1082 HON. LORD M'LAREN ON SYSTEMS OF 



(3) A curve whose equation is strictly symmetrical, and consists of terms of even 

 powers, whereof only one homologous pair have unlike signs, can have only two loops ; 

 but if any of the homogeneous expressions u n _ 2 , ti„_ 4 , &c, has a middle term, the curve 

 may have a number of loops depending on the degree of the equation, because then the 

 angle for which r = depends On a relation between three terms. 



(4) A curve, consisting of loops passing through the centre, is also the result where u n 

 is positive, and w„_ 2 + u 4 , &c, consists of pairs of positive terms and pairs of terms which 

 are both negative ; because evidently there must be definite values of 6 which render 

 r = 0. 



(5) If the terms u n _. 2 4- u n _ i} &c, can be resolved into factors, while u n consists of pairs 

 of unlike terms, the hyperbolic branches may break up into detached ovals sometimes 

 with an infinite branch extending beyond these and within the same angular space. 



These seem to exhaust the possible combinations for symmetrical equations without 

 an arbitrary term. 



(6) If we transform to axes equally inclined to the original symmetrical axes, the curve 

 will be symmetrical about the new axes also, and the new equation will consist of even 

 or of uneven powers of the variables, according to the rule of signs given above (p. 1065). 

 In applying the rule, each homogeneous part of the equation is to be considered 

 separately; so that, if one homogeneous part consist of positive terms, and the other of 

 alternate positive and negative, their equivalents in the transformed equation will consist 

 respectively of even and uneven powers. 



(7) There are limiting parabolic forms where the highest homogeneous part contains 

 only one of the variables, i.e., consists of a single term. 



(8) In the case of axes which do not meet the curve except at the centre, these are, 

 notwithstanding, true diameters, as the form of the equation proves. Accordingly, 

 every such Exterior Diameter, if I may so term it, bisects the intercepts made by the 

 adjacent branches or chords drawn parallel to the conjugate Exterior Diameter, and 

 therefore bisects the Bitangents. 



(9) These results are manifestly true, with the necessary restriction as to angles, for 

 all projections of the curves in question. 



(10) By an easy extension of (8) we have for all symmetrical equations of this type, 

 and their projections, this relation : Each pair of Bitangents is parallel to one axis of 

 symmetry, and is bisected by the axis conjugate to it. 



(16a). Examples of such Curves (Sixth Degree). 



Any of the functions on the left side of the sign of equality may be combined 

 with any on the right ; but of course the terms, when equated to zero, cannot all be 

 positive. 



The limits suited to this paper have been already so far exceeded that I shall not 



