84 



ADDITIONS. 



Scholium. It is obvious that the strength and resilience 

 are in this case in the same ratio as the stiffness. The 

 strength of a tube may be found by deducting from the 

 strength of the whole cylinder that of the part removed, re- 

 duced in the ratio of the diameters. 



JJier article 371. 



ScHOMtM. The Strain produced by the pressure of a 

 fluid on an elastic substance which confines it, may be de- 

 termined from the principles which have been already laid 

 down respecting the flexure of such substances. Thus if a 

 plank placed in a vertical situation, be suiiported at its two 

 extremities only, and exposed to the pressure of a cistern of 

 water of which the surface coincides with its upper end, 

 the curvature will be every where as ax—x", x being the 

 distance from the surface, and will be greatest where the 

 depth is to the length of the plank as 1 to v' 3. If we wish 

 to find the strength of a circular plate, simply supported at 

 its circumference, we must consider the effect of the curva- 

 tures in two directions at right angles to each other ; and 

 we shall find that the second fluxion of the curvature in a 

 direction perpendicular to a radius of the circle at any point, 

 is simply as the curvature in the direction of the radius. The 

 curvature may therefore be represented by the difference 

 between a constant quantity and the ordinate of an elastic 

 curve, the ordinate itself representing the force immediately 

 arising from the curvature ; and since this curve is supposed 

 to deviate but little from a right line, its ordinates become 

 equal to the mean of the ordinates of two logarithmic curves, 

 and the position of its tangent may be determined accord- 

 ingly. Hence it may be shown, that in order to break such 

 a plate, the height of the fluid must be to the height which 

 would break a square plate of the same length, supported at 

 the ends only, as v'S.h.l. (2 + ^/3) or 2.2811 to i. The 

 height required to break a square plate is twice as great, as 

 if the weight of the fluid were collected in the middle of the 

 length of the square (312). 



For article 398. 

 398. Theorem. When a prismatic elas- 

 tic rod is fixed at one end, its vibrations are 

 performed in the same time with those of a 



.9707/* 



«»/♦ 



feet, h will be 1.1907 -^; and if a prismatic 



rod be loosely supported at two points only, 

 the length of the synchronous pendulum will 



and in this case, for a cylindrical rod of 

 which d'ls the diameter, h=- 



did'- 



the time 



ddh 



of vibration being to that of the circumscrib- 

 ing prismatic rod as 2 to the square root 



of 3. 

 We must suppose the form of the curve, in which the rod 



vibrates, to be such, that all its points may perform their vi- 

 brations in a similar manner, and arrive at the line of rest 

 at the same time ; on this supposition we may determine 

 the time in which the rod is capable of vibrating ; and if 

 the time of vibration is the same in all cases, the determi- 

 nation will hold good in all ; if not, the problem is not ca- 

 pable of a general resolution ; but there appears to be little 

 or no difference in the simple sounds excited in various man- 

 ners, this variety arising principally from a combination of 

 secondary sounds. The form of the curve must therefore 

 be such, that the fourth fluxion of the ordinate may be pro- 

 portional to the ordinate itself ; its equation may be found 

 either by means of logarithmic and angular measures, or 

 more simply by an infinite series. 



The conditions of the vibration must determine the va- 

 lue of the coefficients : supposing the loose extremity to be 

 the origin of the curve, the curvature and its fluxion must 

 begin from nothing : for the curvature at the end cannot be 

 finite, nor can its fluxion be finite, since in these cases an 

 infinite force, or a finite force applied to an infinitely small 

 portion of the rod, would be required, and the force could 

 not be proportional to the ordinate ; the initial ordinate 

 must also be independent of the absciss ; in the case of a 

 rod fixed at the end, the ordinate and its fluxion must both 

 vanish at the fixed point ; and in the case of a tod not fixed, 

 the second and third fluxions of the ordinate must also va- 

 nish at the remoter end, and the centre of gravity of the 

 curve must remain in the quiescent line, the whole area, con- 

 sidered as belonging to either side of the basis, becoming 

 equal to nothing ; a condition which will be found identical 



pendulum of which the length is 



I being the length, d the depth, and h the with that of the third fluxion vanishing at the remoter end. 



, . , . , , , I- 1 ^- •, I -c The series for a c^l^ve, in which the fourth fluxion of the 



heieht of the modulus or elasticity : also \t n , . , , v r ,. ^ 



' o •' ordinate is to be as the ordinate, can only be of this form, 



denote the number of complete vibrations in j^^, ,,i„^. „^ i.^x^ 



a second, the measures being expressed in ^""''■^s.3.4.i' 3.3.4.5.6.-. si' ••■■*'T'"*' 2.3.4.5^* 



