384 EEPORT— 1892. 



population, barometrical readings, temperature, &c. Contour lines are 

 also used with Cartesian co-ordinates — for instance, with isothermal 

 curves of air expansion. They are referred to by French writers in various 

 treatises under the title of ' courbes d'egal element,' and have the same 

 relation in graphic diagrams with regard to two dimensions of space as 

 the mere setting up of lengths in one dimension of space illustrated in 

 figs. 4 and 5, requiring some distinguishing mark or notation, as in fig. 11, 

 for each particular line or curve employed. 



As contour lines are thus shown to be a special case of curves formed 

 by the combination of two variable quantities, they need not be further 

 alluded to, although it is worthy of consideration whether in many cases 

 where models have been employed — such, for instance, as the surfaces 

 used by Professor Thurston in his researches on ' Ternary Alloys ' — they 

 might not be conveniently replaced by the use of such contour lines 

 which involve only graphic diagrams. 



There are, it is true, ways in which certain relations of three variable 

 quantities may be represented graphically. Thus the method used by 

 Professor Clerk Maxwell for the composition of colours was suggested ' 

 by Sir G. G. Stokes for representing the results of Dr. Alder Wright's 

 experiments on 'Ternary Alloys.' In this case any triangle being taken 

 (preferably an equilateral triangle), the quantity by weight of each of the 

 three substances used in the alloy may be supposed to be respectively 

 placed at the three apices, and the centre of gravity of all three found. 

 This being always some point in the plane of the triangle, the position of 

 the point on the surface represents the composition of the alloy. A 

 bounding curve, called the ' critical curve,' is then drawn through the 

 points so found, corresponding to the proportion at which the alloys 

 refuse to remain in combination, but separate into two substances, by 

 means of which curve certain important facts are made evident. The 

 diagram, however, does not really represent three variable relations, 

 because it is clear that if every possible composition were tried the surface 

 would be covered with points, those only which lie along the curve in 

 question actually representing the result of the graphical construction. 



The fact that the results of three variable quantities cannot be truly 

 represented upon a plane surface has been insisted on, both to clear the 

 way for what follows and because the references to this are often some- 

 what misleading ; thus Favaro, in his ' Lectures on Graphic Statics,' ^ says 

 truly, p. 147 : ' Every curved surface to really represent with reliefs 

 requires the three dimensions of space which correspond to the three 

 variables of which tbey represent the relation,' but goes on immediately 

 after to say : ' But one can by means of a notation as simple as expressive 

 replace constructions in space by traces entirely contained in a plane,' 

 and then proceeds to describe the system of contours, which it has been 

 shown only partially represent the result of the simultaneous variation of 

 three quantities ; and, indeed, whatever trace is taken on the plane can 

 always be developed into two, being the intersection of surfaces of an 

 irregular cylinder with the curve surface in question. In his translation 

 of the above work M. Terrier proceeds to give in an appendix a number 

 of cases of what he calls stereograms — a term first proposed by Professor 



' Prooeedingt of the Royal Society, vol. xlix. No. 298, p. 174. 



• Leqons de Statique graphique, par Antonio Favaro. Traduites de I'italien 

 par Paul Terrier, deuxieme partie, 1885. Gauthier-Villars, Paris. 



