176 
PROF. H. F. BAKER ON CERTAIN LINEAR 
If next we retain as far as X 3 , we have from 
■ ^ = x. 4 + xz- 1 (r i +^+x 2 ^(r 2 +r)+x^ 3 (r 3 +f 3 ) 
— a x = (p dr 
Jo 
= \hr + \Jc x (— i ~ l + f) + ^X 2 k 2 (— f ~ 2 + £ 2 ) + g~A 3 & 3 (— + f 3 ), 
C x = - V 
Jo 
7. (L 
- — 2 
= Xfc (r 1 - 1 ) + x^ ( -i-r) + \% ( V + 1 - 
r 3 
3 
A 
r 4 ,i r 
3l T +4_ Y 
Hence 
-«i + fci 
r 1 
Thus 
— \h (r + 1 — £) + Afc (-— + 2 H ) + ^2 ( — b11~ K ) + X 3 & 3 ( ~ ]T> + ^ — g 
P-2 c 2 
S , 2 P i 
/ P-3 
r 3 , f t 
0 (-Gfi + fCj) 
= A 2 {A 2 (t+i -£)+A4 K-HK^-i+K-n+V (-K- 2 -r-n 2 +K 2 )} 
+A 3 {M’ 2 (rr 2 +K- 2 -r i +ff+n 2 +K 2 -f) 
+^(-K-*-Tr l +K- 1 +f-f+K > -T^)}. 
This gives 
a 2 = j\/> (-(li + fcjdr 
= ^{/A^ + T+l-^ + M^-T^-f^-T + i + K-K 2 ) 
+h 2 (\r-h 2 -i-bi 2 +u 2 )} 
+A 3 {^ 2 (-K- 2 -tr 2 +r i -t+K+M 2 -K 3 ) 
+tt(ir 3 +Tr i +K- i +|r-i-^+K 2 -W 3 +K 3 )}. 
Similarly, 
Co = (—flj + fa) cZt 
Jo 
- A’ {A*(-Tf- , -«-'-T + 2)+W, (-J^'-K-' + f-' + fr + J-f) 
+y(K- 8 +n- , +4-'-f+if-Tm 
+ A 3 {hJc 2 (— tV£~ 3 +i£ ~ 2 +f r +1+vf—if—K 2 ) 
+^(K-HM- 2 +*r 2 -ir l -T-i+K-M 2 +K 2 )}. 
Forming now — a 2 +fc 2 we obtain 
A 2 {^(-|t 2 -2t-3-t^+ 3^) + M 1 (W 1 + f" 1 + T + i+|rf-f-K S ) 
+ ^ 2 (-M- 2 +lT 2 +T+i-l^f-M 2 )} 
+A 3 {A4(iTr 2 + 1 5 ¥ r 2 -K- 1 +t+K+M+M 2 -K 2 -K 3 ) 
+ lc x k o (— iVf ” —-rH _1 — “ 1 — f t — §■ — n + ^+— 6rf 3 +weT 3 )} • 
