607] 
A MEMOIR ON PREPOTENTIALS. 
415 
or, if for a,.., c, e we restore the values A — L,.., G — L, E — L, then 
a? + ...Jr- + Eo,*-L)» = j> • UK* + 4 - L)... (t + 0-1) (t + B- L))-K 
= —J dt {(t + A)... (t + G)(t + E) (t + ; 
viz. we thus have 
/ 
dS 
{(*$#,.., 0, w, l) 2 }^ 
= (" dt {(< + A ) ... (t + 0) (t + E) (t + 
where (t + A) ... (t + G) (t + E) (t + L) is in fact a given rational and integral function 
of t ; viz. it is 
= — Disct. {(* \X,.., Z, W, T) 2 + t(X*+...+Z 2 +W 2 -T% 
147. Consider, in particular, the integral 
i dS 
J {(a — foe) 2 + ... + (c — hz) 2 + (e — Jew) 2 + ¿ 2 }* s ’ 
here 
(*$X,.., W, T) 2 +t{X 2 + ...+Z 2 +W 2 -T 2 ) 
= (aT -fX) 2 + ...+(cT-JiZ) 2 +{eT-le W) 2 + l 2 T 2 + t{X 2 + ... + Z 2 + W 2 — T 2 ) 
= (f 2 + t) X 2 + ...+(h 2 + t) Z 2 + (Jc 2 +1) W 2 + (a 2 + ... + c 2 + e 2 + l 2 -t) T 2 
- 2afXT - ... - 2chZT - 2ek WT; 
viz. the discriminant taken negatively is 
t+f 2 ,... , -af 
...,t + h 2 , — ch 
— af,... — ch, — (a 2 + ... + c 2 + e 2 + P) + t 
which is 
( fi 2 -f2 X2 fa 2 
t — a 2 - ... — c 2 — e 2 — l 2 + + • • • + 
= {«(* +/>).... (t + (t+*)} (i - ^ -... - ^ -?) 
= (t + ^4)... (t + G) (t + E) (t + L); 
e 2 k 2 \ 
+ t + k 2 ) ) 
and consequently — J.,.., — (7, — E, —L are the roots of the equation 
iJL-t = o. 
t+f 2 t + h 2 t + Jc 2 t 
148. The roots are all real; moreover there is one and only one positive root. 
Hence, taking —L to be the positive root, we have A,.., G, E, —L all positive, and 
therefore d fortiori A — L,.., G — L, E — L all positive: which agrees with a foregoing