INVOLVING THE PRODUCT OF TWO LAPLACE’S COEFFICIENTS. 
585 
When, therefore, one of the quantities n, n' is large and the other not nearly equal to 
it, Q„Q„, is affected with a sign rapidly alternating, and consequently the value of the 
i QnQn'dp is comparatively very small. But if n and n' are nearly equal, the functions 
J 0 
Q„, Q„,, in spite of the rapid alternation, keep together as it were in sign for a considerable 
fraction of the range of integration, and so the value of the integral is largely increased. 
Again, for all cases included in the Table it will be found that 
0 
— I | + 1 dy* 
a relation which is evidently general, although not very easily proved to he so. After 
a good deal of trouble I arrived at the following demonstration 
If in the expression 
(l+ V'ee') (l ~\J ~ Jy 
—~=. log 
V ee 1 & 
vA + < 
■VI 
*</ 1 "h t 
whose expansion gives the integrals under consideration, we put 
ee'=x , 
j—Vi 
we obtain 
_JL } 0O . (i+ l ~ ^y) 
V X & 
/ fJQ 
Vl+xy-Vy\/ 1 + - 
In consequence of the symmetry of this in respect to y and -, it may be expanded in 
a series of positive and negative powers of Vy of the form 
A 0 + A, ( Vy - +^) + A a (y +i) + A 3 (# +r + , 
A 0 , A,, . . . . being functions of x. 
The terms that we are engaged in examining are those in y or so that the 
question reduces itself to the determination of A! as a function of x , or at least an 
examination of its nature. 
Now A, is the term independent of y after differentiation of the series with respect to 
Vy. Hence 
a 4. -i i j. r • d 1 i (1 + vV)(l— Vy) 
A x = term independent of y m log 
Vl+xy— Vy <\/ 
On differentiation and reduction we arrive at the expression 
, X 
1 — 
y 
1 -\-x 
\/x(l —y) ‘ V x 
from which the part independent of y has to be selected. 
