SOLUTIONS OF HOMOGENEOUS AND CENTRAL EQUATIONS. 1085 



of computed values of x and y, although I have not printed the tables of computed 

 places. 



18. The Wave-Surface. 



This surface, as usually given, is of the form 



a 2 x 2 /(r 2 -a 2 ) + by/(r 2 -b 2 ) + c 2 z 2 /(r 2 -c 2 ) = . . . . (1). 



This, when cleared of fractions and expanded, is an equation of the 6th degree, contain- 

 ing all the terms of even powers of the general sextic equation of three variables. 



If, however, the equation be merely cleared of its fractional form, and the terms be 

 arranged in powers of r, it has the form 



{ a 2 x 2 + b 2 y 2 + c 2 z 2 }r i — a 2 b 2 c 2 r* + a 2 b 2 c 2 r 2 = ; 



whence, dividing by a 2 b 2 c 2 r 2 , and writing a, /3, and y, for b 2 c 2 , c 2 a 2 , and a 2 b 3 , we have 



{x 2 /a + y 2 ll3 + z 2 ly}r 2 -r 2 + l = 0; (2), 



where 



r 2 = x 2 + y 2 + z 2 . 

 This may be written 



w- 4 — «- 8 +l = ( 3 ) ! 



where u± and u 2 are homogeneous functions of x 2 , y 2 , and z 2 (or of x 2 and y 2 in the plane 

 curve), consisting entirely of positive terms. 



The generalised form of the wave-surface, or wave-curve of any even degree, 

 evidently is 



u m + u„+ . . —u p — u g +l = (4); 



where u is defined as above. The equation has an arbitrary term. The definition of u 

 implies that each homogeneous part of the equation consists of terms of like signs, and 

 under this condition this equation of different homogeneous parts represents an oval 

 (though it is usual only to consider the semi-oval) entirely concave to the centre. If 

 any of the homogeneous parts u v should consist of a homologous pair of negative terms 

 and a homologous pair of positive terms, the curve would be the inflexional oval (PI. IV. 

 fig. 5); but, as already seen, so long as each homologous pair of terms have like signs, 

 r can neither become or oo ; and the curve or surface is always and necessarily a 

 continuous closed curve of double symmetry. 



Plate III. (fig. 7) is a representation of the limiting form of the 4th degree, obtained 

 from equation (1) by suppressing the 3rd term. The reduced equation is 



a 2 x A + (a 2 + b 2 )x 2 y 2 + b 2 y* — a 2 b 2 r 2 , 



