PARTICULARLY THOSE OF TWO VARIABLES. 
807 
(iii.) Let #= 0 , y— A: the vanishing functions are d 3 , d 6 , $ l3 , d 15 ; hence 
x x =—z, % 2 — 0, and 
K 0 - 
»=f' i 7T fe +Kf* 
fVA'+B'# 7 . 
A =\ n ^7TT dx + 
'OA' + B'a? 7 
(iv.) Let £C=— log pr, y——. log rq : the vanishing functions are $ 4 , 3- 6 , & 8 ,3 g , S 13 , S H ; 
17% 17% 
hence ^= 0 , x 2 =l, and 
£ . f 1 A-t-Ba; 7 
d°^=J.vx & 
K 
7T-2- 
A 
7n 
, fiA' + B'a;, 
a°gr 2 =j j ^ r! fe 
(v.) Let a?=K+^. log r, y= A+— log q : the vanishing functions are 3 0 , d 3 , d 4 , d 
'55 
& 9 ,-9- n ; hence 0 ?^=—^ x 2 =l, and 
K 3 " 
1 
F , K]i (VA + B® 7 fiA + B^ 7 
K+-l°gr=j o cfc 
A , A. r^A' + B'x, , f 1 A' + B'x 7 
A+ S log ? =j o -^ S -&.+j | - 75 -dR 
By the elimination of a, b these ten equations reduce to the following eight 
o=f- 
J 0 
1 A / + B'x 7 1 
ax 1 
K 
=f 
J o 
1 A + Ba? 
dx 
(59) 
\/X ^ j 
Ki 
IT 
•i 4 A + Ba? 7 ~"| 
1 jVA + B 
VX 
A i T f*?A' -f B'a; , 
»=it^ 
i 
(60) 
A 
< 
VA' + B'a? 
1 VX 
5 L 
• ( 61 ) 
MDCCCLXXXII. 
