36 THE REV. F. H. JACKSON ON CERTAIN 
: { » Ay + Pilea + Pi) sss + (as +21) 0 ) 
{p-yw}F=T,(a l+p — SOT ts 
(o-v}F= ma) | (142, (Gie.6: << B 
v (a, + Pi)oy +71) - + + + ee: (a, + Pr) Met at : 35 
(l+p, (R68. ee Brie e (22) 
In the simple case 
P= 1+ p eB 4. 12 Vat PPB Pr oportm ne. 3 . (36) 
DY * PvP + Poy + Po 
the differential equation is 
aB-F + p,{(at+PB+p,)a -y}D.[F +p, po{a" - 1}a@ DOF 
= ap { (1 + pot PB +Pigo - Pins at + PoBt+PyB+Pi+Pogrtm 4. 1 
Diy 
Py Py t+ Poy + Po 
-(1+p a+ BB+Pr ym 4 pert Piet + Pe B+PiB+Pit+Pogprtr 4 5) } . Om 
= iy PP, + Poy-¥ + Po 
Putting «=1 the expression on the right vanishes identically, therefore when 
[D®F}],-, is finite or convergent we have 
(1) 
aSlE si + plete y+r}| So a, —. . (38) 
which is 
ee ap 1200+ PBB +P, , Wee et a 4 pst PB +P 
"Pry PP t PoVY + Ps Y PrY+P2 
+ 7), gat Py a+ p+ Py Bt Py B+ Pi +Ps ca 4 : (39) 
PrP, + Py + Po¥ + Pot Ps 
subject to the convergence of the series. 
When the progression of elements p,.-...- . ISieram Om iaiel ae. 3 
1 
F(a, 6 Te ee ae. a-a+1BB+1 He 
aS aed "Peel a eae yee aa 
7 Pie Beas of, wat IPB+ly a-a+ lat 2-BB+1B + 2° 
Baa; tyae)- 8 ree 14.3 Sat at + 14.94.34 gee Sa c (40) 
In connection with this series we can obtain from (39) the identity 
4 FN es 7, — 12.m — 92 
1 + ap? + aye! cane a awa —la-2? + 
i Poles) m— 1m — 2? 1¢— Vee 2? | a lem 2 8? oe Ve — 2? — 3? } 
(etn) + 1 F 12 92 12.92 792132 1292.32, 9232.42 = (41) 
putting 7= -“2#=m 
tee m*-m*4 — 14 =6 
(1!)4 ze (2!)4 on "ep sd) tele = =— 
In the case when p,p...... form a geometrical progression and F([a][8][y]\ax) 
denotes the series 
y+ LANE) omg Del Tee) VST eae . (42) 
[4] [y [/} [22] - Ly] ly +4] 
the following relations can be obtained : 
