Remarks on Professor Wallace's Reply to B. 301 



had been made by B. and of course it is not necessary to dis- 

 cuss the subject. It may not however be inexpedient to 

 state that every result given by Professor Wallace can easily 

 be obtained from the binomial theorem ; and mathematicians 

 have usually developed such quantities by means of that theo- 

 rem. 



Thus having the identical equation 



A X tn\x 



by substituting the developments of 



^ niAx 



(1-|-W2:)"and (l-i-nz) 



wiven by the binomial theorem, we obtain the equation 



y Til min — n 2 mm — nm— 2n 3 x*^^ 



mAx mkx mkx — n 2 



putting now m=l, s=l, n=0, it becomes like Professor 

 Wallace's formula, Vol. VII. page 283, 



11 1 Aa: Ax k^x^ A^x^ „ 



1 I 



By putting the series in the first member l+y-f r-^+Sic. 



A3^ Ax k^x^ 



= e, it becomes e =1-}-— +-r-^+S2;c. 



A 



and by making e =a, or A^/.c, we get 



■'^' xl.a x'l.a^ 



a =i+-j-4— j;^+&c. 



being the same as Professor Wallace has given in Vol. VIJ. 

 page 284. In like manner, his expression of log. {l-\-x) 

 &c. may be lound, but it is unneciessary here to repeat these 

 calculations. It may however, be proper to observe, that Pro- 

 fessor Wallace is not correct in his assertion that the multi- 



