PEOFESSOE CATLET ON PEBPOTENTIALS. 
747 
these last, with the before-mentioned function 8, are the s-j- 1 new variables of the problem ; 
and for convenience there is introduced also a quantity g ; viz. we have 
d — 
h= s /f+d (3, 
c=\/h 2 -\-Q y, 
g, 
where l=a 2 +3 2 . . . +y 2 +s 2 . 
That is, we have 8 a function of a, b ...<?, e determined by 
a 2 b 2 c~ e 2 .. 
• • • + a 2 Ts + I = 1 ’ 
and then a, (3 ... y are given as functions of the same quantities d,b...c,e by the 
equations 
2 a- n* b 2 2 C 2 
/ 2 + 0’ P — ff 2 +s - ■ • 7 — + 
also g, considered as a function of the same quantities, is 
Z 2 b 2 c 2 
115. Introducing instead of a, b . . . c, e the new variables a, (3 . . . y, 0, the transformed 
differential equation is 
, d 2 \ . n dV 
40 
?X_i_2 — ( s A-2a-\-^ ^ ^ \ _i_ w n 
</0 2 +^ dd + r .. - /i2 + s J+vv_u, 
where for shortness 
VV =W—’- 
h 2 
** • • • -/TTfi 7 2 +! 
1 f r- 
/+H / 2 +s 
* 2 -(3 2 ... 
'id +6 y 
2 +i 
d 2 Y 
7 do. 2 
d 2 Y 
tf/3 2 
■ 1 f / 2 _ 2 _^ A 2 2 , 
20 rf 2 V 
-y— Z5 — &C. 
/ 2 + 0 .^ 2 + 0 
+ /W« {-2?-'2-i» .. . +aqr 9 )} « ^ 
U‘S 
+? ^ Fsi _2^-2 ■“' , • • ■ +to)} 1 3 
rf/3 
MDCCCLXXT, 
