10G6 HON. LORD M'LAREN ON SYSTEMS OF 



degree with unlike signs, then in the expansion of these terms in the transformed equation 

 the sum of the partial coefficients is zero for all terms of even 'powers. 



4. If the sum of the indices in each term be uneven, then the expansion consists of a 

 homogeneous expression containing only the even powers of one of the variables, and the 

 uneven powers of the other variable. In my notation, if the terms are both positive, the 

 transformed expression will consist of even powers of y and uneven powers of x. 



5. If we consider these equations only to be symmetrical where the terms of all the 

 homologous pairs have like signs, or where for all homologous pairs the terms have unlike 

 signs, then in the complete expansion of the transformed symmetrical equation of even 

 degree, the sum of the partial coefficients is zero for uneven powers in the first case, 

 and is zero for even powers in the second case ; in other words, if in the original equa- 

 tion, being of even degree, the homologous terms have like signs, the equation of the 

 curve, when referred to secondary axes, consists entirely of terms of even powers. If in 

 the original equation the homologous terms have contrary signs, the equation of the same 

 curve, when referred to secondary axes, consists entirely of terms of uneven powers. 

 These results are independent of the degree of the curve, and it will hereafter be shown 

 that they are applicable to the projection of any symmetrical equation obtained by 

 writing x/a for x and y/b for y (page 1069). Thus from the order of the signs of any 

 symmetrical equation it is immediately known to which of the previously named classes 

 the equation belongs, i.e., whether the curve represented is elliptic, hyperbolic, or 

 inflexional. 



6. These results are evidently true for any diametral, symmetrical equation, 

 although not homogeneous, because it is only necessary to the proof that the equation 

 should consist entirely of even or entirely of uneven terms. 



12. Diameters in Central Curves of the Fourth Degree. 



I shall now give a proof that every central curve of the fourth degree has two pairs of 

 axes of symmetry, and in general only two such pairs. 

 (1) Let the equation be homogeneous, or of the form 



Asd i + 'Bx' i y i + Cy*=l (1). 



To prove that in general the curve has not a pair of conjugate axes equally 

 inclined to a given line. Let the equation be referred to the given line and an axis 

 perpendicular to that line. It may then be written 



Dx^Exhj + FxY+Gxy^liy^l (2). 



If we now transform to axes equally inclined to the line by formula (2) p. 1064, it will be 

 seen whether the unknown angle 6 can be determined so as to make the uneven terms of 

 the resulting equation disappear, so that the resulting equation should be one referred to 

 conjugate axes. To this effect we are to make the x of equation (2) = (x — y) cosfl, and 



