450 MR. A. B. FORSYTH ON INVARIANTS, COVARIANTS, AND QTJOTIENT- 



= 4' V + Cu V + 4*V" + u, (p.* + 4Q 3/ ,'), 



and thence, by continued differentiation and substitution, 



= 10M 1 V + 10w V+ u\ W - 12Q^') + U + 4 



= 20t*V u + ""i (1 V - 



= 



u 



+ u\ 6/1' - 8 (Q S/A ') - 4Q,> 



- + 4 | 3 (Q 3/A ') - 4 (Q^ 1 

 u\ J21/t' - 20 (Qs/, 1 ) - 40 f t"Q 3 - 

 ) - 8 |(Q 4/ ,') - 40 |(Q 3/ ,") - lOQ^t" - 800,/t" 1 } 

 ) - 4 , (Q^) - 10 (Q,/) - 



(29). 



The determinantal equation which results from the elimination, between these four 

 equations, of the four quantities u lt u\, u\, u [ \ is the equation required ; it is 

 evidently of the seventh order. 



95. Had the initial quotient relation been taken u s = ujj, the equation in p would 

 have been the same as the equation in \L ; and similarly for an initial relation 

 w 4 = Uj\. Hence it is to be inferred that, if X, cr, p be three particular solutions of 

 the //.-equation, its primitive is 



A + EX + Co- + Dp 

 ti ~ A'+B'X+CV + D'/j' 



96. In particular, if in the original equation Q 3 =0, Q 4 = 0, so that the two 

 priminvariants vanish, the equation which determines \L is 



, 21/A 35^ =0 



(30), 



the left-hand side of which may be called the quartic quotient-derivative. Special 

 solutions of the original differential equation are now 



so that 



M! =1, u 2 = z, u 3 z 2 , t* 4 

 [L z, p = z 2 , X = z 8 ; 

 and, therefore, the primitive of the equation (30) is 



