25 
the intensity of magnetic forces, &c. 
ponding to the increment A F of the intensity of the magnets. 
When therefore F becomes F + A F, putting the equation (C) 
in the form 
M — (P -j- Q . cos.^ <p) . F = 0, 
it becomes 
M — {P + Q.cos.*(?> + A9 )^(F+aF) = o, 
whence 
cos.-(? + A^)=^. (E) 
This formula, though sufficiently simple, is not in the most 
convenient form for calculating the values of A cp, corres- 
ponding to different values of cp, owing to the common tables 
of logarithms giving only seven places of figures. The 
values of P and p agree in the first two or three figures, 
so that there will remain only five figures towards the be- 
ginning of the table, and four figures towards the end of it, 
from which the values of (p A(p are to be derived, conse- 
quently they cannot be calculated to the greatest accuracy. 
But if be expanded, the first figure of p ^ — P 
being in the 5th place of decimals, the first figure of f 
F 
(a F '^* 
will be in the 9th place, and the first figure of ^ will be 
in the 13th place; and therefore we should obtain the value 
of p _^^ -p — P true to the 11th place of decimals, or true to 
(a 
7 places of figures when we neglect the term — -f. Now 
in the cases which I had to compute, the first two figures in 
the value of were the same for all the arcs, and conse- 
E 
MDCCCXXV. 
