RECEIPT PROGRESS IN THEORETICAL PHYSICS. 219 



which, if added, and dS be put for the element of surface, give 

 dL d3l dN !/-/.[. . , , , 1 ,. 



_ J_ rff _l. \dya , ^a , dwl] j^ ab dc. 

 2-n -'^'-^ r [ da ^ db ^ dc I 



In the second member of this last equation the factors in paren- 

 theses in each of the two terms, are known to be equal to zero; con- 

 sequently, 



give correctly the relations between the angular velocities of the 

 core of a vortex-filament, and the velocities in the fluid at points 

 outside of, but surrounding the core. 



The values of L, M, N, taken from equations (6), being substitut- 

 ed in equations (4), give certain results, the interp)'etation of which 

 will appear from the next paragraph. These results are 



Am(2 — «) + A^(2/ — ^) + A'''(" — c)=0, (9) 



indicative of a right angle with r; 



«'a A" + «'a A«' + «'a A^ = 0, ( 10 ) 



indicative of aright angle with the axis of the rotating filament; 



, dadbdc 



and 



qr.cosV={x-a)wl+ {y-b)wl^-{z-c)icl (12) 



Where q is the resultant of u'l, ivl, w|, and Y the angle which q 



makes with the radius-vector r. 



I 

 Now it is proven in works on electricity and magnetism* that "the 



vector-potential at a given j^oint, due to a magnetized particle placed 



at the origin of co-ordinates, is numerically equal to the magnetic 



moment of the particle, divided by the square of the radius vector 



to the point, and multiplied by the sine of the angle between the 



* Clerk Maxwell's "Electricity and Magnetism,"' Vol. ii., p. 28. 



