2 
Mathematics. — “/omogeneous linear differential equations of order 
two with gwen relation between two particular integrals.” 
By Dr. M. J. van UveN. (Communicated by Prof. W. Kaprnyn). 
(Stb communication). 
(Communicated in the meeting of April 26, 1912). 
The equations (8) and (29) (see 1st comm. p. 393 and 398) show 
us in the case that the equation f(z, y,z)—=O represents a conic 
(see for the notation: 4° comm. p. 1015): 
cz H Az? prs 
q. g° == € ’ 
Trel 
where c is put equal to 1. 
From this ensues 
A 
Dr 
Let us further put: 
GS TS Er en So rn 
we then find: 
5 zn dl g Bie ae 
beige for we 
or 
5 
. ji Zn 6 ee . « ° . . . . . 73 
: (73) 
The equation (62) (see 4t? comm. p. 1015) runs now as follows: 
2 5 9 2 274 5) 22 2 
I = 385 ~ a, Az? (— daz Anne C -t- 2a,,A¢ 5 —a,,Az ) 
or making use of the notation (59) (4° comm. p. 1003), 
WC? = tek ED 1 re 
so &! is likewise an elliptic function of r. Its invariant has the same 
value (68) as that of the function «= /? (compare (67) *) (4 comm. 
p. 1006). 
We can now deduce out of the equation 
Ate AL V—A,,9? +2Agz-Aa, 2? = Va,, ie) VIG OG (75) 
(see 4th. comm. p. 1005 at the bottom) 
Ase Ay = 82 a, AiG eeN 
') In the 4th comm. in the table on p. 1014 and in the enumeration of the 
cases on p. 1015 3}; =e and ‘y= ei! must be replaced by 3, = eit, 5s = et’. 
