On the Multiplication of Partial Differential Operators, 49 



In like manner, the statement concerning the commutahle 

 operators (/>* and y]r*, made in a footnote, should have been 



the theorem in the text easily enables us to see that 



= F(a, b + ax, c+2bx+ax*, d+3c*+3&a? a +aa? 8 , . . .)» 



which, as remarked by Mr. G. De Morgan and others at the Mathematical 

 Society, may be regarded as a transformation and generalization of the fun- 

 damental law of development in Arbogast's theory, sometimes called by the 

 name of Arbogast's first or unreduced method. The identification with the 

 method in question merely requires the supposition that F(a, b,c,d, . . . ) 

 should become a function exclusively of a single one of the letters within 

 the parentheses ; but of course we must write the left-hand side of the 

 equation under the unreduced form e*Q*F(a, b,c,d . . .). 



The proof, as noticed by my distinguished mathematical friend Mr. Sa- 

 muel Roberts, of the generalized theorem is virtually implied in the method 

 by which I established long ago the partial differential equations of the inva 

 riants to any system of forms ; i. e. it follows from the observation that the 

 effect upon F of altering x into x-\-8x and leaving a, b, c unaltered is the 

 same as the effect of leaving x unaltered and altering a, b,c, d, . . . into 



b+abx, c+2bSx, d+3c8x, ... . 

 Consequently 



£-**. S-cw*,.-.. 



and therefore, by Maclaurin's theorem, 



F(a, b+ax, c+bx-\-2ax 2 , . . . ) = e*P*F{a, b,c,...). 



In memory of the author who appears to have been the first to employ 

 the form which I have called a Protractant, it may hereafter with propriety 

 be termed also an Arbogastiant. 



The equivalence of e*^ with [e (e ~~ ^\]*, when (jb represents an Ar- 

 bogastiant, or rather aform slightly more general, hadbeen previously stated, 

 but in a much less commodious manner, by Professor Cay ley in a memoir 

 contained in Crelle's Journal, vol. xlvii. p. 110. An inspection of this me- 

 moir will satisfy the reader how inarticulate was the language of algebra 

 at the not remote epoch when Mr. Cayley's paper was written, and how, 

 for want of a distinctive abstract symbol of operativeness, she strove like 

 one lame of speech and tongue-tied, to give intelligible expression to her 

 ideas. 



With the star sign the restraining ligament has been cut, and henceforth 

 algebra, as far as yet developed, may revel in unbounded freedom of utter- 

 ance. The rise of this star above the mathematical horizon marks one of 

 the epochs of algebra. It is worth remarking how already it is be- 

 ginning in its turn to assume the attributes of quantity (vide the con- 

 cluding footnote of this paper, where it is used as a divisor) ; so that appa- 

 rently it is destined to run the same course as Newton's fluxional symbol, 

 which is, and of fatal necessity must have been, superseded by the let- 

 tered symbols of Leibnitz, which have now long ago, to all intents and pur- 

 poses, become converted into a new species of algebraical quantity. As 

 Phil. Mag. S. 4. Vol. 33. No. 220. Jan. 1867. E 



