. The Astrophysical journal. ORMER which gives / {p,C) = In (pi-D) I cos 6(ie = o, i Hence Hr = o, H^ = o. and H. = , where ^:o= - 2Trc cos w In ^ —^ t^ij / N {z-h+v {z-hY+p;){z+v z+pt) It will be interesting to obtain a verification. Let h and kbe infinitesimal, and Pz — p+k, Pi = p, (0=0. If we put we have z-h+V(z-h)+{p+ky= L w u {z-\-u)u 2{z-\-u)u^ {z-\-u)u^ z?\ k^-\- 2{z-\-u)u^ Hence z-h+v {z-hy+{p+ky p r,p,, u-\-uz+z,_ In j==^ =7—, ^ k-\-~Jlk -.—; r—— k^-f- .... z—h+V {z—hy-\-p^ {z-\-u)u m3 {z+u)ui This gives z+\ z-\-(p+ky (z-\-u)u (z+u)u^ and finally H,= -27rC ^hk-^ . . M3 RESEARCHES
. The Astrophysical journal. ORMER which gives / {p,C) = In (pi-D) I cos 6(ie = o, i Hence Hr = o, H^ = o. and H. = , where ^:o= - 2Trc cos w In ^ —^ t^ij / N {z-h+v {z-hY+p;){z+v z+pt) It will be interesting to obtain a verification. Let h and kbe infinitesimal, and Pz — p+k, Pi = p, (0=0. If we put we have z-h+V(z-h)+{p+ky= L w u {z-\-u)u 2{z-\-u)u^ {z-\-u)u^ z?\ k^-\- 2{z-\-u)u^ Hence z-h+v {z-hy+{p+ky p r,p,, u-\-uz+z,_ In j==^ =7—, ^ k-\-~Jlk -.—; r—— k^-f- .... z—h+V {z—hy-\-p^ {z-\-u)u m3 {z+u)ui This gives z+\ z-\-(p+ky (z-\-u)u (z+u)u^ and finally H,= -27rC ^hk-^ . . M3 RESEARCHES ON SOLAR VORTICES 369 where c= epv 3X10 If we suppose e and v to be positive, Ehkv electrostatic unitswill pass through the cross-section of the current in one second,which corresponds to a current of ._ ehkv amperes. If we introduce this value, n~= t ——+ .... 10 u^ and this first term represents exactly the magnetic action in thepoint (o,o,s) of the circular current whose section is the infinitesimal. product hk, and where the direction of the current is that ofincreasing (f) (see Fig. 12). VI. SPECIAL CASE IN ^\^^ICH THE THICKNESS OF THE WHIRL IS VERYSMALL COMPARED WITH ITS DIAMETER We will now consider the special case in which the thicknessof the whirl is very small compared with its diameter. Then forpoints {x,y,z) not very near the whirl, we can put jXp,h)-f(p,o) = hf(p,o),gip,h)-g(p,o) = hg{p,o) , 370 CARL STORMER where /(p,o) and g{p,o) are the partial derivatives -. and -| forth.} value f=o. s S Xow .// VN r 2-^ cose dO j t/o (7 o ^^ and _ r COS ^ o where Dl=R-2Rp cos O+p+z^. This must be substituted in the formulae HR=-2ch cos w [/(p2, o)-/(pi, o)] \ fl,f,= 2cA sin to [f{p2, o)-/(pi, o)] > (ii) H:,= 2ch COS w [g(p„ o)-g(pi, O)] / The same result can also be obtained by starting from formula(6), and considering h as infinitesimal. Then rp2 r^ HR= — 2ch COS (D I I dOdp and an
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