MU. L. X. (X FILOX OX AX APPROXIMATE SOLUTIOX FOR BEXDIXG A 
1 10 
Now if, as we have assumed, our expansion of (l — was justifiable, we may 
stop at the term, leaving a remainder less than an assigned quantity, jDrovided v 
])e taken larue enouuh. It will be shown in the next article that this is the case. 
We may then, in the above, write for the upper limit of s' a number v, large but 
finite. The series now consisting of a finite number of terms, we may distribute the 
d-‘ 
integral sign, and further, we can replace by ( —since obviously each of the 
integrals of the type e“'‘" cos iiz chi when A' > 0 allows of being differentiated under 
'■ -0 
the integral siffii. This ffives us, v^hen the several integrals are evaluated, 
[ (As- + _ (W + 3)2ni I 
“ (201 1(40 + 1)2 + (4+ + 3)^ + 
: [ (As- + 1)2^ (4A + 3)2^ | 
.Co 1 (4.5-' + 1)2 + ,r2 (4.f + 3)2 + r2 J 
+ . 
[ (4s-' + If (4.S- + 3)2 1 
.Co U4f + 1)- + .^2 (4s + 3)2 + z^\ 
v" _ (•ff+ 3) 1 
..Co L(4s' + 1)“ + + •■’’)-+ ^xl 
Now writing in the above 
(4k + 1)-^ = {(4k + IF - -A (4k + 3F = {(4k + 3F + 
and remembering that destroys any power of 2 < 2(, we find 
(^0 (— lys'-' + «2 (— + . . . 
4.4 + 1 4.s' + 3 
_ (-1)^/2^ 
(20! 
.Jio l(4k + 1)2 + k (44 + 3)2 + s2 
+ («i (— 1)'^'' + (— + . . 
1 
— a.2(_i2-) S 
o I I I \ J I *)\2 
Co U4k + 1)2 + k ' (4k + 3)2 + k 
But, from Chrystal’s ‘ Algebra,’ vol. 2, p. 338, 
TT 
tanli 
TT Z _ * 
-V ~ 
4.-: 
- k=0 
TT 
sech 
TT.C *' 
— V 
4 
- s'=0 
(44 + 1)2 + k ■ (4,' + 3)2 + , 
4.4 + 1 4s' + 3 
(4.4 + 1)2 + ^2 (4+ + 3)2 ^ ,2 
If, therefore, in our expression for K._,j we now allow v to increase indefinitely, we 
ol Rain 
K.,. =: 
TT (—)' r 4 - 
4 (20! [dz-^ . 
77 •'! 77 
xjj.,, (,s) sech F + ( 2 ) tanh . 
TT.C 
