146 



NATURE 



[June 16, 1892 



the catenary is discussed under a central attraction or 

 repulsion, varying inversely as the square of the distance, 

 when the hyperbolic functions are used in conjunction 

 with the circular functions, we are able to write the 

 equation of the catenary in the form— 



Ijr = I + sec a cos [d sin a), or I + sach a cosh {Q slnh o), 

 including all possible cases ; and it is a curious geo- 

 metrical result that if these curves are rolled on a straight 

 line, the pole will always describe a circle. 



The treatment in § 500 of the catenary curve formed 

 by an elastic rope can also be rendered more elegant by 

 the introduction of the hyperbolic functions. 



The chapter relating to Catenaries is headed " Strings." 

 But siring is used only for tying up parcels ; we use a 

 rope or chain in full scale mechanics, and thread in a 

 model ; the word thread should be used when its own 

 weight is to be neglected, and the words rope or chain 

 when applied to a true catenary. 



A short chapter on Graphical Statics is very welcome, 

 and might with advantage be further developed ; and the 

 final chapter, on Machines, is of the usual academic 

 character. The interest of this chapter would be much 

 increased if the diagrams, particularly of the Balance and 

 of the Differential Pulley were taken from objects actually 

 in existence. 



The author never employs the absolute units of force, 

 the poundal or dyne, which he has defined in Chapter I., 

 but works throughout with the gravitation unit. This is in 

 accordance with the universal practice ; and to satisfy legal 

 and commercial requirements, these absolute units would 

 require to be defined through the intermediate of the 

 gravitation unit, by taking them as one-^^th part of the 

 tension of a thread supporting a pound or gramme weight, 

 the value of g being determined from pendulum experi- 

 ments. There is no apparatus in existence by which the 

 theoretical definition of the poundal and dyne, derived 

 from dynamical phenomena, could be tested with any 

 pretence to accuracy. 



The dyne is the unit of force in the C.G.S. system, but 

 it is a great pity that the commercial units, the metre and 

 the kilogramme, were not adopted ; the unit of energy 

 would then be the joule, and the unit of power the watt 

 or volt-ampere. Merely, apparently, for the purpose of 

 making 



W — sY, instead of looo.fV, 



the Committee of the British Association recommended 

 these niggling C.G.S. units ; but considering that for 

 ordinary substances, metals, &c., variations of texture 

 render it unnecessary to tabulate densities beyond four 

 significant figures, the factor looo is a positive advantage 

 in numerical calculations, as loooj may be replaced by a 

 whole number. 



The " Analytical Statics " is a completely new work, but 



, Dr. Routh's " Dynamics of Rigid Bodies " has been the 



text-book in universal use for thirty years or more, a 



better testimony to its merits than anything that could be 



said here. 



It is a pity that a sufficient working knowledge of the 

 simple ideas of Moment of Inertia is not given in a 

 course of the Integral Calculus, so that the author 

 might start immediately on some familiar problems of 

 the motion of a body which turns as well as advances, 

 NO. T181, VOL. 46] 



and relegate the bulk of Chapter I. to a later chapter, when 

 the motion of bodies in space is considered. This long 

 chapter at the outset chokes off many students, who would 

 be encouraged if the principles were introduced in smaller 

 doses, and only as required. The gentlemanly knowledge 

 of this subject, as Maxwell called it, which does not go 

 beyond motion in a plane, is a very valuable mathemati- 

 cal training, and few students go beyond this stage. 



D'Alembert's Principle is historically important, as a 

 first clear statement of the mode of forming the equations 

 of motion ; but now, in accordance with the modern 

 principle of considering the Third Law of Motion, "Action 

 and Reaction are equal and opposite," as defining 

 a stress composed of two equal opposite balancing 

 forces, D'Alembert's Principle should now be merely 

 looked upon as a convenient mode of writing down the 

 equations of Dynamics in an analytical statical form, 

 when stated in the words, " The reversed effective forces 

 and the impressed forces form a system in equilibrium," 

 while " the molecular, cohesive, or internal forces form a 

 system in equilibrium among themselves." 



The much-abused word " centrifugal force " still sur- 

 vives, and need not cause confusion if used to denote 

 the normal component of the reversed effective force of 

 a body moving in a curve. 



Early methods of argument in Dynamics were very 

 similar to what we now employ in Thermodynamics, in 

 the statement of the Second Law. 



Sir George Airy's commentary on D'Alembert's Prin- 

 ciple, quoted on p. 52, forms a very curious contrast to 

 the corresponding explanation in Maxwell's " Matter and 

 Motion." 



It would be a strange skeleton frame that Sir George 

 Airy would have had to create to propagate the attraction 

 between the Earth and the Moon or Sun ; and an 

 interesting subject of speculation arises as to the modi- 

 fication of Newton's Law of Universal Gravitation when 

 the inertia of the skeleton frame became appreciable. 



The discussion on the Pendulum is very complete ; 

 Kater's pendulum is fully described, but we miss the 

 account of Repsold's pendulum. In this pendulum the 

 effect of the drag of the air is eliminated by making it 

 symmetrical in shape, but unsymmetrical in density. A 

 short account of Repsold's pendulum will be found in 

 the Account of the Great Trigonometrical Survey ; but 

 the pendulum is obviously looked upon with suspicion by 

 our officers, as being employed by their Russian rivals on 

 the other side of the Himalayas. 



The very perfection of the pendulum as a method of 

 determining g is the cause of its defect as a means of 

 recovering the standard of length, so that equally skilled 

 observers would differ to an appreciable extent if set to 

 work to reconstitute the standard yard from the seconds 

 pendulum ; the clause in the Act of Parliament defining 

 the length of the seconds pendulum is in consequence 

 superfluous. 



There is something mysterious and unconvincing in 

 § 109, on the "Oscillation of the Watch Balance" ; con- 

 sidering that the inertia of the spring itself is neglected, it 

 seems that the final equation of oscillation might well be 

 written down immediately, without the introduction of 

 any approximation. 



The Ballistic Pendulum and its theory are fully de- 



