PROFESSOR TAIT ON THE LAW OF FREQUENCY OF ERROR. 143 



If the most probable result, as depending on the several sets of causes, be 

 different for each, the formula (6) becomes, for any one cause, 



3, = A6-^-*> 2 . . . ' . . (9), 



where A is the (small) chance of the most probable result, which is, of course, x = y. 

 The chance of any particular value of x, as due to the simultaneous action of 

 all the causes, is now 



y = A x .... A v €-^ x -^ 2 ~ •■■■ ~i**-v# . . (10), 



which may, of course, be put in the form 



y = % € -M(*-r? (11); 



where the most probable result is now 







X = Y- 



% + A* 2 + • • • • 



■ ) 

 + /*>* 







while 





(where, as 



. . A v e- ( ^> Z+ 

 before, M = fh x + ^ 2 + 



+ fx.,y 2 ) + Mr 2 

 . . . . +/!*,) 







is its 

 If 



probability. 



we take this as our point 



of departure for the error x, 



we 



must write x for 



x— r, 



and we have 





GH^— MX 2 







. . (12), 



for the form of the law of error, which is precisely that of (6) deduced from 

 the simplest conceivable case. 



11. Another remarkable confirmation of the validity of the process suggested 

 above, is to be found in the fact that not only are the curves expressed by equa- 

 tions such as (6) and (9) compounded, by multiplication of corresponding ordinates, 

 into another of the same class, whatever be the positions of their axes of symmetry, 

 but that the same principle holds good in three, four, &c, dimensions also. 



Thus, any number of hills on the plane of xy, represented by equations such as 



f-Ae-tf-^+M 1 ] ..... (13), 



give, by multiplication of their corresponding ordinates, another hill of the same 

 general form, the values only of the constants being changed. 



[Many curious geometrical results may be derived from this construction. 

 One of the most singular is the fact that the projection on x y of the line of inter- 

 section of any two surfaces whose equations are of the form (13) is a circle, and 

 that another such surface (viz., that whose ordinates are mean proportionals 

 between those of the former) can be described, passing through the curve of 



VOL. XXIV. PART I. 2 R 



