OF SIMULTANEOUS ORDINARY DIFFERENTIAL EQUATIONS. 527 
Analytical Connection of Differeni Solutions. (S$ 5-9.) 
Seoumletusiweite J tore (ftyjars e/a) Cl(PisPos- = Pn) and Ny. Ny ea eeetor tine 
minors of Of,/0p,, Ofn/Ops, . . . Ofn/Opn in J. 
The equations /, = 0, jf. =0,...f,=0, J =0 give p,, po, - - - Pa Yn aS functions 
of @, Yj, -- » Yn», and, for the second singular solution, we have to suppose that two 
solutions coincide, that is, we put 
O( fir Se --- Sn I)/0 (Dy Pa - ~~ Pus Yu) = 0. 
Since J = 0, this equation may be reduced, by multiplication by X,, to 

od CHL Clinton os ah 
sat... $A err = 0. 
* Ops a gi 5p) O (Py, + + + Puy Yu) 
(od 
Or >, + 
Now, y, may equally well be replaced by y,, y,..., oY Y,-1, So that the condition 
sought is given by the first factor, which we shall call J,. 
we Grpeinows 75 =O; oseje— 0; J = 0, di— 0 GNG fp fon b oc Dy Uhim Uh OS 
functions of x, y), .. . Yn», and, if the values of p,_;, P,, given by differentiating those 
Of Yn1, Yn, agree with the values given by the solution of the equations, we are to find 
the second singular solution by integrating the equations that give p,, ... Py». 
To find the third singular solution, we have to make the system f, = 0,...f, = 0, 
J = 0, J, = 0 have equal roots. The condition for this is found in the same way* 
to be 

and so we may pass on to the other singular solutions, if any. 
3 ; Ofna : : 
S16. .As Ji= i& +h, He +...+,, s we may write for the integrals that 
Pi Pz Pn 
have to be taken to give the 7™ singular solution 
BECHER ert city eset 
Tha) fu = 0. (Si le eee 2) 
* The condition is ie 
O( fp fa--+ tu J, J) _ 0. 
O( Py, Po: - + Pas Yur Yur) 

Multiplied by X, this becomes 
oJ O(f = + Susi Jj) — 9, 
SNe xX 
Op: @ (pi 2.00 Pr» Yno an) 

The second factor, being unsymmetrical, is irrelevant. 
