Dr . Waring on the 
PROBLEM IV. 
1. Fig. 6. Given an equation exprefiing the relation between 
the two abfcilfe z = AP and x~Pp and their correfpondent 
ordinates y = pm of a folid, to find its folid contents contained 
between two values of itsfirft abfcifiae %. Afiume s as an inva- 
riable quantity, and from the equation refulting find the fluent 
Z of yx contained between the extreme values of x or y ; then 
find the fluent of Z*z contained between the given values of z 9 
and the fluent multiplied into the product of the fines of the 
angles, which the firft abfciflfa makes with the plane of the 
ordinates and fecond abfcifs, and the fecond abfcifs makes with 
its correfpondent ordinates, will be the folid content required. 
2. Fig. 7. Let the firft abfcifs z of a folid be perpendicular 
to the planes of the ordinates, and the fecond abfcifs P p=-x 
perpendicular to the ordinates themfelves pm~y. Firft, aflume 
the firft abfcifs as invariable, and find the increment of the 
arc p'm = (i 2 -by 2 )*, then aflume the fecond abfcifs P p as con- 
ftant, and let mu be the fluxion of the ordinate y or U 9 when 
the fluxion of the firft abfcifs is z = u/, where ul is perpendi- 
cular to the plaue of the ordinates p'pm , and / a point of the 
fur face of the folid ; draw uh perpendicular to the arc p'm , and 
fince ul is conftituted at right angles to the plane pp'm , lb will 
cut the arc p'm at right angles; but uh = = — Ux — \ 
lb ~ [hu -f hpy~ — ( + z'fi the fluxion of the furface wfill be 
lb x s/(y From the given equation exprefiing the rela- 
tion between the two abfciflbe z and x and ordinates y find, by 
afluming 55 invariable px~y 9 and by afluming x invariable qz = 
4 y = 
