Mr. Herschel on various points of Analysis. 465 
be < 1, and n o Id, we shall have e' also < 1. Thus if the curve 
AM be an ellipse, am is also a concentric ellipse. The equa- 
tions of Ex. 2. geometrically expressed afford a property of 
the hyperbola something similar. 
IV. On differential Equations of the first degree. 
Any equation of this species may be reduced to the form 
0 = -j- * A . T>u + ^ A . + . • . . ”A . + X ; . . . ( 1 ) 
u being the unknown, and ‘A, . . . . "A, X, known functions 
of X. To integrate it, assume the following equations 
u + ^ocDll =■ 
u 
(0 
4 - ^ocDu 
(0 
= u 
(2) 
> 
«<”—'+ + X = = 0 . 
(l) («— 1’ 
From these, eliminating successively u , . . . u 
obtain 
= w -f- 
= w + ( -f + ^ctD\) . Dw + *a . 
• • • • • 
o = a'"’=!<+ {'« + “«(! + D'«) + &c| .Dm+ 
•«."a..."a.D"a+X; (3) 
The comparison of the coefficients of this equation with * A, . . . 
^A, gives n equations for determining ^a, . . . into which X 
does not enter. Consequently these functions are independent 
on X, and therefore, the same as if X = 0. Now the successive 
integration of (2 ) gives 
