ADDITIONS. 



SB 



by the removal of a part of the nucleus, CG will be dimi> 

 half as much, and EI one fourth. 



Scholium. If 9^ — 1, or rt~a, this expression fails, 

 the numerator and divisor vanishing : in such cases the 

 value of a fraction is evidently equal to the quotient of the 



ray within the nucleus, and CD a diameter parallel to evanescent incremenu, or of the fluxions. Now' -■ 



AB ; call EF, s, then EG perpendicular to CH will be rs, 



rs 





to -7- ; and 

 b 



and the sine of EHG to that of EAF as 



lince the angles are evanescent, they will be in the same 



ratio as their sines, and the deviation ECH is — — f — — 



j-f 1 rob 



but ECH : EH : EHC : EC 



al- 



?+l 



} ' s.{rb — a) 



, and if EI 1 1 CH, it is obvious that the focal 

 q rb — a 



diitancc EI is half CG or C£ ; and that if AB be diminished 



rab rait 9+1 



rb — a 2 9 rb — a 



of which the latter factor only fails j 



and its value may be found by substituting for r, and mak- 

 ing the exponent q variable 5 thus rb:^( yb and the 



fluxion of ( — y is (h. 1. — J rq, which is to q ai 



( h. 1. — Jr. to' 1; and the focal distance become* 



,,.(h.l.(l) 



2(h.l.r)" 



ADDITIONS. 



jifter article 331. 



331. B. Theorem. The force acting on 

 any point of a uniform elastic rod, bent a 

 little from the axis^ varies as the second flux- 

 ion of the curvature, or as the fourth fluxion, 

 of the ordinate. 



For if we consider the rod as composed of an infinite 

 number of small inflexible pieces, united by elastic joints, 

 the strain, produced by the elasticity of each joint, must be 

 considered at the cause of two effects, a force tending to press 

 the joint towards its concave side, and a force half as great 

 as this, urging the remoter extremities of the pieces in a 

 contrary direction ; for it is only by external pressures, applied 

 so as to counteract these three forces, that the pieces can be 

 held in equilibrium. Now when the force, acting against 

 the convex side of each joint, is equal to the sum of the forces 

 derived from the flexure of the two neighbouring joints, the 

 whole will remain in equilibrium : and this will be the case 

 vrhither the curvature be equal thioughou, or vary uni- 

 formly, since in either case the curvature at any point is 

 equal to the half sum of the neighbouring curvatures ; and 

 it is only the difference of the curvature from this half s m, 

 which is as the econd fluxion Of the curvature, that deter- 

 mines the acceleiating force. 



Jfter article 33S). 

 339. B. Theorem. The stiffness of a cy- 

 linder is to that of its circumscribing prism 

 as three times tlie bulk of the cylinder to four 

 times that of the prism. 



The force of each stratum of the cylinder may be consi- 

 dered as acting on a lever of which the length is equal to its 

 distance x from the axis : for although there is no fixed ful- 

 crum at the axis, yet the whole force is exactly the same as 

 if such a fulcrum were placed there, since the opposite ac- 

 tions of the opposite parts would remove all pressure from 

 the fulcrum. The tension of each stratum being also as the 

 distance x, and the breadth being called 2y, the fluxion of 

 the force on either side of the axis will be ix'yi, while that 

 of the force of the prism is 2,r^x, and its fluent 3x'. But' 

 the fluent of iJfyi, or 2v/(l — xx)x^x, calling tlie radius 

 unity, is ^(j — y'x), » being the area of the portion of the 

 section included between the stratum and the axis, of which 

 the fluxion isyi; fur tlie fluxion of i—-i/'x is i/.i — i/'i — 



2y xi/z^yx^i — 3y 



•<=r> 



:jx*.i:+3i/:i'.i^:4!/x'x; and 



whenx=i, and y::zo, the fluent becomes ii, while the 

 force of the prism is expressed by |. 



