428 Pi-of. Sylvester on a linear Method of Eliminating between 



absurd : if now any one, as [a), be equal to or greater than «, it 

 may be made to supply an integer part to the multiplier of x". 



Here it may be asked what is to be done with such terms 

 as K . .r" .3/* . z'^, when two letters «, b are each not less than 

 their correspondents «, |3 : the answer is, such terms may be 

 made to enter under the multiplier of cz", or oi afi, or to supply 

 a part to both in any proportion at pleasure*. 



From the equations above we get, by linear elimination, 

 F . G' . H" + G . H' . F" + H F. G" 

 - G . F . H" - H . G' . F" - F . H' . G" = 0. 

 This may be denoted thus : IT (a, /3, y) = 0, which equation I 

 call a secondary derivative, and the left side of it a secondary 

 derivee; «, /3, y may likewise be termed the indices of derivation 

 (as ?•, 5, t, etc. are of augmentation). 



Now since « + /3 + y = /« + I, it is clear then the index 

 of n (a, /3, y) is always n -\- n -\- n — [n + 1) ; i, e. 2 ?« — 1 . 



1st. Let any two of the indices of derivation be taken zero, 

 then it is easily seen that all the terms in 11 («, /3, y) vanish, 

 and consequently the secondary derivative equations obtained 

 upon this hypothesis become mere identities, and are of no use. 



2nd. Let any one of them become zero. 



It is manifest, from the doctrine of simple equations, that 



n(«,/3,y)niaybemade equal to < A.U-|-/x.V +v.W>'— ' 



or /a' . U + h/,' . V+ v'. wV -^» 



or/x". U + 1,^". V + v".wl--y' 

 upon the understanding that 



A =G'.H"-G".H', ^ =H'.F"-H".F, v =F.G"-F".G', 

 X' =G".H -G .H", n^' =H".F -H .F", v' =F".G -F .G", 

 x"=G.H'-G'.H, ^."=H.F-H'.F, v" = F .G' -F.G.' 



The three rows of coefficients will be respectively of the 

 degrees (w-/3) + {ii-y\ {n-y) + («-«), (m-«) + («-/3). 



Thus if any one of the indices «, /3, y be zero, 11 («, /3, y) 

 becomes identical with A'^ . U + (j.'' . \ + / . W, where the 

 multipliers of U, V, W are of 2 « - (« + /3 + y) dimensions, 

 i. e. of {n — 1) dimensions, and may accordingly be put under 

 the form 



2A.x\j/''. /". U+%.B..v'./. z'".Y + 2 . C x'./. z'". W, 



* The prefixes of any such terms (say K) may be conceived as made up 

 of two parts, an arbitrary constant, as e and (K - e) ; e will disappear spon- 

 taneously from the final derivee. 



