572 PHILOSOPHICAL TRANSACTIONS. [aNNO IJSQ. 



and the value x of it when x = ahe known, and the value of it when x = 

 a -\' b he required. Find a series of which the first term is x, and which pro- 

 ceeds according to the dimensions of b, if ^ be a very small quantity, and in 

 general at least so small that the series from x=:a to x=za-{-b neither be- 

 comes infinite nor O. In the same manner, if an algebraical or fluxional equa- 

 tion or equations, expressing the relations between x, 3/, z, v, &c. be given, find 

 the correspondent values of z/, z, v, &c. to a? = a, which let be y, z, v, &c.; then 

 find serieses for 3/, z, v, &c. of which the first terms let be y, z, v, &c. re- 

 spectively, and which proceed according to the dimensions of b, but subject to 

 the same conditions as in the preceding case. From fluxional equations may be 

 deduced series which express the value of ^, &c. in terms of x^ and always di- 

 verge, or always converge, whatever may be its value, as appears from the Me- 

 ditationes. 



XVIIL On the Resolution of Attractive Powers. By Edw. fVaring, M, D., 



F.R.S., &c. p. 185. 



1. A force acting at a given point may be resolved by an infinite number of 

 ways into 2, 3, or more (nj forces acting at the same point, either in the same 

 or different planes with the given force and each other; and, vice versa, any 

 number of such forces acting in the same or different planes may be reduced 

 into one. 



Exam. fig. 5, pi. 6. Let a body a be acted on by 3 forces ab, ac, and ad, not 

 being in the same plane; reduce any 2 of them ab and ac to one ae, by com- 

 pleting the parallelogram abec ; then reduce the 2 forces ae and ad to one ap 

 by completing the parallelogram aefd; then the 3 forces ab, ac, and ad, are 

 reduced to the one ap. 



2. If n forces act on the body a at the same time, and any (n — l) of them 

 be reduced to 1, the force resulting will be situated in the same plane with the 

 remaining, and force equivalent to the (nJ forces. 



3. If one force a be resolved into several others x, 7/, z, v, &c. situated in 

 different planes, and the sines of the angles, which the forces 7/, z, f, &c. con- 

 tain with the plane made by the direction of the forces x and a be respectively 

 s, /, s"y &c. then will sy 4; /z + s"v + &c. = 0. 



Prob. 1. fig. 6. — Given the law of attraction of each of the parts of a given 

 line in terms of their distance from a given point p ; to find the attraction of 

 the whole line ab on the point p. — Find the attraction of the line ab on the 

 point p in the 2 directions p/* and fb by the following method. Draw p^ from 

 the point p to any point x of the line ab; then the force acting on the point p 

 by the particle xy will be the given function (determined from the given law of 

 attraction) of the distance into the particle; draw also p^ perpendicular from 

 the point p to the line ab; and let pf=z a, hf=b, and /a? = y; then will the 



