750 PEOFESSOE BOOLE OX THE COMPAEISOX OF TEAXSCEXDEXTS, WITH 
or the form 
^3)+C’ (15.) 
From these solutions the corresponding solutions of the previous and less general pro- 
blem would be obtained, in the one case by making J=l, in the other by imposing 
upon the limits Wi, X3 conditions thereto equivalent. 
6. And not only the solution, but the original statement of a problem may be exhi- 
bited in the two forms above described. Instead of supposing the limits determined by 
an equation involving arbitrary quantities in its coefficients, we may suppose them directly 
connected by symmetrical equations, i. e. we may suppose those relations explicitly given 
which are only implied in the equation determining the limits, and can only thence be 
deduced by eliminating the arbitrary elements. Thus (9.) furnishes us with the three 
folloAving equations : 
^i+^'‘2+^3= 
X^X^ + ^ 2^3 +^ 3'^1 = «• 
1 — 
rp (v* rp - 
cy 
From which, eliminating the arbitrary quantity a, we have 
ep - I - fp t rp rp rp 1. rp /)•» i rp rp _ i 
tt / 2 I ^^2^ ■■tv2w2~ *^ 2^3 I *^3^2 2 
2X1X2XS = 1 — {X1X2 -b ^^ 2^3 +^3^1)^ 
The problem first considered now assumes the following form. Requu’ed the value of 
the integral expression 
dx 
V\+ x'^ 
when the superior limits x^^ x.^-, x^ are connected by the explicit relations (16.). 
The second problem, similarly transformed, would, as there are two arbitrary elements 
to be eliminated, lead to a single equation between ^ 1 , x^-, X 3 , in place of the two equa- 
tions (16.). 
7. We may observe, from the above examples, that Avhen the number of integrals to 
be added is three, the existence of two arbitrary elements in the equation of the limits 
involves the existence of one symmetrical equation among the limits, and the existence 
of one arbitrary constant in the equation of the limits involves the existence of two 
symmetrical equations among the limits themselves. And thus generally if there be n 
integrals to be added, the existence of r arbitrary elements in the equation of the limits 
will involve the existence oi n — r symmetrical equations among the limits. Tlie con- 
verse of this proposition is obviously true also. If any number r of symmetrical equations 
among the limits x^, x ^, . . x„ are given, and' if we regard x^, x^, ■ ■ x„ as roots of the equa- 
tion of the wth degree, 
x'^-\-piX^~^ -\-ppif~^ . . +^„= 0, 
the symmetrical conditions referred to, will, by the theory of equations, establish among 
the coefficients J9i, ^ 2 ? '-Pw a system of relations by means of wffiich we can determhie r of 
