velocity and, from the laws of the penetration of solids into different media, proportional simply to the area of the inden tation. Representing then by a and b the sides of the area of the indentation, supposed rectangular, at the surface of the metal impinged on, d the depth of the indentation, and C the constant ratio of the resistance and the area of the indentation, the following relation obtains between the work expended by the hammer in its fall and that offered by the resistance of the metal

W1(h+h,)=*Cald;

an equation from which C may be determined by experi ment in any particular case.

It will be readily seen that the preceding expressions will be rendered applicable to the cases where the cam catches the hammer on the same side of its axis of rotation as its

dw, MG for + die, MG, and

dt

dt

shock on the trunnions (Arts. 108, 109), and there then obtains, to find the point where the cam should catch the hammer corresponding to this case,

Morin, Suite des Nouvelles Expériences sur le Frottement, p. 67. Paris, 1885

APPENDIX.

NOTE A.

THEOREM.— The definite integral ffxdx is the limit of the sums of the

a

values severally assumed by the product fr. Ax, as x is made to vary by success ve equal increments of Ax, from a to b, and as each such equal increment is continually and infinitely diminished, and their number therefore continually and infinitely increased.

To prove this, let the general integral be represented by Fr; let us suppose that fr does not become infinite for any value of r between a and b, and let any two such values be x and Az; therefore, by Taylor's theorem, F (x + ▲x) = Fx + Arfx + (Ax)'+λM, where the exponent 1 + λ is given to the third term of the expansion instead of the exponent 2, that the dfx case may be included in which the second differential coefficient of Fr, dx

is infinite, and in which the exponent of Ar in that term is therefore a fraction less than 2.

Let the difference between a and b be divided into n equal parts; and let each be represented by Ar, so that

Giving to x, then, the successive values a, a+ ▲x, a + 2 ▲ x . . a + (n−1) Ax, and adding,

F(a+nAr) Fa+sxΣ,"f{a+(n−1) Ax} + ( ▲ x)' + ›Σ M.,

.. Fb-Fa=AxΣ,"ƒ{a+(n−1)Ax} + (Ax)1 +›ΣM.

Now none of the values of M are infinite, since for none of these values is fr infinite. If, therefore, M be the greatest of these values, then is ΣM, less than nM: and therefore

Fb-Fa-Axƒ{a + (n−1) ▲x} < (b—a) M (Ax)^. The difference of the definite integral Fb- Fa, and the sum Σ," (1x)ƒ{a+ (n-1) x} is always, therefore, less than (b— a) M (Ar). Now M is finite, and (b-a) is given, and as n is increased ▲ is diminished continually; and therefore (4x) is diminished continually, a being positive.

Thus by increasing indefinitely, the difference of the definite integral

and the sum may be diminished indefinitely, and therefore, in the limit, the definite integral is equal to the sum (¿. c.)

Fb Fa limit Σ," (Ax).ƒ{a + (n−1) Ax} ;

or, interpreting this formula, Fb-Fa is the sun of the values of Ar. ft, when a is made to pass by infinitesimal increments, each represented by Aa, from a to b.

NOTE B.

PONCELET'S FIRST THEOREM.

The values of a and b in the radical √ a2+b2 being linear and rational, let it be required to determine the values of two indeterminate quantities a and 3, such that the errors which result from assuming √ a2 + b2 = aa + 3b, through a given range of the values of the ratio sible in reference to the true value of the radical; or that aa + 3b — Va2 - ba

or

(7), may be the least pos

Va2 + b2 aa+3b -1, may be the least possible in respect to all that range of √ a2 + b2 values which this formula may be made to assume between two given

a

a

extreme values of the ratio Let these extreme values of the ratio b

be represented by cot. 4, and cot. 42, and any other value by cot. 4. Sub

a

stituting cot. for in the preceding formula, and observing that Va2+¿a

b

= √ b3cot."4+b2=b cosec. 4, also that aa+3b= ab cot. 4+36=(a cos. 4+3 sin. 4) cosec. 4, the corresponding error is represented by

which expression is evidently a maximum for that value 4, of which is determined by the equation

Moreover, the function admits of no other maximum value, nor of any minimum value. The values of a and 3 being arbitrary, let them be

α

assumed to be such that or cot. 4, may be less than cot. 41, and greater

* The method of this investigation is not the same as that adopted by M Poncelet; the principle is the same.

than cot.. Now, so long as all the values of the error (formula 1) remain positive, between the proposed limits, they are all manifestly diminished by diminishing a and 3; but when by this diminution the error is at length rendered negative in respect to one or both of the extreme values, or 4 of 4, and to others adjacent to them, then do these negative errors continually increase, as a and B are yet farther diminished, whilst the positive maximum error (formula 3) continually diminishes. Now the most favorable condition, in respect to the whole range of the errors between the proposed limits of variation, will manifestly be attained when, by thus diminishing the positive and thereby increasing the negative errors, the greatest positive error is rendered equal to each of the two negative errors; a condition which will be found to be consistent with that before made in respect to the arbitrary values of a and 3, and which supposes that the values of the error (formula 1) corresponding to the values 4, and 4, are each equal, when taken negatively, to the maximum error represented by formula 3, or that the constants a and 3 are taken so as to satisfy the two following equations.

1-(a cos. +3 sin. ¥1)= Va2+32—1.

1−(a cos. ¥1+3 sin. ¥1)=1—(a cos. ¥2+ẞ sin. V1).

The last equation gives us by reduction

Substituting these values in the first equation, and reducing,

These values of a and 3 give for the maximum error (formula 3) the expression

with a degree of approximation which is determined by the value of tan. '1(F, — Y1⁄2).

If in the proposed radical the value of a admits of being increased infinitely in respect to b, or the value of b infinitely diminished in respect to a, then cot. = infinity; therefore Y, 0. In this case the formula of

approximation becomes

If the values of a and b are wholly unlimited, so that a may be infinitely small or infinitely great as compared with b, then cot. Y1 = infinity, cot. 2 = 0;

therefore Y1=0, 2=2. Substituting these values, the formula of approx.

imation becomes

and the maximum error

·8284a+8284b..... (10);

1716, or th nearly.

Ifb is essentially less than a, but may be of any value less than it, so

is always greater than unity, but may be infinite, then cot. = in

finity, cot. 42=1; therefore 1=0, 42=4. Substituting these values in the formula of approximation, and reducing, it becomes

•96046a+397836. (11);

and the maximum error

....

03945, orth nearly.

It is in its application to this case that the formula has been employed in the preceding pages of this work.

The following table, calculated by M. Gosselin, contains the values of the coefficients a and 3 for a series of values of the inferior limit cot., the superior limit being in every case infinity.