72 
MR. A. CAYLEY’S SIXTH ^lEMOIR LPOX QrAXTICS. 
a relation which may also be expressed by the equations 
a!\ /3", ll3+m(3', ly+my', 
where m are arbitrary multipliers; aud substitutiug these values in the eq^tion 
„"*+S'V+/ 2 = 0 , we have for the equation of a line subjected to the sl^le conition of 
passing through the point of intersection of the Imes aa;+(3y+72_ , aa+(3y '/. 
the equation l(M;+(3y+y2)+m(o!'*+(3'2/+5''*)=®’ 
which is, in fact, at once obtained by the consideration that the lyues of (x, S'’ 
satisfy simultaneously the equations a*+(3y+r2=0 and cix+(3t/+'/^- , sa is . 
ax-^^y+yz=^0 and and also through the pomt («, b, c), is obvionay 
ocx-^^y-^y^, =^’ 
which, or the equivalent form 
«a;’+ 6 v+Y.g— + + ^ 
afl + j 36 + yc a! a + / 3 'Z> + y'c 
is usually the most convenient one ; but it is to be observed that the equation can also be 
written in the forms 
X , y •> ^ 
a , b , c 
(3y'—^'y, yoi'—y'a, oc(3'—oi'(B 
and 
or in the form 
hz—cy^ cx—az, ay—bx j=0, 
a , (3 , 7 ! 
a' 
m 
(I3y' - yf3')(bz—cy)-h(yoi'—y'(^)(ca;-az)+(ocf3' - f3cz')(ay-bx) = 0, 
= 0 . 
a, (3 , y 
X, y, z 
(3', y' 
a, b, c 
ioZ. io nna tim cuummabco 
(a, b, c), (a', b\ c% with the line ax-\-^y-\-yz=^, we have 
X, y., Z'=-’K(i-\-{^(^ ■> 'Kb-\-yjb\ 
where X, y. are given by y , / , , /ozi , j\—c\ 
’K{aa-\-^b‘\-yc)-\- yj{ot,a -\-^b + 7 ^ ) — 
The preceding are elementary formula; of almost constant occurrence ; it may be proper 
to add to them the formulae which follow. 
