LAW OP THE UNIVERSE. 259 



- Then the differentials of the areas 



of these curves, or u d x and d x, being respectively 



Q d x Qcfdx 



and , and those being 



equal to and , or the sectors which are the differen- 

 2. d _ 



tials of the areas VIC and V X C, the areas themselves are 

 equal to those areas ; and therefore from V X C being given 

 (if the area c D V a be found), and the radius C V being given 

 in position and magnitude, the angle VOX is given; and 

 from C X being given in position, and C V in magnitude and 

 position, and also the area CI V, (if VDba be found), the 

 point I is found, and the curve V I K is known. This, how- 

 ever, depends upon the quantities made equal to u and 

 severally berag expressed in terms of #, for this is necessary 

 in order to eliminate y from the equations to these curves ; 

 and then it is necessary to integrate these expressions ; for 

 else the angle VOX, and the curve V I K, are only obtained 

 in differential equations. Hence Sir Isaac Newton makes the 

 quadrature of curves, that is, first the integration of fydcc, to 

 eliminate y, and then the integration of the equations result- 

 ing in terms of u and x, < and x respectively, the assumptions 

 or conditions of his enunciation. The inconvenience of this 

 method of solving the problem gave rise to the investigations 

 of Hermann and Bernoulli. The equation of the former, 

 involving, however, the second differential of the co-ordinate, 

 is to the rectangular co-ordinates ; that of the latter is a polar 

 equation, in terms of the radius vector and angle at the centre 

 of forces. 



To illustrate the difficulty with which this method of 

 quadratures is applied, in practice take the case of the 

 centripetal force being inversely as the cube of the distance ; 



s 2 



