ERRORS OF OBSERVATION. 417 



fesult, so long as we have nothing by which we can compare the goodness of two different 

 forms of (j). If we want to know the most unsafe value, the one to be avoided above all 

 others, we easily detect (j> (a, a, ...) = a, and we shall conclude that the average of discordant 

 attempts at this determination is the most probable greatest falsehood. If jurisprudence 

 could establish the principle that the corporate guilt of a conspiracy is that of the conspirator, 

 when all the conspirators are equally guilty; then jurisprudence would also be compelled 

 to take the average guilt of variously guilty conspirators as the corporate guilt of the com- 

 bination. 



In the function 0(ai, a.^, ...a^, symmetrical with respect to its subjects, make 



a^ = E + e^, ttz = E + e^, &c., 



E being taken at pleasure. Expanding by Taylor's theorem, and remembering 



<^ {E, E, ...) = E, 

 we have for the function 



E + P2e + Q2e' + R^ee + iS'Se' + TSe'e + U^eee + ... 



where by Eeee, for instance, we mean the sum of all terms of the form e^e^e^, where a, b, c, 

 are different suffixes. Now e, = 62 = ... = e must reduce this identically to ^ + e ; whence, 

 8 being the number of values given, we have 



'2 2 2 3 



Q, R, S, &c. depending on T and s only. Now E + P2e, or E + s~^'2e is (a^ + 0^+ ...) : s. 

 Hence ^(oi, a^, ... a^) is s~'2a augmented by terms of which we have no knowledge what- 

 ever, either as to sign or value, and no means of getting any : we are therefore wholly without 

 reason for supposing that the value of (p {a^, ..,) lies on one side of the average rather 

 than on the other, and must take this average as the most probable value a priori. 



The average, then, is the most probable result so long as we know nothing of the law of 

 facility of error : but this is only so long as the observations are either not made, or not dis- 

 closed. So soon as we see the second observation, we have some information about the law 

 of error : not much, but some. The second blow begins a fray ; the second instance begins 

 an induction ; the second observation begins a law of error. If you love life, says poor 

 Kichard, don't waste time; for time is the stuff that life is made of: if you value the 

 results of the theory of probabilities, don't throw away presumptions, for presumptions are 

 the stuff that results are made of. All information must be used up: and in known obser- 

 vations there must be information. 



Nevertheless, we have arrived at this result. When the observations present little dis- 

 cordance, and differ by terms of the first order, the average can only differ from the most 

 probable result by quantities of the second order at lowest. We may even predict that in all 

 cases the terms of the second order must disappear ; or Q = 0, i? = 0. For if E be the 

 absolute truth, and therefore e,, e^, &c. the real errors, we must suppose that a change of sign 



53—2 



