. The Astrophysical journal. tionof p; and the point (a-,o,s) being outside the whirl, r^ is positiveover the entire field of integration. This field being also finite,the succession of the integrations is indifferent, so that we canwrite more simply 2 cos o) I I i . y. cos tf (3—4) Epv —— ampdi. 0^ r C r t/O t/pj t/O 3X10 I f ) rl t/o «ypi t/o On the other hand, as everything is symmetrical around theZ-axis, we get the components in an arbitrary point ix,y,z) bysubstituting R= Vx+y^ for x; hence 2 coso) i I I (z-C) Epv cos e o ijpx tJo in. r C c Uo tJpi Uot/O t/pj t/o 2 sin (0 I ( 1 (z-C) epv
. The Astrophysical journal. tionof p; and the point (a-,o,s) being outside the whirl, r^ is positiveover the entire field of integration. This field being also finite,the succession of the integrations is indifferent, so that we canwrite more simply 2 cos o) I I i . y. cos tf (3—4) Epv —— ampdi. 0^ r C r t/O t/pj t/O 3X10 I f ) rl t/o «ypi t/o On the other hand, as everything is symmetrical around theZ-axis, we get the components in an arbitrary point ix,y,z) bysubstituting R= Vx+y^ for x; hence 2 coso) i I I (z-C) Epv cos e o ijpx tJo in. r C c Uo tJpi Uot/O t/pj t/o 2 sin (0 I ( 1 (z-C) epv cos 6 . ^ ^^o^. M {Rcos_e-pl^ 3X10 I £>3 where D = R-2Rp cos ^+p^+(s-0= ZTi? is the component normal to the Z-axis, H^ the componentparallel to that axis, and H^ the component normal to Hr and H^.As regards the signs, Hr is positive in the direction from the RESEARCHES ON SOLAR VORTICES 357 Z-axis, H~ is positive in the direction of increasing z, and H,i, posi-tive in the direction of increasing 0 (see Fig. 7).. Fig. 7 III. PROPERTIES OF THE COMPONENTS OF THE MAGNETIC ACTIONEQUATIONS OF THE LINES OF MAGNETIC FORCE IN SPACE Let US, in the formula iorllR, H^, and H^, introduce the function2R $= 3X t/O t/Pi t/o (2) As the point {x,y,z) is outside the field of integration, and 8R 82 8^ 2 1 1 \ 8 i RcosO 5$ 5$ consequently Z)>o, we obtain the derivatives j^ and -j^ as follows: ^R 3X 8$_ 83 3X10 2 f C [s r/?cos dO Rcos6D d6 dpdl, dpdC But t/o t/O t/O cosOde i (p-Rcos6)dd Z)3 r^igp sin ?Jo ^ ede 358 CARL STORAIER Integrating by parts, the last integral in the right memberbecomes equal to the first and we get 8R Further 8 fT^cos^.^ ^C {z-0 cos OdO t/o t/o This gives us 8 2R C C C ^ (p-R cos 6) Epv SR 3X10° III D^ dddpd^ , Jz = -3x^0 ^i dmpdi, t/o «y/>. e/o Pi and we can therefore write cos w 8$^^- -^ Tz sin CO 8$ , _ cos w 8 j^~~~RrJRj From this it is easy to find the lines of magnetic force in spaceoutside the whirl. These are the integral curves of the system dx _
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