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Mathematics. — ''Hoimxjeneous Unedr dljfemitlal eqiadlo/i o/onlei' 

 tiro ivlth (jweii, relation hetireeii tiro particular Integrals". 

 (!'' Cominuiiiccilion). By M. .1. van Uvkn. (CoMiiiimiiciited by 

 Prof. W. Kai'tkyn.) 



(Communicated in Ihc meeting of September 30, 1911). 



We consider tlie lioiiiogeneous linear differential equation of order two 



. d'\v d.v 



;^ + /'(0^H-7(0-=0. . .... (A) 



If we replace the independent variable t by an oilier 7' connected 

 with t by the relation 



^'=/(0' (1) 



then on account of the formulae 



dx __d,v d.f d'x d\v fd.f\- dx d^f 



dt dT dt ' df- dT^ ' ydt J '^ dl' ' df 

 the differential equation {A) passes into 



d'x f"+pt'dx q 



df d-f 



when we |)ut - = /' and — = /". 

 ' dt ' dl- 



Tlie new coefficients are therefore 



ƒ 2 p. 



(2) 



If we put 



we have 



and 



dTT^^ ' ;/^ = ^' 



dt ' dT ƒ' dt 



so that 



q-^2PQ= , 



Q + 2PQ q'-i-2pq ^ 



T~=—T~ = ^- (3) 



Q- q'' 



The differential form J i-emains therefore invariain when ojie 

 independent invariable passes into the other. 



