LINEAR DIFFERENTIAL EQUATIONS. 
29 
mathematics. While the foundations of the theory, as re¬ 
constructed by Picard, avowedly dispense with the quasi- 
geometrical ideas of Lie, yet Lie’s work has paved the way 
for its future extension and application. 
These researches mark the beginning of a new epoch in 
the theory of the linear equation which promises much for 
the future. 
In the attempt to present a connected outline of the theory 
I will first state some well-known properties of the general 
linear group, then state the principal theorems on which the 
general theory is based, and conclude with some applications. 
If y u y 2 , . 2 /a he any system of integrals of 
the general equation (A), it is also satisfied by any linear com¬ 
bination of these integrals with constant coefficients, as, 
F= (h y 1 + a 2 2/2 + « 3 Vs . 2 /a- 
It has been already stated that if there is no linear relation 
with constant coefficients, as, 
Vl + ^2 2/2 + «3 2/3 + Vi ..Vs = (J) 
where s < n, the general integral of equation (A), could be so 
expressed. If, now, we have given n linearly independent in¬ 
tegrals such as y u y. if ..... y a , we can form another system 
of n integrals, Y v F 2 ,.F n , with constant coefficients 
analogous to Y v which will also be linearly independent. 
These Fs, so formed, constitute a system of linear substi¬ 
tutions which form a group—that is, a repetition of the opera¬ 
tion results in a substitution of the same form as the first. 
Take, for example, the case of an equation of the second 
degree. We have 
Y 1 = a n y x + a u y 2 , 
F “ ^21 Vl ^22 2/2* 
Suppose we form a second system, 
Yi + b x 2 Fj 
f 2 =6 2 i f+ b n F-r ; 
5—Bull. Phil. Soc., Wash., Vol. 14. 
