ERRORS OF OBSERVATION. 



421 



pretation, and will remain so until we have discovered* the plusquam impossibile. But the 

 numerical effect is too small to require attention. I find that if we make the hypothesis that 

 the error must lie between ± e, with a modulus of the form Ae'"^' (e^ - a?'), the results, e being 

 greater than unity, or not less, and c as large as it commonly must be, do not differ by 

 anything at all appreciable from those of our second approximation, 



I now proceed to consider the mode of deducing probable results. Let there be a number 

 of functions of the quantities to be determined — say w, y, z, — and of observed constants 

 subject to error. Let Pj, P^, &c. be functions of <r, y, z, and the constants, which would 

 severally vanish if true values of variables and constants were used. Hence all value is error 

 in Pj, Pj, &c. Let it be known that positive and negative values of Py, P^, &c. are equally 

 likely, that is, let nothing whatever be known to the contrary; and let (pP^ be the modulus 

 of P„, with reference to the constants; that is to say, fixed values of x, y, ss, being -used with 

 observed values of the constants, the chance of the nth function lying between P„ and 

 P„+ dP„ is (f)P„ dP„. If then ^dx, rjdy, l^dx, be the probabilities, a priori, of the variables 

 lying between a; and x + dx, &c., we know that the probability o£ this combination, after the 

 observations, is ^rt(,(pPi (pPo...dxdydz dP^ dP^... divided by the complete integral of this 

 differential. The most probable conjunction of antecedents is therefore that which makes 

 0Pi (f)P^... x^i;^ a maximum : and if all values of x, y, x, be a priori equally probable, in 

 which case ^, t], ^, are constants, (pPi ^Pn... is to be a maximum. If <p„P„ be A„6~'^°''", then 

 >|/^,Pj+ \//2P2 + ... is to be a minimum. This appears to me to be the only way in which 

 probable value can be deduced from the acknowledged foundations of the theory. Any 

 other method, however valuable as an illustration, would never have been allowed to impose 

 a result contradictory of any result of this method. Accordingly, and treating of methods 

 as demonstrations only, without reference to accessory value, I am much inclined to speak 

 of all other methods as the slandered Caliph is said to have spoken of the Alexandrian 

 books: " If in the Koran, useless; if not, pernicious: destroy them." 



In practice \^P will always be a function of even form, and rapid convergence. We 

 have then to make a minimum of 2(\|/"o .P-) + -^^ 2(\//"^0. P') +... in which \^"^0 is always 

 small compared with v^"o. We begin by making 2(\|/"o . P-) a minimum; or, P^ being 

 dP : dx, we solve the equations S(\//"o . PP^,) = 0, &c. For a second approximation, sub- 

 stitute the values of a?, &c. thus obtained in ^S(>|''*^0 , P^P^), &c. and solve 



2(x|."0 . PP.) + f the value obtained for 1 E(v//"'o . P'P^)} = 0, &c. 



We shall probably always be compelled to estimate errors by taking the most probable 

 value of each variable as the reputed truth, and taking the reputed errors derived from these 

 values as the real errors committed. It may be worth while, nevertheless, to show the effect 



• The negative probability may no doubt be an index of 

 the removal from possibility of the circumstances, or of the 

 alteration of data which must take place before possibility be- 

 gins. But I have not yet seen a problem in which such inter- 

 pretation was worth looking for. I have, however, stumbled 



upon the necessity of interpretation at the other end of the 

 scale : as in a problem in which the chance of an event hap- 

 pening turns out to be 2,J ; meaning that under the given 

 hypothesis the event must happen twice, with an even chance of 

 happening a third time. 



