212 MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 



tion as those of number. But it may not be amiss to say a few words on the ef- 

 fect of considering them in this light. Many theorems in the differential calculus, 

 and that of finite differences, it was found might be conveniently expressed by 

 separating the S3rinbols of operation from those of quantity, and treating the for- 

 mer like ordinary algebraic symbols. Such is Lagrange's elegant theorem, the 

 first expressed in this manner, that 



a" u^ = (e^dx _l;» u^; 



or the theorem of Leibnitz, with many others. For a long time these were treated 

 as mere analogies, and few seemed willing to trust themselves to a method, the 

 principles of which did not appear to be very sound. Sir John Hebschel was 

 the person in this country who made the freest use of the method, chiefly, how- 

 ever, in finite differences. In France, Servois was, I believe, the only mathema- 

 tician who attempted to explain its principles, though Brisson and Cauchy some- 

 times employed and extended its application : and it was in pursuing this inves- 

 tigation that he was led to separate functions into distributive and commuta- 

 tive, which he perceived to be the properties which were the foundation of the 

 method of the separation of the symbols, as it is called. This view, which, so 

 far as it goes, coincides with that which it is the object of this paper to develope, 

 at once fixes the principles of the method on a firm and secure basis. For, as 

 these various operations are all subject to common laws of combination, what- 

 ever is proved to be true by means only of these laws, is necessarily equally true 

 of all the operations. To this I may add, that when two distributive and com- 

 mutative operations are such that the one does not act on the other, their com- 

 binations will be subject to the same laws as when they are taken separately ; 

 but when they are not independent, and one acts on another, this wiU no longer 

 be true. Hence arises the increased difficulty of solving linear differential equa- 

 tions with variable coefiicients ; but for more detailed remarks on this, as well 

 as for examples of a more extended use of the method of the separation of 

 symbols than has hitherto been made, I refer to the Cambridge Mathematical 

 Journal, Nos. 1, 2, and 3. 



As we found geometrical operations which were subject to the laws of circu- 

 lating operations, so there is a geometrical operation which is subject to the laws 

 of distributive and permutative operations, and therefore may be represented by 

 the same symbols. This is transference to a distance measm-ed in a straight line. 

 Thus if a; represent a point, Une, or any geometrical figure, a (ai) will represent the 

 transference of this point or line ; and it will be seen at once that 



a(x) + aQ/) = a{x+y)', 



or the operation a is distributive. What, then, will the compound operation 

 f) (a (<2?)) represent ? If iz; represent a point, a (a;), which is the transference of a 



