1098 SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 



the curve is of the form last described (PI. V. fig. 5), having the axis of Y for its 



asymptote ; (3) in the forms (e) and (£), or (H 1 — ) and (+ H ), the curve is of 



the same form, having the axis of Y for its asymptote. The linear transformation of the 

 radial equation with negative terms has been found to be a closed curve in all the 

 examples I have tried ; it apparently only becomes asymptotic when the coefficients 

 X and jjl are equal. The oval and the mushroom-like forms of Plate V. fig. 7, are traced 

 from the equation rJ/\ - rl/p = 1 by giving different values to /i and c. 



It is easily seen that in the case of bi-radial curves with diverse coefficients, \ and /x, the 

 transformation to polar coordinates will not give rise to a simplified expression, because 

 the uneven terms of the expansions do not disappear. The equations in x and y contain 

 in general all the even powers of y and all the powers, even and uneven, of x, and are 

 similar in form to those which have been considered as resulting from an oblique section 

 of a central surface parallel to a plane through one of its axes of symmetry. 



