178 Mr Sang on an erroneous Method 



the integral of which may be put 



2. {i?^ + 2^3'4-2jt)r4- (i—1) terms 



-{- q^ 4-2g'r+ (i — 2) terms 



+ r^ 4- (i— 3) terms}. 



p^ will occur in each place; (w — 1) (/i — 2) (n — i + 1) 



times, or altogether i{n — 1) (n — 9) (n — i + 1) times; 



while 2/?^ will occur in each place 9.{n — 9){n — 3) 



{n — i + 1) times; but the number of places is — 2~"' ^^"^^ 

 altogether we have 



^'^ in{n—\) 



1 1 n — i 



= 2 'P^ 



inn — 1 



1 n 



and thus we have this general proposition, that the square of 

 the expected error is proportional to the number of points not 

 examined, divided by the number of those actually compared 

 with the readers; and hence, in the extreme case, when every 

 division has been examined, the chance of error is reduced to 

 2ero. 



I shall now seek the value of the chance given by the me- 

 thod of reading which is above described. In this method, the 

 readings of particular micrometers are multiplied by certain 

 arbitrary numbers : the solution of the general question pre- 

 sents some interesting points. 



Let a, /S, 7, 3, &c. be the arbitrary coefficients of the read- 

 ings; an error will take the form 



ccp + ^q + yr + "^S + &C. 



and hence the squares of the errors will give the sum 



( — ) S. [a p + (i q + yr + ^ s + etc.y = 



( — ) S.{a2p2^2 a(ipq'{'2aypr + etc. 



+ (iq^ + 2fiyq7'+ etc. 



+ yr^ + etc. } 



In this sum, supposing the arrangement of a, /3, 7, &c. un- 

 changed, the errors p, q, r must be permuted in every pos- 



