148 
PROF. II. F. BAKER ON CERTAIN LINEAR 
we obtain 
cr 2 +p + 2i<rP = 0, (cr 2 + g) P-2 ia = 0, 
so that the values assumed by k, k u k. 2 , k s , when X = 0, are the roots of the equation 
or 
Thus 
(cr 2 +_p) (cr 2 + q) — 4o- 2 = 0, 
IT 4 — iT 2 + 4-?7l 2 = 0. 
ct — ± {|-(l-hr/7) 5 ± (l—w)'}, 
and the four values are all imaginary when m > 1, and all real when m < 1. 
Supposing S > E > M, we find at once, from the formula for m, that the least 
possible value of S/(S + E + M) in order that m < 1 is 0‘96147..., but this requires M 
to be very small; but if S/(S + E + M) be greater than 0"9618..., then m is certainly 
< 1 even if E = M. In our solar system the sun’s mass is more than 99‘8 per cent, 
of the mass of the whole system; thus if S in our problem were the sun, and E, M 
were any two planets of the system, the condition for m < 1 would be easily satisfied. 
We shall then suppose m < 1. 
Now compare with the equations (I.)" the equations 
(1—2X cos 9)(<X>" — 2'P / ) = £>d>, 
(1 -2X cos 0) (*" + 2$') = qdq 
(III.) 
obtained from (I.)" by change of the sign of X. They can also be obtained from (I.)" 
by changing 6 into 6 + tt. This last change shows that the characteristic constants k 
belonging to the equations (III.) are the same as for (I.)", while the former change 
shows that the values of k proper to (III.) are obtained by changing the sign of X in 
the constants k appropriate for (I.)". When m is such that the values of k for X = 0, 
namely, the four values of <x above, are all different, a change in the sign of X cannot 
interchange the values of k among themselves. Thus we infer that each k is unaltered 
by changing the sign of X ; for two of the values of o- can only be equal when m 2 = 1. 
In the applications in view of which the question was first considered, S denotes the 
sun, E denotes either Jupiter, or another planet such as Mercury, while M is of 
negligible mass. When E is Jupiter we have 
m 2 = 27 T oW(1+to 1 5o) 2 = 0T257, X = £(0'05) = 0T25, 
and m 2 /x is nearly unity. When E is mercury 
m 2 = 27/5 ‘ 10 6 = 0-0 0 0 0 0 5 4, X = £(0*2) = 0*1, 
and m 2 — 5"4X 6 , m = (2"3)x 3 , nearly. In either case we may regard m as small, and 
the four possible values of a are approximately 
±(l— im 2 ), ±fm, 
