OF A ,SURFACE, AND TIIEIF OEOMETFIC SIGXIFICANt'E. 
PAS 
Substituting for the respective values for the respective arguments, and equiting 
the coefficients of the various derivatives of ^ and r), we have tlie requisite partial 
differential equations. They are ;— 
af= 2 E + F + 2 L + M 4- 3P 4- 4 - K 
OF ^ 0F ^ aL ^ aM ^ ap + ^ 0Q + ap 
4- Ir, J- _L O I S 
+ '‘"0, + ®^3/3 + ’>'fiy + ^S8 
’*■ 0l{, 3E.,„ + 3E,, 3%, 
aiT 
'01 
0F 
io 
aF 
11 
F., -i-D ^.L 
+ 3 Fr .+ si + al ; + 
aF 
11 
^Fo, 
4- G 
^ ao 
10 
+ opj ^/_ _i_ (;| 
+ '/'lo A + 2i^j„ A + <^„ A .f A 24 >,i a _|. ^ A 
oipio 3<p.,„ 3<^n 0<^3(| 3<^,1 
, , 3/ , , 3f , . 3/ 
Vll 
+ 3^30-^+ 2 . 1 .;, 
dl//30 a»//oi dl//i.3 
(I.), 
u/‘ = F + 2 G + M 4 - 4 - O 4 - ‘^Tl 4 - 3S 
aG 
aM 
aN 
ao 
ap 
as 
4- F ^- 
^ “^aFo, 
4- F _J_ OF ^./ 
^”aF,, “^^^''^'aF,: 
~ 1 _ F _!_ OF - 1 - F - 1 - OF _J_ -TF 
^ '''aFjo aFni aF,o aF^^ aF 
10 
+ 2Gio + SG^JI p(- + 2G.0 v/f- + aGji + 4 Go. 2 ./f- 
¥ 
e/ 
¥ 
'aG 
10 
aG, 
01 
aG 
OO 
aG 
11 
aG 
03 
^4 ¥ I A ¥ , .J, ¥ ,, A ¥’ J_ o, ¥ 
+ "Poi y, h 9]1 Wa I" ^ 9 o 3 p”'r' + ^.n Gi- \- - 9 i 3 —i“ ¥03 
drj&oi ^TOi ^Til ^T03 
I , a/’ . , , 0 1 I , a/" . ,,, a/" . o; a/ 
+ '/'oi p'.y + '/'ll v',- + ^¥2 w',~ + lii Ar "F '^'^03 a‘,^ • 
cpoi <^Pii dii/.i dxjj^., di//03 
which come from the coefficients of 17 ,respectively; 
• (F), 
