430 



THE ELECTROMAGNETIC FIELD. [PT. III. CH. XL 



221. Various Resolutions into Elementary Forces. The 



value of the integral will not be changed if we add to the integrand 

 any expression, 



where F is any function of r, for it will disappear when integrated 

 around either circuit. If we put F = r, 



dr _ dr dx^ dr dy dr dz l 

 dsi dx-L dsi di/i ds l d^ l dsi 



If, Fig. 87, we drop a perpendicular from ds 2 on 

 the tangent at ds l} and call the length of the 

 tangent thus cut off p, we see by infinitesimal 

 ^ geometry that 



- ds l cos (r, ds^) = dr, ds 2 cos (ds lt ds 2 ) = dp. 



ds t 



Fm. 87. 

 Accordingly 



(12) 



But 



Consequently 

 (13) cos(ds 

 and 



), 



(14) 



9V 



dp 



-- = COS 



- = co< 



p = r cos (r, 



t 



_9g_ d_ f dr\ dr dr 



1 (dr dr 



= IT~ ^ H cos (^u 



r (9^! 9s 2 



cos (r, ^^ cos (r, ds 2 } cos (c?^ , ds 2 ) 



d*r 



and multiplying this by an arbitrary constant (1 &)/2, and adding 

 to the integrand in ( 1 1 )* 



(15) - W = IJ> ( f i 



JlJ2 r 



(l-k) 



cos (r, 



The value & = 1 gives Neumann's form of the integral, from which 

 may be obtained the resolution into elementary forces already 

 * Helmholtz, Wiss. Abh. Bd. i. p. 567. 



