NATURE 



585 



THURSDAY, OCTOBER 16, 1890. 



ANALYTICAL MECHANICS. 

 A Treatise on Analytical Mechanics. By Bartholomew 

 Price, M.A., F.R.S., F.R.A.S., Sedleian Professor of 

 Natural Philosophy, Oxford. Vol. II. Dynamics of a 

 Material System. Second Edition. (Oxford : Clarendon 

 Press, 1889) 



A SECOND title-page describes the present work as 

 -^ volume iv. of " A Treatise on Infinitesimal 

 Calculus," so that Prof. Bartholomew Price's well-known 

 four volumes may be taken to represent the curriculum 

 of the Infinitesimal Calculus and its applications for the 

 mathematical student at Oxford. 



To one accustomed to the style of the text- books in use 

 at Cambridge, the contrast is very striking ; the Oxford 

 student is much to be envied for the leisurely and 

 luxuriant way in which the subject is here presented, 

 which follows on the lines of Lagrange and Laplace, and 

 utilizes all the resources of analysis. A student who has 

 been through the present work will be prepared to ap- 

 preciate the purely geometrical form in which the New- 

 tonian methods, insisted upon at Cambridge, would 

 present some of the theorems in a more fundamental and 

 incisive form ; but to our mind the Cambridge system is 

 inferior, which ostensibly insists on the purely geometrical 

 methods before allowing the student to make use of the 

 power of analysis. 



Although Newton claims to be one of the inventors of 

 this Calculus, and must have employed its methods in the 

 discovery of his theorems, yet he carefully covered up 

 all traces of the analytical scaffolding, and exhibited a 

 theorem in the " Principia," like a Greek temple, in pure 

 geometrical form. 



His influence on his successors was too great when they 

 attempted to follow in the same lines, with the consequence 

 that our insular school of mathematics lagged hopelessly 

 in rear of Continental progress. 



Although prescribed as the text-book at Cambridge, the 

 " Principia" is not studied in the original Latin, as Newton 

 wrote it, from one end to the other ; but the student 

 makes use of commentaries and selections, which, in 

 accordance with the regulations, he professes to appreciate 

 and apply, before knowing even by sight the supposed 

 mystifying symbols of dyjdx and lydx. 



We might as well send out our soldiers armed with 

 muzzling loading guns, or even bows and arrows, to meet 

 a continental army equipped with the most recent inven- 

 tions of magazine rifles and breech-loading artillery. 



Thus the late R. A. Proctor could write that, although 

 a wrangler, he knew nothing of the Differential Calculus 

 till some time afterwards, when he had to pick it up of 

 himself; however, by a recent regulation, only passed a few 

 weeks ago, a most stupendous change has been made in 

 the Mathematical Tripos, by prescribing a certain very 

 elementary course of Analytical Geometry and the Cal- 

 culus in the First Three Days. 



At Cambridge the large number of candidates for 



mathematical honours acts as a check to change ; and as 



the same papers have to serve for such widely different 



classes as the wranglers and the junior optimes, it may 



NO. 1094, VOL. 42] 



happen that a candidate who merely writes out book-work 

 will beat a better mathematician who is tempted to try 

 the difficult questions. 



The number of students in mathematics at Oxford is 

 much smaller, and the standard for honours is higher ; 

 so that we can take this treatise on Infinitesimal Calculus 

 and contrast it with the extracts from Newton's " Prin- 

 cipia," to illustrate the relative standards. 



Under the enthusiastic influence of a Sylvester we may 



see the mathematical school at Oxford the first in this 



country, as it was two hundred years ago, in the days of 



j Wren, Wallis, Keill, and the founders of the Royal 



1 Society, which had its origin in Oxford. 



At the outset of the Dynamics of a Material System in 

 space, it is necessary to discuss a number of theorems in 

 solid geometry on the distribution of principal axes and 

 the associated theorems of confocal quadrics (chapter i.) ; 

 ; also the kinematics of a rigid body, involving the compo- 

 I sition and revolution of angular velocities, and the trans- 

 formation of co-ordinate axes (chapter ii.). 

 [ The author could simplify the distinction between the 

 two systems of rectangular axes by adopting Maxwell's 

 comparison with the screw, right-handed or left-handed. 

 All specifications of rotation as clock-wise, or counter- 

 i clock-wise, are ambiguous ; because the direction changes 

 as we pass from one side to the other of the clock face. 

 Standing at dusk about a quarter of a mile from a wind- 

 mill, nearly in the plane of the sails, it is possible by a 

 slight mental effort to change the apparent direction of 

 rotation, and back again, as often as we please. 



The author does not permit himself the use of the 

 elliptic functions ; or else he would have found the 

 wonderful chapter i., t. ii., of Halphen's '' Fonctions 

 Elliptiques " of great service in giving the representation 

 of the cosines of the angles which a movable straight 

 line or a movable set of three rectangular axes makes 

 with three fixed rectangular axes. Much of the sub- 

 sequent work on Euler's three angles, the integration of 

 his equations of motion, and of the spherical pendulum, 

 &c., could be completed and the integrations effected by 

 the use of "Halphen's formulas. 



Dynamics proper is introduced in chapter iii., where 



D'Alembert's principle is employed to establish the 



equations of motion of a material system, with the 



subsequent corollaries of the independence of the motions 



{ of translation and rotation, and the principles of the 



1 conservation of momentum and energy. 



It is more the fashion now to dispense with D'Alem- 



! bert's principle, and to refer immediately to Newton's third 



j law of motion ; still, D'Alembert's principle, although a 



j mere corollary, states the thing in such a way as to lead 



\ immediately to the formation of the six equations of 



motion ; and by stating it in such a manner as to reduce 



all dynamical principles to a statical form — " the reversed 



effective forces and the impressed forces form a system 



in equilibrium, the internal cohesive forces (stresses) 



being in equilibrium among themselves " — it was formerly 



considered that a simplification was effected. 



But Maxwell, in his " Matter and Motion," by con- 

 sidering Newton's third law of motion as merely the 

 definition of a stress, has been able to restate all the 

 theorems involved in D'Alembert's principle in a few 

 simple sentences, and in a much more convincing form, 



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