in the Mécanique Céleste. AT5 
S pre F f 
He then says, that since ["/" . Pf (6, V’) sin 6! do! dy’ 
is of the form of Laplace’s coefficients, we may put it 
an 
pe Pee ia 
and hence f (4, ¥) = Yo + Y, + «oes je Vatrtee canes 
In the same manner we shall have 
FOV) = Vo + OV + ence + Y',4, 
the accents denoting that 4! and { are put for @ and y. 
i+] pt pee 
Hence Y;= —— JA ny . Pf (HV) sind dé’ dy by 
the above assumption, 
L284 1 fepte SIEM OY 
= Sf, . P,Y’; sin@'di'dy' by the 
nature of Laplace’s Coefficients. All so far is clear enough. 
But in order to show that (4, ~) cannot be developed in 
another series V,>+V,+ --. V; + --- he says, that if this were 
possible we should have 
eh aN x fin “Vien Ald al ; 
Va ff _P, V/ sind déldy 
by what has preceded ; and then easily deduces the result de- 
sired. But surely this is no less than begging the question, 
All we learn from it is that if we proceed to develop, as above, 
we shall arrive at a series of determinate terms; but it does 
not follow that another method of development cannot be 
discovered which would lead to another series. The following 
demonstration appears to be free from objection. 
In the formula (1.) we may evidently interchange # and #6, 
) and ¥ since P, P,... P; ... are the same functions of 4, p 
and #,/. Hence from that formula we learn that the defi- 
nite integral of the product of any given function of 6 and ¥, 
and the function (1+3aP,+.... + (27+1)¢’P; + ....) siné 
does not vanish between the limits specified above. 
Now, suppose,f (4, ¥) can be expanded in the two distinct 
series Q) + Q, + eee + Qy eveee and Ro+R,+...... R; 4--.e: 
Then by hypothesis Q;—R, does not vanish; and conse- 
quently, 
st eae (1432P, bet dopaaarceds Sater) ial Py sh tesaee) sin 6 
( 
Q,—R,) dédy does not vanish. 
3B2 
