**Px**, Premaxilla.

Of **px**, Premaxilla.

Oc. ' 'r.bs **px**, Premaxilla.

We find that Di must be equal to p x g x for then t x (p x) 3 +, u (g x) 3, Hence, if **px**, qx be the linear factors of the Hessian 64, the cubic can be put into the form A(p x) 3 +ï¿½(g x) 3 and immediately solved.

In the canonical form f=k1(**px**)5 +k2(gx) 5 +k3(rx) 5 .

Hence, from the identity ax (pq) = **px** (aq) -qx (ap), we obtain (pet' = (aq) 5px - 5 (ap) (aq) 4 **pxg** x - (ap) 5 gi, the required canonical form.

His development of the equation x m +- **px** = q in an infinite series was extended by Leonhard Euler, and particularly by Joseph Louis Lagrange.

An imperfect solution of the equation x 3 +-- **px** 2 was discovered by Nicholas Tartalea (Tartaglia) in 1530, and his pride in this achievement led him into conflict with Floridas, who proclaimed his own knowledge of the form resolved by Ferro.

(3), C = ffcos(**px--gy**) dx dy,.

When, as in the application to rectangular or circular apertures, the form is symmetrical with respect to the axes both of x and y, S = o, and C reduces to C = ff cos **px** cos gy dx dy,.

Thus, if x = p cos 4), y= p sin 0, C =11 cos **px** dx dy =f o rt 2 ' T cos (pp cos 0) pdp do.

Over the ellipsoid, p denoting the length of the perpendicular from the centre on a tangent plane, **px** _ pv _ _ pz 1= a2+X' b +A' n c2+A p2x2 + p2y2 p2z2 I (a2 - + X)2 (b 2 +x)2 + (0+X)2, p 2 = (a2+A)12+(b2+X)m2+(c2+X)n2, = a 2 1 2 +b 2 m 2 +c 2 n 2 +X, 2p d = ds; (8) Thence d?

With A' =0 over the surface of the paraboloid; and then' = ZU[y 2 - pJ (x2 + y2) + **px** ]; (9) =-2U p [1/ (x2 + y2)-x]; (io) 4, = - ZUp log [J(x2+y2)+x] (II) The relative path of a liquid particle is along a stream line 1,L'= 2Uc 2, a constant, (12) = /,2 3, 2 _ (y 2 _ C 2) 2 2 2 2' - C2 2 x 2p(y2 - c2) /' J(x2 +y 2)= py ` 2p(y2_c2)) (13) a C4; while the absolute path of a particle in space will be given by dy_ r - x _ y 2 - c2 dx_ - y - 2py y 2 - c 2 = a 2 e -x 1 46.

Is of the form **px** + q.

Is of the form **px** 2 + qx -{- r.

Simpson's two formulae also apply if u is of the form **px** 3 - }- 5x 2 + rx -}- s.

- ~(**Px**) - I(Py) - ~(Pz)

One form of the solution of the equation, and that which is applicable to the case of a rectangular orifice, is z =C sin **px** sin qy.