Mr. Davies Gilbert on the 
z 3 : a 3 \ 
2 Z 
204 
tion and in magnitude by the incremental triangle P rp ; and 
As x : y : : z : a ; as x' 
as x 2 jr \ x 2 : : a 3 z 3 : z 3 ; 
But x 2 -f- y 2 — z 3 universally. 
Therefore, i ; 2 : x 2 :: a 3 + s 2 : s 2 ; and * = 
consequently, x — V a z + z*« — a - 
" No. i. x = Va % + z 1 ^ ^ 
Equation A No, + 
t 2* # 2 
No. 3. a = 
L J 2 j 
Again ; x : y : : s : a v y = Tsubstituting from Eq. A. 
(No. 2. 
V a 1 + Z* 
V201 
+ 
;; and 
w .1 T 
Equ. B. N0.1.) y = 0 x natural log. of = 
a x nat. log. a + x a ~— 5 or hy substituting its value for a 
from Equ. A. No. 3, and dividing by z + x. 
Equ. B. No. 2,)) = «x nat. log. 
or if be substituted for x in 
* Va’ + z 1 
a x 
z 
y = a x 
VV + z 2 
and 
Equ. B. No. 3.) y = ax nat. log. V£+X ±±. _ 
To find x when a and y are given : 
Let N = the number of which ~ (Equ. B. No. 1.) is the 
natural logarithm. ___ __ 
Then ci N — ~ cl x -J” 2 a x + x* > and %/ 2 a x + x* ^ N ci x t 
make a N a =M. Then 2 a x -[- ^ 9 = M 9 — 2 M x + x 3 , and 
M 4 
Equ. C.) x~ ’ 
x being known. 
% is found from Equ. A. No. 2. and 
