650 
PROFESSOR SYLVESTER ON THE REAL 
where 
A=-*±£ b=**- 
10 5 
Hence, by definition, 
L=AB 3 (A 2 — B) 4 = - i 25ho 9 (a a +5 ! )(aV)((a a -i7-16« 2 5 2 ) ; 
and making 
T 
(«2 + c 2) 2 3’ 
L /=i25 1 w(i’- 16 2)--^-P- 16 2)‘- 
(80) Now let us write 
^Z=j?L+^JI)( 63 )+gJ 3 . 
This gives 
5i2Z / =^J / D ; +g(^+4^)J, 3 +?jL, 
or 
4p 4 +128^y+10242y+1638% 3 
= 125(256 i -¥+2V 2 )e+(i)+4 2 )(6 2 )+64 2 )> s +i( # -16 2 ) 4 ) ,), 
by means of which identity we can obtain linear equations for finding the values of e, s, sj. 
Thus, equating the coefficients of y, q 4 ,^p 3 q respectively, we obtain 
4=216 s ++ 
4.64 3 s+-27*j=0, 
which gives jj= — 2 n s (as it ought to do), 
128=(24x125>+(4x216 + 108x64)s+ 64 2 n s 
= 3000e+ 8800s. 
Hence 
] 2 10 
200e=4, .= 35 , 1=-2S’ 
3000^=128 — 176= — 48, e=-^ and + + 
In order to verify the value of e, let jo=— 4, q=-\ ; then, assuming the correctness of 
the above determinations, we ought to find 
4 5 -128.4 3 +1024. 16+16384=125(256. 16-24.64). LL? + . 160000 . -2 U . , 
or 
2 10 (1 — 8+16 — 64)=(— 32.256 + 48.64)— Jj: X160000, 
or 
2 10 (-55)= -5120-25. 2048=2 10 (-5-50), 
which is right. 
( 63 ) Since Z has been proved to be of the form pA + gLD, we know a priori the value of £; but I have 
thought it safer to determine s, rj independently, as an additional check upon the accuracy of the computations. 
