136 THE RATE OF GROWTH [ch. 



and the velocit}^ of a moving body. If ^ = e"^^ be the ordinate of 

 the one, z = le-^'^dx is that of the other. 



There is one more kind of frequency-curve which we must take 

 passing note of. We begin by. thinking of our curve, whether 

 symmetrical or skew, as the outcome of a single homogeneous 

 group. But if we happen to have two distinct but intermingled 

 groups to deal with, differing by ever so little in kind, age, place or 

 circumstance — leaves of both oak and beech, heights of both men 

 and women — this heterogeneity will tend to manifest itself in two 

 separate cusps, or modes, on the common curve: which is then 

 indeed two curves rolled into one, each keeping something of its 

 own individuality. For example, the floating eggs of the food-fishes 

 are much alike, but differ appreciably in size. A random gathering, 

 netted at the surface of the sea, will yield on measurement a multi- 

 modal curve, each cusp of which is recognisable, more or less 

 certainly, as belonging to a particular kind of fish (Fig. 26). 



A further note upon curves 



A statistical "curve", such as Quetelet seems to have been the 

 first to use*, is a device whose peculiar and varied beauty we are 

 apt, through famiharity, to disregard. The curve of frequency which 

 we have been studying depicts (as a rule) the distribution of mag- 

 nitudes in a material system (a population, for instance) at a 

 certain epoch of time ; it represents a given state, and we may call 

 /it a diagram of configuration "f. But we oftener use our curves 

 to compare successive states, or changes of magnitude, as one 

 configuration gives place to another; and such a curve may be 

 called a diagram of displacement. An imaginary point moves in 

 imaginary space, the dimensions of which represent those of the 

 phenomenon in question, dimensions which we may further define 

 and measure by a system of "coordinates"; the movements of our 

 point through its figurative space are thus analogous to, and illus- 

 trative of, the events which constitute the phenomenon. Time is 

 often represented, and measured, on one of the coordinate axes, and 

 our diagram of "displacement" then becomes a diagram of velocity. 



* In his Theorie des probabilites, 1846. 



t See Clerk Maxwell's article "Diagrams," in the Encyclopaedia Britannica, 

 9th edition. 



