MB. J. J. WALKER ON THE DIAXIKTKKS OF A PLANK CUBIC. 189 



giving (36) for the CoTES-point on the u-polar line of that point 



x : y : z = bajfy + My 3 : a^' 8 + o^y' 2 *' : - a,V 8 - S&a^V. . (97) 

 The values give 



...... (98) 



in virtue of the relation (92) 



viz., the above is the equation of the Cotesian for the form (93) of u now used. 



66. The Cotesian v for the forms of u and. s at present employed (93, 94) has been 

 found quite independently of any application of the general equation (37) given 

 above ; but it may be of interest to test the general formula by this result. 



The only terms in the general expression which do not disappear in virtue of 



/ = 0, m=0, F = 0, G = 0, H = 0, 

 are four times 



\ - 



n > 



which, taking the values of u, s, A, B, C (93, 94, 95), for the present case, is identi- 

 cally equal to four times 



8 V (by* + cz 8 + Sa^y + Sa^z + 3b 3 y*z + 3c,2 2 .r + 3c 2 z 2 y) 



+ a^) + a s 2 a 3 6 3 n 4 (by + b s z) -f bafn* fax + c# + cz)} 

 a,6,z 2 ) n* [ba^(a^y + a 3 z) + (- a 2 2 n 2 ) (by + 6 3 z)} 

 = fea z 3 n 6 (by 3 



Rejecting the factor 6a 3 3 H a , and applying the relation (92), 



a 8 &3/& = 03, or 6o3/a 2 = 6 8 , 

 this result is, as before (98) ; or, with the rejected factors restored, 



v = (by 3 -f Sogafy - a^z - b^z) X 4&a 3 s n a , ... (98 bis) 



a cubic having a node at the point 



x = 0, y = 0, 



