140 On a Generalization 0/ Taylor's Theorem, 



solely on the symbols of quantity a,h, c^., , and not on the 



operators -^-j -^j y- . . . ), we shall have 



^ _ (n ! n) ^ _ (O ! n ! II) 

 ^^^~ ^l^T"^ ^^'- 1.2.3" ' 



and so on. Hence the " noteworthy " symbolical equation 

 above written may be put under the hypersymbolical form 



(gAn!_i) 



a suggestive identity that may possibly call forth a sneer from 

 the mathematical cynic, but not from the thoughtful mathe- 

 matician, who, aware that algebra is in its essence a lan- 

 guage which it is the proper business of his art to fathom and 

 develop, is prompt to recognize every step in expression as a 

 gain in power. 



The theorem /i = e^^»/ having, as far as I am aware, been 

 first given by Dr. Salmon in a form, if not quite complete, still 

 sufficient for the immediate purpose to which it was to be ap- 

 plied, ought, I think, in justice to bear his name ; and I see no 

 reason why Salmon's Theorem in its totality should not be 

 expected in the future to bear new fruit in algebraical expan- 

 sions and other uses as important as have flowed from the one 

 familiar and simplest case of it, known as Taylor's Theorem. 

 Thus, ex. gr., for the special case where /i becomes a function 

 of one only of the quantities ^1, Ci, . . . the Salmonian theorem 

 reproduces Arbogast's celebrated one for expanding a rational 

 integral function by the method of derivations, but under a 

 greatly improved form of notation, and with the advantage of 

 a test of convergency supplied by the limit to the remainder 

 given in the text above. Who on a first casual reading could 

 have imagined that Arbogast's problem in the differential cal- 

 culus was virtually solved in an improved form in an article 

 treating " on the symmetrical functions of the differences of 

 the roots of an equation"? " Que diahle allait-il faire dans 

 cette galere laV^ may rise to the lips of many a reader on being 

 made acquainted with the fact*. 



Johns Hopkins University, Baltimore, 

 May 29, 1877. 



* Using Q to denote any rational integral function of x, Salmon's 



dQ d^Q 

 theorem is a theorem for expanding any function of Q, ^j da^f • ^° 



terms of ascending powers of x. 



