SECTION III. 1882. AS? Trans. Roy. Soc. CANADA. 
IT—On an Application of a Special Determinant. 
By J. B. CHERRIMAN, M.A. 
(Read May 26, 1882.) 
Let 4r denote the following determinant of the rth order :— 
ee Mon Daun ee seen screen onesies ed tects oasis session icies eee tee D: 
11; fm flbnt ontasete 99 caencpe cosbeodeuRgaes aosouccdeds. don nue eocdcas 5 fs 
(ML YHE  drooannteousone AO A oder Men PRE TO SSD EEE Pr 
OO SE ee eee eme eue cc ete cer ces Pres 
(D (UU Ba sbaasodbode: ccantnoposaddné baonhood-ecacananecobsaueg vod ace, Pr 
A ES ne + — "pp, 
Tf now pr, po; Psy are the co-efficients in the equation 
Be typi) Dies EDR D Es cee ce reste entre +p, = 0, 
r being not greater than », and ©, denote the sum of the homogeneous products of the rth 
order of the roots, then from the relation 
Capt Olen Da Orne scene oir eee + p, =0, 
it is obvious that 
44 
r 
Again, if s, denote the sum of the rth powers of the roots, it is plain, ftom Newton’s 
expression for these sums, that —’s, is given by the determinant :— 
(Bi, Bin. BYP face cower cobone dcobogkBancesconbaGedn ob noce oct anbAv dbonoe Ds 
Ila flip Yi SoonacsoncoosbendasoocedsononsbboncuconDdc sce RboosnonBcncodar DE 
(L'ILE cocsessocmnvancectoronnonemmotbastotrbietonaceatboncesce Dre 
DOE OS” re anise woes 2 Sea's enr ne me enee > Pr 
which reduces to 
De Ag 2 gt Ang tat Pg. Ans eee ee + —'"t'? p,, 
and it follows, therefore, that 
— S, =D Oy + 2 Po One + 3 Pa Op ge ccceccscseccecsccscers + rp,. 
