290 Mr. J. Cockle on the Solution of 



It is immaterial whether X^. contains x, or not, but it is a con- 

 dition, essential to the solution of (c), that Um+^ should in- 

 volve X. There are of course other conditions, but I shall not 

 here examine them in detail. 



(5.) Let tt^ -f II be of the nth degree in a: ; we will now pro- 

 ceed to consider a few instances in which m given equations 

 admit of what I have (Phil. Mag. S. 3. vol. xxxvii. p. 502, art. 17, 

 and p. 503, art. 19) denoted by the expression a * determination ' 

 of the wth degree. And, first, let us proceed to the system 



U=flP*, V=6P« 



already (Ibid. pp. 372, 373) treated of by Mr. Sylvester. 

 (6.) Let 



-P2=fl, 



then, in the present case*, we have 



and, if we make 

 we have 

 where 



Ai=A^-, = 6U'-«V', 



U=U'^2andV=V'a?^ 



Now, U and V, being homogeneous quadratic functions of x and 

 y, U' and V, are quadratic functions of x~hj and involve no other 

 undetermined quantity. Hence we may satisfy 



Ai=0 

 by means of a relation of the form 



y=px, 



p being known and x left wholly undetermined. Consequently, 

 the relation 



^-iV-P2=:0 (d.) 



being the only one remaining to be satisfied, the problem admits 

 of an 71^^ determination capable in the present instance of being 

 reduced still lower. 



(7.) For, since Y' is a known quantity of the form 



ap^ + ^p-^y, 



the equation (d.) is equivalent to 



±a: ^/FH^Mn^oTt) =P, 

 an equation of ^n dimensions; n being, in this particular instance, 



even. 



* In the present case we might, perhaps advantageously, have made 

 Q=—abF^. So in art. (9.) we might have employed ABC« in place of «. 



