The Rev. C. Graves on the Calculus of Operations. 61 



the supplemental terms necessary to make the equation iden- 

 tically true, viz. 



(1 +^)-"= 1 -na!-\- ''^^Y^ -^c + (- 1)'"-'A„ x^-' 



+ (-l)'"{A„(l+a;')-' + A„_.(l+^)-2 + &c 



+ ^i^±il (1 4.a^)-»+2 4-m(l 4-^)-"+' + (H-a?)-«}a?»'. 



An being used to stand for 



«(w+l). . .. (« + OT — 2) 



1.2 {»»— 1) ' 



or its equal 



m{m-\-\) . . . . (w + w — 2)^ 



1.2 («-l) ' 



and so on for A„_i, A„_2, &c. 



D' 



And, if we write ^yj i" place of ^, we have the identity 



+ (_l)m-iA„D"-«-'»+»D'"»-» + (-l)'"{A„(D" + D')-' 

 D»-n-m+i ^. A„_,(D" + D')-2D"-»-'»+2_^ &c. . . . 



D"-"*-' + (D" + D')-"D"-'"}D''"; 



which continues to hold good when D' and D" are any two 

 symbols of commutative and distributive operation, just as 

 much as if they denoted quantities. 



Using D to denote the operation of taking the diflFereritial 

 coefficient with respect to Xy we have 



D [vu) = vDu + uDv ; 

 from which we derive the symbolical equation 



D = D" + D': 

 understanding that D" shall operate exclusively on m, and D 

 exclusively on v. The symbol D being thus absolutely equi- 

 valent to D" + D', we are entitled to operate on uv with the 

 symbol (D"4-D')~", or its expansion as given above, for the 

 purpose of effecting an w-fold integration. Thus we obtain 

 the complete formula of which we are in search : 



/" , „ dv . n(n+l)d^v . 



