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Proceedings of the Koyal Society of Edinburgh. [Sess. 
then found to be a closer approximation to the true curve for t than the 
curve t 0 chosen arbitrarily. This process may be repeated to obtain t 2 , 
using the curve t x as a new trial curve for t : and so on. When t f differs 
very little from i, t* may be regarded as a very close approximation to 
the true curve for t. 
§ 8. It was found that the best way to carry out this plan of obtaining 
a numerical solution by means of two successive integrations was to choose 
dt 
the interval of integration, from any point at which t and were known, 
CLOG 
sufficiently small that the trial value chosen for t, for the point at the end 
of the interval, could be determined with any degree of accuracy required, 
by means of numerical differences of the values of t already determined for 
the end points of preceding intervals. The arbitrary trial curve is in this 
case a curve which coincides with the true curve in each interval preceding 
the one considered, and in this interval it closely approximates to the true 
curve. It has been found that one application of the process of double 
dt 
integration to this trial curve gives t and at the end of the interval 
CLOG 
with any accuracy desired ; the accuracy depending on the smallness of the 
t* dt 
interval chosen. By taking the curves for — — and as straight lines 
OG (X/OG 
within each successive interval treated, and by making a roughly estimated 
allowance at each step for the error thus introduced, the process can be 
carried out very quickly and quite satisfactorily, as a process of step-by- 
step calculation, without the assistance of carefully drawn curves. 
§ 9. The accuracy of the above process can easily be proved analytically 
for the case of any very short interval ; but when such a process is applied 
to a succession of intervals, there is certain to be a cumulative error, which 
may, or may not, increase without limit as the work proceeds. So far as it 
is possible to judge, however, the process of § 8 seems to be practically 
applicable to obtain numerical solutions of differential equations of the 
form 
^ =/(*>*)> 
provided / has the opposite sign from t. It has been applied with very 
satisfactory results to the two equations — 
dH_ 
dx 1 
( 10 ), 
t 5 
and 
dH 
dx 2 
• ( 11 ), 
