136 Prof. J. J. Sylvester on a Generalization 



We then showed how to calculate approximately the posi- 

 tion of centre of phase for the case of uniform spreading from 

 the end of a pipe both with and without flange. 



We then showed that there are an infinite number of pos- 

 sible forms of divergence, which can satisfy the known nu- 

 merical conditions on the assumption of a suitable law of 

 divergence of reflected elements, and came to the conclusion 

 that the reflected energy-elements diverge within the issuing 

 cone nearly according to the numerical law which would hold 

 if they were reflected as if issuing from an image of the source. 

 We see that the real explanation cannot involve any such 

 reflexion ; and we get a sort of idea of the direction in which 

 the explanation lies. 



[To be continued.] 



Erratum in Note 5 (p. 30, July). 

 The variable terms in tbe density for (i) and (ii) should read, 



r\ DC hr 7 / , X 

 (i) _ ^c^%k(yt^r), 

 r* V 



(ii) - ^^co8 k(vt-^r). 



IT V 



(The r^ in the denominators was omitted in copying.) 



XVIII. On a Generalization of Taylor's Theorem. 

 By J. J. Sylvester*. 



CONNECTED with the study of the Theory of the sym- 

 metrical functions of the differences of the roots of an 

 Algebraical Equation, a theorem presents itself in Dr. Salmon's 

 ' Lessons on Higher Algebra,' 3rd edition, p. 59, art. 63, only 

 partially indicated and insuificiently demonstrated there, which 

 on a closer inspection will be found to be well deserving of 

 notice as containing a true generalization of Taylor's theorem, 

 leading to a development of the same form, subject to a like 

 law of convergence, and easily demonstrable by the same 

 method as that theorem. 



Let / be any function whatever of a, bj c, . . . , and f^ the 

 same function of ai, b^, Ci, . . . , where 



ai^a, bi = b + ah, Cx = c+'2bh-\-ah'^, 

 d^ = d + Sch + 3bh^ + ah'', , , . 

 and let H represent the operator 



db dc a .d 



* Communicated by the Author. 



