352 REPORT—1883. 
First, designating by y;, y2 the two independent particular integrals 
at the point z=oo which are given by developments of the form 
Mm=2 (yot ye +ye2z 7+ ste ei 
—n—l 
mo=2? (yo tye +yeler+ .. -); 
there exists a certain binary quartic 
fm'+gne' +6hm?n2?=$(2), 
which is a function of rational character, not only at the point z=oo, but 
also at, the point z=k-*. Consequently this function is given by one 
single development convergent in all points without a circle of the radius 
1, and with the centre z=0; and the convergence is not disturbed by the 
critical point z=k~°®. According to a general formula given by Mr. 
Brioschi,! the Hessian covariant, viz., the quartic 
h(fny* +9no*) + (fg—4h?) 0,702” 
is a known function of z also; one has therefore two equations from 
which the integrals 7, 7, may be found. 
Secondly, I show that one may obtain two other particular integrals 
of the form ‘ 
ae ee Cf——_ 
y,=GV/ F(z) ee FQVZ P 
3 (a=21—z) 1-2) 
Yo=G! / F(z) ends 
wherein G, G’ are two arbitrary constants, and C designates another con- 
stant chosen in a certain given manner; F(z) is a function of z given by 
a development in ascending powers of z, whose convergence is not dis- 
turbed by the point z=1, but which is convergent for all points within 
the circle with the radius k~*. This series is found directly as a certain 
particular integral of that differential equation of the third order, which 
is satisfied by the quantities y,?, y2, y/o. 
The connection between the functions 7), 9, and yj, 7, is now given 
by two relations of the form 
Yyy=Kn +Ang, Yo=eM+ VN, 
wherein the constants «, \, uw, v may easily be determined by choosing for 
zany point within the circle of the radius 4~*, and without the circle of 
the radius 1. 
So one does not need for the integration of the proposed equation but 
two developments in series (viz., those of the functions ¢ and F), suppos- 
ing that 27 is a whole number. 
An exception will present itself when the constant C (entering in the 
formule for y, and y2) is just equal to zero; this is the particular case 
treated by Mr. Brioschi, loc. cit. 
It is to be remarked that the formule for y,, y. (but not those for 7, 
n2) can be applied whatever the value of » may be. 
This seems to me likely to become useful in certain problems relating 
to the theory of potentials, in a manner that I intend to explain on another 
eccasion. 
1 Annali di Matematica, Serie ii., vol. 9, p. 13. 
