TRANSACTIONS OF THE SECTIONS. 3 



u being considered as a function of x. This theorem implicitly involves the fol- 

 lowing, which was commuuicatd to the author by Chief Justice Cockle in a letter 

 under date Brisbane, Queensland, Australia, October 17-18, 18G5. 

 The differential equation for 



y"-x!f-l=0 (1) 



is 



d 



d-i' 



r — ""i" = p' _-4-!L*— i"i' 



L d.vj Lm dx n J 



m~(n — 7-)x- 



dx I 



^u, . . (2) 



■,■1)1 

 where u=y . 



On this result Chief Justice Cockle, in the same letter, remarks, " The con- 

 ditions under which (2) is immediately depressible by one or two orders are, that 

 one or both of the relations 



m+<T». ^^ (3^4) 



n—r 



should be satisfied ; a and /3 being integers comprised between the limits and 

 « — 1 both inclusive (zero I treat as an integer), and p being an integer comprised 

 between 1 and r both inclusive, and o- being an integer comprised between 1 and 

 n~r both inclusive. If both conditions aro satisfied, but a—^ then (2) is imme- 

 diately depressible by one order. If (only) one condition be satisfied, the same 

 thing ' holds. If botL be satisfied, and «- ^ does not vanish, (2) is depressible by 

 two orders." 



Remarls on Boole's Mathematical Anah/sis of Logic. 

 Bi/ the Kev. Prof. Haeley, F.R.S. 



The author's remarks were arranged under three heads. First, he gave some 

 accoimt of Boole's system as developed in his ' Mathematical Analysis of Logic,' 

 and more elaborately in his great work on 'The Laws of Thought.' Next, he 

 noticed some remarkable anticipations of Boole's views. iVnd in the concluding 

 portion of his paper he pointed out the direction in which he believed Boole's 

 method might be usefuly extended. 



1. He contended that in Boole's system the fundamental laws of thought are 

 deduced, not, as has sometimes been represented, from the science of number, but 

 from the nature of the subject itself. Those laws are indeed expressed by the aid 

 of algebraical symbols, but the several forms of expression are detemiined on other 

 grounds than those which fix the iiiles of arithmetic, or more generally of algebra ; 

 they are determined in fact by a consideration of those intellectual operations 

 which are implied in the strict use of language as an instrument of reasoning. In 

 algebra letters of the alphabet are used to represent numbers, and signs connecting 

 those letters represent either the fundamental operations of addition, subti'action, 

 &c., or, as in the case of the sign of equality, a relation among the numbers them- 

 selves. In Boole's calculus of logic literal symbols (.?•, ?/, &:c.) represent things as 

 Subjects of the faculty of conception, and other symbols (+, — , &c.) are used to 

 represent the operations of that faculty, the laws of the latter being the expressed 

 'laws of the operations signified. For instance, x-\-i/ stands in this system for the 

 aggregation of the classes or collections of things represented by .r and y, and x — y 

 for what remains when from the class or collection .r the class or collection y is 

 withdrawn ; .r X »/, or more simply .vi/, represents the whole of that class of things 

 to which the names or qualities representeed by .r and ;/ are together applicable ; 

 and x = )/ expresses the identity of the classes x and y. The canonical forms of thi* 

 Aristotelian syllogism are really symbolical ; but the symbols are less perfect of 

 their kind than those which are employed in this .system. By adopting algebraical 

 signs of operation, as well as literal symbols and the mathematical sig-u of equality, 

 Boole was enabled to give a complete expression to the fundamental laws of rea- 

 soning, and to construct a logical method more self-consistent and comprohonsi\e 

 than any hitherto proposed. His calculus does not involve a reduction of the ideas 

 of logic under the dominion of number ; but it rests on a fact which its inventor 



1* 



