Compound Law of Error, 209 



simple case * we obtain the differential equations 



dz__ \d& /-.x 



dk~~ 2dx 2 ' { } 



dz__ld^_ /«n 



dm~ 2dy 2 W 



Other differential equations are obtained by supposing the 

 units of x and y altered; substituting for x and y, x{\-\-a) 

 and 2/(1 + /3) respectively. The expression for z thus trans- 

 formed must be multiplied by (1 +a) (1+/3) ; since the 

 measure of the solid contents of the parallelopiped inter- 

 cepted between the surface, the plane of x^, and any two fixed 

 adjacent points in that plane will be increased in that pro- 

 portion. Thus 



s=(l+a)(l+£)<l>0&,y; k,m). 



Regarding a and ft as infinitesimal, expanding and neglect- 

 ing higher terms, we have 



•+-£+»a-<* (3 > 



^| +to £=° ^ 



From (3) and (4) we have 



where <£> and ty are arbitrary functions. Whence 



1 / x y \ 

 m \ vm vm' 



Z = ' 



where % is an arbitrary function. The form of % is restricted 

 by the condition that its value is the same for positive and 

 negative values of x and y, the surface being symmetrical 

 about a vertical plane through each axis. For as we take 

 account only of the second powers in the expansion of each 

 element, we might replace the given system of elements by a 

 new system of symmetrical functions having each the same 

 centre of gravity and mean square of error as the old one f , 



* See the preceding article, Phil. Mag. 1896, xli. p. 90. 



t This does not mean that the given elements must be symmetrical, as 

 is sometimes carelessly said with reference to the simple law of error. 

 The given elements may have any degree of asymmetry, provided that 

 their number is correspondingly great. 



