*54 
Mr. Babbage's essay towards 
_____ 4 /® d:pX 
4 * + 4 ?) 
dx 
which may be thus proved differentiate [a') which gives 
+ ( * + + h+ir 5 ) = *** + *£<!.■*£ 
and by dividing both sides of this equation by 
we have 
+ 1 
' x j 
1^1 
r x | ^ ^ 
1 i]/® d-^x | 
dx 1 
<> (« + 44] 
| dx -f* d\ 
(■ \px dipx) 
{ dx i 
| 
From this it appears, that the subnormal is constant in the 
same series of ordinates, but it does not follow that it must be 
constant in different series; this property, viz. that does 
not change when x becomes x will furnish us with a 
solution of the equation in question; for (^) becomes by put- 
tmg t for j 
WC* + 0.3 , -W'*3 , ='*‘ 
where t may be considered as a constant quantity, the gene- 
ral solution of this equation is 
-fyoc == V xt -j- (pt 
<pt being an arbitrary function of t, therefore the general solu- 
tion of eq. (a) is 
or - 
from w r hich differential equation the curves which satisfy the 
Problem may be found. It ought, however, to be observed. 
