432 Prof. Sylvester on a linear Method of Eliminating between 



press F in terms of the linear functions, out of which it is ob- 

 tained by permutation, i. e. we are able to assign values of 

 A, B, C, and their antitypes, as also of L and its antitype, 

 which shall satisfy the equation 



S(A.:^•^/.^'•".U) + 2.(B..^•^,//'.V) + 2(C.^'^/.;.''.W) 

 + 2(L.n(«,/3,y)) = F.x/3/=\;.'' . . . (1.) 

 where A, B, C, as well as L and all the quantities formed 

 after them, are made up of integer combinations of the original 

 prefixes. 



Now the functions IT (a, /3,y) may be expressed in three ways 

 in terms of U, V, W, as has been already shown. 



We may therefore suppose these functions to be divided 

 into three groups, and make 



^ Q . U + Q' . V + Q" . W 



+ ^-' .y J 



And it is evident that the equations (1.) and ('2.) lead imme- 

 diately to the equation 



X.U + Y.V + Z. W= Fx"+/./ + ^.,t^ + '', 

 if we call «, b, c the greatest values attributed respectively to 



«) /3» y- 



Now if we suppose the first method to be followed, 



f + g + h = '-An — \. 

 And it will always be possible to make a, b, c of what values 

 we please subject to the condition of a + i + c = n — \ ; 

 for one at least of the indices of derivation in IT («, /3, y) must 

 be not greater than its correspondent among a, b, c ; other- 

 wise a + /3 + y would be not less than (a + 6 + c) + 3, 

 but « + ^ + y = « + n ^^^^j^,^ j^ ^^^^^^ 

 a + b -^ c = n — IJ ' 



Hence we can satisfy X . U + B . Y + Z . W = F . *'^/ . z'', 



;j, q, r being subject to the condition of ja + q + r = 3 n — 2, 

 but otherwise arbitrary. 



Moreover, we can not do so if JJ + q + r he less than 3 « — 2, 

 for that would require a + b + c iohe less than n — I. Now 

 if two of the indices of derivation, as « and /3, be made equal 

 io a + \, b+l respectively, the third y = {n + I) — {a + b 

 ^ 2) = {n — I) — {a + b), and is therefore greater than c: 



