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in the heavens an apparent relative motion of rotation 

 from east to west ; but out of countless thousands which 

 obey the rule the Pole Star alone seems to break it. 

 Exact observations indeed show that it also revolves 

 in a small circle, but it might happen for a short time 

 that a star existed so close to the pole that no appreciable 

 change of place would be caused by the daily rotation. 

 It would then constitute a perfect singular exception ; 

 for, while really obeying the law, it would break the terms 

 in which it is usually stated. In the same way the poles 

 of every revolving body are singular points. 



Whenever the laws of nature are reduced to a mathe 

 matical form we may expect to meet with singular cases, 

 and, as all the physical sciences will meet in the mathema 

 tical principles of mechanics, there is no part of nature 

 where we may not probably encounter them. In me 

 chanical science itself the circular motion of rotation may 

 be considered a single exception to the rectilineal motion 

 of translation. It is a general law that any number of 

 parallel forces, whether acting in the same or opposite 

 directions, will have a resultant which may be substituted 

 for them with like effect. This resultant will be equal 

 to the algebraic sum of the forces, or the difference of 

 those acting in one direction and the other ; it will pass 

 through a point which is determined by a simple formula, 

 and which may be described as the mean point of all the 

 points of application of the parallel forces (vol. i. p. 422). 

 Thus we readily determine the resultant of parallel forces, 

 except in one peculiar case, namely, when two forces are 

 equal and opposite but not in the same straight line. 

 Being equal and opposite the amount of the resultant is 

 nothing, yet, as the forces are not in the same straight 

 line, they do not balance and destroy each other. Exami 

 ning the formula for the point of application of the re 

 sultant, we find that it gives an infinitely great magnitude, 



