U2 PROFESSOR TAIT ON THE LAW OF FREQUENCY OF ERROR. 



where the inverse powers of n may be neglected if n is large. For (5) thus be- 

 comes, 



_ A (pn)P n + $ + mx (qn)V n + l- mx 



= A 



(pn + mx)P n + mx + $(qn- mx) 2" ~ mx + i 

 1 



pn + mx + i , N qn-mx+i 



\ pn) \ qn) 



Hence, logy — log A= — (pn + mx + %) log(l + — ) — (qn — mx + %) \os (l — --\ 



, . . ( mx m 2 x 2 m 3 x 3 „ ) , , . { mx m 2 x 2 m 3 x 3 „ ) 



_m 2 x 2 (I 1\ mV / 1 _1\ _m i q i (l 1 \ „ 



~~ 2w Vp ?/ 6n 2 U 2 9 2 j 12n 3 V/> 3 + <zV C ' 



mcc / 1 1\ mV / 1 1 \ 



-o" ( ) +-nr(-2+^t)+ &c - 



Jn \ p <? / 4rr \p~ q~ ) 



The first term of this expression is finite when mx is of the order n h ; and in 

 this case the other terms in the first line are infinitely small, being of the orders 

 n~ *, n~ \ &c. respectively. The latter remark applies to the second line of the 

 expression, which depends upon the ^ in the exponents. When mx is of an order 

 higher than »*, it is obvious from the undeveloped form that the expression 

 must be infinitely large, and negative. Hence, generally, we may neglect all but 

 the first term, and we have therefore 



m 2 x 2 



,/ = Ae~2pT n 



= A€~ fLx2 (6), 



which is the ordinary expression. 



9. This shows that, as is well known, the chance of a result differing x from 

 the most probable combination is, in this very simple case, represented by a number 

 proportional to e — ^ times that of the most probable event. But if we now con- 

 sider, not one but, any number of causes conspiring to produce the observed result, 

 we find that the law is still precisely the same in form, and this whether the most 

 probable event be the same as regards each cause or not. And it is this fact which 

 appears completely to justify the proposed method of regarding the question. 



10. For, if the various causes all tend to produce the same most probable 

 event, its probability will be, by (6), 



a = A t A 2 A 3 . . . . A„ (7) 



while that of a result, whose error is x, will be 



2/=.Ws .*-..* = ae-^ + ^ + • • • • + ^> 2 = a f - >to2 . . (8) 

 (where M=/m 1 + /jl 2 + /jl 3 + .... +/u„) 

 which is the same form as (6). 



