268 STAFF COMMANDER EVANS AND ME. A. SMITH ON THE MAGNETIC 
tion of the ship’s head, Y to starboard, Z vertically downwards or to nadir, and if the 
components of the ship’s permanent magnetism in the same directions be P, Q, and R, 
and of the total force of earth and ship in the same three directions X', Y', Z', then 
Ship’s force to head =X'— X=«X+JY+cZ+P, . . . . (1) 
Ship’s force to starboard =Y'—Y=^X+^Y+/Z+Q, (2) 
Ship’s force to nadir =Z' — Z==</X-f-/ 2 Y+£Z+R, (3) 
a , b, c, d, e, f, g, h, k being coefficients depending on the amount and arrangement of 
the soft iron of the ship. These are Poisson’s fundamental equations, first given in the 
Memoires de l’lnstitut, tom. v. p. 533. 
To adapt these formulae to observation, let 
H be the earth’s horizontal force, 
£ the easterly azimuth of the ship’s head measured from the correct 
magnetic north ; 
6 the dip. 
Then X= H cos £, Y= — H sin £, Z = H tan 6. 
Substituting these values, and dividing (1) and (2) by H, i. e. taking the earth’s hori- 
zontal force at the place as unit, equations (1) and (2) become 
Ship’s force to head =^77^ —a cos £— b sin£-J-ctan0-j-^- • • • (4) 
Ship’s force to starboard=^j^ =d cos £— e sin £+/’ tan • • • (^) 
Dividing (3) by Z, i. e. taking the earth’s vertical force as unit, we have 
Force of earth and ship to nadir = cos £ sin £+ 1 ... (6) 
r Z tan 6 ’ tan 9 ’ Z 
If we resolve the forces (4) and (5) in the direction of the magnetic north, we shall 
find, besides periodical terms, one non-periodical term — which therefore represents 
the mean force of the ship to North, and therefore H=XH, is the “mean 
force to North,” or the mean value of the northern component of the force of earth and 
ship. 
If we take the “ mean force to North,’ or aH. for unit, or, in other words, divide by aH, 
we derive from (4) and (5) the following expressions for the force of earth and ship to 
North and to East respectively, viz. 
H' pos 8 
ToNorth=— jj-=l-fS8cos£— 6sin£+2)cos2£— (Esin2£, ... (7) 
To East =^^- 8 =2t+SSsin£+ecos£+2)sin2£+(Scos2£, ... (8) 
in which H 7 is the directive force of earth and ship on the needle, b the deviation. 
