96 THE PRINCIPLES OF SCIENCE. 



appreciable, and the force of gravity will seem to follow 

 the law of linear variation, being proportional to 1 2 h. 



When the circumstances of an experiment are much 

 altered, different powers of the variable may become pro- 

 minent. The resistance of a liquid to a body moving 

 through it may be approximately expressed as the sum 

 of two terms respectively involving the first and second 

 powers of the velocity. At very low velocities the first 

 power is of most importance, and the resistance, as Pro- 

 fessor Stokes has shown, is nearly in simple proportion to 

 the velocity. When the motion is rapid the resistance 

 increases in a still greater degree, and is more nearly pro- 

 portional to the square of the velocity. 



Approximate Independence of Small Effects. 



One result of the general theory of approximation 

 possesses such great importance in physical science, and 

 is so often applied, that we may consider it separately. 

 The investigation of causes and effects is immensely 

 simplified when we may consider each cause as producing 

 its own effect invariably, whether other causes are acting 

 or not. Thus, if the body P produces the effect x, and Q 

 produces y, the question is whether P and Q acting to- 

 gether will produce simply the sum of the separate effects, 

 x + y. It is under this supposition that we treated the 

 methods of eliminating error (Chap. XV.), and errors of 

 a less amount would still remain if the supposition was a 

 forced and unnatural one. There are probably some parts 

 of science in which the supposition of independence of 

 effects holds rigidly true. The mutual gravity of two 

 bodies, for instance, is entirely unaffected by the presence 

 or absence of other gravitating bodies. People do not 

 usually consider that this important principle is involved 

 in such a simple thing as putting two pound weights in 



