186 Mr. F. Y. Edgeworth on a New Method of 



the left, A!. Let the sum of the 6's, above, be B ; below, B'. 

 Then 



dR=(A'-A)<fo+(B'-B) dy ; 

 while 



adx + bdy = 0, and ds = \/dx 2 + dy' 2 . 



When the starting-point is an intersection, we should put 

 ourselves at the proximate point along the path which is being 



explored. To find ~j- for another line passing through the 



same point, or for a neighbouring point and line, it will not 

 be necessary to evaluate A A', B B' afresh. The coefficients 

 of dx and dy in the expression for dR require alteration only 

 in respect of the coefficients of those lines which have been 

 crossed. 



Continue moving according to these directions, until a point 



is reached at which -j- is positive for every path passing 



through the point. This point constitutes the solution*. It is 

 in general unique. For, if possible, let there be two such 

 minimum-points. Take the line joining them as the axis of y. 

 Then the expression (6jy — v{) + (biy— v*) + & c -> eacn of the 

 bracketed terms taken positively, must be a minimum for two 

 points on the ordinate, Which is in general absurd ; being 

 possible only in the exceptional case when, in the notation 

 above employed, B + b is exactly equal to B', or B' + b to B. 

 In this rare case the solution may be indeterminate, namely 

 any point on a certain line or even areaf. 



In this event common sense seems to dictate that we should 

 adopt the middle of the indeterminate tract as the best point ; 

 and this presumption is confirmed by a formal calculation of 

 utility such as Laplace, in the simplest case of a single un- 

 known quantity, has employed to discover the u most advan- 

 tageous" pointy. 



* The method may be illustrated thus : — Let C — ft. (where U is a con- 

 stant) represent the height of a surface, which will resemble the roof of 

 an irregularly built slated house. Get on this roof somewhere near the 

 top, and, moving continually upwards along some one of the edges or 

 arretes, climb up to the top. The highest position will in general consist 

 of a solitary pinnacle. But occasionally there will be, instead of a single 

 point, a horizontal ridge, or even a ilat surface. 



f This is Mr. Turner's " special case," loc. cit. pp. 468 & 469. 



X Laplace, dealing with a set of observations known to have emanated 

 from a Probability-curve (Theor. Ana/i/t. book 2, chap. 4, art. 23), thus 

 in effect reasons : — In the long run of cases, where we have to do with a 

 set of observations exactly the same as the proposed set, the real point, 

 which is the source from which this grouping emanated, occurs at different 

 points with a frequency which is represented by a certain Probability- 



