ADDITIONS AND CORRECTIONS. 



p. 2. Art. 20, last line, for- read a:b. 

 h 



P. 20. Art. ng, 1. 2, for "planes" read " parallel 

 planes." 



P. 23. Col. 2. L. 35, for " being opposite to them" read 

 since these lines have the same side A B opposite to them in 

 the triangles ABC, A B I, and their equals BC, BI are 

 opposite to the same angle BAG. 



P. 35, after article 2fi5, insert, 



26a. B. Theorem. Supposing the force 

 retarding a pendulum or balance to be to the 

 force of gravitation or of elasticity at the ex- 

 treme point of each vibration as / to 1, the 

 circumference of a circle to its diameter being 

 as c to 1, the time of each vibration will be 



increased in tht ratio of 1 to 1 + S^, or ^ + 



c 



.fi4^ very nearly. 



The nnpulse being supposed to _. 

 be momentary, and to be given at 



CAD 



A, the pendulum, will move to B as if completing a vibration 

 of which C is the middle point, AC being to AB 

 as/to 1 : in its return the middle point will be D, and the 

 extent of the vibration being BE, the space DA, which is 

 equal to C A, will be described in a time as much longer than 

 would have been required for describing C A, as D E or D B 

 IS shorter than C B, that is, in the ratio of i-o/,o i very 

 nearly. But the whole time of describing C A is less' than 

 .f the velocity were equable, in the ratio of the diameter of 

 a circle to its semicircumference, or of i to f, and is there- 



fore to that ofasemivibration as/to f, and to that of a 

 complete vibration as/to c; consequently we have for the 

 retardation -^ the time of a vibration being unity. 



Scholium. If the propelling force of a balance or pen- 

 dulum, and the friction which retards it.'be increased in the 

 same proportion, the extent of the v ibration will remain un- 

 altered, and the motion will be retarded in proportion to the 

 VOL. II. 



square of the fraction expressing the friction. But if the 

 propelling force be increased in a greater proportion than the 

 friction, the extent of the vibration will be increased in the 

 ratio of this excess, and the value of the fraction /will be 

 diminished in the same proportion. Thus, if the friction 

 were doubled, and the propelling force quadrupled, the 

 extent of vibration would be doubled, and the time would 

 remain unaltered; but if the propelling force were only 

 tripled, the fraction/ would on the whole be increased \, and 

 the retardation i. 



P. 54, after art. 359, insert, 



359. B. Theorem. Every compound body 

 has at least three axes of permanent rota- 

 tion, at right angles to each other. 



When a body revolves round any axis, it is necessary, in 

 order that the revolution may be permanent, that the cen- 

 trifugal forces on all sides balance each other, so that the 

 axismay not be urged to revolve rouijd the centre of gravity. 

 The centrifugal force of each particle being pioportional to 

 its distance from the axis, its tendency to turn the axis, in 

 a given direction, being represented by the force reduced to 

 that direction, will be proportional to its distance from a 

 plane passing through the axis, perpendicular to the sup- 

 l)osed direction ; and its effect will also be the greater as its 

 distance from the equatorial plane is greater ; since the axis 

 may te considered as a lever, and the centre of gravity as 

 its fulcrum. Now if a plane be made to revolve on a line 

 passing through the centre of gravity, it is obvious that there 

 is a position in which the sums of all the products of the 

 particles into their distances from this planeand from a plane 

 perpendicular to it, passing through the same line, will 

 be equal on both sides of the plane ; and the plane . 

 remainmg in this position, if another plane be sup- 

 posed to turn round any line out of the first plane until 

 it acquire a similar property with it, it may easily be under- 

 stood that the intersection of these planes vr.ll be an axis of 

 permanent rotation, since any other plane passing through 

 it will possess the same property with respect to the parts of 

 the solid on each side of it. If then two planes perpendicu- 

 lar to each other be supposed to revolve round this axis, 

 until they acquire that position in which either of them 

 b 



