of certain differential Expressions , See. 
®45 
Thus, u’ = U " = . 
If we call, then, x the sine of an arc 9 , 
2 sin. 0. cos. 0 sin. 20 
i-i-e' ’ v(i— sin. 6 1 ) 
+ e" 
V'GS*) * 
+ 0 Vt 1 — f (f— f COS. 20)) 
/ • e* 2e' \ 
(since — = jr; ] 
V 2 {1 + eTI 
sin. z0 
Vf+^+^.T^TST’ similarly, calling a' the 
sine of 20, a" will equal v[ s ,"^ cos ^ ,andso on; consequently, 
expressed in geometrical language, 
f ~~ ( 1 sin. 2 0'-f — (1+e sin. 4$" + &c. 
+ P. <p — PeQ<? } (where q> is the limit to which the arcs in the 
series ( )' , 9 ", 9 "', &c. approach,) and, consequently, since v = z n cp, 
■7C=W) = r ■ d *’ and T- J-Ti —) = p ?- 
Mr, Wallace obtained his formula, following a method given by 
Mr. Ivory, in the fourth Volume of the Edinburgh Transactions ; 
and both these ingenious authors have employed, probably without 
adopting, the substitution of Lagrange, and the principle of his 
transformation, such as that great mathematician uses in finding 
the integral of — 7-77. 
6 V ( *+/* ) [g +*>**) 
Since e — 
See. 
+Vii-Of’ (I + V (I- 
When e is a small fraction, the quantities e e", e r ", &c. decrease 
very rapidly; and, consequently, the preceding form is very com- 
modious for the computation of Jdx when e is any 
fraction between o and It ceases, however, to be commodious 
when e is nearly — 1, or is not equally commodious with the series 
1 + A6 2 +B6 4 + &c. 
-j - |a6 2 -f-iG& 4 -f- &c. }log. b, given page 234. I purpose, therefore, 
now to exhibit a form by which the integral of d. r, v /|' I ~ g _* z ) 
may be conveniently computed, when e is any fraction between 
y' i and 1 , 
