CURTISS. — BINARY FAMILIES IN A TRIPLY 
5 
The differential equation may be read off directly from (39) ; it is 
(54 ) *g + ri-v-vy-«'-»" » n-'ff 
dx 2 x x — 1 x — 8 \dx 
+ + A* A 4 + T + 
a; x — 1 x — x J r (./ 
and is thus given on page 6 of Ritter's paper. From (41 ) we see that A must be a solu- 
tion of the equation, 
(55) ^ 2 - 4«(. - 1) (^—7— + T=t) + •(« - !) ^~ — + **<< + j^J - 0. 
If Q denotes a family which satisfies (54), having one of the solutions, A u of (55) 
for its accessory parameter, the results noted in sections 2 and 3 of Part 11, A, show us 
that the family 
(56) 
X*(x-1)<Q, 
consists of the solutions of the differential equation 
/on &Q . fl-\'-\"-28 , l -z/-„"-2e 1 "Uj? 
(5T) j^" + L 5 " ^1 *-*_]<** 
+ [_ (V + *) (x» + a) + (fi ,_ s _ e) (/ ,_ a _ e) + O^^M + M^zMl!]^ . 0. 
We shall find this result of considerable use in the problem of characterizing our cl. ~i- 
fication of Q families of order 1. 
§ 2. CLASSIFICATION CRITERIA IN THE DOUBTFUL CASES. 
In § 2 of Part II, B, we have seen that the relations in terms of the exponents 
alone, there given, completely determine the class of a family except in cases which we 
have termed doubtful. These we now investigate. 
Type I. No doubtful case, as there is only 
Type II. The only doubtful cases are for families whose exponents satisfy tli 
0. In this case the fact that the exponent sum X' + X 
// 
relation X 
+■/ + y!' + v' + v" is zero gives us also X" + f «+ v" = 0. For these cases we will 
deduce relations between the parameters of the family which we first show to be 
necessary for each class, and then to be sufficient. 
The two roots of (55) will be found to be - X' -/ • and - X" - f% in these doubt- 
ful 
ases. 
Class 1. One member of the family must be W (■ - 1)' , where * is a eon 
stant. (See page 42. The degree of O (*) is - (X' + / + *'), wMoh is here zero 
Similarly refer to the work of pages 42-44 throughout the rest of tins sect.on. 
