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not be determined. This third condition is frequently given in the 

 form of requiring M (a point in RP, produced either way if necessary 

 where RM = . OU and k is scalar) to lie on a given curve. It is 

 then most convenient to introduce two new scalars m and n y so that 

 OM=O+i . n) . OI, when the condition OM=OR+RM, gives the 

 two additional equations m=r-\-ku t n=s+kv, which with the five 

 others will serve to eliminate six of the eight scalars, m, n, p, q y r, s, u, v, 

 and leave the required abstract equation between the remaining two. 

 The eliminations are very simple in a great variety of curves. This 

 theory is fully illustrated by examples. . 



The first problem in the transformation of coordinates, ' given an 

 alteration in the concrete and curve equations, to find the corre- 

 sponding alteration in the assignant equation, so that the curves may 

 remain identical, extent excepted/ is solved thus. Given 



OP=(P 1 +i.P 2 ).OI, , = 0, C 2 =0, A = 0, 

 for the original curve, and 



OP' = (P/+i . P 2 ') . OI, C/=0, C 2 '=0 



for the new equations, where the unaccented letters are formations 

 ofp, q t r, 5, and the accented of p 1 , q f , /', s'. Since OP=OP', we 

 have ?! = ?/, P 2 = P 2 ', between which and C 1 =0, C 2 =0, A=0 

 eliminate p, q, r t s and use the final equation, which will only involve 

 p', c, r', s' as the assignant equation A'=0, which is independent 

 of any particular form of the curve equations C/ = 0, C 2 f =0. The 

 ordinary case of the transformation of coordinates is a particular case 

 of this. The second problem, ' given an alteration in the concrete 

 and assignant equations, to find that in the curve equations,' requires 

 the assignant equation to be put in the form 04-^/=0, which is 

 possible in an infinite number of ways. Then if the locus be that of 



OP=# . 01+ y . OJ, 0(tf, y) + ftar, y) = 0, 

 and we have given 



OP= [P a (p, q,r,*) + i. P a (p, q, r, s}} . OI, 

 <f>'(p, q, r, *) + \l/(p, q, r, s) = 0=A, 



we put 0=0', 4>=^' and hence finding 



x=t(p, q, r, s), y = rj(p, q, r, *), 

 we use P!=$, P 2 =*7 as the two curve equations. The third pro- 



