36, 37] DEFINITE INTEGRALS. 71 



The area of a right section of the four-sided tube thus 

 formed (Fig. 18) is / . 



sin (n K n^ sin (P x P M ) ' A / ds 



and multiplying this by the value 



offl, / +rf x A 



RdS = Px P M dn^ drip d\ dp, FIG. 18. 



which is constant for the whole tube. Consequently we obtain 

 a new proof of the fundamental property of a solenoidal vector, 

 for any tube may be divided up into infinitesimal tubes defined by 

 surfaces of the two families. 



37. Principle of the Last Multiplier. If we have two 

 functions M and N, each of which is a multiplier for the equations 

 (i), they must each satisfy the partial differential equation (7) so 

 that 



dy dz (dx dy dz 



Multiplying the second of these by M, the first by N, and 

 subtracting, 



dz dz 

 and dividing by M 2 , 



++= 



ox oy dz 



That is, the quotient of the two multipliers is an integral of the 

 differential equations (i). This result is of particular importance 

 when we have found one integral X = const, and any multiplier, for 

 we may then find a last multiplier, which shall give us at once 

 the remaining integral. By means of the integral equation 

 X (x, y, z) const, let us, by solving for one of the variables, say z, 

 express z as a function of x, y, X, 



z = z(x, y, X). 



