752 
PROFESSOR CAYLEY ON PREPOTENTIALS. 
the equation to be verified is 
(/*H-9) 
V/ 2 +e] 
' 2 7+ 2 +/4a+/T9-' 
' ’ +A*T"fl] 
i j 
f 9 
9 1 
+ V^+9j 
!~ 2 ^- 2 ~/ + 9 • * 
• “AM^j 
-29 
Again, suppose (value belonging to <p=a/3, see No. 116), a parti- 
cular solution is ? • in fact omitting the constant factor, or writing 
/ 2 +£ 2 ...+ A 2 ’ ° & 
and therefore 
®=s/f*+6*/f+6, 
2 1 V/ 2 +0 vy+flf 
^ 2 Q -| f y^+fl 
+ ■ 
V/ 2 +< 
4 ( (/ 2 + 9)i^ *//“+« vV + 9 (^+«)« 
the equation to be verified is 
a/ + S i 2 V/ 2 4-9 I 
1 (/ 2 + 9)f' r V^ + fl V^ 2 + fl (/ + 0)if 
-[_ / V / ff 2 + 9 _|_ V / / 2 + e \{9f7 I 2-h— — — -1- — 1 
+ v^ti + ~wtq ;h + 2+ / 2 +^ 2 +9- - • +>+9/ 
+ \// 2 +^\A 2 -M |/2"+ 9.^ + 0 +^T9 ( _2 ^ 
2^ 2 ^ 
z ^*+r 
9 ' 
’• “ A 2 -^ 
2/7 2 ^ 
z p+r- 
■•+wTi, 
or putting for shortness Q==y2q^-|-^p0 • • • this is 
_9_vV±9 , A 's/P + Q I / vV 2 + fl | V/'~^ V9 r/ ±9_LO\ 
(/ 2 + 9)* V/ 2 + 6 vy + 0 (£ 2 + 9)* + ^ ; 
2 ^ . I . Vff 2 + 9 / 0/7 q\ i v!/ 2 '+ 9 / Otf 2-4- ^ o\ — 0 
which is true. 
And generally the particular solution is deduced from the value of <p by writing therein 
V7 ¥ T9 _ vy+9 vra 
V/ 2 +£ 2 . • • + A 2 ’ V/ 2 +/...+ A 2 ’ ‘ * ' V/ 2 +/... + A 2 
in place of a, 0, . . . y respectively : say the value thus obtained is @=H, where H is 
what <p becomes by the above substitution. 
124. Represent for a moment the equation in 0 by 
