190 C. F. GAUSS ON THE GENERAL THEORY OF 



contained in each of these elements by d /x, in which the southern 

 fluid is always considered as negative ; call p the distance of d /x. 

 from a point in space^ the rectangular co-ordinates of which may 



be X, y, z; lastly, let V denote the aggregate of compre- 



r 



bending with reversed signs the whole of the magnetic particles 

 of the earth : or say 



v= - r^ 

 J p 



Thus V has in each point of space a determinate value, or it is 

 a function of x, y, z, or of any other three variable magnitudes, 

 whereby we may define points in space. We then obtain, by 

 the following formulae, the magnetic force ■<^ in every point of 

 space, and the components of '>^, parallel to the co-ordinate axes, 

 which we shall call ^, rj, ^, 



d V dV ^ dV .j^2 2_L>^\ 



I shall first develope some general propositions which are in- 

 dependent of the form of the function V, and are worthy of at- 

 tention from their simplicity and elegance. 



The complete differential of V becomes 



^^^ dV J dV , dV , 



d V = T — . dx + -T— . dy + -^— .dz. 

 dx dy dz 



— ^d X + rjdy -{-^d ^. 

 If we designate by d s the distance between the two points to 

 which V and V + d V belong, and by 6 the angle which the di- 

 rection of the magnetic force -\/r makes with d s, we have 

 dV=-<\r cos 6 . ds, 



^ v K 

 because as -v? -r, -t are the cosines of the angles which the di- 

 ■slr vr "ur 



r , 1 -,1 xi 1- X ^^ ^y dz 



rection oi y makes with the co-ordmate axes, so -r-, -j-, -j-, 



d S (Is CL s 



are the cosines of the angles between d s and the same axes. 



/7 IT" 



Therefore -.— is equal to the force resolved in the direction 



oi ds; the same follows from the equation -^ = ^ if we bear 

 in mind that the co-ordinate axes may be arbitrarily chosen. 



