142 



NAIURE 



[December io, 1891 



equations can be obtained by equating to zero a factor of the 

 discriminant. 



Let E = o be the equation of the envelope locus, 

 C = o be the equation of the conic node locus, 

 B = o be the equation of the biplanar node locus, 

 U = o be the equation of the uniplanar node locus. 



Now at any point on the locus of ultimate intersections — 

 (I.) TJtere may be one system of values of the parameters 

 satisfying the fundamental equations. 



In this case there may be envelope, conic node, or biplanar 

 node loci ; and the corresponding factors of the discriminant are 

 E, C^, B^ respectively. 



(TI. ) There may be more than one system of distinct values of 

 the parameters satisfying the fundamental equations. 



In this case the effect of the distinct values is additive. Thus 

 if there be p systems of values at a point on the envelope locus, 

 the factor E would occur to the/th power. 



(III.) Two or more systems of vahies of the parameters satisfy- 

 ing the fundamental equations may coincide. 



The results must be stated differently in the cases (o) where 

 the degree in the parameters of the equation of the system of 

 surfaces is greater than two ; (;8) where the degree in the para- 

 meters of the equation of the system of surfaces is two. 



In the case (a) it will be shown that there may be envelope 

 loci in which theenvelopehasstationary contact with each surface 

 of the system ; conic node loci, which are also envelopes ; 

 biplanar node loci, in which the edge of the biplanar node always 

 touches the biplanar node locus ; and uniplanar node loci : and 

 the corresponding factors of the discriminant are E-, C, B*, U^ 

 respectively. 



The case ()3) always falls under the next case — 



(IV.) The values of the parameteis satisfying the fundamental 

 equations may become indeterminate. 



If the equation of the system of surfaces be of the second 

 degree in the parameters, and the analytical condition hold which 

 expresses that the fundamental equations are satisfied by two 

 coinciding systems of values, then this condition requires to be 

 specially interpreted. For now the second and third fundamental 

 equations are of the first degree in the parameters, so that if they 

 are satisfied by two coinciding systems of values, they must be 

 indeterminate. 



It is, however, possible to determine a single system of values 

 of the parameters satisfying them. In this case the three surfaces 

 represented by the fundamental equations intersect in a common 

 curve (which is fixed for fixed values of the parameters) lying on 

 the locus of ultimate intersections ; whereas in the previous 

 cases they intersect in a finite number of points lying on the 

 locus of ultimate intersections. 



The surface of the system, corresponding to the fixed values 

 of the parameters, touches the locus of ultimate intersections 

 along the above-mentioned curve. 



In general, there are two conic nodes of the system at every 

 point of the locus of ultimate intersections. The parameters of 

 the surfaces having the conic nodes are determined by two 

 quadratic equations, called the parametric quadratics ; and in 

 general the roots of each parametric quadratic are unequal. In 

 this case the corresponding factor of the discriminant is C-. If 

 the roots of both parametric quadratics are equal, the two surfaces 

 having conic nodes are replaced by one surface having a biplanar 

 or uniplanar node. In this case the corresponding factors of the 

 discriminant are B^, U*, respectively. i 



If the parameters of one of the surfaces having a conic node I 

 become infinite, this surface may be considered to disappear, I 

 and there is but one conic node at each point of the locus of | 

 ultimate intersections. In this case the corresponding factor of 

 the discriminant is C^. 



If the parameters of both surfaces having conic nodes become 

 infinite, both these surfaces may be considered to disappear, and 

 the locus of ultimate intersections is an envelope locus (touching 

 each surface of the system along a curve). In this case the 

 corresponding factor of the discriminant is E*. 



If the parameters of both surfaces having conic nodes become 

 indeterminate, then there are at each point an infinite number of 

 biplanar nodes, and each surface of the system has a binodal 

 line lying on the locus of ultimate intersections. In this case 

 the corresponding factor of the discriminant is B*. 



Physical Society, November 20.— Prof. W. E, Ayrton, 

 F.R.S., President, in the chair. — Dr. Philippe A. Guye gave 

 a short account and discussion of the various forms which have 



NO. 



II 54, VOL. 45] 



been given to the general equation cxpiessing the behaviour of 

 liquids and gases under different conditions of volume, tempera- 

 ture, and pressure, by Van der Waals, Clausius, Sarran, Violi, 

 Heilborn, and Tait. He first considered the equation of Van 

 der "Waals, which, although only an approximation to the true 

 one, may be made to lead to numerous important deductions. 

 He then showed that, of the various more exact formulae 

 proposed, that of Sarran is the simplest, and may be used 

 with less expenditure of time and trouble than any of the 

 others. In conclusion, he insisted on the necessity of experi- 

 mental researches as the only means of arriving at a definite 

 conclusion as to which of the various formulffi is the true one ; 

 such researches should involve determinations, as exact as 

 possible, of the critical constants, and of isotherms at high tem- 

 peratures and great pressures. Prof. Ramsay inquired whether 

 the constants in the formula of Clausius had any physical 

 meaning, or were they merely numbers ? M. Guye, in reply, 

 said that, although some of the constants in the improved formulae 

 nad physical interpretations, Van der Waals's equation was the 

 uniy one in which all the constants had precise physical signi- 

 fications. Prof. Riicker said it was only necessary to look at the 

 formulae to see how important a factor Van der Waals's expres- 

 sion had been in later developments of the subject. Although 

 it did not agree with experiment under all conditions, par- 

 ticularly at small volumes, yet it was a close approximation 

 over a considerable range, and was the only formula in which 

 all the constants had definite physical meanings. Prof. Tait 

 had pointed out that the number of constants were too few to 

 fully represent the facts, for, by following Andrews's reasoning, 

 he had shown that about the critical point a straight line cuts 

 the isotherm in five points. Nevertheless, during the last twenty 

 years all the so-called improved formula* were modifications of 

 Van der Waals's expression, and this, he thought, showed how 

 valuable the original formula was. Prof. Fitzgerald said he once 

 tried how far Llausius's formula agreed with the experimental 

 results published by Messrs. Ramsay and Young, and after 

 several months' work, relinquished it on account of the tremen- 

 dous labour involved. He thought that such complicated 

 formula; retarded rather than advanced science ; simple ones 

 (even if less accurate) were likely to lead to greater advance- 

 ment. Prof. Carey Foster remarked that the expression 

 pv = RT, which is nearly true for gases, was the starting-point 

 of all subsequent advances. Van der Waals had arrived at a 

 still closer approximation by taking into account the volume 

 occupied by the particles and their mutual pressure. The 

 President said Van der Waals's memoir had been adversely 

 criticized because of its supposed insufficient recognition of 

 Andrews's investigations on the subject. Better acquaintance 

 with the work had, however, shown this criticism to be un- 

 deserved. — Dr. C. V. Burton read a paper on a new theory 

 concerning the constitution of matter. It is assumed that it 

 is possible to have in the ether a distribution of strain which is 

 itself in equilibrium. Such a distribution is called a "strain- 

 figure." An atom is looked upon as an aggregation of strain- 

 figures, the possible varieties of strain-figures (and hence of 

 atoms) being limited by the conditions of equilibrium, and the 

 sizes of possible strain-figures dependent on the coarsegrained- 

 ness of the turbulent motion or other structure of the ether. 

 The motion of matter is considered to be merely the trans- 

 ference of a strain distribution from one portion of the ether to 

 another. This the author illustrated by causing a loop to travel 

 along a rope, the loop being regarded as a strain distribution 

 which is propagated along the rope, whilst the rope itself is not 

 transferred. Such transference may occur without encountering 

 any resistance, and the strain-figure will retain the same form, 

 provided the velocity is small compared with that at which 

 gravitation is propagated. The equations of motion of a strain- 

 figure are deduced, and are shown to be identical with those of 

 ordinary matter, provided certain conditions of symmetry are 

 realized. It is also shown under what conditions an atom consist- 

 ing of strain-figures would have a finite number of degrees of free- 

 dom, and some attempt is made to examine how gravitation and 

 other attractions might follow from a distribution of stress in the 

 strained ether. An inquiry is also made into the reason why 

 elements have fixed properties, and their transmutation is dis- 

 cussed. Prof. Fitzgerald, referring to the elastic-solid theory 

 of the ether, said Sir W. Thomson's more recent papers had 

 thrown considerable doubt upon it. The propagation of strain- 

 figures was, he thought, a case of wave motion. In his lectures 

 he had likened the passage of matter through space to that of a 

 drop of water through ice, the ice in front melting, and the rear 



