Analytical Forms called Trees. 173 



this to the questiou in hand, PU consists of a single term repre- 

 Fig. 1. Fig. 2. Fig. 3. 



1 



sented by fig. 1 {bis) ; QPU consists, as above, of two terms 

 represented by the two parts of fig. 2 {bis), viz. the first part 

 represents the term (Q x P)U, and the second part represents the 

 term (QP)U. And it is obvious that fig. 2 {bis) is at once formed 

 from the figure 1 {bis) by adding on a branch terminated by Q at 

 each of the knots of the single part of fig. 1 {bis) . In like manner 

 RQPU consists of six terms representedby the six partsof fig.3 (6w), 

 Fig. 1 {bis). Fig. 2 {bis). Fig. 3 [bis). 



V ru uuuuuu 



B Qf 



(J R Q 



Q P 



« p e 



and this figure is at once formed from fig. 2 {bis) by adding on 

 a branch terminated by R at each knot of each part of fig. 2 {bis). 

 It is hardly necessary to remark that the first part of fig. 3 {bis) 

 denotes what, in the notation first explained, would be denoted 

 by (R X Q X P)U, the second term what would in like manner be 

 denoted by (RQ x P)U, and so on, the last part being the term 

 which would be denoted by ((RQ)P)U ; viz. R operates upon 

 Q, giving the operandator RQ, which operates upon P, giving 

 the operandator (RQ)P, which finally operates upon U. 



The figures 1 {bis), 2 {bis), &c. contain the same trees as are 

 contained in the corresponding figures 1, 3, &c. ; only on account 

 of the different modes of filling up, trees are considered as so 

 many distinct trees in a figure of the second set which are con- 

 sidered as one and the same tree in the corresponding figure of 

 the first set. A difference in the number of trees first occurs in 

 the figures 3 and 3 {bis), the first of which contains only four, 

 while the latter contains six trees, viz. the first tree, tlie second, 

 tliird and fourth trees, the fifth tree and the sixth tree of fig. 3 {bis) 

 correspond respectively to the first tree, the second tree, the 



