Mr. J. E. Drinkwater on Simple Elimination. 27 



We thus get, /•rvr/T' f.. IM 



/fxYZT...(n)} = ± X„/{YZT ... (n-1)] 

 -Y„/{XZT...(n-l)}-Z„/[YXT ... («-l)}-&c.} 

 (8.) If any factor in/{XYZT...(«)}, as X, be divided into 

 two parts, X = V + W, the function may be similarly divided, 



so that . r„ , M 



/{(V+W).YZT ... in)} =/{VYZT ... («)} +/{WYZT ... («)} 

 placing each part of X in the same relative position (which in 

 this example is the first) which X itself occupied before the 

 division. „ 



(9.) If any quantity which does not vary from one equation 

 to the other, and which therefore is not liable to be affected 

 with an index, is found under the symbol, it may be consi- 

 dered a constant coefficient of every terni of the developed 

 function ; and written as such on the outside of the symbol: ot 

 this nature are the unknown quantities themselves, so that tor 

 instance, /{XYorZT ... (w)} = a;/{XYZT ... («)} 

 and so of like quantities. 



(10 ) Premising these properties of this function, it is easy 

 to show that the final equations derived from those proposed 

 in (2) are/{AYZT ... («)} + ^/{XYZT ... {n)}=0 

 /{XAZT ... in)} + 3//{XYZT ... («)} = 

 &c. 

 (11.) In the proposed equations let A + X^ = B, and sup- 

 pose the theorem proved for n-l equations, we have, 

 B, +\\i/ + Z,z +T,t + ...{n-l) = 

 B2 +Y^7j + Z^z +T^t +...(«-!)= 



fi^i + Y_,j/ + Z^i« + T„_ii+ ... {n-l) =0 



B„ +Y„7j +Z„z +TJ +...(«-!)= 



From the n-l first of these we get, by the supposition, 



/{BZT ... (n-l)} +i//{YZT ... (n-l)} = 



/{YBT ... {n-l)} + zf {YZT ... {n-l)} = 



Multiplying the first of these by Y„, the second by /„, &c., 

 we get, by comparison of the result with the last ot the pro- 

 posed equations, 

 B„/{YZT...(«-l)}-y„/{BZT...(«-l)}-Z„/{YBT...(«-l;}-&c. = 



We have already seen (7) that this expression is equivalent 



to ±/{BYZT ... («)} 



• /^BYZT ... {n)\ = 0, or restoring the value of B 



=/{( A ^■Xx). YZT. ..(«)} 



dividing 



