652 THE PRINCIPLES OF SCIENCE. [CUA? 



Among the most interesting of apparent exceptions are 

 those which I call singular exceptions, because they are 

 more or less analogous to the singular cases or solutions 

 which occur in mathematical science. A general mathe- 

 matical law embraces an infinite multitude of cases which 

 perfectly agree with each other in a certain respect. It may 

 nevertheless happen that a single case, while really obeying 

 the general law, stands out as apparently different from all 

 the rest. The rotation of the earth upon its axis gives to 

 all the stars an apparent motion of rotation from east to 

 west ; but while countless thousands obey the rule, the Pole 

 Star alone seems to break it. Exact observations indeed 

 show that it also revolves in a small circle, but a star 

 might happen for a short time to exist so close to the pole 

 that no appreciable change of place would be caused by the 

 earth's rotation. It would then constitute a perfect singular 

 exception ; 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 encounter them. In mechanical 

 science the motion of rotation may be considered an ex- 

 ception to the 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 re- 

 sultant 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 

 (p. 364). 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 

 -esultant is nothing, yet, as the forces are not in the same 



