37. 
NOTE ON SCHWARZIAN DERIVATIVES. 
[.Messenger of Mathematics, xv. (1886), pp. 74—76.] 
Reading with great pleasure and profit Mr Forsyth’s masterly treatise on 
Differential Equations (in my opinion the best written mathematical book 
extant in the English language), it occurred to me to find an easy proof of the 
fundamental and striking identity concerning Schwarzian derivatives, from 
where one of which is, it may be observed, that (y, x) like y" has the property 
of remaining a factor of what it becomes when x and y are interchanged ; a 
persistent factor, so to say, of its altered self. I will return to this point subse 
quently, my present concern is to give a natural proof of the above striking 
identity; to do this, it will be sufficient to show that (considering y, z, x, the 
two former as fixed, and the last as a variable function of a common variable) 
(y, x)-(z, x) 
does not vary when x becomes x + e(f> (x) where e may be 
regarded as infinitesimal* *. For then this must remain true by successive 
accumulation when x becomes any function whatever of itself, and accordingly 
making x — z we obtain (y, z) as the value of the invariable quotient as was 
to be shown. Calif eS<£# «= 6, then using dashes to denote differentiation qua 
x, and a parenthesis to signify the augmented value of the derivatives, we 
obtain 
(;y') = y'~ e v'> 
(f) = y"-2dy"-d'y' ) 
(y , ") = y'' , -My , "-MY-6"y / . 
* It is easy to see a priori that if the theorem is true, it can only be so in virtue of (y, x) 
when x receives an infinitesimal, becoming of the form 
(1-26) (y, x) + \0", 
as is subsequently shown to be the case in the text. 
[+ Cf. p. 306 below.]