of certain differential Expressions , &c. 257 
ferently expressed, in the methods of Legendre, of Mr. Ivory, 
and of Mr. Wallace, who have learnedly and ingeniously written 
on this subject. 
In order to deduce the relation between three ellipses, Legendre, 
Mem. de V Academic, 178 6, p. 650, assumes 
f t/\ • c' z . sin. <b\ cos. ffl' 
( 1—6 ) sin. <p = — 
' ' r VC 1 — c a sin. <p ) 
Now, according to this author’s notation, c n — , and 1 — 6'= 
2 C 
7+7 
; consequently, sin. = which is P recisel y 
the same substitution as id— x ^ f — L— r ]. 
i + e V \ i — e 7 - x 1 I 
In the Edinb. Trans. Vol. IV. p. 183, sin. (t{y — <p) is assumed 
= c . sin. 4/ ; but sin. (1 (/— p) = sin. ^ • cos. <p — cos. ip . sin. <p t 
.-. sin. ^ 
i+c — 2 C .COS. 
I +C 1 — 2C (2 (COS. — j — 1) 
consequently, puttting = e 
— , the same substitution as 
sin. 4^ = , 0 ,. 
(i+ c )V(i — eV(cos, — ) ) 
Again, in Edinb. Trans. Vol. V. p. 272, sin. 2 <p' is made — 
sin. 2 a , 1 2 . sin. <p . cos. (p 
717 + 7 ^ 77 -—) =■ “ nse q uentl y. V.+^+^O-TST-rt = 
sin. ffl . COS. 
4 e 
I, the same substitution as u'= 
i-fe'" V(i— si n. a <p) ' (i + O 2 
It appears, then, that the preceding substitutions, although, by the 
aid of geometrical language, differently expressed, are all reducible 
to the algebraical substitution of u'= ,.x J ( 1 7-g" ) ’ * n ^ ie 
LI 2 
