xxi.] THEORY OF APPROXIMATION. 475 



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- f 

 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 theory of approximation possesses such 

 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 x, and Q produces y, the question is whether P 

 and Q acting together will produce 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 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 is entirely 

 unaffected by the presence of other gravitating bodies. 

 People do not usually consider that this important prin- 

 ciple is involved in such a simple thing as putting two 

 pound weights in the scale of a balance. How do we 

 know that two pounds together will weigh twice as much 

 as one ? Do we know it to be exactly so ? Like other 

 results founded on induction we caHnot prove it absolutely, 

 but all the calculations of physical astronomy proceed 

 upon the assumption, so that we may consider it proved 

 to a very high degree of approximation. Had not this 

 been true, the calculations of physical astronomy would 

 have been infinitely more complex than they actually are, 

 and the progress of knowledge would have been much 

 slower. 



