522 Prof. F. Y. Edgeworth on the Law 



transformed curve, are in the neighbourhood of the maximum- 

 points Q 1? Q 2 . 



The question may now be raised: whether one of these 

 types — especially the more distinctive one, where F' ( X) = — 

 is likely to be realized in the (exceptional) case of the action 

 of numerous independent agencies not resulting in the pro- 

 bability-curve. The following problem may have wide ana- 

 logies. Let a body be moved from rest in a right line by the 

 simultaneous action of numerous small impulses, each as likely 

 to be in the positive as in the negative direction. The law 

 of frequency for the momentum of the body will be the proba- 

 bility-curve ; for the energy of the body is the curve represented 

 in fig. 2. 



V. So far we have supposed the " elements " to assume 

 different values in virtue each of a single variable x } or x 2 . 

 Let us now generalize this supposition by supposing that each 

 element fluctuates according to a law of frequency which in- 

 volves two (or more) variables. Thus let the rth element, I , vary 

 according to the law of frequency z=f r { x, y); it is required to 

 determine approximately the law of frequency for ¥(l l9 l 2 , &c), 

 where F is a function which fulfils the conditions above 

 prescribed. 



Now the law of frequency for ! r , given in terms of x and y, 

 is capable of being expressed in terms of x only. For the 

 (proportionate) number of the values of l r - which pertain to a 

 particular value of x (which occur between x and x -f Ax) are 

 obtained by integrating between extreme limits with regard 

 to y, while x is constant. Thus the law of frequency for 

 l r may be written : — 



z =\ fr{xy)dy\ 



if we put b r , fir, for the limits of y in the ?'th element. 

 But, by the preceding section, if l u l 2 , &c. each vary ac- 

 cording to a law of frequency involving x only, F(Z ]5 l 2 , &c.) 

 is approximately an error-function in x : that is, of the 

 form ~Ke~ h2x2 . Now the function in x of which this 

 form is an approximation should be obtained by integrating 

 F with regard to y between extreme limits, x being treated as 

 constant. And by parity a similar conclusion is obtained by 

 integrating with respect to x, — y being treated as constant — 

 on the one hand the elemental functions and on the other hand 

 the compound function F. Thus F is approximately such a 

 function that if it is integrated with regard to ?/, between ex- 

 treme limits, x being treated as constant, the result is approxi- 

 mately of the form lle~ h2x2 ; and if it is integrated with regard 



