dotible^ treble, and other Systems of Algebraic Equations. 431 



number 7- is greater than three, if«4-/3 + y + 8 = (r— 2)« + 2; 

 but if a + /3 + y + 8 = (r — 2) ?^ + 1 only possible for the 

 case of ?z = 2, and for that case alone. 



Particular method applicable io four Quadratics. 



Let U = 0, V = 0, W = 0, Z = 0, be four quadratic equa- 

 tions existing between ^, y^ z, t: 



Make ^.U = x .\ = jr.W = .r.Z = 

 3/.U = ?/.V = 7/.W = 3/,Z = 



x.U = s.V = z.W = 2.Z = 



/.U = ^.V = ^.W = t.Z = 0. 



Also write U = a;^ . F + ?/ . F' + z . F" + ^ . F" = 

 y = x\G + 7/.G' + Z.G" + t. G'" = 

 W ^ x^ .H + 7/ .W + z .H" + t . H'" = 

 Z = xKK+y.K' + ^.K" + t. K'" = 0. 

 By eliminating linearly we get 



2{FSG'. (H"K"'- H'". K")} = 0, which will be of the third 

 degree, since the factors represented by the unmarked letters 

 F, G, H, K are of zero, and all the rest of 7i?iit dimensions. 



Similarly we may obtain other equations, so that besides the 

 sixteen augmentatives already written down, we have four se- 

 condary derivatives, namely, 



n(2iii) = o n(i2ii) = o n(ii2i) = o n(iii2)=o. 



Thus we have tixienty equations and as many arguments to eli- 

 minate, since a perfect cubic function of four letters contains 

 twenty terms. 



The final derivee will contain 16 + 4''4 letters, i. e. 32, 8 

 or 2^ belonging to each system of original prefixes in each 

 member, and will therefore be in its lowest terms : for one of 

 the canons of form teaches us, a priori, that every member of 

 the derivee deduced from any number of assumed equations 

 must contain in each member as many prefixes belonging to 

 one equation of the system as there are units in the product 

 of the indices of all the rest taken together. 



Corollary to Case (A). 

 Either of the two methods given as applicable to this case 

 enables us to determine integer values of X, Y, Z, which shall 

 satisfy the equation 



X.U + Y.V + ZW= F.a;^/. ^^ 

 where F is the final derivee and p + q +- ?• = 3 « — 2. For 

 by the doctrine of simple equations we know how to ex- 



