172 Mr. A. Cayley on the Theory of the 



greatest power is to be obtained by coucentrating the greatest 

 amount of light on the highest degree of artificial heat. The 

 conibiuutiou of the two may perhaps have important practical 

 applications. The chemist may possibly produce new results by 

 adding to the highest resources of artificial heat the powerful 

 agency of concentrated light. 



The subject is unfinished, and it is my intention to resume it 

 on some future occasion. 



XXVIII. On the Theory of the Analytical Forins called Trees. 

 By A. Cayley, Esq."^ 



A SYMBOL such as AB, + BB^+ .., where A, B, &c. con- 

 tain the variables x, y, &c. in respect to which the 

 difi'erentiatious are to be performed, partakes of the natures of 

 an operand and an operator, and may be therefore called an Ope- 

 randator. Let P, Q, K . . be any operandators, and let U be a 

 symbol of the same kind, or to fix the ideas, a mere operand ; 

 PU denotes the result of the operation P performed on U, and 

 QPU denotes the result of the operation Q performed on PU; 

 and generally in such combinations of symbols, each operation 

 is considered as affecting the operand denoted by means of all 

 the symbols on the right of the operation in question. Now con- 

 sidering the expression QPU, it is easy to see that we may write 



QPU = (QxP)U + (QP)U, 

 where on the right-hand side (Q x P) and (QP) signify as fol- 

 lows : viz. Q X P denotes the mere algebraical product of Q and 

 P, while QP (consistently with the general notation as before 

 explained) denotes the result of the operation Q performed upon 

 P as operand; and the two parts (QxP)U and (QP)U denote 

 respectively the results of the operations (Q x P) and (QP) per- 

 formed each of them upon U as operand. It is proper to remark 

 that (Q X P) and (P x Q) have precisely the same meaning, and 

 the symbol may be written in either form indifferently. But with- 

 out a more convenient notation, it would be diflicult to find the 

 corresponding expressions for RQPU, &c. This, however, can be 

 at once effected by means of the analytical forms called trees 

 (see figs. 1, 2, 3), which contain all the trees which can be formed 

 with one branch, two branches, and three branches respectively. 

 The inspection of these figures will at once show what is meant 

 by the term in question, and by the terms root, branches (which 

 may be either main branches, intermediate branches, or free 

 branches), and knots (which may be either the root itself, or 

 proper knots, or the extremities of the free branches). To apply 

 * Communicated by the Author. 



