lated blood. There was also found a quantity of blood, partly florid, partly dark coloured and coagulated, in a portion of the jejunum, as well as in the ileum, colon, and rectum, in which last the blood was in a very putrid state.

On the Action of Springs, By James Jurin, M. D, F. R. S. N° 472, p. 46. The theory of springs is not only of use in the more curious parts of mechanics, as the structure of watches, &c. but may give light to many operations of nature, there being few substances but what are endued with some degree of elasticity; and particularly the bodies of animals, and even of vegetables, being known to consist, in a great measure, if not wholly, of organs strongly elastic.

For which reason it is not to be wondered, that this theory has engaged the thoughts of several eminent mathematicians of the last and present age; as Dr. Hook, Mr. John Bernouilli, M. Camus, &c.

But, as all hitherto done on this subject goes no further, than to compare the effects of different springs with each other, in one case only, where they are supposed to be bent to the same degree, and that without showing how the effect of any of them may be reduced to, or compared with, that of any other natural cause, the general proposition following may merit attention, both on account of its simplicity, and of its comprehending all possible cases of a body acting on a spring, or a spring on a body, where no other power intervenes; and also of its reducing the effect to that most known and simple one, the effect of gravity on falling bodies.

1. By a spring is meant a body of any shape perfectly elastic.-2. By the natural situation of a spring, is meant the situation it will rest in, when not dis turbed by any external force.-3. By the length of a spring, is meant the greatest length, through which it can be forced inwards. This would be the whole length, were the spring considered as a mathematical line; but in a material spring is the difference between the whole length when the spring is in its natural situation, and the length or space it takes up when wholly compressed or closed.— 4. By the strength of a spring, is meant the least, force or weight, which, when the spring is wholly compressed or closed, will restrain it from unbending itself. -5. By the space through which a spring is bent, is meant that space or length through which one end of the spring is removed from its natural situation.-6. By the force of a spring bent or partly closed, is meant the least force or weight, which, when the spring is bent through any space less than its whole length, will confine it to the state it is then in, without suffering it to unbend any farther.

Principles-Ut tensio, sic vis; that is, if a spring be forced or bent inwards, or drawn outwards, or anywise removed from its natural situation, its resistance is proportional to the space by which it is removed from that situation.

Thus, if the spring CL, fig. 4, pl. 1, resting with the end L against any immoveable support, but otherwise lying in its natural situation, and at full liberty, shall, by any force p, be pressed inwards, or from c towards L, through the space of 1 inch, and can be there detained by that force p, the resistance of the spring, and the force p, exactly counterbalancing each other; then the force 2p will bend the spring through the space of 2 inches, 3p through 3 inches, 4p through 4 inches, &c. the space cl, fig. 5, through which the spring is bent, or by which the end c is removed from its natural situation, being always proportional to the force which will bend it so far, and will detain it so bent.

And if one end 1 be fastened to an immoveable support, fig. 6, and the other end c be drawn outwards to 1, and be there detained from returning back by any force p, the space cl, through which it is so drawn outwards, will be always proportional to the force p, which is able to detain it in that situation. The same principle holds in all cases, where the spring is of any form whatever, and is, in any manner whatever, forcibly removed from its natural situation.

Here it may be noticed, that the elastic force of the air is a power of a different nature, and governed by different laws, from that of a spring. For supposing the line LC, fig. 4, to represent a cylindrical volume of air, which, by compression, is reduced to Ll, fig. 5, or, by dilatation, is extended to Ll, fig. 6, its elastic force will be reciprocally as Ll, fig. 5 and 6; whereas the force or resistance of a spring will be directly as cl.

Dr. J. now proceeds to his general proposition, and its corollaries; in which he remarks, that if he happen at any time to express himself with less accuracy, as in making weights, times, velocities, &c. to become promiscuously the subjects of geometrical or arithmetical operations, he desires, once for all, to be understood, not as speaking of those qualities themselves, but of lines, or numbers, proportional to them.

Theorem. If a spring of the strength P, and the length CL, fig. 7, lying at full liberty on a horizontal plane, rest with one end L against an immoveable support; and a body of the weight, moving with the velocity v, in the direction. of the axis of the spring, strike directly on the other end c, and thereby force the spring inwards, or bend it through any space cв; and a middle proportional CG, be taken between the line CL X, and 2a, a being the height to which a heavy body would ascend in vacuo with the velocity v; and, on the radius RCG, be erected the quadrant of a circle GFA, it will be,

M

1. When the spring is bent through any right sine of that quadrant, as Cв, the velocity v of the body м, is, to the original velocity v, as the co-sine to the

2. The time of bending, the spring through the same sine CB, is to т the

t

time of a heavy body's ascending in vacuo with the velocity v, as the corresponding arch to 2a: that is t = T X

U

Gr 2a

Demonstr.-1. While the spring is bending through the space CB, let the space, through which it is at any time bent, be called a, the time of bending it through the space x, and the velocity of the body at the time ; and let CL =L, CB. Then, if p be the force, with which the spring, when bent through the space x, resists the motion of the body; by Dr. Hook's principle, L:x:: P: p ===

L

And since, in the case of forces that act uniformly, the quantities of motion generated are proportional to the generating forces, and the times jointly, if Mu be the nascent quantity of motion taken from the body by the resistance

Pri
L

MLT

Also, since, in the same case of forces acting uniformly, the spaces are pro

portional to the velocities, and the times jointly,

TVX

: 2a :: v7: VT, or += Σαν

But the former of these was v2, when x, and

12

=

v2 or v2

v2 Pre 2M La

2M La
P

= r2; therefore, v2 = v2 - 1,

;' and,

x2; and, when a becomes equal to 1, and v to v, v2 = v2

Now let CD, fig. 8, be equal to x; and draw the co-sine DE, the radius CE, and the perpendicular ed; then will the triangle DEC be similar to the nascent triangle deɛ; and consequently DE: de :: CE: CE =

Therefore, &= XeE, and T = TX And when

T 2a

2a

a becomes equal to CB, and to t, the arch GE becomes equal to the arch GF:

Under this theorem are comprehended the 3 following cases:

In case 1, the spring is bent through its whole length, or is entirely compressed and closed, before the moving force of the body is consumed, and its

motion ceases. In case 2, the moving force of the body is consumed, and its motion ceases before the spring is bent through its whole length, or wholly closed. In case 3, the moving force of the body is consumed, and its motion ceases at the instant that the spring is bent through its whole length, and is entirely closed.

For this reason, and in order to make the following corollaries of more ready use, Dr. J. distributes them into 3 classes; the first of which are as general as the theorem itself, extending to all the 3 cases, but are more particularly useful in case 1. The 2d class of corollaries extends to both the 2d and 3d case; but are more particularly useful in case 2. The 3d class extends only to case 3, and by that means are much more simple than either of the former. CLASS 1.-General corollaries, but of more particular use in case 1; wherein the spring is wholly closed, before the motion of the body ceases. Corol. 1.-When the spring is bent through any right sine cв, fig. 7, the loss of velocity is to the original velocity, as the versed sine to the radius, or v— v=v × G. For, since v = v X v X

R

BF

BF
R

R-BF
R

Corol. 2.-When the spring is bent through any right sine CB, the diminution of the square of the velocity is to the square of the original velocity, as the square of that right sine to the square of the radius, or v2 — v2 = v2 x For, since v = v × 1, v2 = v2 × and v2 — v2 — v2 — v2 × 2p

CB2

BF

R

= v2 x ca2. R2

BF2

Corol. 3.-When the spring is bent through any space l, v the velocity of the body is equal to v X

2M La

2мa-pl. and is proportional

2M La P/2

or to v X √

2ма

Hook's principle, L:/:: P:p, or el pL, vv X √

2MLa

× 2ma pl. But by Galileo's doctrine, is a constant quantity; and there-

Corol. 4.-The time t of bending the spring through any space l, is propor-tional to the arch GF divided by a; 1 being the right sine of the arch, and

2M La

GA

R, =√, being the radius. For, by the theorem, TX; and is

P

a constant quantity..

Corol. 5. The diminution of the product of the weight of the body into the square of the velocity, or, to use the expression of some late writers, the diminu tion of the vis viva, that is, Mv2- Mv2, by bending a spring through any space †, is always equal to 22, or to ; where A is the height from which a heavy body will fall in vacuo in a second of time, and c is the celerity gained by that fall. For, by corol. 2, v2 — v2 = v2 ×

2LA

struction, being equal to

c2pl

2A

CB2 12/2
i
R2

; and R2, by the con

But, by Galileo's theory,

Corol. 6.-The diminution of the vis viva, by bending a spring through any space ↳, is always proportional to, or to pl: and if either the spring be given,

P

or

L

be given in different springs, the loss of the vis viva will be as 12, or as p2.

2A

Corol. 7.-The loss of the vis viva, by bending a spring through its whole length, or by wholly closing it, is equal to CPL, and is proportional to PL: and if PL be given, the loss of the vis viva is always the same. This is evident from corol. 5 and 6, for I is now equal to L.

CLASS 2.-Corollaries of more particular use in Case 2; wherein the motion of the body ceases before the spring is wholly closed.

or to c

2MLA

pl 2MA

Corol. 8.-If the motion of the body cease when the spring is bent through any space 1, the initial velocity v is equal to cl For, by And here, the motion of the body ceasing, c2pl ; or vcly/ 2MA

Corol. 10.-If the motion of the body cease when the spring is bent through any space, the time t, of bending it, is equal to 1" of time, multiplied by Ml or to 1" X where m is to 1, as the circumference of a circle to 2 2PA'