ON LINEAR DIFFERENTIAL EQUATIONS OF THE SECOND ORDER. 669 
| and, since 
| log P(1:38) =1:94868 log I'(1-88) = 1-98004, 
| we find (105-0176 ; 1) = — 0-231467 ; S(105:0176 ; 1)= +.0:008964 
I need scarcely say that in this case a satisfactory calculation by means of the series 
is quite impossible. For smaller values of c the computations are of course still more 
convenient. If, for instance, c= 14°96, or n=4'4, we have 
sin 36° 
(14-96 ; 1) = na = 028654. 
T(2:2) sin36°_ 12 T(1-2 
TT) jean paleniGr 
We notice also the relation 
1 
n(n —1)C(e, 1) - 
S(c, 1)= _ Sin tr (42) 
Note added on June 30.—The relations (41) and (42), which follow here immedi- 
ately from a well-known property of the hypergeometric series F(a,8,7; 1), were 
originally derived by Professor CurystaL from a different point of view. (See §§ 25 
and 41 of the Hydrodynamical Theory.) His important relation (53) in § 41 may be 
also obtained from the equation 
Le, =Fin,1—m, 2; =U +nyna—nm= — a, 
Mn — T 
and, in consequence of (42) : 
L(e, 1)=Ci(e, 1). Sé, 1). 
aaa 
At the request of Professor Curystrau I subjoin tables from which the numerical 
values of C(c,1) and S(c, 1) may be taken for any value of c. If we write 
aa bth yerd 
Oe T(a) sin ar 
T(ia+4) <a 
Sa) = _ T(a+4) cos ar 
T(a) Jr’ 
we find easily from the above formule the following relations: 
dod 
O<a<1:C(e,1) = - a @(a +1); 
a 
ee GS 
c 
2 
1<a<2:C(c, 1) z @(a) 5 Se, i= (a) 
Gee =3: Cle, ly) = - “1 e(a-1); E8(e,1) = - - 2233 (a- 1); 
mae) 
(a= 2); £8(c, 1) = Sa Dig), 
Scat: Oley) = Cae e @=2\G=) 
-3)(a- ia 
_ where the values of 6 and > are to be taken from the following Table I. 
