417.] On the measure of curvature of a surface at a given point. Gauss in his celebrated memoir "Disquisitiones Generales circa Superficies Curvas," has introduced a definition of curvature of a surface which is derived analogously from the means of measuring the curvature of a plane curve; and from his definition has deduced a mathematical estimate of curvature.

Suppose as to be the finite arc of a plane curve commencing at a point p; and at the extremities of as let two normals to the curve be drawn. In the same plane take a circle whose radius is unity, and through its centre let two radii be drawn parallel to the two normals at the extremities of As, and let the radii dy include an angle or arc ay; then the limit ds

ΔΥ

AS

towards which

converges, when the arc of the original curve is infinitesi

mal, is, according to the definition of Art. 281, the curvature of the curve at the point P.

Imagine now upon a curved surface a finite area enclosed by a contour, within which is a given point P; and also imagine a sphere whose radius is unity; and suppose normals to the surface to be drawn at every point of the enclosing contour, and radii of the sphere to be drawn parallel to these normals; by this process a spherical area will be enclosed on the surface of the sphere. Let as be the area enclosed by the contour, and a the area of the enclosed figure on the surface of the sphere; then

dz

ΔΣ

the limit towards which converges, when the contour

ds

AS

becomes infinitesimal but still encloses the point P, is the curvature of the surface at the point P.

Let the area ▲ on the given surface be a rectangle contained by four lines of curvature; and let A1, A2 be the angles subtended at the centres of principal curvature by two adjacent sides of the rectangle ;

and the curvature of the surface at any point is equal to the

product of the curvatures of the principal normal sections at the same point.

The truth of the result is manifestly independent of the form of the small area; for whatever its form be, it can always be divided into a number of infinitesimal rectangles, for every one of which the result of equation (116) will be true; and therefore by simple addition the aggregate of all, which is expressed by equation (116), will be true also. The curvature then will be affected with a positive or a negative sign, according as the radii of the principal normal sections have the same or different signs.

In the case of developable surfaces, one of the principal normal sections has an infinite radius of curvature; it would therefore follow from (116), that the curvature of a developable surface is zero: but such is the case only with a plane. We must therefore retrace our steps and modify the process in the following manner, by operating on a right circular cylinder whose radius is unity instead of on a sphere:

Let the two containing sides of the rectangular area on the given surface, and which are coincident with the lines of curvature, be as1 and ▲ 82; of which let as lie along a generating line of the developable surface, and ▲s, be at right angles to it; then in the limit, if do1 be the angle subtended by ds, at the centre of principal curvature, ds1 = pido1. Let the axis of the cylinder be parallel to the generating line of the developable at the given point, and from a point in the axis of the cylinder let normals be drawn to the cylinder parallel to normals drawn to the developable surface along A81, and let so, be the intercepted arc of the circle on the surface of the cylinder; then ▲σ1 = 491; also let a line equal to as be taken on the surface of the cylinder and perpendicular to ▲ through one of its extremities; by this process therefore

and therefore the curvature of a developable surface at a given point is equal to the curvature of the principal normal section which is perpendicular to a generating line.

I must in conclusion observe that a different definition of curvature of a surface has been given by Madame Sophie Germain; for which I must refer the reader to her own memoir on the Curvature of Surfaces; see Crelle, VII. 1; and Récherches sur la Théorie des Surfaces Élastiques, Paris, 1821 and 1826.

CHAPTER XIX.

THE CALCULUS OF OPERATIONS.

418.] MANY theorems, which occur in common Algebra and elsewhere, are true, not because the subjects of them have any special nature as quantities, but because they conform to certain laws and combine in certain manners; and these theorems therefore are equally true of all symbols which are subject to these laws. For the sake of an example let us take the binomial theorem, and its ordinary proof: n simple factors of the form x-a, x-b, ... are multiplied together; and the law of their combination having been detected, we are thence led to a general form; and if we assume ɑ= = b = c = we obtain the general expansion of (x-a)". Now this theorem is true, not because x, a, b, ... are symbols of quantity, but because being symbols of quantity they satisfy certain laws, in conformity with which the said theorem is. All other symbols therefore which satisfy these laws are subject to the same theorem. Similarly other algebraical theorems of expansion, and expansions such as those which have been given in the preceding pages for er, sin x, tan-1x, and, in short, Maclaurin's and Taylor's Series, are true of all symbols which satisfy certain laws.

A distinction, and a division consequent upon that distinction, has been made of symbols; some are called symbols of operation, others symbols of quantity: those which indicate processes to be performed on subjects, such as symbols of differentiation; those of trigonometrical operations, as sin, tan, sec-1; those of logarithmic affection, as loga, e, &c. are called symbols of operation: whereas the subjects on which these operations are performed are called symbols of quantity. Thus if a, b, x, y are symbols of the ordinary quantities of Algebra, they are called symbols of quantities. The distinction however is scarcely accurate; for may not be the same as a × 1? in which case a symbolizes a process, that of multiplication, performed on unity, and is a symbol of operation; but it is useful enough for our purpose.

419.] Let us investigate and defiue two or three laws to which symbols whether of operation or of quantity may be subject. Let,

be symbols of operation; and let u and v be symbols of quantity, and of subjects on which and

And let us suppose 4, v, u to be such that

are performed.

successively on u, one on the back of the other, let us assume the result to be the same whatever is the order in which they are performed: two such symbols of operation are said to be commutative, and to satisfy the law of commutation.

Similarly, again, if ø, v, x are such that

these symbols are commutative. Quantities multiplied into each other, satisfy this commutative law: thus if a, b, c are constants

a x bx c x u = b x c x a x u = c x b xa x u. (3) Symbols of differentiation are also subject to the same law: thus, if u is a function of x and y,

d d dy dx

u =

d d dx dy

u;

(4)

and a similar theorem is true if u is a function of three or more variables. The form in which the law of commutation has been heretofore expressed as to differentiation is, "the order of differentiation is indifferent." See Art. 79. Trigonometrical functions are not generally subject to this law; thus, sin 2 x is not equivalent to 2 sin æ; neither is sin tan-1 equivalent to tan-1 sin æ. Again, let be a symbol of operation, and u and v two symbols of quantity; and let o be such that

$ (u + v) = $ (u) + $ (v) ;

(5)

then the operation expressed by 4 is said to be distributive, and is said to satisfy the distributive law. Similarly the number of subjects of the function may be n; and we may have

......

+ Un) = $ (U1) + $ (U2) +

+ $(u). (6)

$ (U1 + Uz + Let us at present take only two subjects, as in (5); and let us operate on them again with the symbol ; then, if we symbolize

PRICE, VOL. I.

4 H

two such operations, performed successively on a subject by 42, we have

p2 (u+v) = ${$ (u + v)}

= ${$(u) + 4 (v)}

= p2 (u) + p2 (v) ;

and if the operation is performed n times successively

p" (u+v) = p" (u) + $” (v).

(7)

Symbols of multiplication are in conformity with this law; thus, if a is a constant,

a (u + v) = au + av,

a" (u + v) = a*u + a^v.

Symbols of differentiation are also subject to it; thus, if u and v are functions of x,

trigonometrical operations are not subject to it; thus, sin (u+v) is not equivalent to sin u + sin v.

Constants and other symbols of multiplication are also subject to a law of notation; or rather a law of notation has been framed with respect to them to which other operative symbols may be subject, when they are repeated successively on the back of each other. The origin of this law which is analogically extended is the following; let a be a constant multiplied into u; then

am anu = am+nu;

that is, if u is multiplied by a first n times, and then m times, the result is the same as if u had been multiplied m+n times by a. So if is a symbol of operation which is performed on u first n times and then m times, and if the result of these successive operations is the same as if the operation of which is the symbol had been performed m+n times, then

this law of notation is called the iterative or the repetitive law. Evidently differentiation is a process subject to it; because it has been shewn in the preceding pages that

Now the calculus wherein these and other similar laws of