ON LINGUISTIC RESEMBLANCES. 



71 



Vk Fl-i.(8,)» .a 



-(8,;, 



-cra,^ 



*-i -(;; :l)l _. 



(8) 



"2. Equation (4 I2 ) has the copula ==, because our knowledge that there arc no coinci- 

 dences at (iSt'u. .Iv'i) slightly increases the probability q of one at i&' i+ iO+i- But if 2 

 be a pretty continuous function of - (/:, 0), - (/;, 1), &c, (below), any knowledges that 

 affect q only through them, affect q and (n = prob. number coincidences at (9iv'i, • ■ H',:) ) 

 almost proportionally. The knowledge of at least h such coincidences, all specified, 

 would have changed simultaneously 



n = ('30, to n = h +('3,)__ ; ,., and q = (l,) o to q = (l a )_»; writing ('3,), %, &c, 

 for what (8j), p k , &C, become when 6' is changed to i. Now n = 0, 



.'. 3 =f (!,)• ( 9 ) 



C'8.) 



" Or thus. When we know'that at least h [or just Te\ specified words in (|fr'„ . . &',.) 

 afford coincidences, let ? /.: [or ^ A] be the probability of a certain event, or the mean 

 value of a certain number, and let - (7c, I) [or 0'] be the probability that (JX\, . . M\) afford 

 just (k+l) coincidences. We have 



» (*> = ('?/— 'B ]-0-7,:, 



, & s • (/.-, 0). <» 7,: I - (A, 1). ^ (As + 1) + . ., .-. <p {Jc + 1) = &c, 



.-. 4>k = 



<p h 

 00 



01 



"00. 10 



01 02 

 10 11 



9 (ft +S 



101 02 08 

 10 II 12 

 20 21 



• 00.. 30 



+ .., 



" 00.. 20 



where hi stands for «• (&"+ 7*, ?)• Using (Z -f- 1) terms we nearly allow for the others^ 

 if <P (Jfc -| 7,)=' $ (k + Z„ + 1) =F • ., by changing • (k + h, l—l>) to 1— • (* | h, 0)— . .— 

 w a. | / /? l o — ],, — 1). When the 'event' is a coincidence at JitYi ./> 9 (& + = (is)-*-u ancl 

 (9) is reproduced by averaging the sub-indices by ('7 U )* and the equation 



.X X" 



(I 

 1 1 



2 



1! 2! 



x 



n 



(w— 1)! 



e"— 1 



- (n— 2)1 



= — (-),;+!)!• ^ io) 



n - tf 

 The error of (4 28 ) is of the same order ; as is seen by writing (1 8 ) for x, and b> for y, in 

 n— z y — e ***&-> = 6-"» ,{l+ S + 'W(l— f . x— . .).e-^ (11) 



" The error of (7,,), being of the second order when h is small, will be neglected. 

 "Thus, while (1), (2 ia 3), (5,,), (6 2S ) are already accurate; (4 13 ), (5 U ),(6 S1 ), (7 U ), become 

 as follows, if we retain but one order of infinitesimals, 



(4,) = (i—n.,) ).. (1— (l y ) )=(1— (1.) ) 6 '=(4 8 ) , 



* "Or rather, in my notation of Amir. Acad., 1864, by <M' = (*) C 7 m)_7 C -a == I ('8,)'.e _( ''' &0." 



