of Action between Magnets. 127 
Nature of Coeficients P, Q, R. 
§ 14. The law of action between two magnets in Lamont’s 
first position—the position met with in deflexion experi- 
ments—was investigated by Lamont himself, and more 
recently with greater completeness by Dr. Borgen *. Let us 
first suppose the cross sections of the magnets negligible, 
and that their axes lie in the same horizontal plane, and 
are strictly perpendicular to one another. Let du, dp’ be 
quantities of “magnetic matter” in the deflecting and 
deflected magneis, at distances «, 2! from their centres, and 
let r be the distance between the centres. Then the couple 
L arising from the action of the one magnet on the other is 
given by 
L =|\(r —w)al{ (r—2) +a? 2d dp’, 
where the integration extends throughout both magnets. 
Using the Lamont-Borgen notation 
\edp= M, ja'dp! =i 
ede MA lain!’ — My, 
Borgen finds 
: Peete My) 7M, | MyM,’ | 45 M, 
= Iy--3 any (Neen ae: Same epee ys ae a il 
L=2MM"% [1+ -a(2 M cole na (3 M 15 ee eae x) 
iy, M, M; M,’: 105 M,M,’ 35M,’ 
Se ee ee eee ei ie NS ee 2) 
+o 2p wet poate ~ ao) t JO) 
M and M' represent the magnetic moments of the magnets, 
and so M is equivalent to the more ordinary notation m used 
earlier in the paper. The coefficients of 7—*, r-*4, and r-§ 
inside the square bracket represent the P, Q, R of (1). If 
the magnets are replaceable by poles, at distances 2A, 20’ 
apart respectively, then according to Bérgen 
M/M=., M!/M'=a!-1, 
§ 15. When the areas of the cross sections are not neglected 
small additional terms are introduced. These must partly 
depend on the distribution of magnetism throughout the 
thickness of the magnets, which is unknown. In thin-walled 
circular cylinders the uncertainty cannot, however, be large. 
Taking the result which Borgen approves for this case +, we 
have in full 
P=2n?—3n? + (3/4) {3(a2 +b) —2(a2 +82) $, . (33) 
* Terrestrial Magnetism, vol. i. 1896, p. 176; Archiv. der Deutschen 
Seewarte, 1891, &c. 
+ Terrestrial Magnetism, 7. c. p. 189. 
