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zero to ¢ into n parts h,... h,; let the values of ¢ in the division 
points be respectively ¢,...¢,; then the equation (1) must be 
considered as limiting case of the following set of differential 
equations: 
dT'—_>S> bn, irs Yrs (t,) dy, A ie te (1) 
th rs 
= = | = KA sYrs (¢,) ale Wehirs (ty t1) Yrs (¢,) h,| dy ch ~ (2) 
th rs 
dT'— > > { bihjrs Yrs (¢;) = Ue Wi), [rs (t, t,) Yrs (t,)h, of ) 
the Ts (3) 
+ Wir frs (t, t.) Yrs (¢,) h,| AY ch ) 
at == = | = [bis Yrs (4) + Wih/rs (ty t,) Yrs (t,) h, a ) 
hor 
th s Sr (n) 
FU) [rs (tnt) Yrs (ty) Age AW frs (tn Enzi) Yrs (tn t)hn—1 ayn ) 
As is known, the conditions of integrability of (1) may be derived 
as follows: 
Call its solution 7” (yin (¢,)), then: 
or 
ME s 
th Oyu (t,) 
this must be identified with (1); which gives: 
GE Det, (t ) (A 
— Nhirs Yrs oe Sade, ERS). Gh PF. 
Oy: (¢,) rs : ) 
hence 
or’ : 
Se sh 
Oy:h OY rs 
in the same way: 
5 : 
Ty Ze Or sfchs 
Onda 
hence as was known: 
Ds brs the 
Integration of (A) then yields as usual 
Td = = bAjrs Yih Yrs 
Now we get to (2). What are here in the righthand member the 
independent variables? The idea is that now 7” is no longer only 
a function of y,;(t,), but also of y,s(t,); these latter must, therefore, 
be added to the others and considered as new independent variables ; 
the meaning of the differential of y,,, as it is in (2), is clear: it is 
dyin (t,). We get: 
Conditions of integrability : 
Call the solution again 
