DIFFERENTIAL EQUATIONS OF ASTRONOMICAL INTEREST. 133 
the case in which each k is unaltered by change of the sign of A is the case in which 
only even powers enter in this series. The case in which the two values of k are 
interchanged by change of the sign of A may arise when the differential equation is 
such that for A = 0 the two values of k are equal or differ by an integer ; in this case 
e KT /e KT is a periodic function for A = 0, and the factors e K ' r , e KT do not individualise the 
functions with which they are associated. 
In the present case, the equation reduces when A = 0, to 
d 2 x , o7 dx , A 
J 2 "b 7 "h 0)*^ — 0, 
dr~ dT 
which, if a 0 is not zero, has the two solutions e aT , e aT , where o-, a have the values 
[-b 0 ±(b 0 2 -a 0 c 0 Y]/a 0 . 
Thus if we suppose not only that a 0 is other than zero, but also that 
2 (/7 u 2 -« (J C 0 )'V«o 
is not zero or a positive or negative integer, we can assume 
Then putting 
K — IT + K 2 A J + K. t A 4 + . . . . 
X = 1 + \(f>i + A"0 2 + • • • , 
where is a linear function of £ r , .... £ 2-r , £~ r , the differential equation for X 
can be compared with that of § 1. In the present case there is an unknown 
quantity k entering into the coefficient A/c + B of dX/dr, but it will be seen that in 
the equations obtained by taking the successive powers of A, each unknown coefficient 
in k in this A/c + B is determined at an earlier stage as entering in the coefficient 
A/c 2 +2B/c+C, and so enters as a known coefficient. We have 
A/c+B = [a 0 + A (cq^+a.^ 1 ) + A“ (cqA + ^-2^ “ + C£ 20 ) +...] [ct+k 2 A“ +...] 
+ &U+A (&i£+ ') + A 2 (h 2 £‘ + b_ 2 £ ““ + 5 2u ) + ... 
— a 0 <x + 6 0 +A [<T (ctjC, + 1 ) + &X+ 
+ A - [<7 (cs 2 ^' + ct_ 2 £ + ck 20 ) T <^0^2 + b 2 <," + b^ 2 % " + ^20] 
+ ..., 
and similarly, ♦ 
A/c” + 2B/c + C — c^qit” + 2& 0 cr + Cq 
+ A [<r“ (cq£ + 1 ) + 2o-(& 1 £+-&_ 1 £ _1 ) + c 1 f+c_ 1 £ 
+ A“ [o-' (cq£ 2 + a_ 2 £~"'+'a 20 ) + 2<r (b 2 £~+ b_ 2 £~~ + b 20 )+ 2 k 2 (cQt + b (l ) 
