4 ON A DIFFERENTIAL EQUATION [889 
But this being so, from the assumed equations (3) and (4) we have 
X= [ r 3 dd, Y=l r 3 cos Odd, Z= f r 3 sin Odd, 
J e J e J 6 
and further, by integration of (4), 
L cos 6 + M sin 6 = a cos 6. X — § ( Y cos 6 + Z sin 6). 
Here L and M denote properly determined constants : viz. the conclusion is that 
r, X, Y, Z admit of being determined as functions of 6 and of an arbitrary constant 
r 0 , in such wise that 
a cos 6. X — | ( Y cos 6 + Z sin 6) 
shall be a function of 6, of the proper form L cos 6 + M sin 0, but not so that it 
shall be the precise function b 3 cos (a + 6). To make it have this value, we must 
have L = b 3 cos a, M = — b 3 sin a (where L, M are given functions of a, /3, r 0 ), i.e. we 
must have two given relations between a, b, a, ¡3, r 0 : or treating r 0 as a disposable 
constant, we must have one given relation between a, b, a, /3. 
The equation d6 =—— a C ° S f dr gives r" — 2ar cos 6 = G, where G = r 0 2 — 2ar 0 cos /3. 
— ar sin 6 ° 
There would be considerable difficulty in working the question out with r 0 arbitrary, but 
we may do it easily enough for the particular value r 0 = 0 or r 0 = 2a cos ¡3, giving 
(7 = 0 and therefore r = 2a cos 6 : and we ought in this case to be able to satisfy the 
given equation not in general but with two determinate relations between the constants 
a, b, a, ¡3. 
We have 
Jcos 2 Odd = ^0 + \ sin 26, 
j cos 4 Odd = §6 + £ sin 2 0 + -^ sin 4$, 
cos 3 d sin Odd = — 4 cos 4 d. 
J 
And thence 
a cos d. X — | ( F cos d + Z sin d) 
= 4a 3 cos d (¡3 — d) + 4 (sin 2¡3 — sin 2#)} 
— -L6 a 3 cos d {§ (/3 — 6) + l (sin 2/3 — sin 2d) + ^ (sin 4/3 — sin 40)} 
— Jg 6 -a 3 sin d { 
= — ^ a 3 cos d (sin 2/3 — sin 2d) 
— )ra 3 cos d (sin 4/3 — sin 40) 
+ |a 3 sin 0(cos 4 /3 —cos 4 0), 
J (cos 4 /3 — cos 4 0)} 
8 8 9 J 
where the 
the terms 
which may 
viz. this 
I m 
segment 
is the c. G 
And 
be on the