January 17, 1895J 



NATURE 



28 = 



tions exists, so far as we know, throughout space. A view of 

 this kind entirely gets over any difficulty of a body acting where 

 it is not ; for all bodies arc everywhere, and if we consider 

 matter to be the cause of motion of other matter, there seem; 

 no very imperious necessity for imagining another cause which 

 we call force. 



There are two assumptions that Hertz makes which he con- 

 siders can only be proved by their success. One is that all the 

 connections in nature can he represented by linear difterenlial 

 equations. There are plenty of cases imaginable in which 

 this would not be true, as, for example, connections depending 

 on the curvature of the path. The other assumption is that 

 forces can be represented by force functions. This, again, may 

 not be a complete representation of nature. 



Following this introduction comes the book itself, which is 

 divided into two parts. The first part is purely kinemalical, 

 the second deals with the deductions from Hertz's fundamental 

 postulate of motion in the straightest possible path. 



The first part begins by explaining what is meant by the path 

 of a system of points. To get at this we calculate the mean 

 square of the displacements of a system of pomts when they are 

 displaced : the square root of this. Hertz calls the displacement 

 of the system of points. If there is a mass at each point, then 

 the displacement of the system is the square root of the mean 

 squares of the displacements of the points, each multiplied by 

 the mass at it. Thus, if s be the displacement of the system, 

 and s-^s.,. Sec, the displacements of each point of masses m^m.,, 

 &c. Then 



(w/j + OTj + . 



)i'- = >n^s{- + >n„s^ + . 



By taking ij, /„, &c., as the displacements in the element of 

 time, we evidently get a similar expression for the velocity of the 

 system, and for its acceleration. The mean square of the 

 velocity of the parts of a system is well known in connection 

 with the principle of least action. P'urther than this, however. 

 Hertz defines the angle between two displacements. This is 

 defined by the equation 



(;«i + OT2 + . . . . )w'cose 



^(wiijJi'cosoi + m..s..^./ cos a., + ....) 



s and s' being the two displacements of the system as calculated 

 above, and ^i^i', &c., the two displacements of each point and 

 tti, o.., the angles between these latter, then e is the angle 

 between .( and s'. Hertz remarks that these can all be very 

 interestingly expressed in terms of space of multiple dimensions, 

 in which analytical diagrams are supposed to be drawn. This, 

 however, represents the real by the unattainable. There follow, 

 then, several chapters expressing these displacements in terms 

 of various systems of coordinates, and discussions as to the con- 

 ditions that the connections of a system should fulfil in order 

 that they may be represented by equations not involving 

 differentials. The curvature of the path is here studied. It 



is defined as c = —, and from this it follows that, representing 



.A- 



by -v", &c. 



.).-^ = 2i'{«iAi"- + j'i'"-^-t-=i"=). 



The problem then of making the path of the system 

 straightest, is to make c a minimum consistently with the 

 connections of the system. Now, in accordance with his 

 assumption that the connections of the system are linear 

 ditTerential equations of the foim 



2,'P,.v\ = o, 



whose differentiation gives 



we are to determine the minimum value of 



•zr^.x" 



vhen 



in = /«! I- /«o + . . . . 



In determining the variations of these, we must recollect that 

 the positions and direction of displacement, i.e. the first 

 differentials of the system, are supposed given, and that it is 

 only the second differentials that can be varied in order to make 

 c a minimum. Calling, then, a system of indeterminate co- 



NO. I316, VOL. 51] 



efficient \, y., &c., corresponding to the equations of condition, 

 we evidently get a system of equations of the form 



"'•.<"i H- 2,'P,\ = o, 

 m 



which are sufficient to determine the second differentials re- 

 quired. 



From this form of result one can see how the ordinary 

 equations of moiion are derivable from the conception of the 

 straightest path, and how, when dealing with part of a system, 

 these indeterminate coefficients introduce what are equivalent to 

 forces. This method of deducing the equations of motion lends 

 itself particularly well to the deduction of the principles of least 

 action, and the other general methods in dynamics. So far, he 

 deals with (ree systems subject only to internal constraints. It 

 is where he investigates how to deal with parts of systems that 

 he requires to consider the nature of the constraints joining one 

 part to another. For this purpose he defines two systems as 

 coupled when coordinates can be so chosen that one or more of 

 them are the same for both systems. Force is then defined as 

 the action one system has on another. Now, when a co- 

 ordinate is the same for two systems, one of the equations of 

 condition is/ = p , p and/' being coordinates of the coupled 

 systems, and for this equation the coefficient P becomes the 

 same in the two systems, being uniiy for each, so that the 

 equations of motion involve the indeterminate coefficient 

 \ corresponding to this equation equally wth reference 

 10 each system. It is thus that the equality of action 

 and reaction appears, being thus bound up with the 

 constant equality of the common coordinate. This seems to 

 be where the assumption that the connections are rigid 

 is introduced. When rigid bodies act upon one another by 

 non-slipping contact, certainly the coordinites of the point of 

 contact are common to the two systems. It is also quite evident 

 that if we assume rigid bodies acting upon one another by 

 contact only, we can have no potential energy, and all necessity 

 for talking about the forces disappears. In Hertz's system there 

 are no forces like -N'ewlon's acting between bodies which have 

 no common coordinate, like the earth and the sun. We would 

 have to invent connections to explain the motion before we 

 could be certain that action and reaction are equal in this case. 



The proof of the principle of virtual velocities by substituting 

 for the forces between parts of a system a number of pulleys 

 which produce the same effects, is quite analogous to Hertz's 

 supposition that the actual connections are by rigid bodies. It 

 is not, however, liable to the ol'jeciion that the connections may 

 become tangled, for it is only applied to the case of infinitesimal 

 virtual displacements, while Hertz postulates the possibility of 

 his connections existing as the real ones for all time, and 

 throughout all finite displacements of the system. 



The work considers many other matters, and shows how all 

 the general methods in dynamics are deducible from his funda- 

 mental postulate of the >traightest path. It includes discussions 

 on how best to deal with systems whose connections do not in- 

 volve differentials, how to treat cyclical coordinates, and many 

 other matters. It is most philosof,hicaland condensed, and gives 

 one of the most — if not the most — philosophical presentations of 

 dynamics that has been published. It is worthy of its author : 

 what more can be said? G. F. Fitzgerald. 



PSEUDO-SATELLITES OF JUPITER IN THE 

 SEVENTEENTH CENTURY. 



TN the New York Nation for January ii, 1S94, Dr. D. C. 

 ■*■ Oilman, President of the Johns Hopkins University, called 

 attention to an ineresting letter from John Winthrop, jun., to 

 Sir Robert Moray, concerning the satellites of Jupiter. In this 

 letter, which was written Irom Hartford, Connecticut, on 

 January 27, 166J, Winthrop described an observation of 

 Jupiter which he had made on the night of the previous 

 6th of August, when he had very distinctly seen five satellites 

 about that planet. He was naturally " not with out some con- 

 sideration wheiher that fifih might not be some fixt star with 

 which Jupiter might at that tyme be in neare conjunction," 

 and expressed the wish that more frequent observations might 

 be made upon ihat planet with a view to ascertaining whether 

 it is not impossible to discern a fixe<l star, when it is so near 

 to the planet as to appear '" within the periphery of that single 

 intuitus by alube which taketh in the body of Jupiter," and if 



