NUMBER OF UNKNOWN QUANTITIES. 60 



In the example above taken, we have supposed that the number 

 of equations and unknown quantities were the same, but if we supposed 

 that following the same law as in that example, the number of equa- 

 tions were n + m, then the numerator N which was shown to be of 

 n dimensions, ought to vanish when x is any number of the series 



1, 2, 3 n + m; that is, the equation A^=0 has more roots than it has 



dimensions, which is impossible ; it is therefore equally impossible to 

 satisfy all the given equations. 



On the other hand, if the number of the given equations was 

 only n — m, then n would by the preceding reasoning have a factor 



{x — l){x — 2) {x-7i-{tn), 



and since it is of n dimensions, it must have another factor of m dimen- 

 sions, as C {x - a^) {x — a-i) (x — a„). 



Hence - -\ — ^ H "* ^ ^- 



X x + 1 x + 2 x + n 



_ C(x—'l){x—2) .{x~ tl + m){x — ai) jx — a-^ { x~a,„) ^ 



■~ ''"'xT{x + l){x + 2) [x + n) ' 



following now the same steps as before, we find 



^^ cj-iy.a.az g. . g^( ly "■('»-^) jn-m + i) 



«.(« — 1) {n-m + 1)' '' ' ' ai.a-i a,„ 



c(-l)".(l+ai)(l+a.> (Ifg.) ^ (l+aiXl+aa) (!+«„,) w n-m + l 



'~ (w-1)(m — 2) {n — m + 2) o, . a^ a,„ '1' 1 



^. ., , (2 + a,)(2+a,) {2+a^) n.{n + l) (ti-m+l) {n-m+ 2) 



Similarly, %■^= - ~ — • ., ^ • i ^ • 



The quantities a„ a.^ a„ are evidently arbitrary, and each of the 



required quantities », z-., he. x„_„, are here determined in such a manner, 

 as to contain the m arbitrary constants. This is therefore the most, 

 complete solution of the problem. 



