130 ANNUAL REPORT SMITHSONIAN INSTITUTION, 1909. 
that; but of what use are you then?” More often, however, there is 
a better understanding. The engineer does not need his integral in 
finite terms. He needs only a rough value of the integral function, or 
perhaps only a certain numerical result which he could easily deduce 
from such a value of the integral if he had it. Ordinarily we could 
get this numerical result for him if we knew just how accurate it 
must be—that is, with what approximation. 
Formerly an equation was not considered solved except when the 
solution was expressed by means of a finite number of known func- 
tions; but that is possible scarcely once in a hundred times. What we 
can always do, or rather what we may always try to do, is to solve 
the problem qualitatively, so to speak—that is, to find the general 
shape of the curve which the unknown function represents. 
It remains, then, to find the quantitative solution of the problem; 
but if the unknown can not be determined as a finite result it can 
always be represented by means of an infinite convergent series which 
will allow the numerical calculation. May we regard this as a true so- 
lution? It is related that Newton once communicated to Leibnitz an 
anagram something lke this: 
aaaaabbbeeeeti, ete. 
Leibnitz naturally was wholly at a loss as to its meaning; but we who 
have the key know the signification of that anagram and translat- 
ing it into ordinary language it becomes: I know how to integrate all 
differential equations; and we are led to say to ourselves that Newton 
had strange good luck with such a singular illusion. He would have 
said all simply, that he could form (by the method of undetermined 
coefficients) a series of powers satisfying formally the given equation. 
Such an apparent solution would no longer satisfy us to-day; and 
that for two reasons, because its convergence would be too slow and 
because the terms would follow one another according to no definable 
law. On the other hand, the series © seems to us to leave nothing to 
be desired, first, because it converges very rapidly (and that because the 
engineer wishes his result as quickly as possible), and then because 
we may see at a glance the law of its terms (that, for the satisfaction 
of the esthetic needs of the mathematician). 
But there are no longer some problems which are solved and others 
which are not; there are only problems more or less solved accordingly 
as they are represented by a series converging more or less rapidly and 
following a law more or less harmonious. It occurs sometimes that 
an imperfect solution leads to a better one. Sometimes the series con- 
verges so slowly that calculations from it are impracticable, and we 
have shown only the possibility of a solution. And then the engineer 
thinks the solution only derisory, and he is right, as it will not allow 
him to finish his construction on the given date. He cares little 
