230 
THE EEV. GEOEGE SAXMOX OX QUATEEXAET CTBICS. 
a. (B, y, Every contravariant, then, in five letters will be a function of the differences 
between a, /3, y, a. 
It is easy to show now that «, (3, y, s are cogredient vrith ^ respect- 
ively. We know, in fact, that when U is expressed in terms of four letters, we may sub- 
stitute in any contravariant, for a, /3, y, 
d d d d 
dx' dy' dz’’ du 
When, then, U is transfoimed 
into a function of five letters, we must substitute for which (in vii’tue of 
(Lx (lx dv dx 
the relation connecting oc, y, 2 :, w, v) is equal similarly for &c. But 
these are the very transformations by which the contravariant in four letters is expressed 
in terms of five. 
It is equally easy to show that in a covariant we may substitute for x. y. z. u. f, 
d d d d d 
du W Jy W Te' 
Let us commence by applying these principles to calculate the discriminant and reci- 
procant of a quaternary quadric when expressed in the form 
ax^ + + cz'^ + dii^ -f - ev^. 
These functions are known when the quadric is expressed in the general form 
ax^ + ly"^ -f- cz^ -j- did 
-\-2lyz-\- 2mzx + 2nxy + 2])xu + 2qyu ■\-2rzu. 
In this case the discriminant is — - 
(No. I.) 
ahcd—{ add + hdnd -f cdn^ -f hcp^ -f cficf -{-ahi^) 
hnpr + cnpy -\-dImn) 
+ dyd -f - -f- — 2mnqr— 2nlrp—2l mpy. 
The reciprocant (which is also called the bordered Hessian, since it is obtained by bor- 
dering with contragredient variables the determinant Avhich expresses the Hessian) is — 
(No. 2.) 
od{l)cd~hf~ — cq^ — dd-\-2lqr) 
-f- (3^(cda — cqf — dm^ — af- -f 2pmr) 
-j- y‘^(f? ab — dn^ — aq"^ — bp^ 2nqp) 
-\-}d{abc—ad — bird— crd -f - 2 Imn) 
-f 2 /3y( — adl + aqr-\- dmn + — nrp — mpq) 
-\-2'yci{—bdm-\-bpr-{-dln-{-m^—nqr— Ipq) 
-\-2a^{~cdn-{-cqiq-\-dIm-\-nr^—mqr—Irq)) 
-\-2al{ — bcp-\- bmr + mq -\-dp—nJ)'— Imq) 
+ 2/3S ( — caq -\-alr-\- cnp -\-iifq — mnr — Imp) 
-\-2yh{ — abr-\-alq-{-bmp-\-fdr—m'nq — nIp). 
