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A( x > y) must be all scalar. Thus, in the case of the parabola 

 derived from a circle, by substituting for the ordinate of a point in 

 the latter its distance from a known point in the circumference of 

 the same, we know that it is impossible to derive more of the para- 

 bola than can be obtained by taking the diameter of the circle as 

 the ordinate. The algebraical process gives first 



(1) OR=o7 . OI+ V 0* 2 +2/ 2 ) OJ, y 2 +# 2 =2a# ; 

 and then, putting p= x t <?=-f V(# 2 +^ 2 ) we find 



(2) OR=p . Ol + q . OJ, <f=2ap. 



In this case q is always positive and =+ >J(2ax). But x, and 

 therefore p, always lies between the limits and 2, and hence q 

 must lie between the same limits. Consequently the only part of 

 the curve (2) represented by the equations (1) is the semi-parabola 

 contained between the origin and the ordinate 2 . OJ. 



This general form of the concrete equation, therefore, furnishes 

 an elementary method of representing curves or parts of curves. 

 Thus 



OR=(A + a cos CL + X cos a) . OI + (^:( + a sin a + x sin ri)).OJ 



are the equations of a line 



HK=2 (cos a . Ol + sin a . OJ), 

 and having one of its extremities determined by the equation 



OH=A.OI-M.OJ, 



so that its length is 2 times that of OI and the angle (OI, HK)=a. 

 To determine the intersections of two such finite curves, given by 

 the equations 



OP-rtfo y) . OI +/,(*, y) . O J, 0<>, y) = 0, 

 and OF=//(ar',y').OI+/ 8 '(^y').OJ, 0'<V, y')=0, 



we have OP=OP', and consequently f i =f 1 ' and / 2 =/ 2 f , which, 

 with 0=0 and 0'=0, give four equations to determine x, y y x' t y 1 . 

 The curves will, however, not intersect, unless not only four such 

 scalars exist, but they make f v / 2 , //, // all scalar. 



The transformation of coordinates may now be investigated more 

 generally in the form of the two problems : ' given a change in the 

 concrete (or abstract) equation, to find the corresponding change in 



