218 



WISCONSIN ACADEMY SCIENCES, ARTS, AND LETTERS. 



Equations which are analogous to equations (H),and which suggest 



that, since * here takes the place of + — in those equations, 



M, or the magnetic permeahirdy, may be equal to m'\ lo"^ tv^, divided 

 by 9, generally, while here definitely equal to the constant 2 n*. 



That equation (2) is satisfied by the values of w, t\ and to given 

 above in eqaations (4) is shown by differentiating equations (4) and 

 adding; we thus get 



^"^^^ 'dl^l^' 'di~~di^"' ~d7'"d?"^ 



then, on adding 



dti dv dw 



dx dy dz 



That equations (3) are likewise satisfied, is also shown by differen- 

 tiating equations (4), and then making the necessary subtractions; 

 noticing the values 2jf\ 2ii;^, 2zr, given by equations (5). 

 We thus get: 



dv dw n , 

 dz dy 



1 f 1^ 4. ^ 4. ^ 1 



dz y dx, dy dz J 



dvj du _^ ^ d {dL dM .dN\ 

 dx dz dy [ dx dy dz J 



(7) 



du dv 9 .3 _ ^ ( ^ _L ^ . '^1 

 dy dx dy \^dx * dy dy J 



which, if the second terms of the second members are zero, show 

 equations (3), and likewise equations (Q), to be completely satisfied. 

 That these second terms are zero is shown by first differentiating 

 equations (6) with respect to rr, y, and z successively; thus getting 

 results of the form 



dL 



dx 



1 



fj-f ''^ ^l "^ dadb dc 



for each coordinate. And then, on integrating these latter results 

 by parts, we get the following three equations: 



dL 1 ,^ wi .. . I „„r I dwl 



dx 



J_ jY "f^Aidc — L yyY 1. ^Zi da db dc 



dy 2ti "^-^ r 211 '''^^ r 



db 



da db dc 



dN 

 dz 



1 /y!!i dadb - — fff --. ^ da db dc 



2 11 



*The letter n represents the ratio 3.1416. 



(8) 



