554 
Mr. Airy on the 
o, 1,2, 3, See. and to i the values o, i, 3, 4, &c. For the 
value i = 2, we must suppose log. R expanded into the 
series C Q y 1 ^ c c z y ' ^ + &c. and must take the 
r ^ • c l r r (2) (&) . . 
sum of the quantities r 2 J p — J J Q . y‘ , making 
k successively = 0, 1,2, &c. The integrals with respect 
to a must be taken from a = a to a = a. 
(7.) Now Laplace has shown ( Mec . Cel. liv. 3. n°. 12). 
that f f Q^. Z ,( ' ^ from p' = — - 1 to p 7 = -J- 1, and from 
J J J y! ~ 
(J = o to = 2 -7T, will always be = o, except & = i. And 
it appears also (liv. 3. n°. 11.) that U^= -777 a + ? Y^; 
but (see n°. 17. ) is there = a . a + 3 f, J , Y /( ^ be- 
0) [A 
tween the same limits, being the value of Y' ^ when p 
and ea are put for p' and u : hence ^ t 0^ ? Y^ 777^ Y * ] 
where Y /( ^ is any function of p' and satisfying the equa*T 
tiono= — 1 + *"77^+ z * 2 + 1 - Y • 
^ iff rrft) 4 7r „(*) r r n (o ,co 
Consequently JJQ.Z = *< + . Z ■JJ /2 * = 
?r ca , r r' ^( 2 ) /( 2 ) 4* ( 2 ) t w 
77 z () ; and J ^ J , Q . y =— y where Z , z , 
r (0 <*) 
2 2 + 
and y (2) are the values of Z' W , 2/ W , and/ (z) , when p' is 
changed to p. 
(8.) Since we propose to include only the second order 
of e, and since to that order no powers of p' beyond the 
fourth are found in R or any function of R, it follows, that 
qO) £/( 4) an d q« ^ w ill be the last terms to be integrated, 
