276 Wilder's Algebraic Solution. 



+ S'lS'm~l ^ 4-S'lS'm-I '| 



+p8m-2 j ^p'^'m-2 j 



+ S''lS"m-l 1 _fSl"-3(l)S«-3(l),„_l ^ 



r ^ 



+p"S"wi~2 

 +(^"S"w-3 



4-«"S"m-(w~2) 



-\-u^S"m-{n-\)j -{-t<"-iS«-3Ci)m-(?i-l) j 



for the general expression of (A') ; the accented letters hav- 

 ing the same relation to a[3-\-ay-}-aS-{-f3y etc., a/3y-f a/3(5 

 ^ayS-\-j3y8 etc., a^y^+etc, that Si, S7/i-l,p, S??^-2, 

 etc., have to a+l3-j-y^S etc., a'"-' 4-/3'"-i -f-y'"-! +5"^-' etc., 

 al3-\-ay-\-aS-^^y etc., a'«-2_{_/3'»-2 _}_y'n-2 _|_(Jm-2 gj^,^ 



Proposition second. Let cp{xypq etc.), and cp'{xypq etc.), 

 be two functions of the independents xypq etc., and let (p be 

 a factor of 9', then I say that if any function (?"{xpq etc.), 

 written for y in (p', makes it identically nothing, it will also, 

 when written for y, make 9 identically "nothing. For if not, 

 we shall have by putting y'=(p"{xpq etc.), 

 ^'{xii'pq etc.) 



"H^'pg^ii<)"=K^^i;i^' "«• nothing divisible by some- 

 thing. And, reciprocally, if t" when written for y in *, makes 

 it identically nothing, it will when written for y in ^', make 

 it identically nothing. For if not, we shall have 

 ^'{xy'pq etc.) ^'{xy'pa etc.) 



^^5^^= 0^ ' ^''O' ^f^^to'" of *', which is 



impossible. 



Proposition third. If the function ¥'{xpq etc.), written for 



y, makes both ^ and ^' identically nothing, then I say that 



either 1) is a Victor of*' or $' is a factor oft, for we either have 



¥{xy'p q etc.) p{xi/pq etc.) 



IfZ77Z. — I77r\=ni OJ" I'r — ^ ^7 \=7^; and, it the division 



^[xypq etc.) 0' ^{xypq etc.) 0' ' 



does not take place independently of //', or what is the same,?/, 

 we shall have a relation between x^p, q, etc., which is contra- 

 ry to the hypothesis. We either have therefore ^{xypq etc.), 

 a factor off'^xtjjyq etc.), or ¥{xypq etc.), a factor of f{xypq 

 etc.), according as ^'> or < 4). 



