double, treble, and other Systems of Algebraic Equations. 435 



equal number of Vs, but of W's + 2 ' 



now n [n — \), n{n — I) and n^ exactly express the number 

 that ought to appear of each of these respectively : hence the 

 final derivee is clear of irrelevant factors. 



Ternary Systems. 



Case (C). Tim of the indices equal ; the third one, greater by 



a unit. 



Here, calling (??) the highest index, the augmentees must each 



be made of the degree (2 n—3,) their number will evidently be 



in-^)in-l) ^ („ »-!) » + (!i^^, making the sum 



of the indices of derivation now, as before, equal to {n + 1); 

 it will be still possible to form integer equations of decompo- 

 sition, which will give rise to augmentatives of the degree 

 {71— a) + {n — 1) —^ + {n — l)- y, i. e. of (2«— 3) dimensions. 

 The total number of equations, what with augmentatives and 



1 1 • .• -ii , r (» - 2) (» - 1) . (n-l)n 

 secondary derivatives, will be <. n — *" 2 



{n-l)n\ 71 {n- I) _ 4?t^— 4m + 2 _ (2 n-2) (2?t-l) 



+ 2 J + 2 ~ 2 2 ' 



i. e. is equal to the exact number of distinct arguments con- 

 tained between them. 



Also the final derivative will contain in each member 



{n~2){n- l) n.{7i-l) j. e. (?j - 1) (w - 1), letters be- 

 2 "^2 



^(n -^ l)n n. {n — 1) . 

 longing to the first equation, and 1- ^ » i- e. 



n .{n — ]) belonging to those of the second and of the third, 

 and will therefore be in its lowest terms. 



Corollary to Case (B) and (C). 

 It is not necessary, alter all that has been already said, to 

 do more than just point out that the processes applicable to 

 these cases enable us to determine X, Y, Z, which satisfy the 

 equation X.U + Y.V + Z.W = F.^.y^.z'^^ 



where f + g + h = 3 ti - 3 for case (B). 

 and f+ g + h = 3 ?« - 4- for case (C). 



22 Doughty Street, Mecklenbuigh Square, April 2i), 1841. 

 [To be contiiuicd.J 

 2 F2 



