118 ON MAGNETISM. 



where for any special locality on the earth, u, X, and r, 

 must have the proper numerical values, but i v i v i z 

 must for the present be left in a symbolical form. The 

 expansions of p*, p* } &c., will introduce other integrals 

 or unknown constants z 4 , i 5> i 6 , &c. multiplied by other 

 functions of u, \, and r. And thus the forces N, W, C, 

 can be exhibited for every locality, in expressions which 

 involve these unknown constants: then the westerly 



W 

 declination, whose tangent = j=, can be so expressed: 



the total horizontal magnetic force = J(N* + TF 2 ) can 

 be so expressed: and the angle of dip, whose tangent 



' can be so ex P ressed - 



The number of integrals or undetermined constants 

 thus introduced is large. Limiting the order (as above 

 mentioned) to P 4 or to the fourth power of p, 24 con- 

 stants are required. In order to obtain these numeri- 

 cally, 24 observations of some kind are necessary. Any 

 determinations of magnetic elements will suffice : for 

 instance, determinations of western declination, horizon- 

 tal force, and dip, at each of eight stations. Gauss, refer- 

 ring generally to Sabine's map of Total Intensity in the 

 Seventh Report of the British Association, and to Barlow's 

 map of Declination, Phil. Trans. 1833, and to Homer's 

 map, Physikalisches Worterbuch, Band VI, but without 

 giving numerical details of his process, has obtained the 



y 

 following value for ^ . It is to be remarked that the 



numbers have all been adapted to give horizontal force 



