Integration of Differential Equations. 279 



Since z+^,=y, we get ^=y-^=/ d^[Ke ^-^ 



+ B/ c?9e" "'^"^ (log.M-{-21og. sin.9'), or if we still denote 



Aa^6'^ (for brevity) by A, ■z== ^nfl^^ (^j^y^^-cos.v _ i) ^3 



/ c??)(log. w+21og. sin.?'] . e" '^•''"^•'P^ ^py jf ^yg p^^; ]3=0, 



A=l,'wegetz—-jr7^ / dcpie^ a.cos.(p_^\ ^jjjch is a particu- 

 lar value of z that satisfies (m), and agrees with the value used 

 by Laplace in the supplement to the 10th book of the Mecanique 

 Celeste, Yol. lY, 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, Yol. lY. 



Again, if^=0, (oris indefinitely small,) (2) becomes t-j + 



\-q dy 



-J- —q^a'b~u^^~^y=^Oj (q), which is not satisfied by La- 



l-c /^ -p 



croix's integral, but our integral u / dv{a^—v'^) cos.bu^v 



•v^ — 1, which (since 1—c = g', (a^ — v^) —1, rejecting indefi- 

 nitely small quantities,) becomes / u'^dv cos. bu^v "/ — 1 = 



y -abul ahui abul -ahvH 



sm.aOM^v— I e —e e e 



bolic base,) will satisfy it ; it is also evident that (q) ought to be 



abul -abul 



satisfied by each of the values y = e , y = e , which on trial 

 will each be found to satisfy [q), hence if A and B denote two 

 arbitrary constants, the complete value of y that satisfies (5') is 



abul - abul 



y=ke +Be 



Also, if^ = l, (or if 1— jpis indefinitely small,) (2) becomes 



d-y i-\-q dy 



-7-7+ -i- — q-a^b^ii^^~'^y = 0, (r), which is not satisfied by 



our integral, but Lacroix's integral will satisfy it, that is, (since 



abul -abul 



{a^-x'~f =1,) y=jl ^^cos.6it^^v/-l=2^-2^, will 



