OO 
Integration of ae 279 
yi? , 
Since z al Y) We get z=y — a =/f* de (Ae or a) 
+B af “dge" ~/2al o00.9 (i sre sin. +), or if we still denote 
Aa‘b’x (for brevity) by A, z= aE aaa fas (ad 2al.cos.p 1) 4B 
Sd (t08. w+ 2log. sin.) ag el 8? (p). If we put B=o 
A=1, we get z= aie f. dv(¢ A a 1), which is a particus 
lar value of z that satisfies (m), and agrees with the value used 
by Laplace in the supplement to the 10th book of the Mécanique 
Céleste, Vol. IV, p. 60, (of the supplement,) and he expresses it 
as his opinion that the complete value of z cannot be found by 
any of the known methods ; we will add that the same particular 
value is to be found in the profound. commentary by Dr. Bow- 
ditch, at p. 973, Vol. IV. 
d? 
Again, if p=0, (or is indefinitely small,) (2) becomes at 
l-qg d ‘ 
—4 —q?a7b?u2?-*y=0, (¢), which is not satisfied by La- 
l-c ‘ss 
eroix’s integral, but our integral w Os dv (a? —v’) *cos.bul 
Vv =I, which (since l—c = 4, (a?—v?) "=, rejecting indefi- 
hitely small quantities,) becomes I, uidv cos. bux VW — 1 = 
-abul abut abud " 
é 4 
sin.abud 
ees Slt gts =r oP (where e= the hyper- 
bolic base,) will satisfy it; it is also evident that (q) ought to be 
abu ~abul 
Satisfied by each of the valuesy=e ,y=e — , whichon trial 
will each be found to satisfy (q), hence if A and B denote two 
arbitrary constants, the complete value of y that satisfies (¢) is 
fk: me 
Also, if p=1, (or if 1—p is indefinitely small,) (2) becomes 
d? 
eyeiss on — grat bias? y=0, (r), which is not satisfied by 
Our integral but Lacroix’s integral will satisfy ws. tag bes \aame 
(a2 '1,) y= [* de cos. bu'e V— “ania 
