July 21, 1898] 



NATURE 



267 



have a multiplicity of one dimension, which may be 

 called an orientation of the first order. In general by 

 means of <^ = o, the relations dp = rdx + sdy, dq = sdx 

 + tdy, and those derivable from them by differentiation, 

 it is possible to find a definite expansion for s-, which 

 formally satisfies the differential equation and also con- 

 tains the given orientation : if the expansion is con- 

 vergent in a certain domain, this defines z as an analytical 

 function of x, y. Geometrically, if we take x, y, z as 

 point coordinates, the assumed orientation consists of an 

 arbitrary curve, with an arbitrary, but continuous, distri- 

 bution of tangent planes along it, enveloping a develop- 

 able surface ; if we like, we may regard it as a thin 

 ribbon cut out of a developable. The process sketched 

 above is equivalent to finding an integral surface contain- 

 ing the aforesaid ribbon, or in other words containing 

 the given curve, and touching at each point of it the 

 ^nven associated tangent plane. The problem of Cauchy 

 for an equation of the second oi-der is to find a solution 

 capable of being specialised, by the choice of arbitrary 

 constants or arbitrary functions, or both, so as to contain 

 any given orientation of the first order. Such a solution 

 is said to be general in Cauchy's sense, as distinguished, 

 for example, from one that is general according to 

 Ampere's celebrated definition. 



It may happen that the orientation of the first order, 

 Mj say, is such that the relations ^ — o, dp = rdx -\- sdy^ 

 dq = sdx -f- /dy are, for every element of it, equivalent to 

 only two independent equations ; in this case Cauchy's 

 problem becomes indeterminate, and there are an infinite 

 number of integral surfaces containing Mj, which is then 

 said to be a characteristic of the first order of = o. It 

 is an exception for an equation of the second order to 

 admit of a multiplicity Mj ; since 



.y, -,A?. 



s^, s, ^ 

 dU-' ' dy 



dx\ 



has to be satisfied identically for all values of s, and this 

 leads to a number of distinct relations, not generally 

 compatible. One of these is always 



d<P 



dy^ 



^4dxdy + ^4dx'^ 

 ds -^ 3/ 



on every integral surface this equation defines a system 

 of characteristic curves. 



Throughout the whole treatise the theory of character- 

 sties plays a predominant part. Thus in Chapters i.-iii., 

 >\hich deal with the equation of Monge and Ampere 

 Hr + 2Ks + U + M -h x\(r/- j2) = o), it is shown with 

 admirable clearness how the success of Monge's method of 

 integration depends upon finding integrable combinations 

 of the differential equations of the characteristics. The 

 cases of partial or total failure are discussed as well as 

 those of success ; and the reader thus becomes familiar 

 with the rationale of the process, instead of merely 

 acquiring facility in applying a method which, in some 

 way that he hardly understands, leads (with good luck) 

 •o the required solution. Chapter iii., in particular, con- 

 -ains a large number of important applications very fully 

 worked out. 



M. Goursat's first volume concludes with an important 



chapter on the general theory of characteristics and on 



intermediate integrals. The notion of characteristics is 



extended to the second and higher orders, and it is 



NO. 1499, VOL. 58] 



shown, among other things, (i) that every equation of 

 the second Qrder possesses in general two distinct 

 systems of characteristics of the second order ; (2) that 

 two characteristics of the second order belonging to two 

 distinct systems, and having in common an element of 

 the second order determine one, and only one, integral 

 surface (p. 193). All equations of the second order may 

 be arranged in four classes according as they have (i) 

 two different systems of characteristics, each of the 

 second order (this is the general case) ; (2) two systems, 

 one of the first order, one of the second ; (3) two 

 systems, usually distinct, each of the first order ; (4) one 

 system of the first order. 



The second volume begins with an account of La- 

 place's method of treating linear equations, which maybe 

 profitably compared with the discussion of the same 

 subject in Darboux's " Theorie des Surfaces." After this 

 come two chapters, of the highest interest and im- 

 portance, on systems in involution and on Darboux's 

 method of integration. The first of these deals with 

 systems of equations which admit of solutions involving 

 an infinite number of arbitrary constants, and introduces 

 us to ideas of great value and generality which have 

 been developed by various mathematicians, including 

 M. Goursat himself. The chapter is, to a great extent, 

 introductory to the one on Darboux's method, which 

 immediately follows, and which will probably be found 

 the most engrossing part of the work. The leading idea 

 is that of finding integral combinations of the differential 

 equations of characteristics ; not necessarily of the first 

 order, as in Monge's method, but of the second, third, or 

 higher order : thus, for instance, Liouville's equation 

 J = r^ is completely integrated by proceeding as far as 

 the characteristics of the second order. M. Goursat 

 very justly remarks that Darboux's method is the most 

 powerful as yet available, and includes most others, for 

 instance those of Monge, Ampere, and Laplace, as par- 

 ticular cases. In order that it may succeed, it is neces- 

 sary that every integral of the proposed equation should 

 also be an integral of another partial differential equation 

 which has in common with the given equation an in- 

 finity of integrals depending on an arbitrary function, 

 while at the same time the second equation must not be 

 satisfied by all the integrals of the first (II. p. 190). The 

 main practical difficulty is that it is generally impossible 

 to say beforehand whether a given equation admits of 

 solution by this method or not. By means of Lie's 

 theory of transformation-groups it is, however, possible 

 to construct a variety of equations to which Darboux's 

 method may be successfully applied. 



The next chapter deals with equations of the kind 

 called by Amptire those of the first class ; this is followed 

 by one on transformations ; and the treatise concludes 

 with a somewhat miscellaneous chapter containing 

 various generalisations of the preceding theory. 



A work so attractive as this, and written by an author 

 so well known, is assured of the favourable reception 

 which it thoroughly deserves ; taken with M. Goursat's 

 previous work on equations of the first order, and M. 

 Darboux's "Thdorie des Surfaces," it will provide 

 mathematical students with an excellent guide to what 

 has been done in this part of analysis. One way, 

 amongst many others, in which M. Goursat's treatises 



