PRESENT PROBLEMS OF GEOMETRY 585 



This is equivalent to the tensor field which arises in studying the 

 moments of inertia of an arbitrary distribution of mass. The more 

 general case actually arises in Maxwell's theory of magnetism. 



4. As a final domain of application we mention the class of ques- 

 tions which have received systematic treatment, under the title of 

 nomography , only during the past few years. This subject deals with 

 the methods of representing graphically, in a plane, functional 

 relations containing any number of variables. Thus a function of 

 two independent variables, z=f(x, y), may be represented by the 

 system of plane curves /(z, y) =c, each marked with the correspond- 

 ing value of the. parameter. This " parametered " system is then 

 a cartesian graphical table, which is the simplest type of abacus or 

 nomogram. 



By means of any point transformation, one nomogram is con- 

 verted into another which may serve to represent the same functional 

 relation. The importance of this process of conversion (the so-called 

 anamorphosis of Lalanne and Massau) depends on the fact that it 

 may replace a complicated table by a simpler. The problems which 

 arise (for example, the determination of all relations between three 

 variables which can be represented by a nomogram composed of 

 three systems of straight lines 1 ) are of both practical and theoretical 

 interest. The literature is scattered through the French, Italian, 

 and German technological journals, but a systematic presentation 

 of the main results is to be found in the Traite de Nomographie 

 of d'Ocagne (Paris, 1899). 



We return to the abstract theory of transformations. The type 

 of transformation we have been considering, converting point into 

 point, is only a special case of more general types. The most im- 

 portant extension hitherto made depends upon the introduction of 

 differential elements. Thus the lineal element or directed point 

 (x, y, y'} leads to transformations which in general convert a point 

 into a system of elements; when the latter form a curve, every curve 

 is converted into a curve and the result is termed a contact trans- 

 formation. Backlund has shown that no extension results from the 

 elements of second or higher order: osculation transformations are 

 necessarily contact transformations. The discussion of elements of 

 infinitely high order, defined by an infinite set of coordinates (x, y, 

 y') y") )> ma y perhaps lead to a real extension. The question may 

 be put in this form: Are there transformations (in addition to or- 

 dinary contact transformations) which convert analytic curves into 

 analytic curves in such a way that contact is an invariant relation? 

 The idea of curve transformation in general will probably be worked 



1 The case of three systems of circles has also been discussed. See d'Ocagne, 

 Journal de I'Ecole Poly technique, 1902. 



