142 
we obtain 
where 
PROF. H. F. BAKER ON CERTAIN LINEAR 
dU 
C It 
— <?U = — £(u- Vf), ^ -gV = 
m cLt 
*(U£--V), 
m 
<p — A {h x + hiWi) + A" {h 2 + k 2 w 2 ) + ..., 
w r = £ r +£~ r . 
We may then further substitute 
leading to 
W = U£~ m -V, U* = mTJ, 
dU x 
dT 
dW 
dr 
-q\J 1 = -</>£ m W, 
-qW = —£ -m U 1 , 
where 
x = — e imr - gr W. 
m 
These equations can be solved by waiting 
q = \q x + X 2 q 2 + , 
IX i — IT \u x -t-A 2 u 2 T ..., XV — 
A+r 
m 
T \w x T A w 2 + .. 
where A is a constant, and w 1} w 2 , w x , w 2 , ... are polynomials in £, £~ l . 
For m — 2, in particular, we find that if h x = 0, the quantity A is required, and 
determined in the course of the work, and q x = 0. But if h x is not zero, we must 
take A = 0, and obtain q x = \h x , the succeeding q 2 , g 3 , ... being real. In fact, as far 
as A 3 , 
q = iKx-dK+W-ih) \ 2 + {*L +HW- 1 x s +.... 
which gives 
g * = ihX-ih (iK+W-ih)* 3 
+ {A*, 4 + ah; V- A *, 4 ■+P ,%■- i ti-K (f K -+ W) + iV+ivy X‘ +.... 
We know, as is shown in Part II. of this paper, that the form of q 2 is valid even 
when h x = 0. Then we have 
q 2 = iV {h 2 +k-W) {h-h+W) +..., 
which, when h 2 = 0, is only positive, provided 
5 k x 2 > 3 k 2 > k x 2 . 
