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



Take the product XYZT («) of all the letters coeffi- 

 cient of the unknown quantities, and affect it with the succes- 

 sive terms of this series (with their proper signs), considered 

 as a series of indices. We thus form the series 



+ X1Y2Z3T4 {n) 



-X^YiZgT, (;0 



-hX^YgZiT^ {71) 



&c. 

 which we shall denote by y{ XYZT ... (m)} 



(6.) It will be observed in all the terms of/{XYZT... (ji)} 

 that the order of the factors remains unchanged, the indices 

 alone being permuted. If any change is made in the order of 

 the factors under the symbol, it is obvious that^the only possi- 

 ble change produced will be a change of sign, which will be- 

 come positive or negative according to the same rule of signs 

 already explained among the indices. For instance 

 /{XZYT ...(«)} = -/{XYZT... (n)} 

 and so of any other. An odd number of single steps by the 

 requisite number of letters requires the negative sign; an even 

 number, the positive sign. The entire order of the factors may 



be inverted in —^ single steps, and accordingly as this 



number is even or odd,/{XYZT...(n)} = +/{(n)...TZYX}. 



(7.) The function of n factors/{(«) TZYX} may be 



expressed in a series of functions of w— 1 factors, by consi- 

 dering that the permutations of n numbers are obtained by 

 successively adding each at the end of the permutations of the 

 11 — 1 others. Attending also to the change of sign necessary 

 to bring each factor in succession to the end of the product 

 under the symbol, which is necessary in order to keep the 

 products of n factors in the same order in each term so ob- 

 tained, 



f{{n) ... TZYX} = X„/{(«-l)...TZY} 



-Y^{(/^-l)...TZX} + Z„/{(7^-l)...TYX}-&c. 



or inverting the order of the factors under the symbol in every 

 term, 



/{XYZT...(«)} = ±X„/[YZT ...(«-!)} 

 -Y„/{XZT...(7i-l)}+Z„/{XYT...(n-l)}-&c.} 

 The sign of the first term on the right-hand side of this equa- 

 tion may be made different from that of all the others, by 

 making X under the symbol successively occupy every place 

 from the first to the last, the order of the others remaining 

 unchanged. 



We 



