and Magnetic Field Constants. 207 



constant stated above, equal to the value of R produced by 

 the volume distribution at any point of that circle. 



5. By means of the results which have been obtained 

 above, and various dynamical considerations, we can evaluate 

 certain elementary Bessel function integrals, and obtain some 

 other results which seem of interest. I am aware of course 

 that these, and a large number of other definite integrals of 

 greater complexity, have been evaluated analytically. 



Taking the value given by (7) above for the axial force F, 

 at the point P, produced by a uniform disk of radius a, the 

 axis of which is OC, we see from (16), if coincide with 0, 

 so that r = 0, that the surface integral of normal force over 

 the two circular ends of a concentric disk-shaped surface of 

 radius b, tightly enclosing the disk, is 



87r-W>f J 1 (A.a)J 1 (X&)^. 

 Jo * 



But the mass (of electricity or matter, as the case may be) 

 enclosed by the surface specified is nraa 2 . Hence we have 



Sir 2 aab | Ji(Xa) J x (\b) ~ = 4tt . nraa\ . . (20) 



that is 



i 



1 

 J 1 (Xa).J 1 (X6)rfX = q y. (1>>a). . . . (21) 



o 

 It follows that, if b = a. 



2 b 



j; 



J\*(\a)d\ = i. . (22) 



Multiplying now (6) by 2irydy, and integrating with 

 respect to y from to a, we obtain the surface integral 

 of potential over the disk. It is 



2tt \ydy = Waa 2 \ J^Xa)^. . . . (23) 

 Jo Jo x 



If, to fix the ideas, we suppose the disk to be composed of 

 attracting matter, the exhaustion of potential energy involved 

 in increasing a by da is the expression just found multiplied 

 by da. Hence the total exhaustion of potential energy, 

 involved in building the disk up from density zero to 

 density a, is 



P ^girVatJ J! 2 (Xa)~ (24) 



