MECHANICS. 23 
semi-ellipse, whose semi-major axis is a, and semi-minor axis =A. Should 
the least height not equal 0, but a quantity, CC’ —c; then if M’N be taken 
=—y,and MN =y’, y’ + will equal the height, and the equation becomes 
(y' +c)? vast (a? —x?); and for the points, A and B, beyond which the 
a2 
; ' - 
height remains unchanged, where y’ thus = 0, we will have z a 
V/h? —c?. 
It is often desirable to determine the amount of flexion which precedes 
the fracture of any elastic body; in this case it is necessary to determine 
the shape of the elastic line formed by the neutral axis. Suppose ( fig. 17) 
BZ to be the natural condition of a fibre attached at one end, B, and this 
fibre loaded at A by Q, and uniformly along its whole length by a weight, 
which, for a single unit of length, amounts to p; the fibre takes the form 
of the elastic line, AB. Let AC be the axis of abscissas, A the origin of 
co-ordinates, and for any given point, E, of the curve, whose radius of cur- 
vature is e, take AF =z, FE =y, the greatest ordinate, BC =u, and 
AC =a; let W also be the moment of elasticity, and for the elastic 
line we will have the co-ordinate equation, yo (Ge a) +o 
3 
(a? —-}%°), and the greatest ordinate, uw (where x =a), =" + ie. 
If p =o, or the fibre be loaded only at the end, then will uae, 
and y a (3a* —x?); and if Q —0, or the fibre be loaded only uni- 
j iinpet et Pad = a A 
formly along its whole length, uv = SW? dud yf == > 1W (4a Hi) Mam 
cording to the above formule, the co-ordinates are as 8:3, thus the depres- 
sion is much greater when a weight hangs at the extremity of the fibre, than 
when it is distributed along its whole length. 
If the elastic fibre rest, as in fig. 18, at both ends, the weight Q being 
applied in the middle, the equations answering to these conditions result 
from the preceding. Let Q be the weight applied to the middle, pL that 
distributed along the whole length, L; then each support receives a pres- 
sure — 4(Q + pL). Suppose, however, the fibre to be fastened at C, and 
the pressure at A and B to act upwards, then, in the preceding co-ordinate 
equation, 4 (Q + pL) must be substituted for Q: the second part of that 
equation must be taken negatively, as it contains p as a factor, and this 
must necessarily act vertically downwards, or in the opposite direction to 
$(Q+4>pL). As, moreover, } L = a, we obtain the new co-ordinate equa- 
tion y = oe (= i x) u— a a e —_ oe The greatest ordinate, 
1 Dw 
Iso, wi r=iL,! ‘ = SSS > 5 See ===, th — 
also, when 7 = 3 L, becomes u saaw 824 5pL). If p—0, then y 
197 
