SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1059 



In equations (1) and (2) p is the variable parameter of the section parallel to xy, 

 and is evidently the ordinate in the principal plane xz. Hence, x y = p; x m = p n =f(xy). 



Also the value of z in the two sections is the same. 



Substituting in (2) the value of x above found, we have for the equation of the 

 surface 



x n + A 1 x n ' 1 y + A 2 af - Y =F =F Ky n = z n T B^"-' ± . . . + B n z n±p ; 



whence values of x and y can be found by homogeneous variation to any argument z. 

 For, we have only to suppose z constant, and to state the sum of the terms of z as an 

 arbitrary quantity in the form P re , and the problem is reduced to that of the introductory 

 proposition. 



In the example given, the plane XY is a plane of symmetry, and the axis z is an axis 

 of symmetry ; but these considerations are insufficient to determine similar parallel 

 sections unless the quantities x and y are combined homogeneously. It is easy to see 

 that in the example given the sections parallel to XZ and YZ are not similar, because 

 they are neither homogeneous functions of the respective pairs of variables, nor of these 

 quantities and the parameter. 



On trial it will be found that no other generating curve, except a homogeneous curve, 

 gives similar parallel sections. If possible, let the sections of the unknown surface taken 

 parallel to the following symmetrical equation be similar curves 



x n + A. 1 x n ~ 2 f+ A 2 af-"V =F =FA 2 ccy- 4 + A 1 x 2 y n ~ 2 +y n = P* . 



In order that the sections parallel to the plane of xy may be similar, their equations 

 must be homogeneous functions of x, y and P, as has been proved. Hence the condition 

 is satisfied by substituting z = P or f(z) = Y. 



It thus appears that a central surface, other than a conical surface, will not furnish 

 sections parallel to xy which are similar curves, unless the third quantity, z, be given 

 explicitly, so that the terms of x and y alone constitute a homogeneous equation. In 

 treating of equations in this perfectly general form, one is apprehensive of some possible 

 exception or flaw in the demonstration ; I have accordingly taken pains to verify this 

 conclusion, by endeavouring to find values of x and y from the surfaces generated by 

 various non-homogeneous curves, choosing the most symmetrical forms referred to con- 

 jugate axes as being those which were most likely to give results. 



In every instance the values of x and y, found on the assumption that the parallel 

 sections were similar, failed to satisfy the original equation, although they must 

 necessarily have done so had the hypothesis been correct. 



I consider it then demonstrable that the sections of homogeneous surfaces are only 

 similar curves when the sections are homogeneous functions of x and y equated to powers 

 of z uncombined with x or y. In other words, the homogeneous function must be of the 

 form f(x, y) =f(z, P) ; otherwise parallel sections will be dissimilar. 



VOL. XXXV. PART IV. fao. 23). 7 Y 



