64 on laplace’s method foe, the atteaction of ellipsoids 
[173 
In fact, assuming this equation for any particular value of i, we find first 
„ d nn d , d \ . f d 2 _ d 2 d 2 V -1 / „ cZ 2 d 2 „ d 2 \ 
a “^ + + r Ki ~ 1 ( a dP + 13 db 2 + 7 dp) ( a da 2 + i3 'db 2 + 7 dd) 
-> 2 ^ + ^ 2 i + 72 S 
cZ 2 
db 2 
¿Zc ; 
cZ 2 
(Z 2 
cZa 2 
„ cZ 2 
+ 055 + 7 
d ! \"> 
V (a 2 + Z> 2 + c 2 ) 
1 
db 2 ' (ZcV V( a2 + 6 2 h- c 2 ) 
^ a2 cZa 2 + /32 (Z6 2 + 72 rfc 
Ki-v 
Now putting for shortness 
. d 2 n d 2 d 2 
A ~ a ~ + # + V ^> 
da 2 ^ cZ6 2 ' eZc 2 
we find, replacing Aif i+1 and AJi^ by their values K i+2 and J5T i+1 , 
A ( a 2 + Z> 2 + c 2 ) if i+1 = ( a 2 + b 2 + c 2 ) if i+2 
d 
A (“* da + 6/3 db + 07 dc) A ' ~ 
A (“ a s + ^| +7, i)^ = 
A ( a + /3 + y ) Ki = 
+4 (“s + W;s +, »as) 
+ 2(a +/3 +7 ) iT i+1 , 
{ m 3a + d0 Tb + Crf lc) Ki+ ' 
n ( „ iZ 2 cZ 2 „ cZ 2 
+ 2 V a da 2 + ^db 2 + ry dc'- 
K, 
„2 d , 03 ^ , o d 
a ^ + ^^ + 7 
- a 
cZa 
iZ 2 
+ ^^2 + 7’ 
K u 
Ki, 
dyj ~ l+] 
d 2 
da 2 ' ^ cZ6 2 ' 1 dc 2 ) 
(a + /3 +7 ) Ki +1 ; 
hence operating on the equation of differences with the symbol A, we obtain 
(a 2 + Z> 2 + c 2 ) Ki +2 + (4Z 4- 7) f act -j- + Z>/3 jr + 07 ■j-') K{ +1 
V iZa 
(Z 
<ZZ> 
(Z 
dc 
d 
+ (4i + 2) ( a* - + ^ +y ^) K , 
+ {2 (4* + 3) - (4i + 2)) (*» £ 2 + /3’ + y d ' 
clb 2 
d& 
K 
+ (2i + 3) ( a. + /3 + 7 
)K i+ 1 = 0,