XXV 



where a', //, c' depend upon a, b, c and Z, in, I', m'. The two values of 

 y determined by this equation are equal if (and only if) the quantity 

 a'c 6' 2 vanishes. But the equality of the two values of y depends 

 upon and is determined by the equality of the two values of x, the 

 latter equality being secured if the quantity ac b" vanishes. It fol- 

 lows that the vanishing of either of the quantities a'c'b' 2 and acb* 

 requires the vanishing of the other; and it is therefore inferred that, 

 when neither of them vanishes, one of them contains the other as a 

 factor. When the actual calculation is made, it is found thataV -b'- 

 is the product of ac b z and (lm! Z'/) 2 , the latter being a quantity 

 that depends solely upon the transforming relation. Consequently it 

 appears that a combination of the coefficients in the original equation 

 exists, such that when the equation is transformed by any relation of 

 the type indicated and exactly the same combination of the new 

 coefficients is constructed, the two combinations are equal to one 

 another save as to a factor depending solely upon the transform- 

 ing relation. Such a combination of the coefficients is called an 

 invariant. 



Again, it is known that every curve (of degree higher than two) pos- 

 sesses a number of points where a tangent to the curve not merely 

 touches it but, having contact of one degree closer, crosses it ; and it 

 is found that all these points, called points of inflexion, also lie upon 

 another curve uniquely derived from the first. When the curves are 

 represented by means of equations, the statement is that the points of 

 inflexion of a curve U = are given as the intersections of this curve 

 with a curve H = 0, the latter equation being uniquely derived from 

 U = 0. Now suppose that the axes, to which the curves have been 

 referred, are changed to another system, so that new co-ordinates x' t y 

 are connected with the former co-ordinates by relations 



. 



a z x + b 2 y + c^ ax + by + c 

 A new equation U' = 0, obtained by eliminating x and y between 

 these relations andU = 0, will now represent the curve. The change 

 thus made does not affect the geometrical properties of the curve ; 

 its points of inflexion are still given as its intersections with the curve 

 H = 0. But the points of inflexion of the curve represented by 

 U' 3= are the intersections of this curve with another curve repre- 

 sented by H' = 0, an equation derived from U' = in exactly the 

 same way as H = is derived from U = 0. It therefore appears 

 that the associated curve H' = cuts the given curve in precisely 

 the same points as the associated curve H = 0, a result which suggests 

 that the associated curves are the same. Now H' = has been 

 derived from U' = ; but actual calculation shows that, if the rela- 

 tions between x', y' and a*, y be used to eliminate x, y from II = 0, the 

 resulting equation is H' = 0; in other words, the relations between 



