the Stability of Ships. 2 Sg 
dinates (expressed by Z, a general term or function of z), a, b, c, 
d, &c. are erected at right angles, and at intervals each of which 
is = r, so that when z = 0, Z .= a : when z = r, Z = b: when 
z = 2 r, Z = c, and so on. If innumerable ordinates or values 
of Z be supposed drawn between each of those which are given, 
at the common very small interval z, the sum of the products 
arising from multiplying each of the ordinates into the incre- 
ment z, that is, the fluent of Z z, will be found, by approxima- 
tion, according to the three following rules ; which may be not 
improperly termed, rules for approximating to the integral va- 
lues of fluxional quantities. According to 
rule 1. 
Fluent ofZi; = P— 
in which expression, 
P = the sum of all the ordinates a -f- b -f- c -f- d, &c, . ’ 
S = the sum of the first and last ordinate. 
r = the common distance between the ordinates. 
RULE II. 
Fluent ofZz = S-(-4P-j-^QxY; 
in which expression, 
S == the sum of the first and last ordinate. 
P = the sum of the 2d, 4th, 6th, 8th, &c. ordinate. 
Q — the sum of the 3d, 5th, 7th, 9th, &c.- ordinate, (the last 
excepted. ) 
r = the common distance between the ordinates. 
and hi. ; and the positions of the centres of gravity are thus determined accordihg to 
the methods of computation employed in the subsequent pages. The position of the 
centre of gravity in solid bodies, is determined by a similar application of these rules. 
