursue this branch of the 



D Li 



