THEORY OF APPROXIMATION. 95 



rnence 'by discovering linear, and afterwards proceed to 

 elliptic or more complicated laws of variation. The ap- 

 proximate curves which we employ are all, according to 

 De Morgan's use of the name, parabolas of some order 

 or other ; and since the common parabola of the second 

 order is approximately the same as a very elongated 

 ellipse, arid is in fact an infinitely elongated ellipse, it 

 is convenient and proper to call variation of the second 

 order elliptic. It might also be called quadric variation. 



As regards many important phenomena we are yet only 

 in the first stage of approximation. We know that the sun 

 arid many so-called fixed stars, especially 61 Cygni, have 

 a proper motion through space, and the direction of this 

 motio.n at the present time is known with some degree 

 of accuracy. But it is hardly consistent with the theory 

 of gravity that the path of any body should really be a 

 straight line. Hence, we must regard a rectilinear path 

 as only an approximate and provisional description of the 

 motion, and look forward to the time when its curva- 

 ture will be ultimately detected and measured, though 

 centuries perhaps must first elapse. 



On the surface of the earth we are accustomed to 

 assume that the force of gravity is uniform at all ordinary 

 heights above or below the surface, because the variation 

 is of so slight an amount that we are scarcely able to 

 detect it. But supposing we could measure the variation, 

 we should find it simply proportional to the height. 

 Taking the earth's radius to be unity, let h be the height 

 at which we measure the force of gravity. Then by the 

 well-known law of the inverse square, that force will be 

 proportional to 



^ w , or to </(! -2h+ 3 K'-4h>+ ). 



But at all heights to which we can attain h will be so 

 small a fraction of the earth's radius that 3 h 2 will be in- 



