492 
THE ASTRONOMER ROYAL’S EXPERIMENTS 
proportional to the distance from the centre of the magnet, which includes also the laws 
that there is a gradual increase of red magnetism from one end and a gradual increase of 
blue magnetism from the other end. Putting l for the half-length of the magnet, a and b 
for the longitudinal and transversal ordinates of the attracted point, x for the longi- 
tudinal ordinate (measured from the centre) of any attracting point, and supposing the 
magnet to be a line, it is easily seen that the quantities to be integrated are : — 
Longitudinal ^ Transversal — — — — 
° {{x-af + b*}? {{x-af + b^ 
and the results of integration are 
l 
Longitudinal force = 
{( l + af + b {(*_«)* + J*}* 
-f hyp. log {(l + af+fif—l—a}— hyp. log{(7— af-\-b^-\-l— a}. 
Transversal force 
—al—a*—P 
~ b{{l+aY + b*}^ 
— al + a 2 + i 2 
b{ ( l-af + b 2 }^’ 
I have computed these numbers for each of the eighteen stations. For comparison 
with observation, I have taken the experiments with the fiat side towards the small 
magnets, which represents most nearly the case of a linear large magnet ; and, for 
facility of comparison, I have multiplied the experimental numbers by 6. The fol- 
lowing is the comparison : — 
Experimental. 
Theoretical. 
Longitudinal. 
Transversal. 
Longitudinal. 
Transversal. 
— 1500 
0 
-1849 
0 
— 1560 
+ 822 
-1750 
— 1089 
— 1416 
+ 1656 
- 1441 
21 } o 
-1092 
+ 2664 
- 827 
-2928 
— 216 
+ 3240 
+ 155 
-3180 
+ 948 
+ 2706 
+ 1126 
— 2283 
+ 1960 
+ 1 890 
+ 1589 
-1389 
+ 3312 
+ 1104 
+ 2395 
— 622 
- 960 
0 
-1029 
0 
- 894 
+ 432 
- 971 
+ 517 
— 738 
+ 816 
- 776 
+ 960 
— 474 
+ 1110 
— 428 
+ 1267 
- 72 
+ 1302 
— 2 
+ 1319 
+ 306 
+ 1140 
+ 335 
+ 1066 
+ 552 
+ 924 
+ 409 
+ 801 
+ 720 
+ 708 
+ 668 
+ 633 
+ 942 
+ 492 
+ 805 
+ 380 
+ 1140 
+ 240 
+ 984 
+ 251 
The agreement is not satisfactory ; but I am unable to suggest the nature of the 
change that ought to be made in the assumed law. 
I shall add only one remark, of a somewhat practical character. In a paper published 
originally by Dr. Lamont in Poggendorfe’s ‘ Annalen,’ vol. cxiii. p. 239 &c., and of which 
