356 Mr. Bappace on the Influence of Signs 
disappear, and in the other, that the series shall terminate. 
Euler has taken advantage of the former of these circumstances, 
to discover curves whose indefinite quadrature should depend 
on a given species of transcendent, whilst the areas of particular 
portions of them are susceptible of an Algebraic expression *. 
The integration of the equation is not always sufficient for a 
complete analysis of a question, for in some cases besides: the 
general integral, there exists another not included in it, which 
is known by the name of a particular solution; in order to 
be secure of not overlooking any such, it must be observed that 
a change in the magnitude of an index may cause the intro- 
duction of such a solution. When the complete integral as well 
as all the particular solutions are found, the interpretation of 
them according to the circumstances of the question is not always 
an easy task, nor are any general rules yet established to which 
we can refer for information. In the theory of curves the in- 
terpretation of particular solutions is sufficiently well known: 
they represent the curve which touches all those formed by the 
complete integral when its parameter varies. In mechanical 
questions a considerable degree of uncertainty prevails relative to 
these kind of solutions, as in some instances they seem to have 
no reference to the preblem which gave rise to them, whilst in 
other cases its solution can only be fully represented by their 
assistance; some light has been thrown on this subject by 
M. Poissont+ in a memoir in which he has explained the theory 
of particular solutions with great perspicuity. 
Of whatever kind the equation to which our question con- 
ducts us, may be, it ought to be regarded, merely in an analy- 
tical point of view; and all its various roots or solutions, should 
* (Euler Acta Acad. Petrop.) + Journal de Ecole Polyteenique, Cah. 13. p. 60. 
