MinorDete^'mmantsofLimarlyEqidmlentQuadraticFunctions. 301 



viduakai^bitrai-ily., selected .out , cdf ,tbev4»'— Vi)i nupxli;>er§,?»-i;L 

 ?« + 39.y.h»»)ilK ii'^i'ni sBfl 8B ,'XBOfri[ >ii -xofno smsg -jifHo Y JjriB IT 



,a*)OitTmfj:. /fl.i -. :.r.,';. ■,"'!- ..,;:i i , ! , - i // l>TM((foi(J; sdT • 



I'TUAif '■'f'fj i''''i<i/-,„. M j, ' ' : ' ,■■< — ^i) "}<! rtouiw.a ii .'''ii-i-v'd 

 If we make n=37anaw} = 7^ anaffy+^.a-y+s— V tor all positive 



values of either r or s, and «y_iff„+e=0 for all values of i and e 

 differing from one another, and for equal values ay_e-Oy+c^=^ — Ij 

 it will readil}' be seen that tliis last theory reduces to the one 

 first considered ; and on careful inspection it will be found, that 

 the solution given of the genei'al question includes within it that 

 presented for the particular case in question. Such inclusion, 

 however, I ought in fairness to state is far from being obvious ; 

 and to demonstrate it exactly, and in general terms, requires the 

 aid of methods which my readers would probably find to exceed 

 their existing degi-ee of knowledge or familiarity with the subject. 

 The theory above enunciated was in part suggested in the 

 course of a conversation with ]\Ir. Cayley (to Vv hom I am indebted 

 for my restoration to the enjoyment of matliematical life) on the 

 subject of one of the preliminary theorems in my paper on Con- 

 tacts in this Magazine.' ■M,,;iri;i!. 1.' ii'j// ■!'■ 'Mil- ,-iM|.;,i- ijiiC 



It is wonderful that a theory so purely analyiical shoidd ori- 

 ginate in a geometrical speculation. JNIy friend JNI. Hermitehas 

 pointed out to me, that some faint indications of the same theory 

 may be found in the Recherches Arithmetiques of Gauss. The 

 notation which I have employed for determinants is very similar 

 to that of Vandermonde, with Avhich I have become acquainted 

 since writing the above, in Mr. Spottiswoode's valuable treatise 

 "On the Elementaiy Theorems of Determinants." Vandermonde 

 was e\idently on the right road. I do not hesitate to affirm, 

 that the superiority of his and my notation over that in \ise in the 

 ordinaiy methods is as gi'cat and almost as important to the pro- 

 gress of analysis, as the superiority of the notation of the differ- 

 ential calculus over that of the fluxional system. For what is the 

 tlicor}'^ of detemiinants ? It is an algebra upon algebra ; a cal- 

 culus whicli enables us to combine and foretell the results of 

 algebraical operations, in the same way as algebra itself enables 

 us to dis))ense with the performance of the special operations of 

 arithmetic. All analysis ^jiust ultif\j\ately, (jlothc itself under this 



* FerliJips the most remarkable imlirect qtiestion to wliich the method 

 of ilf termiiiants has been hitherto apphcd is Hesse's problem of rechicihg 

 a cubic function of .'1 iettei-s to another coimisting onlv Of 4 terms In' Unoav 



