Mr. A. Cayley on the Analytical Forms called Trees. 377 



symbols ; viz. if A, B, C, J), &c. are symbols capable of succes- 

 sive binary combinations, but do not satisfy the associative law, 

 what is the number of the different significations of the ambi- 

 guous expressions ABC, ABCD, ABODE, &c. respectively? 

 For instance, AB has only one meaning; ABC may mean either 

 A . BC or AB . C. In like manner ABCD may mean A(B.CD), 

 or AB . CD, or (AB . C)D, or (A . BC)D, or A(BC . D) ; the 

 numbers 1, 2, 5 being those of the trees in the last three figures 

 respectively ; and similarly for any greater number of symbols. 

 Let 0?M be the required value corresponding to the number m ; 

 then we may in any manner whatever separate the number m 

 into two parts m', m", and then combining inter se the m' knots 

 (or symbols) and the m" knots (or symbols) respectively, ulti- 

 mately combine the two combinations ; hence a part of cfini is 

 <}>7}i'. ipm". The assumed definition of <^in docs not apply to the 

 case ?« = ! ; but if we write 01 = 1, then the foregoing conside- 

 ration shows that we have 



0?H= (f)l(f){m — l) 



+ (p2(f>{m-2) 



+ cf>{m-l)^l); 



from which it is easy to calculate 



01=l,<^2 = l,(/>3 = 2,(/>4=5, (/)5 = U, 06 = 42, 07 = 132, &c. 



But to obtain the law, consider the generating function 



M = 01 + .r02 + «203 + &c. ; 

 we have 



M2 = 0l<^l + ,^.(c^l<^3 + <^30]) + a^2(0l 03 + 02 02 + 0301)+ &c., 

 which is 



= 02 + 0703 + *2<^4 + &c. J 



and we have therefore 



and consequently 



,^^ 1- \/l-4x 

 2x 

 But 



^/Y=^=^\-\4.x+ k^ {4.XY- ^ 7y ~^ (4.r)H &c. 



= \-2x- 2.r2 -4a^ - lOa;" + &c., 

 and therefore 



1— \/l — 4a? , - « o ^ ', ,, 



u = = 1 + 1 A- + 2x^ + 5*'' + &c., 



iX 



