35, 36] 



DEFINITE INTEGRALS. 



36. Representation of Solenoidal Vector. Multiplier. 



We have obtained in 32 a means of representing a lamellar 

 vector- function by means of the level surfaces of its ' potential 

 function. By means of Jacobi's multiplier we may find a some- 

 what similar representation for a solenoidal vector. If we suppose 

 the curves drawn whose tangent at every point has the direction 

 of the vector function R whose components are X, Y, Z, since the 

 direction cosines of the tangent are 



dx dy dz 



~ds' ds' ds' 



the curve is defined by the differential equations 

 (I) dx:dy:dz = X:Y:Z. 



The integrals of these equations will each contain an arbitrary 

 constant. Let us suppose that an integral is of the form 



X (x, y, z) = const. 



Then we must have 



ax 7 ax 7 ax 7 -. 



^- dx + ~- dy + dz = 0, 

 dx dy ' dz 



and since dx, dy, dz are proportional to X, Y, Z, 



(2) 



~ 



dx 



- -- \- 



dy 



= U. 



dz 



This partial differential equation may serve as a definition of an 

 integral of the system of differential equations (i). Geometrically 

 it shows that the vector R is perpendicular to the normal to the 

 surface \ = const., that is, is tangent to the surface. If //, = const, 

 is a second integral, then 



(3) 



o 



ox 



^ 



dy 



TT- 



dz 





and since R is tangent to a surface of each family X = const., 

 //, = const., the lines of the vector R are the intersections of the 

 surfaces X with the surfaces p. From (2) and (3), linear equations 

 in X, Y, Z, we may determine their ratios. We obtain 



(4) 



X:Y:Z= ~ 



