274 Integration of Differential Equations. $e 
_(1)is taken from Vol. III, p. 537, second edition, of the Traité 
dn Calcul Differentiel et du Calcul Integral, par S. F. Lacroix ; 
and (2) is deduced from (1) by changing the sign of the last 
term, or which comes to the same thing, by changing 5? in (1) 
into —b?, that is, using bV —1 for b. We shall put 2pq—q+1 
=c, (a), and shall use the characteristic, /*, when prefixed to any 
differential expression, to signify that its integral is to be taken 
with reference to the variable, from its value m, to the value m; 
or m and m are two values of the variable, and the integral is sup- 
posed to be taken from the first limit », to the second m. 
Lacroix remarks that y=, oon(a? —x? ii cos. bu?z, is a par- 
ticular value of y, which eainien (1); in which u! is to be re- 
garded as constant in taking the integral, x and its functions be- 
ingthe only variables, and the integral is supposed to be taken 
from ¢=0 to «=a, which are the first and second limits of z. 
We have also found by integrating (1) (by the method of se- 
sies) that it is satisfied by the particular value of y, which is de- 
noted by y=u'~f delat —v?) ” cos.bute, when p is positive, 
and such that 1 —p is positive, and not an indefinitely small quan- 
tity, and it is to be noted that v and its functions ‘are considered 
. as the only variables in taking the integral, uw? being regarded as 
constant. Hence if we use A and.B to denote two arbitrary con- 
stants, we shall have (by the well known theory of integrals) 
the complete value of y, the expression y=A if; dz(a? — —02) 
cos.butr + Bu’ ay dv(a? —v?) *cos.buty, (b), in which p must 
not be considered as an indefinitely small quantity, and 1—p is 
positive and finite ; since the limits of z and v are the same in 
(6), we may change v into x, and then the value of y may be 
put under the form y = hy, dx(a? —22) ‘cos.bulc (A+ Bu 
ho ime 
(a? eo) a (¢). If we put l-~c=e, 1-2p=f, or c=1—@, 
2p=1—F, (d), we get uw “(a2 —29) =w(a* — 22), and if we 
substitute the oe oe in(a), we get by a 
slight reduction bee ou u‘(a? —; 7 i- os (u(as —zx*) *), hence, 
when ¢ is very small, using” hyperbolic logarithms and rejecting 
