766 
PROFESSOR CATLET ON PREPOTENTIALS. 
146. For a,b...c,e write herein a + 6, b + Q . . . c+5, e-\-6 respectively, and multi- 
plying each side by Q 2 ~\ where q is any positive integer or fractional number less than 
% s , integrate from 5=0 to 5=oo . On the left-hand side, attending to the relation 
| 2 +^ 3 . . . -]-£ 2 + 0 2 =l, the integral in regard to 5 is 
r* dt 
Jo ^ 2 +0p +1) ’ 
where g 2 , =a^ 2 -{-bif . . . -j-c£ 2 -j-ea 2 , as before is independent of 5; the value of the 
definite integral is 
__r(K* + l)-g)r( 0 ) 1 
r±o+i) gs+1-22’ 
which, replacing by its value and multiplying by dS, and prefixing the integral sign, 
gives the left-hand side ; hence forming the equation and dividing by a numerical factor, 
we have 
+ c£ 2 + eco 2 )- 
o/Tu.y+i p 00 
= r^r4(s + 1 ) - q 1 o . . . t+c . 
and in particular if q— — then 
dl, _2(ri) s 
+ct? + eWf s ~ I> 
( dt. . . . t-\-c. t-\-e)~\ 
or, if for a ... c, e we restore the values A— L. . . C — L, E— L, then 
C d s 
J(A0 2 ...+C? 
f "■ ^+ A - L • ■ • *+ c - L • ‘+E-in 
f dt ■ ("+ A • • • i+C . <+E . t+ L)-» ; 
viz. we thus have 
^w - =2 -W' L dt(t+A ■■■ t+c - (+E • <+L ^ 
where t-\- A . . . t-\-C . tf-j-E .£+L is in fact a given rational and integral function of t ; 
viz. it is 
= -Disct.{(*XX . . . Z, W, Tf+t(X\ . . +Z 2 + W 2 -T 2 )}. 
147. Consider in particular the integral 
dS 
here 
{' diS _ 
J { (a- faY ...+ ( c-hzy + ( e - kwy + Z 2 p s ’ 
(*JX . . . Z, W, T) 2 -K(X 2 . . . + Z 2 +W 2 -T 2 ) 
= («T-/X) 2 . . . +(cT-hZy+(eT-my+l 2 T 2 
+t(X 2 . . . +Z 2 +W 2 -T 2 ) 
:(/ 2 -K)X 2 . . . +(h*+t)Z 2 +(Z?+t)W 2 +(a\ . . +c 2 +e 2 +l 2 -t) T 2 
— 2«/XT . . . 2<?AZT-2^WT; 
