TRANSACTIONS OF THE SECTIONS. 



mtist be independent of t. Similarly, the limits of the variable s may be functions of 

 all the variables x, y, z. . . . which precede it, and so on up to the variable x whose 

 limits are independent of all the variables. 



The result of these operations will be a definite expression, which, for brevity, may 

 be represented by D V. It is required to find the derived function of QV with 

 respect to a parameter a, which may at the same time be contained in V and in all 

 the limits of the variables. 



For brevity we employ the symbols 



dx 

 dx 



dx 

 dx 



as defined by the equations 



Idx 

 da 



dx 



L 



dy 

 dx 



Vi 



d JL, 

 dx 



dx 



_ iy^ 



dx 



dx ' 



Vi 



dx 



d - ! '-, 

 dx' 



Vi 



dy 

 dx' 



y* 



dx 

 dx 



dy 

 dx 



dx, 

 dx' 



_ ^ 



dx' 



dy 

 dx 



i& & c 



dx ' 



This will clearly give rise to no error, since the expressions 



dx dx dy 

 dx dx dx 



have, in themselves, no meaning whatever. 



This being admitted, the general rule at which I have arrived may be thus 

 enunciated. 



In order to differentiate any definite expression with respect to a variable para- 

 meter x, neglect, in the first place, all the symbols of substitution and take the 

 derived function of the remaining integral, treating the variables to which the above 

 substitutions refer, as if each were a function of all the preceding ones and of x ; in 

 each term of the result restore the symbols of substitution before withdrawn. From 

 the symbolical form which is thus obtained the true expression of the required de- 

 rived function may be immediately obtained. 



In order to illustrate the application of this rule, let U9 seek the derived func- 

 tion, with respect to x, of the expression 



DV 





Vdy. 



Neglecting, in the first place, the symbols of substitution, we must differentiate the 

 integral 



j 



y-2 



Ydy; 



its derived function with respect to x is, according to the formula of M. Sarrus, 



<tyA_ 





'%}). 



\dxi 



Now since * and x. are the variables to which the neglected symbols of substitu- 

 tion refer, we must, on differentiating, proceed as if each of the three variables 

 x, x, z were a function of those which precede it; that is to say, we must consider 



