of certain differential Expressions , &c. 
259 
z + tf' rdit 
T'J I- 
or into 
d+O (1+0 
rdti" 
J Vf—u"*) {\—e" z u" 
-u' z ) [i—e- u‘); 2 . 2 {i-e"*u"*)‘ 
&c - but X”(.lx ~ (i_ 7 - ~ ej ( m a constant quantity) expresses the 
time of vibration of a pendulum in a circular arc; consequently, 
the time of vibration of one pendulum may be estimated from the 
time of vibration of another pendulum, vibrating in a different 
arc; and, generally, corresponding to relations established be- 
tween abstract quantities f, f, f", Sac. will be found properties 
subsisting between those subjects, of which, in particular appli- 
cations,^ /', f", Sac. become the exponents and expressions. 
A certain method for computing the integral of dx 
(df) being obtained, in a systematic treatise, the next business of 
the analyst would be, to show what differential forms depended for 
their integration on that of df. Such differential forms are many ; 
and, by the introduction of geometrical language, with consi- 
derable embarrassment to the computist,. varied in their expression. 
■, dQ . y/ (i-f m . cos. 0 ), 
dx 
dx 
v^(i —x*) (1— e z x 7 )-z' 
^(i—x 2 ) . ( 1 — e 2 x*)~ 
2 ra+ 
cos. nQ.dQ.\/ (i-j-772 . cos. 0 ), dx J j— - p) •(*, x greater than 1) 
may be reduced to depend for their integration, on 
h and fdx J ( — jZx 2 - ) 9 (*»•£ less than 1). Amongst 
these, dx J ( g -^"f j merits some attention. In an analytical 
point of view, there is nothing curious or remarkable in the reduc- 
tion of such a form to dx \/ (— and other quantities that 
can be integrated; but, with certain conditions, J j g | ■ ) 
represents the arc of an hyperbola ; consequently, announcing the 
analytical result in geometrical language, the hyperbola may be 
