194 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



103. Equations of Lines of Force. In 36, we have 

 considered the integration of the differential equations of the lines 

 of any solenoidal vector-function and have found that the lines 

 may be represented as the intersections of two families of surfaces. 

 We shall now consider the same subject in terms of generalized 

 orthogonal curvilinear coordinates. Let us call the components of 

 the force at any point q lt q 2> q s in the directions of the coordinate 

 axes at the point, R 1} R 2) R 3 , which by 16 are 



7? ft dV K /, 8F 7? I, W 



B,.-A,g-, .= -4,^, ft.-A,, 



If now ds be an element of a line of force, its projections on the 

 three axes being 



fa _ d fc fa - <%' fa _ <%* 



**-.--, dS *'^> ^-I7> 



we have 



(2) ds l : ds 2 : ds 3 = R l : R 2 : R 3 , 



or dq : : dq 2 : dq 3 = h^ : h 2 R 2 : h 3 R 3) 



so that the differential equations of the line of force are 



(3) 4*'*-vg:Vg:.vg. 



or, dividing by hjiji 3 , 



h, dV h 2 dV h s dV n n n 



( 4 ) *-* s *-ife s ia s ^.5"*^^ sft 



while we have by Laplace's equation the relation, ( 87 (5)) 



dq 3 \h-Ji 2 dq. 

 that is 



dq l dq 2 dq s 



We may now use the principle of the last multiplier demon- 

 strated in 37, replacing x, y, z, by q l} q 2 , q 3 and X, F, Z by 

 Qi, Qz, Qs- That is to say, if we have found an integral 



^ (2i> #2> #3) = const., 

 we may obtain the other at once by a quadrature as 



(6) 



