Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

SUPPOSED HETEROGENEOUS. 159
must add together the former, and neglect the latter. The
forces are - F, - G, - H, in the directions of f, g and h:
the sum of the forces in the direction of Pp is
-(F.cospPq + GcospPr + HcospPt)
+ G
Fs sf+s Gs+H
And the matter upon which they act is pSs (the section of the
canal being supposed = 1): therefore the pressure which they
produce ultimately (see the Note to Art 25)
= -p(FSf + G3g + HIh)
-~P((F+G f+H2)
If then we can find a quantity V such that
(V) = F + G dG + H
dfj~-' r+ df + df '
V taken between the proper limits will be the pressure.
If now the values of p, F, G, H, be such that the expression
for -- cannot be integrated without expressing g and h in
df
terms off*, we must, in order to find the pressure, substitute
for p its value in terms of f, g, and h: and for g and h their
d(V)
values given by the equation to the canal. Then f will
= - ( (f), and V = - f (f), where the form of < and consequently that of 4r is different for every different canal. Let f0,
'This would be the case, if, for instance, p=k, F= Cg, G —Cf: for then
df g) Ck (g-f -): which cannot be integrated without expressing g in terms
off.