Mr. Sang on the Product f)ftxoo Monome Functions. 263 



a primary and twice with the primary z, may be easily ex- 

 pressed by -j^ ^u\ wliile that indicated by the second can be 

 denoted by 5im(j^x)^ or rU ^ ar. 



The series of quantities 



"' i^"» 2."' 3;."' 4^"' &c- 

 being similarly derived, each from that which precedes, the 

 derivatives of any one of them may be found by increasing its 

 numerical subponent. Thus the second derivative of _ %i is 

 c u. and in general {%)■=■,,. ^ u. 



Again, each member of the series being the derivative of 

 that which precedes, may in turn be regarded as the primi- 

 tive of that which follows, so that the second primitive of 5^" 

 is ~u. The operation of integrating or going back in the 

 series, may therefore be conveniently expressed by prefixing 

 the negative sign to the subponent, so that _.u means the 

 primitive of the function ?/, the primary variable being z\ «, 

 the second primitive, and so on. We have thus in general 



and also ,u = u. In this manner an advantage accrues to 



the notation similar to that which follows from the employ- 

 ment of negative exponents. 

 The quantities 



&c., _^.}h _2,"» _u"> «> i^Wj 2/^' Zz^^ &c. 



thus form an uninterrupted series, any one of which may be 

 regarded as the original from which the others were deduced ; 

 thus, if J jz = t), the same quantities might have been written 



&c- -i^p-> _3,^» -i^i -\~yi '^j x^i Q,y» &c. 



This resemblance of primitive and derivative functions is 

 not the mere consequence of an artificial notation ; they possess 

 ))roperties in common, one of the most remarkable, and at the 

 same time most easily investigated of which, I proceed to de- 

 monstrate. 



It has been shown (Bossut, Traitc de Calcul, page 103; 



Lacroix, Diff. Cal. Transl. p. 1 1 ) that ^^ =u~ +v^ 



' '^ f- ' dz (Is ^ dz 



or (Lagrange, Thcorie des Fonctions, page 26) that {uv)' = 

 «'u + u-J, which theorem, in our notation, is expressed by 



Takinjr 



