578 



NA TURE 



[April 23, 1896 



when attempting to explain the effect of environment on 

 man, as the following extract will also prove. 



" It has been shown that the precursor was most prob- 

 ably furry, with a woolly under and a sleek outer coat, 

 and it is conceivable that in a volcanic environment like 

 that of Java, it might have been advantageous to shed the 

 wool and retain the sleek hair, together with all the other 

 physical characters of the primitive Negrito." 



The white race {Homo Caucasicus, as Mr. Keane 

 delights to term it) is held by the author to have evolved 

 in, and dispersed from, North Africa ; but he strangely 

 omits to refer to Dr. D. G. Rrinton, who, in his " Races 

 and Peoples" (1890), had already promulgated that view. 



It is evident that Mr. Keane is a very diligent and 

 widely-read literary man, but he is decidedly weak on the 

 scientific aspects of his subject. Lastly we must criticise 

 those figures which were copied from the author's " Types 

 of the Races of Mankind," in Longmans' New Atlas. The 

 process-blocks from these lithographs have a very coarse 

 appearance, and offer a marked contrast to those taken 

 from photographs. On the whole, the selection of the 

 illustrations of racial types is well made. 



Although there is a good deal of what may be termed 

 contentious matter, besides numerous errors, in Mr. 

 Keane's book, we can recommend it as a most useful 

 introduction to a very complicated study; and as the 

 author has brought together and abstracted a large 

 number of references, the student can use the book as a 

 point of departure, and thus it will serve as a base for a 

 more extended or detailed survey of this really important 

 branch of science. A. C. Haddon. 



RIGID DYNAMICS. 

 An Elementary Treatise on Rigid Dynamics. By W. J 

 Loudon, B.A., Demonstrator in Physics in the Uni- 

 versity of Toronto. Demy 8vo, pp. ix -f- 236. (London : 

 Macmillan and Co., 1896.) 



THERE are few mathematicians who do not vividly 

 recollect the difficulties they experienced when 

 reading " Rigid Dynamics " for the first time. Mr. 

 Loudon's treatise does much to smooth away these diffi- 

 culties ; and if it still leaves undone much that might 

 have been done in simplifying the subject for beginners, 

 it nevertheless fills a gap the existence of which has long 

 been felt among teachers. 



From a purely mathematical standpoint, we have none 

 but praise to offer. As a digest of the earlier matter 

 of Dr. Routh's treatise up to, but not including, La- 

 grange's generalised equations of motion, it will be wel- 

 comed by all students whose primary object is to master 

 the equations of motion of a rigid body without diving 

 too far into higher applications. 



The order of treatment is essentially based on " Routh," 

 with the exception that Mr. Loudon gives no separate 

 chapters on " Motion in Two Dimensions," " Momentum," 

 and " Vis Viva." Thus the first two chapters deal with 

 "Moments of Inertia" and "Ellipsoids of Inertia," and 

 are followed by chapters on " D'Alembert's Principle " 

 and on " Motion about a Fixed Axis." After the latter 

 problem has been considered both for finite and "im- 

 pulsive" forces, the same is done for motion about a 

 fixed point. In this connection, the equations of motion 

 NO. 1382. VOL 53] 



of a top, and of a body moving under no forces, are 

 discussed as far as they can adequately be treated with- 

 out using elliptic functions. The book concludes with 

 a chapter on the " Gyroscope," in which the experi- 

 mental proof of the earth's rotation is figured and de- 

 scribed at some length. 



One very commendable feature is the large number of 

 diagrams. To represent on paper three planes at right 

 angles in a rigid body is a task which previous writers 

 have shirked ; but M r. Loudon's large and bold figures 

 will do much to assist the reader in forming a concrete 

 idea of the motions he is dealing with. We might in- 

 stance more especially Fig. 50, illustrating the motion 

 of a top spinning on a horizontal plane, and Fig. 58, 

 illustrating how the motion of a rigid body under no 

 forces is completely represented by the rolling of the 

 momental ellipsoid on a fixed plane. 



To our mind the book's chief drawback, considered as 

 an ^/^w<?«/ary treatise, lies in the author having, no doubt 

 unconsciously, followed Dr. Routh's analytical methods 

 too closely instead of striking out in simpler lines of 

 treatment. That it is a useful exercise to start every 

 problem by writing down the fundamental equations 



cannot be doubted, but the ordinary beginner often finds 

 it hard to proceed from these equations to the final 

 solution. What he now chiefly requires is a thorough 

 grasp of the nature and significance of "angular 

 momentum." We by no means wish to overrate the 

 educational value of the familiar type of Tripos rider, 

 whose solution merely involves writing down the equations 

 of conservation of angular momentum and energy, and 

 eliminating between the two ; at the same time, we do 

 think that much may be learnt from problems of this 

 class, especially by the beginner. For a similar reason 

 we are sorry not to find " Motion in Two Dimensions" 

 treated earlier. Again, in deducing Euler's equations 

 of motion, it seems a pity that the author has adopted 

 Dr. Routh's laborious proof, a proof which is always found 

 very hard to grasp. Its difficulty is largely due to the 

 necessity of proving the relation 



'~dt In 



connecting the rates of change of the angular velocities 

 about fixed and moving axes respectively. The author 

 gives two proofs of this identity, occupying four pages of 

 difficult mathematics ; but the result is, after all, only a par- 

 ticular case of the general property of moving axes, 

 which, when applied to any other vector quantity (angular 

 momentum, for example), assumes the far more intelligible 

 and suggestive form 



dt 



= — J - h. 

 dt 



\ + ^3'^2 



and thus leads to a far shorter proof of Euler's equations. 

 In a few respects the book slightly lacks in finish. A 

 tyro might easily complete the chapter on " D'Alembert's 

 Principle" without having his attention drawn to what 

 that principle really is, or might even mislead himself 

 into the impression that the principle consisted in the 

 mere equations 



2(/,)=2(/,) = 2(/3) = o. 



