( 1043 ) 



XXIII. — On Systems of Solutions of Homogeneous and Central Equations of the nth 

 Degree and of two or more Variables ; with a Discussion of the Loci of such 

 Equations. By the Hon. Lord M'Laren. (Plates I.-VI.) 



(Read 6th July 1888.) 



1. Principle of Homogeneous or Linear Variation of a 



Homogeneous Function, ..... 1044 



2. Application of the Principle to finding Solutions of 



Homogeneous Equations of one part. (Case I.), . 1045 



3. Solution of Equations of two Homogeneous parts of 



different Degrees. (Case II.), . . . . 1049 



4. Another mode of Solution ; viz., by expressing each 



of the r Variables in terms of r - 1 New Variables, 1050 



5. Solution of certain Homogeneous Equations by the 



introduction of a New Variable. (Case III.), . 1052 



6. S luble Cases of the Homogeneous Function F„(a. , J y,z) 



= 0, 1057 



7. Solution of Homogeneous Equations of Functions 



of the Variables. (Case IV.), .... 1057 



8. To find the Condition under whicli Parallel Sections 



of a given Surface may be similar Curves, . . 1058 



9. Classification and Forms of Curves considered as Sec- 



tions of Surfaces whose Equations are Homogeneous, 1060 



10. Classification of Central Curves of the Form 



F(a;, y)" = A", 1062 



11. Transformation to Secondary Axes — Rule of Signs, . 1064 



12. Diameters in Central Curves of the Fourth Degree, . 1066 



PAGE 



13. Diameters in Central Curves of Higher Degrees, . 1068 



14. Sextic Curves of the Homogeneous Form F(a;, y) 6 = A 6 , 1070 

 14a. To find the Equations of the Equiaxial Curves re- 

 ferred to Secondary Axes, ..... 1070 



146. Limiting Forms of the Equiaxial Curves, . . 1072 

 14c. Form and Variations of the Equiaxial Curves (a) 



and (0), 1073 



lid. Examples of the other Equiaxial Curves, . . 1075 



15. Determination of Contour-Lines of Homogeneous 



Surfaces, 1079 



16. Central Curves whose Equations are of the Form 



F^x, yy = ¥. 2 (x, },)»-?, 1081 



16a. Examples of such Curves (Sixth Degree), . . 1082 



17. Contour- Lines of Surfaces derived from Central 



Curves passing through their Centres, . . 1084 



18. The Wave-Surface, 1085 



19. Curves Symmetrical about One Axis, . . . 1086 

 19a. To find a Symmetrical Expression for the Oval of 



Single Symmetry, 1088 



19&. Examples of Curves of Single Symmetry, . . 1089 



20. Parabolic Limiting Forms, 1090 



21. Biradial Coordinates, 1091 



The purpose of the present paper is to ascertain how far it is possible to find, exact 

 solutions or values of x, y, &c., in equations between variables, so that the forms of plane 

 curves and contour-lines of surfaces may be exactly determined. No approximate 

 methods have been admitted, and only those methods have been used which are applicable 

 to algebraic equations of every degree and any number of variables. In the examples 

 given I have generally selected equations of even degree and even powers of the variables. 

 But every such solution evidently includes the solution of the non-central equation of 

 half the degree having corresponding terms and equal coefficients. The methods of 

 solution employed are founded on the following introductory theorem or principle, which 

 may be described as that of Homogeneous or Linear Variation of the quantities. 



The paper, as laid before the Royal Society in July 1888, embraced only the solution 

 of homogeneous equations in which one of the quantities was given explicitly in terms of 

 the others. The preparation of the paper for the press having been interrupted by my 

 absence abroad for a considerable time, I have resumed the investigation from a more 

 general point of view. 



VOL. XXXV. PART IV. (NO. 23). 7 U 



