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Ex. 1. To find the orthogonal trajectory of a series of spheres touching a given plane at a given point.

Let the given point be taken as the origin, and the given plane for the plane of (y, z); then the equation to the spheres is

22 — 2 ax + y2 + z = 0 = F(x, y, z), where a is variable; and therefore (270) becomes

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+(1+(?)2 = (g = f(C);

:. x2 + y2 + x2 = xf (*), where f expresses an arbitrary function.

Ex. 2. Find the equation of the orthogonal trajectory of

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a’ log x— 62 log y = ;

62 log y-c2 log z = Cz; .a2 log x — 62 log y = f (b2 log y-co log 2), is the equation to the trajectory, where f represents an arbitrary function.

430.] The following geometrical problems involve total differentials of three variables.

Ex. 1. Find the equation to the surface whose tangent plane is

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comparing this with the general equation of the tangent plane

we have

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and as the numerator of this last fraction = 0, we have also

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the equation to an ellipsoid.

Ex. 2. Find the equation to the surface whose tangent plane is

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xyz = k). 431.] The following geometrical problems also involve partial differential equations of the first order.

Ex. 1. Determine the surface whose tangent planes pass through the same point.

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the general equation to conical surfaces.

Ex. 2. To determine the surface such that the intercept of the axis of x by the tangent plane is proportional to a.

The differential equation which expresses this property is

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Ex. 2. Determine the equation to the surface in which the coordinates of the point where the normal meets the plane of (x,y) are to each other as the corresponding coordinates.

As the equations to the normal are


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z = Ci; 22 + y2 = cz;

.z = f (x2 + y2), where f represents an arbitrary function.

432.] Also let us consider the differential equation of the first order and of the second degree, which expresses the lines of curvature of an ellipsoid. Let the equation to the ellipsoid be

1. x2 22 22
+ 32 + 2 = 1;

(271) then by the general equation (64) Art. 409, Vol. I, the equation to the lines of curvature is

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which is an equation of Clairaut's form; and of which, if k is an arbitrary constant, the general integral is


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(a? 62) k
*. * a2 ? = (a? — c2)+k (62—(2)

(277) which represents a cylinder of the second degree, the axis of which is perpendicular to the plane of (x, y); and consequently the lines of curvature are determined by the intersection of these cylindrical surfaces with the ellipsoid.




SECTION 1.-General Properties of Differential Equations of

Higher Orders. 433.] We are now just on the outskirts of our science, and are unable to give any general theory for the integration of differential equations of higher orders; almost all that deserves the name of philosophical treatment has been exhausted; and it only remains for us to insert such discussions on isolated topics as are useful either in the way of extending the boundaries of our knowledge, or for the purposes of subsequent application.

The most general forms of differential equations of the nth order are (1) (2) (3) (4) in Art. 364; the last two of these are partial, and the discussion of them is reserved to a future Section of the present Chapter : we shall confine our researches at present to an equation of the form

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dx' ? ' "* dx"

(1) which contains only two variables, and wherein one of these is equicrescent. Of such equations we have in Art. 365 pointed out the geometrical meaning; and in Art. 367 have shewn that the general integral involves n arbitrary constants. If a function satisfies (1) and does not contain n arbitrary constants, it may be either a particular integral or a singular solution; but it is not the general integral. And it will be either a particular integral or a singular solution according as one or more of the arbitrary constants has been replaced by particular constant values or by functions of the variables : and it is manifest that such substitutions may take place, at any one, or at more than one, of the successive integrations.

434.] Now with reference to general properties of differential equations of the form (1), if (1) admits of being expressed explicitly in the form

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