358 Mr. Baspace on the Influence of Signs 
From this equation it appears to me, to be the direct course 
to deduce the form of ¢: its general solution should first be 
shown, and then from the peculiar circumstances of the problem, 
that particular one which belongs to it should be pointed out. 
In the work, to which I refer, the particular form of ¢ has been 
deduced at once by properties peculiar to the problem, without 
any reference to the general solution of the equation. A similar 
objection may be made to other demonstrations of this celebrated 
theorem: the equation to which the inyestigation conducts, is 
usually solved in a manner not sufficiently general. This is the 
case in a work deyoted to the analytical exposition of the elements 
of Geometry, pp. 53, 54;* the substitutions employed, although 
satisfying the conditions, not containing all possible — solutions. 
M. Poisson has given an investigation of this theorem not quite 
so open to the objections just stated; by the introduction of 
two variables and. the employment of one sign of function, the 
solution is necessarily more restricted in its extent. Equations 
of that class are frequently contradictory, although in the case re- 
ferred to, a fortunate property leads directly to the solution. See 
Poisson, Mecanique, p.14. I cannot conclude this slight criti- 
cism on a detached passage of the Mecanique Ceeleste, without 
expressing that respect for its illustrious author, which is shared 
with all those, who are capable of appreciating the important 
additions he has made to mathematical science, or who have the 
happiness of being personally acquainted with him. 
When any question leads to an algebraic equation, it is usual 
to resolve it generally, and then to point out amongst its roots 
that particular one which is sought; if the individual root re- 
where ¢, is perfectly arbitrary, and X is any symmetrical function of @ and s —0. 
* Precis d'une nouvelle methode pour reduire 4 de simples procedés Analytiques la 
demonstration des principaux theorémes de Geometrie. Par. I.G.C. Paris, An. vi. 
