70 



MATHEMATICAL NOTE 



pendent languages for which (a!, a", b', b", rn, r,) are the same, the square of the difference 

 between the assumed and the actual number of coincidences averages least when the: 

 assumed number is (3). 



"From (lis),* tne probability of wo coincidence at M'i is 1 tt^t/i, so at W-z, &c. ; .-. 



prob. of no coincidence between (W ,,..l.tV) and (W\,.-W'y) 



-a-: 



-1/ I/' m r 



(4) 



" To know that (iit'iv-MV) afford at least k coincidences, is to know that either (W<> 



Wo ,, ,, ) or (Wo-2, u • • M'o 2 , /,:) or &c, do so, — (o :i ,, 



0; 



i,k) 



being some k of the indices 



(1, . . b') ; in which case k roots and k meanings, which were reckoned in (a', a.", //, b", m, r) , 

 are already disposed of, so that if either one selection (3K-V; ;<]1 , . . Wo jo ^) were known to afford 

 k coincidences with (M"i,- • iX" !,")■> (4 18 ) would be changed to 

 prob. of no (k+1) 'th coincidence, when those k are known i 



W- 



(b'—k) [l/'—lc) (m—k) (r—k) 



= (4:,)-,; 



(5) 



(a'—kf (a"—kf 



.-. it might seem that (4 3 )_ t was the probability of no (k+1) 'th coincidence whenever 

 we knew there were some Jc, even though ignorant which k. But in reality, though ~~ 

 of the different cases under any one selection (UfcV^. ,,. . %i f o ,,„,,) give each but k coinci- 

 dences ; yet, when thus ignorant, we must consider successively all the (* = , .. . . . , ,-, ) 

 possible selections made by variously choosing and assigning to each other k roots and k 

 meanings of the r and m common to JL' and 3L" ; and we shall thus obtain each distinct 

 case of just k coincidences but once, each of just (k+1) coincidences -y— times over, each 

 of just (k+2) coincidences — ' \" + times, &c. ; so that 



prob. just k coincidences 



(4.)- ^(4,) 

 —» —ft 



prob. just k -f 



Jfe+I 



I 



prob just (k [-1 ) -|- 



and not 



prob. just k coincidences 



prob. just k + prob. just (k -|- 1 ) + ..=■ 1 



" Let p k — prob. of at least k coincidences, .\ p k — p,, , , 



(6) 



prob. just k coincidences, 



between the sub-languages (Wi, ■ ■ 1&V) and ($£ 



a' ." a" ! 2 



. KV)- Of these sub-languages there 

 are , _ h//) ; , y/J possible pairs, formed by variously taking and as- 



signing to one another b' [and If] roots and as many meanings from those in?L' [and 2,"]. 



To each of the * selections, there correspond x'- k a"-, c = j^zjPy nb'— k)i x X a"—b")i*{b"—k)i 

 different sub-language-pairs with k or more coincidences each; .-.nearly (4,)_ /c . -*'„,/'_,, 

 with just k coincidences; and all the x x nearly (4 ;; )__ fc 



l '- k x "-k pairs thus found are distinct ; 





p" (3i), . . (3i)i-*- c 



A'/l" 

 -(8Jui 



1 m * 



. e 



(?) 



* See note on page 69. 



