Mr. J. Cockle on Differential Covariants. 225 



ought to yield with sulphuric acid /3-sulphamylic acid instead of 

 producing di- and tri-araylene. But to this surely it might be 

 sufficiently answered, that too close a parallel must not be drawn 

 between the members of homologous series, and that much in 

 the amyle series may depend upon the remarkable tendency 

 which it has to produce polymeric defines. It appears to me 

 that, placing isethionic acid* aside, as constructed on a different 

 and mixed type, in which ethylene functions as a diatomic radi- 

 cal, the difference in the sulphethylic acids may be structurally 

 expressed by supposing in one the existence of the molecule 

 ethyle, and in the other of the body C 2 H 4 . H, the subhydride 

 of ethylene, both playing the part of monatomic molecules. 



XXXIX. On Differential Covariants. By His Honour Chief 

 Justice Cockle, M.A, F.R.A.S., F.C.P.S., %c 3 President of 

 the Philosophical Society of Queensland^ . 



IN my paper entitled " Correlations of Analysis," published 

 in this Journal i, I showed that in the theory of linear dif- 

 ferential equations the factorial substitution leads to results which 

 are analogous to those obtained by the linear substitution in 

 algebra. That we may pursue these analogies, take a symbol S, 

 such that 



*d^l_ d m ~h/ 

 dc^^dti^ 

 and consider the differential quantic 



(1, b, cX l, Ify, 



whereof the quadratic critical function is 



b 2 -c + b', 



the accent denoting differentiation with respect to the indepen- 

 dent variable x. Then, as in algebra the evanescence of the dis- 

 criminant shows that a quadric and its first derived function are 

 simultaneous, so the condition 



b*-c + b' = 



indicates that the above differential quadric and its derived 

 function 



8(1, b, *X*t l) 2 * 2(1, 6j|, 1) 



vanish simultaneously. 



* Gorup Besanez, vol. ii. p. 167. 

 t Communicated by the Author. 

 X Phil. Mag. S. 1. vol. xxiv. p. 531. 



Phil Mag. S. 4. Vol. 27. No. 181. March 1864 Q 



