164 Mansfield Merriman — List of Writings relating 



selves. The arbitrary nature of this assumption is sliown by Glai- 

 SHER, Mem. Astron. Soc. Lond., 1872, Vol. XXXIX, pp. 75-81, where 

 the proof is analyzed and regarded as " very slight and inconclusive." 



The second proof occupies pages 96-97, and not having been 

 alluded to in the reprint noted above, I give it in Adrain's own 

 words as 77ie Analyst is quite rare : 



" Suppose that the length and bearing of A£ are to be measured ; 

 and that the little equal straight lines JBb, Be are the equal probable 

 errors, the one Bbz^Bb' of the length oi AB, and the other Bc=zBc' 

 (perpendicular to the former) of the angle at A, when measured on a 

 circular arc to the radius AB: and let the question be to find such a 

 curve passing through the 

 four points b, c, ^,' c , which 

 are equally distant from B, 



that, supposing the meas- 



urement to commence at ^, ^ 

 the probability of termina- 

 ting on any point of the 

 curve may be the same as 

 the probability of terminating on any one of the four points b, 

 c, b\ c'." 



Then follows trivial reasoning which ends by concluding that " the 

 curve must be the simplest possible " and " must consequently be the 

 circumference of a circle having its centre in ^." This established, 

 the proof is the following : 



'•Now let us investigate the probability of the error Bm=x, and 

 of nin-=.y. Let ^and I^be two similar functions of x and y denot- 

 ing those probabilities, X\ Y\ their logarithms, then A^X 3"r= con- 

 stant, or -X''+ Y' z=z constant, and therefore JC' -{- Y' ^ 0, or Jl"x+ 

 Y"y= 0, whence X "x = - X"y. But x^ -]-y^ =r^ = Bb^ , therefore 



. . " A^" Y" 



XX = — yy, by which dividing X"x — — Y"y, we have — =z — ; 



and therefore, by a fundamental principle of similar functions, the 



X" Y" 



similar functions '^— and must be each a constant quantity : put 



X y 



X" . . . ' 



then := n, and we have X"x = nxx, that is X' =. nxx, and the 



X 



7ix^ nij^ 



fluent is A"' 1= c -| — ^ — ; in like manner we find l"' = c-j — ^ and 



therefore the probabilities themselves are e 2 and e "^ 2 ^ m 

 which n ought to be negative, for the probability of x grows less as 

 X grows greater." 



I have seen no allusion to this proof by any subsequent writer. It 

 is essentially the same as given in 1850 by Hkrschel and usually 

 called Herschel's proof. I regard it as defective in taking "A"X Y 

 = constant," or in considering the probabilities of the x and y devi- 

 ations as independent. See 1850 Eli^is. See Boole's Finite Differ- 

 ences (Cambridge, 1860), pp. 228-229. 



