528 Prof. V. Karapetoff on Divergence and 



This formula and others giving still closer values of the 

 roots are especially useful when m is small in comparison 

 with n. The formula} given by McMahon * and others for 

 the roots of J n (z), Y n (^) are limited in their application, and 

 cannot be employed in finding the earlier roots of J B (^), &c, 

 when n is large. Similar expressions are found for the 

 roots of the two Neumann functions. 



-LXIV. Divergence and Curl in a Vector Field in terms of 

 curvature and tortuosity. By V. Karapetoff i\ 



r pHE usual expressions for divergence and curl, in Car- 

 JL tesian coordinates, convey a somewhat indirect picture 

 of the physical nature of these quantities, particularly to the 

 beginner. The expressions given below are based directly 

 upon the characteristics of a field near a given point, namely, 

 the geometric shape of the lines of force, and the rate of 

 change of the field density along a line of force and in the 

 directions perpendicular to it. No fixed origin of coordinates 

 or projections with respect to this origin are introduced into 

 the discussion. For the sake of simplicity, and as a sort of 

 introduction to the method, a two-dimensional field is 

 considered first. 



1. Two-Dimensional Field. 



Let AB, A'B', A"B" (fig. 1) represent lines of force of a 

 vector field in the vicinity of a point 0. Let these lines lie 

 in the plane of the paper, and let all the lines of the field 

 lie in parallel planes, so that the field may be called two- 

 dimensional. The field at and near point is determined 

 by the following quantities and characteristics : — 



(1) The magnitude and the direction of the flux density 



DatO. 



(2) The scalar rate of change of flux density, dD/^s, 



along the line of force AB. 



(3) The scalar rate of change of flux density, dD/d?z, 



along the normal to AB. 



(4) The radius of curvature, P0=p, of AB. 



(5) The radius of spread, Q0 = R, at 0. 



* Annals of Mathematics, vol. ix. (1895) ; Proc. Physical Society, 

 vol. xxiii. part iii. (1911). 



t Communicated by the Author. 



