In addition, he wrote a number of scientific memoirs and papers, including two on the integration of partial differential equations (Jour.
He proves, by means of the six linear partial differential equations satisfied by the concomitants, that, if any concomitant be expanded in powers of xi, x 2, x 3, the point variables-and of u 8, u 2, u3, the contragredient line variables-it is completely determinate if its leading coefficient be known.
It has been shown above that a covariant, in general, satisfies four partial differential equations.
Operating with 5l-xidxlwe find S2C 0 =o; that is to say, C ° satisfies one of the two partial differential equations satisfied by an invariant.
By considering only the particles of air found in a right line, he reduced the problem of the propagation of sound to the solution of the same partial differential equations that include the motions of vibrating strings, and demonstrated the insufficiency of the methods employed by both his great contemporaries in dealing with the latter subject.