37, 38] DEFINITE INTEGRALS. 73 



f a function /<&, and 

 ( Ydx Xdy) const. 



makes it the differential of a function /<&, and 

 M 



I 



is a second integral of the equations (i). X, Y and =-- must of 





course be expressed in terms of x, y, \. 



Consequently if we have the system of differential equations 

 dx : dy : dz = X : Y : Z, 



and we have found one integral X = const, together with a mul- 

 tiplier satisfying the partial differential equation 



\J 



ox dy oz 



then the expression 



M 

 8X 

 dz 



is an integrating factor, or last multiplier* for the equation 



When X, Y, Z satisfy the solenoidal condition, the last multi- 

 plier is 



J_ 

 8X* 



dz 



This result will be used in 103. 



38. Variation of a Multiple Integral. In illustrating 

 the method of the Calculus of Variations we have found the varia- 

 tion of a single integral, and in the example taken the functions 

 varied were the coordinates x, y, z, of points of a curve, the variable 

 of integration being t. We may in a similar manner vary a 

 surface or volume integral, by causing the functions entering into 

 the integrand to change their forms by an infinitesimal trans- 

 formation, while the variables of integration are unchanged. For 

 instance let 



Vdxdydz 



* Jacobi, Vorlesungen tiler Dynamik, p. 78. 



