Partial Differential Operators. 55 



where in the summation which gives the exponent of e, it is to 

 be understood that the natural order of P, Q, R, S in the nume- 

 rators is to be maintained. 



The formula from which this result has been thus simply de- 

 rived is of that fundamental character which entitles it to be 

 regarded as a master theorem, i. e. rather as a method than an 

 ordinary formula. As already observed, it essentially consists iu 

 the union of two known theorems ; but these combined and, as 

 it were, duly adjusted and focused, constitute together an instru- 

 ment of research as unlike either of its separate elements as a 

 telescope differs in its powers and functions from the pair of 

 lenses out of which it has been formed. And truly the formula 

 in question has a telescopic power in the sense of bringing the 

 remote results of calculation close up to the mental vision. 



The very first application made of this instrument, directed to 

 the algebraical firmament, has been rewarded by the discovery of 

 the beautifully simple and general expansion given in the text 

 above — a result in beauty and the feeling of wonder it awakens 

 fairly to be paralleled with the spectacle which gladdened the eyes 

 of Galileo when for the first time he pointed his telescope to the 

 skies. 



and write 



Sl — _+—+... 

 0')Qa* , mv, 



then I think there can be little doubt, or, at all events, there is a strong 

 presumption that the following ultra-general theorem holds good : — 



SlS 2 +JlS2*3+ • ; ' 



If we suppose all the P's inter se, all the Q's inter se, &c. to coincide, 

 the above expansion is certainly true, as may be inferred from the expan- 

 sion proved in the text, conjoined with the known theorem in factorials, 

 that ((*) + (i)+ . • . ) n is identical with what 0'+i+ . . . ) n becomes when, 

 in the development of the latter expression for any power of any element, 

 we substitute the corresponding factorial product, i. e. when in it for 

 *?, jv, ... we substitute («)?, (j)? . . . 



Even if on examination the above equation should turn out not to be 

 exact, the mere statement of it will be useful in indicating the kind of 

 expression that is applicable. According to the conservative maxim that 

 my universally lamented friend the late Mr. Buckle used to be fond of ci- 

 ting, in science even a wrong rule is preferable to anarchy and confusion. 



