of the Third Order, 517 



the node and the line (x— mz=0, y— nz=0). In fact, assuming 



(a 3 + / S 3 )(X-wZ)(Y-«Z)Z(7wY-wZ) 



-«/3(/3X-«Y) {(X-™Z) 3 + (Y-nZ) 3 } 



= {£(X-»iZ)-a(Y-nZ)}<S>(X,Y,Z), 



it will presently be shown that <3> is an integral function. Hence, 

 omitting the factor in question, we have 



(u 3 + j3?){X-mZ){Y-nZ) + <Z>{X, Y, Z)=0, 



which is the equation of the surface. It only remains to find <I> : 

 writing for this purpose X + mZ, Y + nZ in the place of X, Y, 

 respectively, and putting for a moment 



<J>(X + wZ, Y+nZ, Z) = &, 

 we have 



(« 3 + /9 3 )XYZ(mY-/iZ)-«/3{/3(X + mZ)- a (Y+wZ)}(X 3 + Y 3 ) 



= (/3X-*Y)<£>'; 



that is, 



(/3X-«Y)^=Z{(a 3 + / e 3 )XY(7wY-nZ)-(X 3 + Y 3 )«/3(?w/3-/ia)} 



-(/3X-aY)«/3(X 3 + Y 3 ); 

 or, effecting the division 



0> / =Z{(X 2 a-Y 2 / S) (arc-/3m)-XY(a 2 m-f $H)\ -a/3(X 3 + Y 3 ), 



and then writing X— mZ, Y—nZ in the place of X, Y respect- 

 ively, we have 



3>(X, Y, Z) = Z{((X-mZ) 2 «-(Y-wZ) 2 /3)(aw-/Sm) 



-(X-mZ)(Y-nZ)(a 2 »i-f/3 2 w)}-^{(X-mZ) 3 + (Y-7iZ) 3 }, 



And, finally, the equation of the surface is 



(a 3 + /3 s )(X-mZ)(Y-nZ)-u{3{{X-mZ) 3 +(Y-nZ)*} 



+ Z{((X-mZ) 2 a-(Y-nZ) 2 /3)(a?i--,&w) 

 -(X-™Z)(Y-7iZ)(a 2 m+/3 2 rc)}=0, 



which is, as it should be, of the third order. 



Arranging in powers of Z and reducing, the equation is found 

 to be 



(a 3 + y3 3 )XY-a / 8(X 3 +Y 3 ) 



+ Z { - (a 3 + J3 3 ) (wY + nX.) + (X 2 a + Y 2 /3) (m(3 + na) 



+ affimX 2 + rcY 2 ) - (a 2 77Z + /3 2 rc)XY} 



+ Z 2 {m7i(a 3 + j Q 3 -^X-^ 2 Y) + ( / 3/i 2 -a7W 2 )(/3X-«Y)}=0. 

 The first form puts in evidence the nodal line 



(X-mZ=0, Y-nZ = 0), 



