NEWTON S PRINCIPIA. 85 



severally being expressed in terms of x, 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 Y C X, and the curve V I K, are only ob 

 tained in differential equations. Hence Sir Isaac Newton 

 makes the quadrature of curves, that is, first the inte 

 gration of/y d x, to eliminate y, and then the integration 

 of the equations resulting in terms of u and x 9 &amp;lt;p and x 

 respectively, the assumptions or conditions of his enun 

 ciation. The inconvenience of this method of solving the 

 problem gave rise to the investigations of Hermann and 

 Bernouilli. The equation of the former, involving, how 

 ever, 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 ; then y = ~ and the curve B L F is 



quadrable. If we seek the circle Y X Y by rectangular 

 co-ordinates X O, O Y, we find the equation to obtain 

 O Y = D in terms of x, is of the form 



Q a 2 d 



(c being the constant introduced by integrating fy d x\ 

 Now there is no possibility of integrating these two quan 

 tities otherwise than by sines, and we thus obtain, nor 

 can we do more, the following equation to D in terms of x\ 



* o 3 



